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+{"text":"\\section{Introduction}\n\n\nThe next generation wireless communication system, typically referred to as the fifth generation (5G), is currently under\nintensive research and development. Millimeter wave (mmWave) communication is regarded as one of the enabling components for 5G cellular communication systems, thanks to the large amount of available frequency spectrum \\cite{rappaport2013mwave,andrews2014will,demestichas20135g,Ruisi_1, Ruisi_2,Ke_2}. However, mmWave  transmissions suffer from high propagation loss and blockage, unlike sub-6GHz legacy frequency band. High antenna gain (typically realized with a large-scale antenna structure) is required to overcome the radio propagation loss and achieve the good signal-to-noise ratio (SNR) at mmWave bands. Therefore, mmWave transmissions will be potentially ultra-wideband (UWB) (e.g. with an absolute bandwidth larger than 500 MHz) and involve large-scale antenna systems at both communication ends \\cite{rusek2013scaling,larsson2014massive}. \n\nIt is of importance to understand how the radio signal propagates in the given scenario. A new air interface system  design typically starts with the understanding of the deployment scenario, where channel models should be developed to reflect the physical transmission.\nAccurate knowledge of the delay and angle information of multipath components is essential for many applications in mmWave systems.\n\\added{The extracted multipath components can be further utilized for analysis, clustering and modeling of wireless channels, which is fundamental for system design and performance evaluation \\cite{gustafson2014mm,he2018clustering}. }\nA popular channel sounding setup at mmWave bands is to utilize a nonreal-time channel sounder based on the virtual antenna array concept \\cite{dohler2002vaa,medbo2015vcua,allan2018vcua,Ke_1}. This strategy is simple, flexible and cost-effective, though it is not suitable for dynamic propagation scenarios due to slow mechanical movement of antennas and channel sounding time for each virtual array location. Furthermore, the mutual coupling effects between the array elements are not presented, which is beneficial since it can further simplify the channel estimation. Virtual uniform circular array (UCA) is particularly popular, since it only requires radial mechanical movement of the positioner and presents approximately constant beam patterns over $360^o$ azimuth angle, regardless of the steering angle \\cite{davies1983uca,long1967uca}. \n\n\\replaced{Channel estimation algorithms proposed in the literature for power-angle-delay profile (PADP) estimation were mainly under the far-field and narrowband assumptions. The far-field assumption holds when the distance between the scattering source and the antenna array is larger than the Fraunhofer distance $2D^2\/\\lambda$, with $D$ the antenna array aperture and $\\lambda$ the wavelength. The far field assumption can be violated in short-range mmWave propagation scenarios. The narrowband assumption holds when the condition $D\/\\lambda \\ll f\/B$ is satisfied, where $f$ is the frequency and $B$ is the bandwidth. When the system bandwidth becomes wide (e.g. in the mmWave context) that the narrowband assumption does not hold anymore for a given array aperture and frequency, the propagation delay of each path can be resolved at different delay bins between array elements. The existing algorithms in the literature would fail to work under near-field and wideband scenarios, as briefly summarized below.}{} \n\\begin{itemize}\t\n\t\t\\item[1)] \\added{Beamforming techniques such as classical beamforming (CBF) and Capon beamforming.  As demonstrated in \\cite{fan2016eurasip}, The beam pattern of the CBF method is highly sensitive to near-field conditions (which would introduce a power loss in the target direction) and UWB bandwidth (which would introduce a joint sidelobe in the delay and angle domain and difficult to suppress \\cite{fan2016eurasip}). Capon beamformer degrades as well in near-field and UWB conditions, since the steering vectors are typically calculated at the center frequency and under far-field conditions.}\n\t\t\\item[2)] \\added{Subspace methods. The conventional subspace angle estimation methods are based on the element-space covariance matrix, where the steering vector is the function of center frequency. Thus, it implied the narrowband assumption. UCA is attractive in angle estimation also thanks to its circular symmetry, where beamspace transform of UCA based on the phase mode excitation principle can be applied \\cite{tewfik1992bs,gentile2008fi,chan2002design}. Basically, we can transform the UCA array manifold (in the element space) into uniform linear array (ULA) manifold (in the phase mode space), which enables us to develop computationally efficient and high-resolution subspace algorithms \\cite{zoltowski1992direction,zoltowski1996closed,pesavento2002direction,belloni2003unitary,mathews1994_2d}. However, the aforementioned beam-space multiple signal classification (MUSIC) or estimation of singal parameters by rotational invariance techniques (ESPRIT) algorithms were only investigated under narrowband and far-field assumptions. Subspace methods were also developed for joint angle-delay estimation (JADE). With the JADE strategy, the JADE-MUSIC \\cite{van1997jade_music} and the JADE-ESPRIT \\cite{van1997jade} algorithms exploited both space and time properties by stacking the array element channel impulse responses into a high dimension vector. After performing the eigen-decomposition on the covariance matrix of the constructed vector, JADE can be obtained either with MUSIC or ESPRIT methods. However, both algorithms are unfavorable for the UWB large-scale antenna systems, due to the high complexity introduced by the high dimension of the stacked space-time vector.\t}\n\t\\item[3)] High resolution parametric methods. Maximum likelihood estimator (MLE) is a popular high resolution channel estimation algorithm. However, the MLE is well-known for its high computational complexity due to its joint estimation mechanism, particularly when the channel  parameter dimension is large. An attempt to reduce the MLE computation complexity was proposed in \\cite{ji2017_mle,ji2018channel}. The space-alternating generalized expectation-maximization (SAGE) algorithm is a relatively low-cost expectation-maximization (EM) algorithm, where the multi-dimensional search is replaced with several one-dimensional searches \\cite{bernard1999sage}. However, a prerequisite to utilize the SAGE algorithm is that the likelihood function needs to be independent between different parameter spaces, which might be violated when narrowband and far-field assumptions do not hold, e.g. for the UWB large-scale antenna systems \\cite{bernard1999sage}. A general spherical wave model, i.e. including the distance to the scatterer, can be introduced in the MLE type algorithm implementation, which would, however, further increases the computation complexity. \n\\end{itemize}\n\n\n\\added{There is a strong need for a low-cost and generic channel estimation algorithm for joint angle and delay profile estimation, which works in practical propagation scenarios, regardless of the antenna system scale (i.e. large or small) and frequency bandwidth settings (i.e. narrow or wide). That is, it can be applied for the near-field UWB scenarios without increasing the algorithm computation complexity. However, such algorithm is missing in the literature, to the best knowledge of the authors. In this work, a novel low-cost beamspace UCA beamforming algorithm with a successive cancellation scheme is proposed to jointly detect the angle and delay of the multipath components. The proposed algorithm is insensitive to the system bandwidth and distance to the scatterers, making it suitable for UWB large-scale antenna systems. In this paper, we firstly demonstrated the performance deterioration of CBF of the UCA in the element space when far-field assumption does not hold. To solve the issue, we resort to the beamspace transform of UCA based on the phase mode excitation principle. The beamspace UCA is shown to be robust to phase errors introduced by the spherical wavefront. A novel beamspace UCA algorithm with the successive cancellation of the detected paths is detailed. Both numerical simulations and experimental results at mmWave bands are provided to demonstrate the effectiveness and robustness of the proposed method in channel parameter estimation for UWB large-scale antenna systems.}\n\n\n\n\\section{Problem Statement}\n\\textcolor{black}{Assume a UCA is distributed in the x-y plane and its center is located at the origin of the coordinate system. The UCA consists of $P$ isotropic antenna elements with radius $r$. The angular position of the $p$-th element is $\\varphi_{p}=2\\pi\\cdot(p-1)\/P$, $p\\in[1,P]$. Suppose there are $N$ paths impinging at the UCA, the channel frequency response at the $p$-th UCA element is the superposition of the channel responses of the $N$ paths,\n\\vspace*{-.01in}\n\\begin{equation}\nH_p(f)=\\sum_{n=1}^{N}  \\alpha_n \\exp(-j2\\pi f \\tau_n) \\cdot  a_p(f,\\mathbf{\\Theta}_n),\\label{eq:element frequency response}\n\\end{equation}\nwhere $\\alpha_{n}$ and $\\tau_{n}$ represent the complex amplitude and delay of the $n$-th path, respectively. $\\mathbf{\\Theta}_n = [D_n, \\theta_n, \\phi_n]$ denotes the scatterer location vector of the $n$-th path, where $D_n$ is the distance between the $n$-th scatterer and the UCA center, $\\theta_n$  and $\\phi_n$ the elevation and azimuth angle of the $n$-th path, respectively. $a_p(f,\\mathbf{\\Theta}_n)$ is the transfer function between the $n$-th scatterer and the $p$-th UCA element, which is normalized by the transfer function between the $n$-th scatterer and the UCA center, as\n\\vspace*{-.01in}\n\\begin{equation}\na_p(f,\\mathbf{\\Theta}_n)=\\beta_{p,n}\\exp(-j k \\varPsi_{p,n}).\\label{eq:a_p}\n\\end{equation}\n $\\beta_{p,n} = D_n\/d_{p,n}$ denotes the relative path loss term at the $p$-th element with respect to (w.r.t) the UCA center, where $d_{p,n}$ represents the distance between the $n$-th scatterer and the $p$-th UCA element. Under far-field assumption, we have $\\beta_{p,n}=1$.\n $k = 2 \\pi f \/c$ is the wave number with $c$ denoting the speed of light.\n $\\varPsi_{p,n}$ indicates the relative propagation distance to the $p$-th UCA element w.r.t UCA center, i.e.,\n\\vspace*{-.01in}\n\\begin{equation}\n\\varPsi_{p,n} =d_{p,n} - D_n,\\label{eq:r_dis}\n\\end{equation}\nwhere the distance term $d_{p,n}$ is given by\n\\begin{equation}\nd_{p,n}= \\sqrt{D_n^2 + r^2 - 2rD_n\\sin{\\theta_n}\\cos(\\phi_n-\\varphi_p)}.\\label{eq:dis}\n\\end{equation}\n Following Taylor series expansion, we can approximate $d_{p,n}$ as\n \\begin{equation}\n d_{p,n}= D_n - r\\sin{\\theta_n}\\cos(\\phi_n-\\varphi_p) + \\varDelta_{p,n},\\label{eq:dis_a}\n \\end{equation}\n where $\\varDelta_{p,n}$ is the term introduced by the near-field condition. When the plane-wave assumption holds, we have $\\varDelta_{p,n}=0$.\n By substituting (\\ref{eq:dis_a}) into (\\ref{eq:r_dis}), $\\varPsi_{p,n}$ can be simplified as \n \\begin{equation}\n \\varPsi_{p,n}=  - r\\sin{\\theta_n}\\cos(\\phi_n-\\varphi_p) + \\varDelta_{p,n}.\\label{eq:r_dis_a}\n \\end{equation}}\n \n In this paper, the objective is to  detect path parameters $\\{\\alpha_n, \\tau_n, \\phi_n\\}$ with arbitrary elevation angle $\\theta_n$ present in the near-field condition for $n\\in[1,N]$. The performance of CBF and frequency-invariant beamforming (FIBF) based on UCA for a general 3D multipath scenario \\added{under far-field assumption} has been discussed in \\cite{zhang2017fi}. Below, we limit our discussions on the beamforming properties of CBF and FIBF under near-field conditions, assuming that all impinging paths are confined to the plane of the UCA, i.e. with elevation angle $\\theta_n = \\frac{\\pi}{2}$ for $n\\in [1,N]$ for simplicity. Though the proposed algorithm works for arbitrary 3D propagation scenarios as later discussed in section \\ref{simulation}  and \\ref{measurement}. \n\n\n\n\\subsection{CBF in the UCA element space}  \\label{cb_ff}\n\\textcolor{black}{Using the CBF under plane-wave assumption, the steering weight of the $p$-th UCA element can be written as\n\\begin{equation}\nw_{p}(f,\\phi)=\\frac{1}{P} \\exp[-k r \\cos(\\phi-\\varphi_p)].\\label{eq:w_cb}\n\\end{equation}\nTherefore, the UCA array beam pattern can be obtained by taking the coherent summation of the element responses as\n\\begin{equation}\n\\begin{split}\nB(f,\\phi)& = \\frac{1}{P}\\sum_{p=1}^{P} w_p(f,\\phi) \\cdot H_p(f).\\\\        \n\\end{split}\n\\label{eq:b}\n\\end{equation}\nBy taking (\\ref{eq:element frequency response}) into (\\ref{eq:b}), we have\n\\begin{equation}\n\\begin{split}\nB(f,\\phi)& = \\sum_{n=1}^{N} \\alpha_n \\exp(-j2\\pi f \\tau_n)\\cdot \\sum_{p=1}^{P} w_p(f,\\phi) \\cdot a_p(f,\\mathbf{\\Theta}_n)\/P   \\\\   \n         & = \\sum_{n=1}^{N}  \\alpha_n \\exp(-j2\\pi f \\tau_n) \\cdot v_n(f,\\phi)  \\\\\n         & = \\sum_{n=1}^{N} B_n(f, \\phi),    \n\\end{split}\n\\label{eq:b_1}\n\\end{equation}\nwhere $B_n(f, \\phi)$ represents the beam pattern of the $n$-th path, i.e. the beam pattern of a single path. As seen from (\\ref{eq:b_1}), the linear superposition of the CBF beam patterns of all paths gives the beam pattern of the multiple paths.}\n\\textcolor{black}{$\\arrowvert v_n(f,\\phi)\\arrowvert$ denotes the unit beam pattern term of the $n$-th path with CBF. In ideal case, $\\arrowvert v_n(f,\\phi)\\arrowvert$ mimics a Dirac delta function in $\\phi$ domain, i.e. $\\arrowvert v_n(f,\\phi)\\arrowvert  = \\delta(\\phi-\\phi_n)$. Thus the peak location of $\\arrowvert  v_n(f,\\phi) \\arrowvert $ gives the estimate of $\\phi_n$, $\\alpha_n$ and $\\tau_n$ can be obtained via inverse Fourier transform (IFT) of $B(f,\\phi)$ at $\\phi = \\phi_n$.\nAccording to (\\ref{eq:b_1}), (\\ref{eq:w_cb}) and  (\\ref{eq:a_p}), $v_n(f,\\phi)$ can be given by\n \\begin{equation}\n \\begin{split}\n v_{n}(f,\\phi)=& \\frac{1}{P} \\sum_{p=1}^{P}\\beta_{p,n} \\cdot \\exp(-jk \\varDelta_{p,n}) \\\\\n               &\\cdot \\exp\\{-jkr[\\cos(\\phi-\\varphi_p)-\\cos(\\phi_n-\\varphi_p)]\\}.\\label{eq:v_cb}\n \\end{split}\n \\end{equation}\nUnder the far-field condition (i.e. $\\beta_{p,n}=1$ and $\\varDelta_{p,n} = 0$), we have $\\arrowvert v_{n}(f,\\phi_n)\\arrowvert = 1$ and $ \\arrowvert v_{n}(f,\\phi_{n'})\\arrowvert <1$ with $\\phi_{n'}\\neq \\phi_{n}$, as illustrated by the blue line in Fig.\\ref{bp_cb_dis}. Thus the peak location in the beam pattern gives the angle estimate of the $n$-th path. }\n\n\\added{To investigate whether $\\arrowvert v_{n}(f,\\phi)\\arrowvert$ can still maintain this property under near-field condition, we reduce the distance $D_n$ and its effect on the beam patterns of $v_{n}(f,\\phi)$ is shown in Fig. \\ref{bp_cb_dis}, i.e. the red and black curves, as examples. The results show that the beam patterns are highly susceptible to phase errors introduced by the spherical wavefront in the near-field condition.\nThe near-field condition results in the main beam distortion of the beam pattern, including the power loss (the red curve) and the concave pattern in the target direction (the black curve). The power loss results in underestimating the amplitude $\\alpha_n$ and the concave pattern would result in fake path angle detection.}\n \n\\added{To further investigate the power loss in the target direction under near-field condition, the distance from $3$ m to $70$ m was set and the result is shown by the blue dotted line in Fig. \\ref{cbf_fibf_dis2}. The figure shows that with CBF method, a power loss more than $35$ dB might exist in the target direction for typical indoor scenarios.}\n \nTherefore, besides the joint sidelobe problem introduced by the UWB bandwidth in PADP as shown in \\cite{fan2016eurasip}, the CBF of UCA in the element space would also suffer from significant power loss in target directions or even failure of detecting the true paths, due to phase errors introduced in the near-field condition, which makes it unsuitable for channel parameter estimation of mmWave UWB large-scale antenna systems.  \n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 260bp 695bp 565bp,clip,scale=0.6]{bp_cb_dis_new.pdf}}\n\t\t\\par\\end{centering}\n\t\n\t\\caption{The unit CBF beam pattern $v_{n}(f,\\phi)$ in the UCA element space for different $D_n$ with $\\theta_n = \\pi\/2$, $\\phi_n = \\pi$, $r$ = 0.5 m, $f$ = 29 GHz and $P=720$ are set for the UCA. The calculated Fraunhofer far-field distance $D_{far} = 2(2r)^2\/\\lambda$ is around 193 m. }\n\t\\label{bp_cb_dis}\n\\end{figure}\n\n\\subsection{Beamformer in the UCA phase mode space}\n\n\\textcolor{black}{The frequency response of the UCA element space can be converted to phase mode space as\n\\begin{equation}\n\\begin{split}\n\\underline{H}_m(f) &= \\frac{1}{P}\\sum_{p=1}^{P}\\hat{G}_m(f) \\cdot \\exp(jm\\varphi_p) \\cdot H_{p}(f),\\\\\n\\end{split}\n\\label{eq:hm}\n\\end{equation}\nwhere $\\underline{H}_m(f) $ denotes the $m$-th mode response of the UCA in the phase mode space.\nAs discussed in \\cite{zhang2017fi}, $\\underline{H}_m(f)$ can be approximated to $\\underline{H}_m(f) = \\exp(jm\\phi_n)$ under far-field condition and with all paths confined in the UCA plane, i.e. $\\theta_n = 90^o, n\\in[1, N]$, where the compensation filter was defined as $\\hat{G}_m(f) = 1\/[j^m J_m(kr)]$ with $J_{m}(\\cdot)$ denoting the Bessel function of the first kind with\norder $m$. However, in practical propagation scenario, it is very unlikely that the incident paths are strictly limited to the UCA plane. Therefore, we modified the compensation filter for 3D propagation scenarios as \\cite{zhang2017fi},\n\\begin{equation}\n\\frac{1}{\\hat{G}_m(f)}= 0.5j^{m}[J_{m}(kr)-jJ_{m}^{'}(kr)],\\label{eq:Gm}\n\\end{equation}\n where\n  $(\\cdot)^{'}$ denotes the differential operator.}\n\n\\textcolor{black}{By taking equation (\\ref{eq:element frequency response}) into (\\ref{eq:hm}) and changing the order of the two summations, we have\n\\begin{equation}\n\\begin{split}\n\\underline{H}_m(f) =&  \\sum_{n=1}^{N} \\alpha_n \\exp(-j 2 \\pi f \\tau_n)\\\\\n                  & \\cdot \\bigg\\{\\frac{1}{P}\\sum_{p=1}^{P}\\hat{G}_m(f) \\exp(jm\\varphi_p)\\cdot a_p(f,\\mathbf{\\Theta}_n) \\bigg\\}\\\\\n                  =& \\sum_{n=1}^{N} \\alpha_n \\exp(-j 2 \\pi f \\tau_n) \\cdot \\underline{a}_m(f,\\mathbf{\\Theta}_n),\n\\end{split}\n\\label{eq:hm_1}\n\\end{equation}\nwhere $\\underline{a}_m(f,\\mathbf{\\Theta}_n)$ is given by the $\\{\\cdot\\}$ term in the above equation.\nComparing the above equation with equation (\\ref{eq:element frequency response}), we can see that for the $n$-th path, $\\underline{a}_m(f,\\mathbf{\\Theta}_n)$ is the UCA manifold of the $m$-th mode in the phase mode space and $a_p(f,\\mathbf{\\Theta}_n)$ defined in (\\ref{eq:a_p}) is the UCA manifold of the $p$-th element in the element space.  }\n\n\\textcolor{black}{The beam pattern of the FIBF can be written as\n\\begin{equation}\n\\underline{B}(f,\\phi)= \\frac{1}{2M+1} \\sum_{m=-M}^{M}  \\exp(-jm\\phi) \\cdot \\underline{H}_m(f), \\label{eq:aoa_fi}\n\\end{equation}\nwhere $\\exp(-jm\\phi)\/(2M+1)$ is the steering weight of the $m$-th phase mode.\nSubstituting the array phase mode response $ \\underline{H}_m(f)$ defined in (\\ref{eq:hm_1}) into the above equation and rearranging the summation order, we have\n\\begin{equation}\n\\begin{split}\n\\underline{B}(f,\\phi)&= \\sum_{n=1}^{N}  \\alpha_n \\exp(-j 2 \\pi f \\tau_n) \\cdot \\underline{v}_n(f,\\phi)\\\\\n                     &= \\sum_{n=1}^{N} \\underline{B}_n(f,\\phi), \\label{eq:aoa_fi_1}\n\\end{split}\n\\end{equation}\nwhere $\\underline{B}_n(f,\\phi)$ represents the FIBF beam pattern of the $n$-th path, i.e. the FIBF beam pattern of a single path. The FIBF beam pattern of the multiple paths is obtained by the linear superposition of the FIBF beam patterns of all paths. Similar to the unit beam pattern term $v_n(f,\\phi)$ of CBF,  $\\underline{v}_n(f,\\phi)$ indicates the unit FIBF beam pattern term of the $n$-th path, which is expressed by\n\\begin{equation}\n\\underline{v}(f,\\phi)= \\frac{1}{2M+1} \\sum_{m=-M}^{M} \\underline{a}_m(f,\\mathbf{\\Theta}_n) \\exp(-jm\\phi), \\label{eq:v_fi}\n\\end{equation}\n where $\\underline{a}_m(f,\\mathbf{\\Theta}_n)$ is given in (\\ref{eq:hm_1}).}\n \n \\added{Similar to the discussions about the unit beam pattern term $v_n(f,\\phi)$ of CBF in section \\ref{cb_ff}, we study the property of $\\arrowvert \\underline{v}_n(f,\\phi)\\arrowvert$ for various distance $D_n$. With the same simulation setting as for CBF in Fig. \\ref{bp_cb_dis}, we can plot the unit FIBF beam pattern $\\underline{v}_n(f,\\phi)$ in Fig. \\ref{bp_fi_dis}. The results show that the beam pattern is insensitive to phase error introduced by the spherical wavefront in the near-field conditions, where an approximately constant beam pattern is achieved for different distance $D_n$ within a large dynamic range.}\n \n \\added{We further investigate the power loss in the target direction under near-field conditoins for FIBF and the results are shown by the red dotted curve in Fig. \\ref{cbf_fibf_dis2}. It shows that the power values of the target direction are approximately unchanged with different distances $D_n$. } \n \n \\added{The unit FIBF beam pattern peaks in the target direction and the peak value keeps approximately constant (approximates to 1) for various distances setting, as shown in Fig. \\ref{cbf_fibf_dis2}. It indicates that the path can be accurately detected with FIBF method under either near-field or far-field conditions.}\n\n\\added{As mentioned earlier, the beamformer in the UCA phase mode space with the modified compensation filter works in 3D propagation scenarios when the elevation angle is not restricted to the UCA plane. As detailed in \\cite{zhang2017fi}, the beam patterns are approximately constant with different elevation angles. When the elevation angle gets away from the UCA plane, i.e. with $\\varDelta \\theta_n = |\\theta_n - 90^o|$  getting larger, the main beam peak drops slightly and the sidelobes of the beamforming pattern at $\\phi = \\phi_n \\pm \\pi$ becomes broader. }\n\nThe simulation results indicate that the beamformer in the UCA phase mode space with the modified compensation filter $\\hat{G}_m(f)$ can also be applied in near-field \\added{3D} scenarios, since the beamformer pattern is insensitive to the introduced phase errors. As discussed in \\cite{zhang2017fi}, the joint PADP can be directly obtained with the modified FIBF, which is simple, effective and robust. However, the resulting PADP suffers from high sidelobes,  as shown in Fig. \\ref{bp_fi_dis}. In the next section, a FIBF algorithm with the successive cancellation of the detected paths is detailed, with the objective to eliminate the high sidelobes of the dominant paths.\n\n\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 260bp 695bp 565bp,clip,scale=0.6]{cbf_fibf_dis2.pdf}}\n\t\t\\par\\end{centering}\n\t\n\t\\caption{The main beam peak varies with $D_n$ for CBF and FIBF, where $\\theta_n = \\pi\/2$ and $\\phi_n = \\pi$, \\added{$f=29$ GHz and $P=720$ are set for the UCA}.}\n\t\\label{cbf_fibf_dis2}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 260bp 695bp 565bp,clip,scale=0.6]{bp_fi_dis_new.pdf}}\n\t\t\\par\\end{centering}\n\t\n\t\\caption{The unit FIBF beam pattern $\\underline{v}_{n}(f,\\phi)$ in the UCA phase mode space for variant $D_n$ with $\\theta_n = \\pi\/2$, $\\phi_n = \\pi$, $r$ = 0.5 m, $f$ = 29 GHz and $P=720$. \\added{The calculated Fraunhofer far-field distance $D_{far} = 2(2r)^2\/\\lambda$ is around 193 m.}}\n\t\\label{bp_fi_dis}\n\\end{figure}\n\n\n\n\n\\section{Proposed beamspace UCA with the successive cancellation scheme for near-field scenarios}\n\nAs discussed, the beamspace UCA with the modified compensation filter $\\hat{G}_m(f)$ can achieve constant beam patterns, insensitive to  the distances between the scatterers and the UCA center, which makes it suitable for angle estimation in near-field scenarios. In this section, a novel  FIBF  based on successive cancellation principle to estimate azimuth angle, delay and power of each multipath component is proposed for near-field scenarios, where the paths are detected one by one with the power values in a  descending order. \n\nAs mentioned in paper \\cite{zhang2017fi}, the PADP with modified FIBF can be directly obtained by performing IFT of $\\underline{B}(f,\\phi)$ in (\\ref{eq:aoa_fi_1}) as,\n\\begin{equation}\n\\begin{split}\n\\underline{b}(\\tau,\\phi)&= \\sum_{f=f_1}^{f_L}\\underline{B}(f,\\phi) \\exp(j2\\pi f \\tau)  \\\\          \n&= \\sum_{n=1}^{N}\\sum_{f=f_1}^{f_L}\\underline{v}_n(f,\\phi) \\cdot \\alpha_n \\exp[j2\\pi f (\\tau-\\tau_n)]\\\\\n&=\\sum_{n=1}^{N} \\underline{b}_n(\\tau,\\phi),\n\\end{split}\n\\label{eq:padp}\n\\end{equation}\nwhere $\\underline{b}_n(\\tau,\\phi)$ is the PADP of the $n$-th path.\n\nIn the ideal case, the PADP mimics the Dirac delta function which peaks at the unique angle-delay positions, i.e. $(\\phi_n, \\tau_n)$ with desired power values. However, as discussed, $\\underline{v}_n(f,\\phi)$ presents strong sidelobes at angle around $\\phi_n \\pm \\pi$ for $n  \\in [1,N]$. Therefore, weak desired paths might be buried by undesired sidelobes of strong paths. Below, a novel algorithm following the successive cancellation principle is proposed to tackle this problem.\n\n\\textcolor{black}{The array element response vector $\\mathbf{H}(f)$ is defined as\n\\begin{equation}\n\\mathbf{H}(f) =  [H_1(f);...;H_P(f)],\\label{eq:h_el_v}\n\\end{equation}\nwhere $H_p(f)$ was defined in (\\ref{eq:element frequency response}).}\n\n\\added{\tIn the following algorithm description, we add the superscript numbers to the array element response vector $\\mathbf{H}(f)$ and PADP $\\underline{b}(\\tau,\\phi)$ to indicate that these terms need to update for each iteration. The superscript number $q$ denotes that $q$ path(s) are removed. For example, $\\mathbf{H}^{1}(f)$ denotes the array element response vector with 1 path removed and $\\underline{b}^2(\\tau,\\phi)$ represents the PADP with 2 paths removed. Typically, the original array element response vector and PADP are represented with superscript number $q=0$, i.e. $\\mathbf{H}^{0}(f)$ and $\\underline{b}^0(\\tau,\\phi)$, respectively. }\n\nThe procedure is detailed as below:\n\\begin{itemize}\n\\item[1.] Based on the \\added{current array element response vector $\\mathbf{H}^{0}(f)$}, we apply equations (\\ref{eq:hm}), (\\ref{eq:Gm}), (\\ref{eq:aoa_fi}) and (\\ref{eq:padp}) to obtian the current PADP \\added{$\\underline{b}^0(\\tau,\\phi)$}.\n\\item[2. ] \\added{We find the peak location in the current PADP $\\underline{b}^0(\\tau,\\phi)$, which gives the delay and azimuth angle estimation of path 1, i.e. $\\hat{\\tau}_1$ and $\\hat{\\phi}_1$, respectively. By taking the estimated delay and azimuth angle back to the original PADP $\\underline{b}(\\tau,\\phi)$, we can obtain the amplitude estimation of path 1, i.e. $\\hat{\\alpha}_1 = |\\underline{b}(\\hat{\\tau}_1,\\hat{\\phi}_1)|$. } \n\\item[3.] We remove \\added{path 1} from the array element response vector \\added{$\\mathbf{H}^0(f)$} to obtain the updated array element response $\\mathbf{H}^{1}(f)$ as detailed below.\n\\begin{itemize}\n\t\\item[3.1.] The frequency response vector of the UCA elements corresponding to path 1, can be synthesized based on the detected path parameters under plane-wave assumption as: \n\t\\begin{equation}\n\t\\hat{\\mathbf{H}}(f) =  \\hat{\\mathbf{a}}(f,\\hat{\\phi}_1)\\cdot \\hat{\\alpha}_1  \\exp(-j 2 \\pi f \\hat{\\tau}_1),\\label{eq:h_el_1}\n\t\\end{equation}\n\twhere $\\{\\hat{\\alpha}_1, \\hat{\\phi}_1, \\hat{\\tau}_1\\}$ are parameters estimated in step 2 and $\\hat{\\mathbf{a}}(f,\\hat{\\phi}_1) \\in \\mathbb{C}^{P\\times 1}$ is the array manifold under plane-wave condition for path 1. The $p$-th entry of $ \\hat{\\mathbf{a}}(f,\\hat{\\phi}_1)$ is given by\n\t\\begin{equation}\n\t\\hat{a}_{p}(f,\\hat{\\phi}_1) = \\exp[jkr \\cos(\\hat{\\phi}_1-\\varphi_p) ].\\label{eq:a_s_1}\n\t\\end{equation}\n\t\n\t\\item[3.2.] The synthetic channel impulse response (CIR) vector \\added{$\\hat{\\mathbf{h}}(\\tau)$} corresponding to \\added{path 1} over array elements and the \\added{current CIR vector ${\\mathbf{h}}^0(\\tau)$} can be directly obtained via performing IFT of \\added{$\\hat{\\mathbf{H}}(f)$ and ${\\mathbf{H}}^0(f)$}, respectively.  \n\t\n\t\\item[3.3.] Generate a label vector \\added{$\\mathbf{s}( \\tau)$} with the same size as $\\hat{\\mathbf{h}}(\\tau)$, where the $p$-th entry is obtained as\n\t\n\t\t\t\n   \\begin{equation}\n s_{p}(\\tau) = \\left\\{\n\t\\begin{array}{ll}\n\t0, & \\arrowvert\\hat{h}_{p}(\\tau)\\arrowvert > \\hat{\\alpha}_1 \\cdot 10^{\\frac{-\\eta_t}{20}} \\\\\n\t1, & otherwise \\\\\t\n\t\\end{array} \\right.\n\t\\label{eq:s}\n\t\\end{equation}\n\twhere \\replaced{$\\eta_t$} {} denotes the \\replaced{threshold value}{} in decibels \\added{ and $\\hat{h}_{p}(\\tau)$ the $p$-th entry of synthetic CIR vector $\\hat{\\mathbf{h}}(\\tau)$}. The entries of $\\hat{\\mathbf{h}}(\\tau)$ with dominant power values are labelled to $0$ in the label vector $\\mathbf{s}(\\tau)$.\n\t\n\t\\added{The objective of the label vector $\\mathbf{s}(\\tau)$ is to mark the trajectory of the estimated path in the CIRs over array elements. As explained in step 3.1 and 3.2, we can reconstruct the path trajectory over array elements by using the estimated delay, azimuth angle and amplitude of the path under plane-wave assumption. The label vector would enable us to remove the estimated path from the CIR over array elements in Step 3.4.}\n\t\n\t\\item[3.4.] \\added{Remove path 1 from the current CIR vector $\\mathbf{h}^0(\\tau)$ to obtain the updated CIR vector $\\mathbf{h}^1(\\tau)$ by}\n\t \\begin{equation}\n\t \\mathbf{h}^1(\\tau) = \\mathbf{h}^0(\\tau) \\odot \\mathbf{s}(\\tau), \\label{eq:h_el_11}\n\t \\end{equation}\n\t where $\\odot$ denotes element-wise multiplication.\n\t \n\t \\item[3.5.] By performing Fourier transform (FT) on the updated CIR vector $\\mathbf{h}^1(\\tau)$, we can obtain the updated array element response vector $\\mathbf{H}^{1}(f)$. \n\\end{itemize} \n \\item[4.] Repeat the above steps until the estimated power of the path is not within the preset dynamic range.\n\\end{itemize}\n\n\\added{Note that the superscript numbers of the array element response vector and PADP increase 1 for each iteration. For instance, in the $n$-th iteration, the current array element response vector and PADP are $\\mathbf{H}^{n-1}(f)$ and $\\underline{b}^{n-1}(\\tau,\\phi)$, respectively. After step 3.5, we obtain the path parameters $\\{ \\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}$ and the array element response vector is updated to $\\mathbf{H}^{n}(f)$. }\n\n\\added{It is also noted that in step 2, the amplitude estimation of the path is based on the original PADP $\\underline{b}(\\tau,\\phi)$ without superscript, which is kept unchanged for all iterations. Therefore, the amplitude estimations of all the paths are obtained based on the origianl PADP instead of the updated PADP.  The reason is that, when the paths e.g. path $n_1$ and $n_2$ with $n_1>n_2$, have similar or same delays, the trajectories of the two paths are overlapped. As a result, the trajectory of path $n_2$ will be partly removed as we intend to remove path $n_1$ in step 3.4. Thus the amplitude estimation of path $n_2$ based on the updated PADP will be underestimated. This will be further illustrated in the simulation section.}\n\nThe whole procedure of the proposed path estimation algorithm is summarized in Algorithm \\ref{euclid}. \\added{Note that $\\eta$ is the preset dynamic range in decibel.}\n\\begin{algorithm}\n\t\\SetKwInOut{Input}{Input}\n\t\\SetKwInOut{Output}{Output}\n\t\\caption{The proposed algorithm}\\label{euclid}\n\t\\Input{$\\mathbf{H}(f)$}\n\t\\Output{$\\{ \\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}, n \\in [1, N]$}\n\t\\BlankLine \n\t$n \\coloneqq 1$\\; \n\t\\replaced{$\\hat{\\alpha}_{n-1} \\coloneqq 1 $ , $\\hat{\\alpha}_{max} \\coloneqq 1 $ }{};  \n\t\t  \n    $\\mathbf{H}^{n-1}(f) \\coloneqq \\mathbf{H}(f)$, $\\mathbf{h}^{n-1}(\\tau) \\coloneqq  IFT(\\mathbf{H}(f))$\\;\n    \t\\While{ $\\hat{\\alpha}_{n-1} > \\added{(\\hat{\\alpha}_{max} \\cdot 10^{-\\eta\/20})} $}{\n  \t\tPerform phase mode space beamforming based on $\\mathbf{H}^{n-1}(f)$ and obtain the PADP $\\underline{b}^{n-1} (\\tau,\\phi)$.  \/\/ Eqs. (\\ref{eq:hm}), (\\ref{eq:Gm}), (\\ref{eq:aoa_fi}) and (\\ref{eq:padp})\\; \n    \tDetect the strongest path of PADP $\\underline{b}^{n-1} (\\tau,\\phi)$ and obtain the path parameters $\\{ \\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}$\\; \n    \tRemove the detected path from CIR vector $\\mathbf{h}^{n-1}(\\tau)$ and update the CIR vector to $\\mathbf{h}^{n}(\\tau)$ \\; \n    \t$\\hat{\\alpha}_{max} \\coloneqq  \\hat{\\alpha}_{1}$, $n \\coloneqq  n + 1$. \n    }\t \n\\end{algorithm}\t  \n\nThe basic principle of the low-cost successive cancellation scheme is that the propagation delay value among UCA array elements are insensitive to $D_n$ (i.e. distance between the array center and the $n$-th scatterer location) \\added{and elevation angles}. Therefore, we can remove the detected path based on the azimuth and delay values under the plane-wave assumption. To investigate the effectiveness of the idea, \\added{a single path scenario is considered here.} We can utilize the residual power rate $R_p$ \\added{to evaluate how effective the detected path is removed from the CIR,} which is defined as\n\\begin{equation}\nR_p = \\frac{ \\big\\lVert vec{ \\big\\{\\mathbf{h}^1(\\tau)  \\big\\}} \\big\\lVert ^2}{\\big\\lVert vec{ \\big\\{\\mathbf{h}(\\tau) \\big\\}} \\big\\lVert ^2}\\times 100 \\%,\\label{eq:r_p}\n\\end{equation}\n\\added{where $vec\\{\\cdot\\}$ represents vectorization of a matrix and $\\lVert \\cdot \\lVert $ indicates the Euclidean norm of the vector.}\n\n\\added{The high residual power rate means that the deviation between the trajectory of the synthetic CIR vector $\\hat{\\mathbf{h}}(\\tau)$ and the true CIR vector $\\mathbf{h}(\\tau)$ is large, and the trajectory of the detected path will not be properly removed in the updated CIR vector $\\mathbf{h}^1(\\tau)$. The consequence is that we will estimate the residual trajectory as the fake path if the residual power is within the preset dynamic range.}\n\n\\added{Below, we simulate a single path case to illustrate the residual power rate $R_p$ w.r.t bandwidth, distance $D$ and the elevation angle $\\theta_o$. In the simulation, a single path with a fixed incident azimuth angle $\\phi_0 = 180^o$ impinging at a UCA of radius $r = 0.5$ m is set. The UCA consists of 720 elements with half-wavelength spacing. Besides, we set the center frequency to $29$ GHz, the bandwidth from $400$ MHz to $3$ GHz, distance $D$ from 3 m to 30 m and elevation angle from $90^o$ to $120^o$. }\n\n\\added{The impact of system bandwidth, distance $D$ and elevation angle $\\theta_o$ on the residual power rate is shown in Fig. \\ref{b_dis}.  }\n\\textcolor{black}{\\begin{itemize}\n\t\\item For a given bandwidth and elevation angle $\\theta_o$, $R_p$ decreases as $D$ increases due to the fact that the larger the $D$ is, the closer we approximate the plane-wave condition, resulting in a smaller $R_p $.\n\t\\item For a given distance $D$ and elevation angle $\\theta_o$, $R_p$ increases as bandwidth broadens. The wider the bandwidth, the higher the delay resolution we have. As a result, the larger reconstructed CIR error will be detected as the bandwidth becomes wider, which leads to a larger $R_p$.\n\t\\item For a given bandwidth and distance $D$, $R_p$ increases when elevation angle $\\theta_o$ gets further away from the UCA plane, i.e. with $\\varDelta \\theta =\\arrowvert\\theta_o -90^o \\arrowvert $ becoming larger. This is due to the fact that the synthetic CIR was calculated with $\\hat{\\theta}_o = 90^o$ in (\\ref{eq:h_el_1}) and (\\ref{eq:a_s_1}). The larger the elevation angle away from the UCA plane is, the larger error will be introduced to the reconstructed CIR. The maximum $R_p$ are $0.06\\%$, $0.07\\%$, $0.16\\%$ and $1.1\\%$ when elevation angle $\\theta_o$ are set to $90^o$, $100^o$, $110^o$ and $120^o$, respectively.\n\\end{itemize}\n}\n\n\n\\added{For a practical measurement setting with bandwidth less than\n\t$3$ GHz, the scatterer distance larger than 3 m and the elevation angle $\\arrowvert\\theta_o -90^o \\arrowvert \\leqslant 30 ^o$, the residual power rate is up to $1.1 \\%$, as shown in Fig. \\ref{b_dis}. Therefore, the proposed algorithm works well for practical measurement settings. The proposed cancellation scheme is low-cost, since it only requires the estimated delay and azimuth angle values to remove the detected path. It is effective and robust, as demonstrated in the numerical simulations. }\n \n \n \\added{As a summary, the basic principle of the low-cost successive cancellation scheme is that the propagation delays among the UCA elements are insensitive to the elevation angle of the path and the distance between the UCA center and the scatterer location. Therefore, we can effectively remove the detected path based on azimuth angle and delay of the path in the updated delay profile among elements. Due to the sparsity of mmWave channels, typically only a few iterations are needed to extract all multipath components. As shown, for each iteration, we obtain the delay and azimuth angle estimate based on the power spectra, which essentially is calculated from the one-dimensional beamforming operation in (\\ref{eq:aoa_fi}) and IFT operation in (\\ref{eq:padp}). Therefore, the computational cost is significantly lower, compared to typical high resolution algorithms, where expensive joint search in multiple parameter domains is required.}\n \n \\begin{figure}\n \t\\begin{centering}\n \t\t\\textsf{\\includegraphics[bb=105bp 275bp 477bp 568bp,clip,scale=0.69]{el_D_B.pdf}}\n \t\t\\par\\end{centering}\t\n \t\\caption{\\added{The residual power rate $R_p$ varies with bandwidth, distance $D$ and elevation angle $\\theta_o$, where $f_c = 29$ GHz, $\\phi_o = \\pi$ and $\\tau_o = 0$ ns.}}\n \t\\label{b_dis}\n \\end{figure}\n \n\n\\section{Simulation results} \\label{simulation}\n\\added{In the simulation, we consider a UCA composed of 720 isotropic antennas with radius $0.5$ m and half-wavelength element spacing. The frequency band is from $28$ - $30$ GHz with $750$ frequency points. We simulate a representative yet critical scenario for the channel estimation algorithm, where path 1 and path 3 have the same impinging azimuth angle $\\phi$ at $90^o$. Path 2 has an incident angle of $270^o$, yet it has the same delay as path 3. In addition, the elevation angles are not strictly confined to the UCA plane as detailed in Table \\ref{tab:table1}.}\nThe critical scenario is intentionally set to demonstrate the robustness of the algorithm.\n\n\\replaced{Three beamforming algorithms, i.e. CBF \\cite{fan2016eurasip}, FIBF \\cite{zhang2017fi} and the proposed algorithm, are compared in Fig. \\ref{ad_sim}.}{} As shown in Fig. \\ref{ad_sim} (top), though target paths can be \\added{roughly} detected in the PADP, the CBF algorithm under far-field assumption presents two major drawbacks as explained in Section \\ref{cb_ff}, i.e. susceptible to strong joint side lobes and main lobe distortions. The power loss of the main lobes up to around \\replaced{$13$} {} dB and the concave main lobe (path 1) can be observed in the figure due to the far-field CBF applied in the near-field scenario. The detailed of the concave main lobe can be observed in Fig. \\ref{bp_cb_dis} for $D_n = 3$m. Note that the joint side lobes for the weak path (i.e. path 3) are not shown in the figure due to the limited dynamic range ($40$ dB) set in the simulation. The PADP with the FIBF algorithm is shown in Fig. \\ref{ad_sim} (middle). It shows that the main lobes are not distorted and the peak values are accurate as explained in Fig. \\ref{bp_fi_dis}. However, as discussed, the FIBF suffers from strong side lobes around azimuth angle $\\phi_n \\pm \\pi$ for $n \\in [1,N]$ with a shifted delay. The estimated parameters $\\{\\hat{\\alpha}_n, \\hat{\\phi}_n, \\hat{\\tau}_n\\}, n \\in [1,N]$ with the proposed algorithm are shown in Fig. \\ref{ad_sim} (bottom). The estimated parameters agree well with the targets, with a deviation in path power within $0.3$ dB. The small power deviation is caused by the power variance over the UCA elements in the near-field scenario. \n\\added{Based on the Friis free space propagation equation, the maximal power deviation over the UCA elements for a single path case can be evaluated by the ratio of the maximal and minimal power across the UCA elements, i.e., $\\rho = \\frac{D+r}{D-r}$,  where $r$ denotes the radius of the UCA and $D$ the distance between the scatterer and the UCA center with $D>r$. In the far-field case (i.e. $D\\gg r$), we have $\\rho \\approx 1 $. However, the power over the UCA elements varies in the near-field case. }\n\nBelow, we detail the procedure how the path parameters are estimated with the proposed algorithm.  \n\nTo detect the most dominant path, we can follow the procedure below: \n\t\\begin{itemize}\n\t\t\\item[1.] We can perform the phase mode beamforming based on $\\mathbf{h}^{0}(\\tau)$ (i.e. the raw CIR vector $\\mathbf{h}(\\tau)$), as shown in Fig. \\ref{el_delay} (top), and the obtained PADP $\\underline{b}^0 (\\tau, \\phi)$ is shown in Fig. \\ref{bp_fi} (top).\n\t\t\\item[2.] From the PADP, we can detect the strongest path, i.e. path 1 as shown in Fig. \\ref{bp_fi} (top) with path parameters $\\hat{\\alpha}_1 = 0.2$ dB, $\\hat{\\phi}_1 = 90^o$ and $\\hat{\\tau}_1 = 16.6$ ns.\n\t\t\\item[3.] Then, we remove path 1 from the original CIR vector $\\mathbf{h}^0(\\tau)$ and obtain the updated CIR vector $\\mathbf{h}^1(\\tau)$ as shown in Fig. \\ref{el_delay} \\added{(upper-middle)}.\n\t\t\\item[4.] In the end,  $\\hat{\\alpha}_{max} = \\hat{\\alpha}_1$ is set.\n\t\\end{itemize} \n\\added{We can repeat the above procedure to detect the second and third paths. The detected path parameters of path 2 and path 3 are $ \\{\\hat{\\alpha}_2 = -2.8$ dB,  $\\hat{\\phi}_2 = 270^o,\\hat{\\tau}_2 = 40.1 $ ns $\\}$ and $ \\{\\hat{\\alpha}_3 = -18.1$ dB,  $\\hat{\\phi}_3 = 90^o,\\hat{\\tau}_3 = 40.1$ ns $\\}$, respectively.}\nIn the end, a path with power value within $40$ dB dynamic range can not be found based on $\\mathbf{h}^{3}(\\tau)$ and therefore the channel estimation procedure is complete.\n\n\\added{We can clearly see that the detected path is removed in the updated CIR, e.g. path 1 is removed in the updated CIR vector $\\mathbf{h}^{1}(\\tau)$ as shown in \\ref{el_delay} (upper-middle). Thus the influnce of the path is also eliminated in the updated PADP $\\underline{b}^1 (\\tau, \\phi)$ as shown in \\ref{bp_fi} (upper-middle). In this simulation, path 2 and 3 have the same delays and therefore the trajectories of the two paths are overlapped, as shown in Fig. \\ref{el_delay} (upper-middle). When we remove path 2, the trajectory of path 3 will be partly removed, as illustrated in Fig. \\ref{el_delay} (lower-middle). As discussed in section III, to avoid underestimating the amplitude of path 3 $\\hat{\\alpha}_3$, we estimate the amplitude based on the original PADP $\\underline{b}(\\tau,\\phi)$ as shown in Fig. \\ref{ad_sim} (middle).}\n\n\n\n\\begin{table}\n\t\\caption{Path parameters}\n\t\\label{tab:table1}\n\t\\begin{centering}\n\t\t\\begin{tabular}\t{|c|c|c|c|}\t\t\n\t\t\t\\hline \n\t\t\tPath & $1$  & $2$  & $3$  \\\\ \n\t\t\t\\hline \n\t\t\t$\\alpha$ [dB] & 0 & \\added{-3}   & \\added{-18} \\\\ \n\t\t\t\\hline \n\t\t\t$\\phi$ [deg] &90  &270   &90\\\\ \n\t\t\t\\hline \t\n\t\t\t$\\theta$ [deg] &90  &95  &100 \\\\ \n\t\t\t\\hline\t\n\t\t\t$D$ [m] &4.98   &\\added{12}   &12\\\\ \n\t\t\t\\hline\t\n\t\t\t$\\tau$  [ns] &16.6  &\\added{40.0} &40.0   \\\\ \n\t\t\t\\hline\n\t\t\\end{tabular} \n\t\t\\par\\end{centering}\n\\end{table}\n\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=95bp 225bp 500bp 630bp,clip,scale=0.65]{angle_delay_1_revised.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{The PADPs with CBF (top), FIBF (middle) and the proposed successive FIBF (bottom).}}\n\t\\label{ad_sim}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=80bp 180bp 500bp 665bp,clip,scale=0.58]{element_delay_1_revised_new.pdf}}\n\t\t\\par\\end{centering}\t\n\t\\caption{\\added{The CIR vectors of $\\mathbf{h}^{n-1}(\\tau), n \\in [1,4]$, where the superscript denotes the $(n-1)$ path(s) are removed.}}\n\t\\label{el_delay}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=70bp 185bp 530bp 675bp,clip,scale=0.58]{bf_fi_1_revised_new.pdf}}\n\t\t\\par\\end{centering}\t\n\t\\caption{\\added{The PADPs based on $\\mathbf{h}^{n-1}(\\tau), n \\in [1,4]$, where the superscript denotes the $(n-1)$ path(s) are removed.}}\n\t\\label{bp_fi}\n\\end{figure}\n\n\n\n\n\n\n\\section{Measurement Results}  \\label{measurement}\n\n\\subsection{Introduction}\n\nTo verify how well the proposed algorithm works in practice, we need\nto validate it with practical measurements. The detailed description\nof the measurement campaign was given in \\cite{fan2016eurasip} and only\noutlined here. The measurements were conducted in a typical indoor\nbasement \\added{with the floor dimensions of $7.7$ m $\\times$ $7.9$ m. The basement was empty with few objects including a metallic heater and a metallic ladder leaned against the wall.}\nBoth line-of-sight (LOS) and obstructed LOS (OLOS)\nscenarios were considered. The OLOS scenario was created by placing\na $1.2$ m $\\times$ $1.2 $ m metallic blackboard to block\npaths in LOS directions.\n\n\\added{A wideband biconical antenna with a gain of $6$ dB at 28-30 GHz was used at\n\tthe transmit (Tx) side. The Tx antenna was mounted $0.84$ m above the floor. While an identical biconical antenna was exploited at the receive (Rx) side.\n\tThe Rx antenna was mounted on a turntable with the same height as the Tx antenna. The distance between the Rx antenna and the rotation center was adjusted to $0.5$ m. Then a virtual UCA of radius $r=0.5$ m at Rx side were obtained by rotating the Rx antenna on the turntable. The frequency response of the $p$-th UCA element was measured when the Rx antenna was rotated to the angular position $\\psi_p = 2\\pi\\cdot(p-1)\/P, p\\in [1,P]$ with $P = 720$. \n\tFor each virtual UCA element, the frequency\n\tresponse was measured with a vector network analyzer (VNA) from 28-30\n\tGHz with 750 frequency points. }\n\n\\added{The rotational horn antenna\n\tmeasurements were used as a reference. In the rotational horn antenna measurements, the biconical antenna at the Rx side was replaced by a horn antenna with a gain of $19$ dB at 28-30 GHz. The horn antenna was positioned\tat the rotation center of the turntable (i.e. $r=0$ m) with the same height as the Tx antenna (i.e. $0.84$ m above the floor). The same measurement settings (the same Tx antenna, the same frequency sweep and orientation sweep) adopted in biconical antenna UCA measurements were used in rotational horn antenna measurements for comparison purpose. }\n\nNote that mutual coupling effect is not present with the virtual array measurement, which is desirable\nfor the  channel characterization purpose. The distance between the Tx\nand the center of the Rx array is around 5 m, while the far-field\ndistance at 30 GHz for the UCA is around 200 m. The system bandwidth\nin the measurement is larger than 500 MHz and the UCA array aperture\n(i.e. 1m) is  much larger than the delay resolution multiplied by\nthe speed of light (i.e. 0.15 m). Therefore, for the measurement data,\nboth the far-field assumption and narrowband assumption are violated.\nNote that \\added{the antenna gains of both Tx and Rx antennas are de-embedded in measured CIRs. Further, to focus on the specular and dominant multipath components detection,} a dynamic range of 30 dB is set in the measurement section. \n\n\\subsection{Measured results}\n\nThe measured CIRs over virtual UCA elements (i.e. measured locations) for the LOS scenario are shown\nin Fig. \\ref{LOS_synthetic} (top), where a few specular paths can\nbe clearly detected besides the dominant LOS path. \\added{The mmWave channels are more sparse compared to sub-6GHz channels. In our measurements, the measurement was performed in an empty indoor basement, with no furniture, which also results in sparse channel profiles.} The synthetic CIRs\nover UCA elements  for the LOS scenario are shown in Fig. \\ref{LOS_synthetic}\n(below). The synthetic CIRs over virtual UCA elements are reconstructed based on detected multipath component parameters  $\\{\\hat{\\alpha}_{n},\\hat{\\phi}_{n},\\hat{\\tau}_{n}\\}$\nfor $n\\in[1,N]$ under the plane-wave assumption. The synthetic results agree well with the measured data,\nindicating a consistent estimation result. The \\added{trajectories of the paths} over UCA elements match well with the measured ones, even for weak multipath components. The measured PADP\nwith rotational horn antenna is shown in Fig. \\ref{LOS_horn} (top).\nAs shown in the measured results, paths having the same impinging angle\nyet different delays exist due to the path  bouncing in the LOS direction.\nThe estimated PADP based on the virtual UCA with\nthe proposed algorithm for the LOS scenario is shown in Fig.\n\\ref{LOS_horn} (below), \\added{where in total 10 paths are detected}. An excellent match of the measured PADPs between the rotational\nhorn antenna and virtual UCA in terms of the number of propagation paths,\nazimuth angle, delay and power of each path can be observed. However, the\nmeasured results based on rotational horn antenna suffer from wide\nantenna beam-width, as expected. The proposed algorithm presents consistent\nparameter estimation, with high resolution in the angle and delay\ndomains. \n\\added{ Note that the antenna gains of the horn antenna and biconical antenna are calibrated out in the power spectra. Thus within the same power range, the same number of paths and approximately same path parameters can be observed in the plots. Furthermore, the estimated power of the LOS path (strongest path) is $-76.5 $ dB, which matches well with the calculated path loss according to Friis equation with $D = 5$ m and $f_c = 29$ GHz, i.e. $-75.7$ dB. The deviation might be introduced by the inaccureate data in antenna gains and measurement uncertainties.}\n\nThe measured CIRs and synthetic CIRs over virtual UCA elements for the\nOLOS scenario are shown in Fig. \\ref{OLOS_synthetic}. A good agreement\nbetween the synthetic and measured CIRs can still be observed,\nthough \\added{there exists many weak multipath components in }the measured CIRs. The measured PADPs with rotational horn\nantenna and virtual UCA with the proposed algorithm for the OLOS scenario\nare shown in Fig. \\ref{OLOS_horn}, \\added{where $27$ paths in total are detected within the dynamic range of $30$ dB. Within the same power range, the channel parameters, e.g. the number of paths and path paramters, agree well between the rotational horn and the virtual UCA for\n\tthe more critical OLOS scenario. }\n\n\\added{As observed in the measured CIRs over array elements, channel non-stationarity exists where different channel profiles can be observed by different array elements. This can be caused by several reasons, e.g. the power variation over the UCA elements in the near-field scenarios as explained; the coherent summation of unresolved multipath components due to limited system bandwidth; limited angle of view from the near-field scatterers and the measurement system non-idealities. The channel non-stationarity is not addressed in the proposed algorithm, as seen in the synthetic CIRs over UCA elements. Though channel non-stationarity has been considered in channel modeling works, e.g. \\cite{he2018mobility}, it has not been considered in the channel estimation in the literature so far due to high computation complexity.  }\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 567bp,clip,scale=0.58]{los_syn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured CIRs (top) and synthetic CIRs (below) over virtual UCA elements\n\t\tfor the LOS scenario. }}\n\t\\label{LOS_synthetic}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 550bp,clip,scale=0.58]{los_horn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured PADPs with rotational horn antenna (top)\n\t\tand virtual UCA with the proposed algorithm (below) for the LOS scenario. }}\n\t\\label{LOS_horn}\n\\end{figure}\n\n\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 567bp,clip,scale=0.58]{nlos_syn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured CIRs (top) and synthetic CIRs (below) over virtual UCA elements\n\t\t\tfor the OLOS scenario. } }\n\t\\label{OLOS_synthetic}\n\\end{figure}\n\n\\begin{figure}\n\t\\begin{centering}\n\t\t\\textsf{\\includegraphics[bb=100bp 270bp 480bp 550bp,clip,scale=0.58]{nlos_horn_30db_1.pdf}}\n\t\t\\par\\end{centering}\n\t\\caption{\\added{Measured PADPs with rotational horn antenna (top)\n\t\tand virtual UCA with the proposed algorithm (below) for the OLOS scenario.} }\n\t\\label{OLOS_horn}\n\\end{figure}\n\n\n\n\n\n\n\\section{conclusion}\nAccurate knowledge of the radio propagation parameters is important for system design, applications and performance evaluation of the 5G systems. However, multipath parameter estimation  for UWB large scale antenna systems are challenging,  due to the fact that the well adopted plane-wave  and narrowband assumptions might not hold. In this paper, a novel beamspace UCA algorithm based on phase mode excitation principle is proposed. The proposed beamformer can maintain approximately same beam patterns, independent of distance between the array and scatterer location, and of the system bandwidth, making it suitable for UWB near-field scenarios. The proposed algorithm has low computational cost since it avoids expensive joint estimation in multiple parameter domains. To remove the strong sidelobes of the proposed beamspace beamformer, a novel algorithm based on the successive cancellation principle is proposed. The path cancellation is based on the fact that propagation delays among array elements are insensitive to system bandwidth and near-field effects \\added{for a given elevation angle range, i.e. $\\varDelta \\theta = \\arrowvert \\theta - 90^o\\arrowvert \\leq 30^o$}. The cancellation scheme is effective and robust. For example, the residual power rate of less than 0.2\\% can be achieved for the UCA with radius $r = 0.5$m, system bandwidth 2 GHz and measurement range  $D\\geqslant 3$ m.  To demonstrate the proposed algorithm, a critical scenario is selected in the numerical simulation \\added{with frequency band set the same as in the measurement campaign, i.e. 28-30 GHz}, and the results showed that all the paths can be accurately detected with less than 0.3 dB deviations for the path powers. To validate the algorithm, we applied the proposed algorithm in the practical virtual UCA measurement data in both LOS and OLOS scenarios. The synthetic CIRs obtained based on the detected multipath parameters matched well with the measured CIRs for both measurement scenarios. The detected parameters were further validated against the horn antenna reference measurements. As a summary, both numerical simulations and experimental measurements demonstrated the effectiveness and robustness of the proposed algorithm. The proposed algorithm is a general low-cost channel estimator, since it works in both near-field scenario and UWB system.\n\n\\added{Further, due to the fact that the proposed beamformer pattern is not sensitive\n\tto the elevation angle and scatterer location, the proposed\n\talgorithm would fail to detect the elevation angle and scatter\n\tlocation. A high resolution propagation parameter estimation\n\talgorithm, which is capable of estimating all\n\tpropagation parameters in the 3D near-field conditions is missing\n\tin the literature due to the fatal computation complexity.\n\tOur proposed algorithm, which offers high resolution azimuth\n\tangle, delay and complex amplitude estimation in a low-cost\n\tmanner, can be utilized in the initial stage of the full parameter\n\testimation algorithm, e.g. the maximal likelihood estimator,\n\twhich can significantly reduce the computation complexity\n\tdue to the reduced searching space. This work will be carried out in a future work. }\n \n\n\n\n\n\\bibliographystyle{IEEEtran}\n\\addcontentsline{toc}{section}{\\refname}","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplemental Material}\\setcounter{page}{1}\n\nStarting from the Lindbladian $\\mathcal{L}_\\text{full}$ given in\nEq.~\\eqref{eq:lfull} in the main text, we show how to derive the effective\nmodel describing the slow dynamics. The first step is to define a set of\n(super-)operators, ${O}_q = {O}_+ - {O}_-$ and ${O}_c = ({O}_+ + {O}_-)\/2$\nwith ${O}_+(\\rho)$ = ${O}\\rho$ and ${O}_-(\\rho)=\\rho {O} $ where $\\rho$ is the\ndensity matrix of the system.  Using these definitions for the ladder\noperators, we obtain the commutators $[a_q,a_c^\\dagger] =  [a_c,\na_q^\\dagger]=1 $ (while the rest of the commutators of $a_q, a_c, a_q^\\dag,\na_c^\\dag$ vanish).\n\nAt first, we diagonalize the quadratic part of the Lindbladian\n$\\mathcal{L}_\\text{quad}$  for $s=0$ and without the stabilizing potential in\nthe Hamiltonian $H$ by a symplectic diagonalization to conserve the bosonic\nstructure \\cite{Prosen:10}.  The Lindbladian assumes the form\n\\begin{equation}\n  \\mathcal{L}_\\text{quad} = -\\tfrac12(\\gamma-\\epsilon) v_su_s-\\tfrac12(\\gamma +\\epsilon)v_fu_f\n\\end{equation}\nin terms of ladder operators $v_j, u_j$ such that $[u_j, v_k]=\\delta_{jk}$.\nExplicitly, they are given by\n\\begin{align}\\label{eq:basis}\n  u_s &= x_c +i  \\frac{\\gamma (2\\bar n+1)}{2(\\gamma-\\epsilon)}y_q,&  v_s &= -iy_q, & u_f &=  y_c -i \\frac{\\gamma (2\\bar n+1)}{2(\\gamma+\\epsilon)}x_q,  &v_f &=ix_q,\n\\end{align}\nwhere we introduced the two quadratures $x_{q,c} = (a_{q,c}^\\dagger + a_{q,c})\/\\sqrt{2}$ and\n$y_{q,c} = i(a_{q,c}^\\dagger - a_{q,c})\/\\sqrt{2}$ with $[x_{c},y_{q}] =[ x_{q}, y_{c}]=\ni$. Note, however, that the creation and annihilation operators, $v_j$ and $u_j$, are\nnot the adjoint of each other which is due to the fact that the Lindbladian $\\mathcal{L}_\\text{quad}$ is not Hermitian. \n\nThe slow dynamics of the system is given by the $u_s, v_s$ (or $x_c,y_q$\nrespectively) while $u_f,v_f$ correspond to a mode that decays with the rate\n$\\gamma$ at threshold. The spectrum of the system is given by $\\lambda =\n-\\tfrac12(\\gamma-\\epsilon) n_s -\\tfrac12(\\gamma+\\epsilon) n_f $ with $n_s, n_f \\in \\mathbb{N}_0$. The corresponding right eigenstates are the product states\n$|n_s, n_f\\rangle = |n_s\\rangle |n_f\\rangle$ with $u_j |n_j\\rangle =\n\\sqrt{n_j} |n_j-1\\rangle$ and $v_j |n_j\\rangle = \\sqrt{n_j+1} |n_j+1\\rangle$.\nThe left eigenstates  similarly are characterized by $\\langle n_j| v_j =\n\\sqrt{n_j} \\langle n_j-1|$ and $\\langle n_j| u_j = \\sqrt{n_j+1} \\langle\nn_j+1|$. Note that in the superoperator formalism $|0,0\\rangle$ corresponds to\nthe stationary state $P_s$ and $\\langle 0,0|$ is the trace operation.\n\nNext, we can take the nonlinearity given by the stabilizing potential $V$ into\naccount. It produces the additional term $\\mathcal{L}_V = - i(V_+ - V_-)$ with\n$\\mathcal{L}_\\text{full} = \\mathcal{L}_\\text{quad} + \\mathcal{L}_V + s f\\gamma\na_+ a_-^\\dag $.  The small  parameters $\\alpha$ allows to treat\n$\\mathcal{L}_V$ perturbatively. In particular, we are only interested in the\neffect of $\\mathcal{L}_V$ on the slow mode as the fast mode remains in the\nstationary state $|n_f=0\\rangle$.  Since we are close to threshold, we\nalso set $\\epsilon = \\gamma$ in $\\mathcal{L}_V$.\n\nFor the Josephson oscillator, we can simply project $\\mathcal{L}_V$ onto the subspace with $n_f =0$. Using the expressions \\eqref{eq:basis}, we obtain\n\\begin{equation}\n  \\mathcal{L}_V \\approx \\langle n_f =0 | \\mathcal{L}_V |n_f = 0\\rangle= \\tfrac1{16} \\kappa \\gamma iy_qx_c + \\tfrac1{48}\\kappa\\gamma\\bigl(i y_qx_c^3 + \\tfrac14i y_q^3x_c\\bigr)\\,.\n\\end{equation}\nThe first term corresponds to a small shift of the threshold. This shift,\nwhich vanishes for $\\alpha \\to 0$, can be seen in Fig.~\\ref{fig:fano}. The\nmodel is then given by $y_q \\mapsto p\/x_*$ and $x_c \\mapsto\nx_* q$ with a rescaling factor $x_* \\gg 1$ that is specified in the main\ntext. As discussed in the main text, we have $y_q \\simeq x_*^{-1}$ and $x_c\n\\simeq x_*$ and thus $y_q \\ll x_c$ for $\\alpha \\to 0$. Due to this,  the term\n$i y_qx_c^3$ is the most relevant nonlinearity. We can thus neglect $y_q^3\nx_c$ which allows to identify $m=2$ and $\\alpha = \\kappa\/48$ for the Josephson\noscillator.\n\nThe case of the Duffing potential needs more careful treatment due to the\nrotational symmetry as noted in the main text. In particular, we have $\\langle\nn_f =0 | \\mathcal{L}_V |n_f = 0\\rangle= 0$ due to this\nsymmetry.\nIn order to find the stabilizing effect, we need to include virtual\nexcitations of the fast mode in second-order perturbation theory. We obtain ($\\bar n =0$ for simplicity)\n\\begin{align}\\label{eq:lduff}\n  \\mathcal{L}_V &\\approx \\langle n_f =0 | \\mathcal{L}_V |n_f = 0\\rangle -\\sum_{n_f'> 0}\\frac{\\langle n_f =0 | \\mathcal{L}_V |n_f'\\rangle\\langle n_f'| \\mathcal{L}_V |n_f=0\\rangle}{ \\gamma n_f'} \\\\\n                &=  \\chi^2 \\gamma\\bigl( -\\tfrac{31}{16} iy_qx_c - \\tfrac{13}{4} iy_qx_c^3  + iy_qx_c^5 + \\tfrac{5}{16} y_q^2  - \\tfrac{5}{8} y_q^2x_c^2 + \\tfrac{1}{4} y_q^2x_c^4  + \\tfrac{13}{16} iy_q^3x_c    - \\tfrac{5}{32} y_q^4   + \\tfrac{1}{8} y_q^4x_c^2 + \\tfrac{1}{64} y_q^6  \\bigr).\\nonumber\n\\end{align}\nAs before, the rescaling $y_q \\mapsto p\/x_*$ and $x_c \\mapsto\nx_* q$ leads to $y_q \\ll x_c$ in the limit of weak nonlinearities. Because of this, the most relevant nonlinearity is the term $i \\chi^2 \\gamma y_q x_c^5$. This allows to identify $m=4$ and $\\alpha = \\chi^2$ for the Duffing oscillator. Note that the term $\\propto y_q x_c^3$ has the wrong sign and does not stabilize the mode. Moreover, it is subleading as $y_q x_c^3 \\propto x_*^2$ while $y_q x_c^5 \\propto x_*^4$. While the result \\eqref{eq:lduff} is shown for $\\bar n=0$ for simplicity, the relevant term  $i\\chi^2 \\gamma y_qx_c^5$ is in fact independent of $\\bar n$. \nThe counting field leads to the additional term\n\\begin{equation}\n  s f \\gamma a_+ a_-^\\dag = \\frac{sf\\gamma}2 \\Bigl[ (x_c + \\tfrac{i}2 y_q)^2 + (y_c -\\tfrac{i}2 x_q)^2 \\Bigr]\\,.\n\\end{equation}\nIn the limit $\\alpha \\ll 1$, the dominant term is given by  $\\frac12sf\\gamma \nx_c^2 = \\frac12sf\\gamma  x_*^2 q^2 $ as stated in the main text.\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\\noindent\nA Gause type predator-prey model with response\nfunction $f(x)$ is given by\n\\begin{equation}\\label{ppnodelay}\n\\left\\{\n\\begin{aligned}\n&\\dot x(t)=rx(t)\\bigg(1-\\frac{x(t)}{K}\\bigg)-y(t) f(x(t)),\\\\\n&\\dot y(t)=-sy(t)+Y y(t) f(x(t)),\n\\end{aligned}\\right.\n\\end{equation}\nwhere $x(t)$ denotes the density of the prey population and $y(t)$\nthe density of predators. Parameters $r,\\ K,\\ s$, and $Y$ are\npositive constants denoting the intrinsic growth rate and the\ncarrying capacity of the prey, the death rate of the predator in the\nabsence of prey, and the growth yield constant for the conversion of prey\nto viable predator density, respectively.\n\nIf  $f(x)$ is\nof Holling type I form in model (\\ref{ppnodelay}) (i.e. $f(x)=mx$ where\n$m$  is a positive constant denoting the maximal growth rate of\nthe predator), it is well-known (see e.g.\n \\cite{Bazykin,Freedman1980}) that  either\nthe predator population approaches extinction and the prey population\napproaches its carrying capacity, or the predator population and the\nprey population coexist and their density approaches a positive\nequilibrium. Hence, for\nall choices of the parameters, all solutions of this system approach a\nglobally asymptotically stable equilibrium, and so any nontrivial\noscillatory behaviour that arises due to the introduction of delay in\nthe model\ncan be attributed solely to the delay. For this reason, we choose\nHolling type I response functions instead\nof the more realistic Holling type II form, since the Holling type II\nform  results in a model that gives rise to  nontrivial period solutions  without delay (see Rosenzweig \\cite{Rosenzweig1971}).\nOne would also expect that any  exotic dynamics that the model with  Holling\ntype I form admits due to the introduction of delay would  be shared by\nthe model with Holling type II form.   Li et al. \\cite{MR3180725} studied this model with the Holling type II\nresponse function of Monod form and showed that stability switches caused by\nvarying the time delay are accompanied by bounded global Hopf branches, and they proved that when multiple Hopf branches exist, they are nested and the overlap produces coexistence of two or possibly more stable limit cycles.\nHowever, they did not go on to discover the even richer  dynamics that\nour analysis suggests exists in that case.\n\nIncorporating a time delay in (\\ref{ppnodelay}) to model the time between the\ncapture of the prey by the predator and its conversion to viable\npredator biomass, in the case of Holling type I functional response,\n$f(x)=mx, \\ m>0$, we  obtain the following system:\n\\begin{equation}\n\\left\\{\\begin{aligned}\n&\\dot x(t)=rx(t)\\left(1-\\frac{x(t)}{K}\\right)-m y(t)x(t),\\\\\n&\\dot y(t)=-sy(t)+Ye^{-s\\tau}my(t-\\tau)x(t-\\tau). \\label{ppreydelay}\n\\end{aligned}\\right.\n\\end{equation}\nThe term $e^{-s\\tau}y(t-\\tau)$ represents those predators that\n survive the $\\tau \\geq 0$ units of time required to process the\n prey captured at time  $t-\\tau$  in the past. Thus, we have incorporated the delay in the  growth term of the predator equation in a manner that is\nconsistent with  its  decline rate given by the model, as described in Arino,\nWang, and Wolkowicz \\cite{Gail2006Alter}.\n\nWe define\n$\\mathbb{R}_+\\equiv\\{x\\in\\mathbb{R} : \\, x\\geqslant 0\\}$,\nint$\\mathbb{R}_+\\equiv\\{x\\in\\mathbb{R} : \\, x>0 \\}$.   We denote by\n $C([-\\tau,0],  \\mathbb{R}_+)$,  the Banach space of continuous functions   from\n the interval $[-\\tau,0]$ into  $\\mathbb{R}_+$, equipped with the uniform norm.\n We assume initial data for model (\\ref{ppreydelay}) is taken from\n \\begin{equation}\\label{eq:X}\n X= C([-\\tau,0],  \\mathbb{R}_+) \\times  C([-\\tau,0],  \\mathbb{R}_+).\n\\end{equation}\n\nModel (\\ref{ppreydelay}) also has other interpretations.\nGourley and Kuang \\cite{Kuang2004} studied\na stage-structured predator-prey model\nin which they included an equation for the juvenile\npredators and assumed a constant maturation time\ndelay, i.e.,  they assumed that the juvenile predators\ntake a fixed  time to mature.\nUsing the approach developed in Beretta and Kuang\n\\cite{Kuang2002},\nthe authors considered the\npossibility of stability switches, and concluded that\nthere is a range of the parameter modeling the time delay for which\nthere are periodic solutions.\n If the juveniles in their model suffer the\nsame mortality rate as adult predators, their model decouples and\nyields model (\\ref{ppreydelay}).\nForde \\cite{forde2005delay} also considered this model and conjectured\nthat there are periodic orbits whenever the interior equilibrium exists\nand is unstable. He also noted that if the interior equilibrium\nexists and is asymptotically stable without delay, then for small\ndelays it remains globally asymptotically stable. We show that whenever the\ninterior equilibrium exists and is unstable, the system is uniformly\npersistent. Gourley and Kuang\n\\cite{Kuang2004} had already showed that a Hopf bifurcation eventually occurs\nif the delay is increased, destabilizing this equilibrium and  giving\nbirth to a nontrivial periodic solution. Forde  \\cite{forde2005delay}\n left as an open question whether more than one periodic orbit is\n possible and provided a numerical example suggesting chaos is possible\n but does not consider the route to chaos.\n We give numerical evidence that there is a range of parameters for which\n two stable periodic orbits and an unstable periodic orbit all exist and\nwe show a period-doubling route to chaos followed by a\n period-halfing route back to stability of the interior equilibrium.\n\n Cooke, Elderkin, and Huang \\cite{Cooke2006} considered a model\nsimilar to the one in Gourley and Kuang \\cite{Kuang2004}, and\nobtained results concerning Hopf bifurcation of a scaled version.\nThe scaling they used  eliminated the parameter modelling the time\ndelay, the parameter that we focus on and use as a bifurcation\nparameter.  This simplified their analysis,\nsince then, unlike in our case, the components of the coexistence\nequilibrium are independent of the time delay.\n\n\n\n\nIn this manuscript we show that the introduction of\ntime delay cannot only destabilize the  globally asymptotically stable\ncoexistence equilibrium of model (\\ref{ppreydelay}),  it can also be responsible for exotic dynamics for\nintermediate values of the delay as well as the  eventual   disappearance of\nthe coexistence equilibrium with the extinction of the predator\nfor large enough delays.\nAlthough chaotic dynamics has been observed in other models of\npredator-prey interactions, the other models either require at least\nthree trophic levels, or the  response functions  are not as simple and\nso the models admit oscillatory behavior even in the absence of delay,\nor the other models incorporate the delay  in such a way  that   the\npredators   still contribute to\n population growth even if the time required\n to process the prey is longer than the life-span of the predator\n (i.e., the factor $e^{-s\\tau}$ is missing in the $\\dot{y}$ equation),\n or  the delay is used to model different mechanisms (see e.g.,\n\\cite{Ginoux2005,Hastings1991,Morozov2004,JWang2012}).\nThe observation that the resulting strange attractor   resembles the\nstrange attractor for the Mackey-Glass equation  \\cite{Mackey:2009} is also new.\n\nThis paper is organized as follows.  In section~\\ref{basic}, we scale the\nmodel and show that it is well-posed. In section~\\ref{analysis}, we\nconsider the existence and stability of equilibria.\nIf parameters are set so that it is possible for the predator to survive\nwhen there is no delay, it is well-known that the\nequilibrium at which both the prey and the predator survive  is globally asymptotically stable with respect to positive\ninitial conditions   (i.e. solutions\napproach this equilibrium for any choice of positive initial data).\nIn the case of delay,\n  the components of this coexistence equilibrium depend on the\ndelay.\nWe prove that for positive delay,  when this equilibrium exists, both the predator and the prey\npopulations persist uniformly. However, a sufficiently long delay\nresults in   the disappearance of this equilibrium, resulting in  the extinction\nof the predator and convergence to a globally asymptotically stable\nequilibrium with the prey at carrying capacity.\nWe give criteria which when satisfied imply that there are at least two Hopf\nbifurcations that occur before the extinction of the predator, resulting\nin sustained oscillatory behaviour for intermediate values of the delay. Finally, in\nsection~\\ref{sec:example},  by means of\ntime series, time delay embeddings, and orbit (bifurcation) diagrams we show\nthat there are saddle-node bifurcations of limit cycles resulting in\nbistability as well as sequences of period doubling bifurcations leading to\nchaos,\nwith a strange attractor resembling the strange attractor for the\nMackey-Glass equation \\cite{Mackey:2009}.\nWe conclude with a brief discussion.\n\n\n\\section{Scaling and basic properties of solutions}\\label{basic}\nIn order to simplify the analysis, we introduce the following change of\nvariables:\n\\begin{equation}\n\\begin{aligned}\n&\\breve{t}=rt,\\qquad \\breve{x}(\\breve{t})=x(t)\/K,\\qquad\n\\breve{y}(\\breve{t})=m y(t)\/r, \\\\\n&\\breve{\\tau}=r\\tau,\\qquad \\breve{s}=\\frac{s}{r},\\qquad\\qquad\\qquad\n\\breve{Y}=YKm\/r.\\label{rescale}\n\\end{aligned}\n\\end{equation}\nWe drop the $\\breve{} \\ $'s for convenience and study the\nequivalent scaled version of model (\\ref{ppreydelay}):\n\\begin{equation}\n\\left\\{\n\\begin{aligned}\n&\\dot x(t)=x(t)(1-x(t))-y(t)x(t),\\\\\n&\\dot y(t)=-sy(t)+Ye^{-s\\tau}y(t-\\tau)x(t-\\tau),\\\\\\label{dimenless}\n& (x(t),y(t))=(\\phi(t),\\psi(t))\\in X, \\, \\mbox{for} \\,\\, t\\in[-\\tau,0],\n\\end{aligned}\\right.\n\\end{equation}\nwhere $X$ was defined in  (\\ref{eq:X}).\n\n\nFirst we address well-posedness of system (\\ref{dimenless}).  For positive delay $\\tau$, the existence and uniqueness of solutions of\nsystem (\\ref{dimenless})\nwas shown in Gourley and Kuang \\cite{Kuang2004}. The following\nproposition, proved in \\ref{app:bounded}, indicates that for positive\ndelay the solutions remain\nnonnegative and provides an  upper bound for each component.\n\n\\begin{proposition}\\label{bounded\n Consider model (\\ref{dimenless}) with initial data in $X$.\n\\begin{enumerate}\n\\item The solutions  exist, are unique, and  remain\n  nonnegative for all $t\\geqslant 0$.\n\\item  $ \\, \\limsup_{t\\rightarrow \\infty}x(t)\\leqslant 1 \\, $ and\n$ \\, \\limsup_{t\\rightarrow \\infty}y(t)\\leqslant\n\\frac{1}{4s}Ye^{-s\\tau}(s+1)^2$.\n  \\item Consider model (\\ref{dimenless}) with initial data in $X^0$ where\n  \\begin{equation}\\label{eq:X0}\n    X^0=\\{ (\\phi(t),\\psi(t)) \\in X : \\, \\phi(0)>0 \\,\\,\n    \\& \\,\\, \\exists \\,  \\theta \\in[-\\tau,0] \\, \\mbox{s.t.} \\, \\,\n  \\phi(\\theta)\\psi(\\theta)>0.\\}\n\\end{equation}\nThen,  $x(t)>0$ for all $t>0$ and there exists $T\\geqslant 0$ such that\n$y(t)>0$ for all $t>T$.\n\\end{enumerate}\n\\end{proposition}\n\n\n\n\\section{Existence and stability of equilibria and uniform persistence} \\label{analysis}\n\n Model\n(\\ref{dimenless}) can have up to three distinct\nequilibria:\n\\begin{eqnarray}\\label{notation1}\n  E_0=(0,0), \\  E_1=(1,0), \\     E_+=(x_+(\\tau),y_+(\\tau))= \\left(\\frac{s}{Y}e^{s\\tau},\n1-\\frac{s}{Y}e^{s\\tau}\\right).\n\\end{eqnarray}\n\nThe components of $E_+$ are nonnegative and $E_+$ is distinct from\n$E_1$, if,\nand only if, $0\\leqslant \\tau < \\tau_c$, where\n\\begin{equation}\\label{notation2}\n\\tau_{c}=\\frac{1}{s}\\ln\\bigg( \\frac{Y}{s} \\bigg).\n\\end{equation}\nThus, when $\\tau_c>0$, i.e., when $Y>s$, the components of $E_+$ are both positive, and   $E_+$ is referred to as the coexistence equilibrium.\n\nWhen there is no delay, i.e. $\\tau=0$ in (\\ref{dimenless}), $E_0$ is\nalways a saddle attracting solutions with $x(0)=0$.\nIf $Y<s$, one of the components of $E_+$ is negative and so it is not\nrelevant, and $E_1$ is globally asymptotically stable with\nrespect to initial conditions satisfying $x(0)>0$ and $y(0)\\geqslant 0$. When $Y=s$,  $E_1$ and $E_+$ coalesce  and are globally\nattracting provided $x(0)>0$. If $Y>s$,  then $E_1$ is  a saddle\nattracting solutions with $x(0)>0$ and $y(0)=0$  and\n$E_+$ sits in int$\\mathbb{R}^2_+$ and is global asymptotically stable\nwith respect to initial conditions in  int$\\mathbb{R}^2_+$.\n\nWhen $\\tau>0$, to determine the local stability of each equilibrium\nsolution, we use the linearization technique for differential equations with\ndiscrete delays (see Hale and Lunel  \\cite{Hale1993}). After\nlinearizing (\\ref{dimenless}) about any one of these equilibria,\n $(x^{\\star},y^{\\star})$, the characteristic equation,\n $P(\\lambda)|_{(x^{\\star},y^{\\star})}=0$,\n is given by,\n\\begin{equation}\n  (\\lambda+s)(\\lambda+y^{\\star}-(1-2x^{\\star}))\n+Ye^{-(s+\\lambda)\\tau}x^{\\star}(1-2x^{\\star})-\\lambda\nYe^{-(s+\\lambda)\\tau}x^{\\star}=0 .\n\\label{chareqn}\n\\end{equation}\n\nWe summarize the results on local and global stability of\nthe equilibrium points and uniform persistence of the populations in the following theorem.  The proof can be found\nin \\ref{app:gasEi}.\n\\begin{theorem\n  \\label{th:gasEi}\nConsider (\\ref{dimenless}).\n\\begin{enumerate}\n  \\item  Equilibrium $E_0$ is always unstable.\n  \\item  Equilibrium $E_1$ is\n    \\begin{enumerate}\n      \\item unstable if\n$0\\leqslant \\tau <\\tau_{c}$, and\n\\item globally asymptotically stable (with respect to $X^0$) if $\\tau>\n  \\tau_{c}$, (i.e. if $\\frac{se^{s\\tau}}{Y}>1$).\n\\end{enumerate}\n  \\item Both components of $E_+$ are positive (i.e., $E_+$ exists), if,\n    and only if,  $0\\leqslant \\tau < \\tau_c$,  (i.e. $\\frac{se^{s\\tau}}{Y}<1$).\n    \\begin{enumerate}\n  \\item When $E_+$ exists and $\\tau=0$, $E_+$ is globally asymptotically\n    stable with respect to int$R^2_+$.\n  \\item When $E_+$ exists, and $\\tau\\geqslant 0$, model (\\ref{dimenless}) is uniformly\n    persistent with respect to initial data in $X^0$, i.e., there exists\n    $\\epsilon>0$ independent of $(\\phi(t),\\psi(t))\\in X^0$ such that \\\\ $\\liminf_{t\\rightarrow\\infty}\n    x(t)>\\epsilon$ and $\\liminf_{t\\rightarrow\\infty}\n    y(t)>\\epsilon$.\n\\end{enumerate}\n\\end{enumerate}\n\\end{theorem}\n\n\n\nThus, for any fixed time delay $\\tau$, if $\\frac{s}{Ye^{-s\\tau}}>1$,  only the prey\npopulation survives and it converges to a steady state. On the other\nhand, if the inequality is reversed,  for appropriate initial data both\nthe prey and the predator populations are uniformly persistent, i.e.\nsurvive indefinitely.\nHowever, we have not yet addressed what form the dynamics takes in the latter\ncase.\n\n\n\\subsection{Local stability  of $E_+$} \\label{HB}\n\nWhen $E_+$ exists, by Theorem~\\ref{th:gasEi} both populations survive\nindefinitely.  To address the possible forms the dynamics can take, we begin\nby investigating the local stability of $E_+$ when it exists, i.e., when\n$0\\leqslant\\tau<\\tau_c$, and hence both components are positive.\nEvaluating the characteristic equation (\\ref{chareqn}) at $E_+$ gives\n\\begin{equation*}\nP(\\lambda)|_{E_+}=\\lambda^2 +\\lambda\ns\\left(1+\\frac{e^{s\\tau}}{Y}\\right)+\\frac{s^2}{Y}e^{s\\tau}+\ne^{-\\lambda\\tau}s\\left(-\\lambda +\\left(1-\\frac{2 s\ne^{s\\tau}}{Y}\\right)\\right)=0.\n\\end{equation*}\nTherefore, $P(\\lambda)|_{E_+}=0$ is of the form\n\\begin{equation}\nP(\\lambda)|_{E_+}=\\lambda^2+p(\\tau)\\lambda+(q\\lambda+c(\\tau))e^{-\\lambda\\tau}+\\alpha(\\tau)=0,\\label{CharE+}\n\\end{equation}\nwhere\n\\begin{equation} \\label{eqn:coeffs}\np(\\tau)=s\\left(1+\\frac{e^{s\\tau}}{Y}\\right),\\ \\ q=-s,\\ \\ c(\\tau)=\ns\\left(1-2\\frac{s e^{s\\tau}}{Y}\\right),\\ \\ \\mbox{and} \\ \\\n\\alpha(\\tau)=\\frac{s^2 e^{s\\tau}}{Y},\n\\end{equation}\nFirst assume that $\\tau=0$. Then (\\ref{CharE+}) reduces to\n\\begin{equation*}\n\\lambda^2 + ( p(0) + q )\\lambda + (\\alpha(0) +c(0) )=0.\n\\end{equation*}\nSince $\\alpha(0)+c(0)=s\\left(1-\\frac{s}{Y}\\right)=s y_+(0)>0$\nand $p(0)+q=\\frac{s e^{s\\tau}}{Y}>0$, by the Routh-Hurwitz\ncriterion  \\cite{Gantmacher1959b},\n all roots of (\\ref{CharE+}) have negative real part.  Therefore, $E_+$ is locally\nasymptotically stable when $\\tau=0$ and hence also for $\\tau>0$ sufficiently\nsmall.\n\nWe  consider the stability of $E_+$ as $\\tau$ varies in the interval\n$0<\\tau<\\tau_c$. Here,  $P(0)|_{E_+}=\\alpha(\\tau)+c(\\tau)=s\\\ny_+(\\tau)>0$ and so   $\\lambda=0$ is not a root of (\\ref{CharE+}).\nTherefore, the only ways that $E_+$ can lose stability is:  (i)\n  when one of the characteristic roots equals zero.  This only occurs\n  when $\\tau=\\tau_c$.  This gives rise to a\n  transcritical bifurcation\nwhere $E_+$ coalesces with $E_1$   and then\ndisappears  as  $\\tau$ increases\nthrough $\\tau_c$; (ii)\nif characteristic roots bifurcate in from infinity; or (iii)  if a pair of\ncomplex  roots with negative real parts and non-zero imaginary parts cross the imaginary axis as\n$\\tau$ increases from $0$, potentially resulting in  Hopf bifurcation.\n In \\ref{app:imaginarycross} we prove that (ii) is impossible to obtain\n the following lemma.\n\n\\begin{lemma}\\label{lem:imaginarycross}\nAs $\\tau$ increases from zero,  the number of\nroots  of (\\ref{CharE+}) with positive\nreal part can change only if a root appears on or crosses the\nimaginary axis as $\\tau$ varies.\n\\end{lemma}\n\n\nIn order to determine when Hopf bifurcations occur, we first\n determine for what values of $\\tau$\npure imaginary roots of (\\ref{CharE+})  exist so that (iii) can occur.\nWe will also be interested in secondary Hopf bifurcations.\n\nSuppose that $\\lambda=i\\omega$ $(\\omega > 0)$ is a root of\n$P(\\lambda)|_{E_+}=0$, where $i=\\sqrt{-1}$. Then\n$$\nP(i\\omega)|_{E_+}=-\\omega^2+ i p(\\tau) \\omega +(i q \\omega +\nc(\\tau)) e^{-i\\tau\\omega}+\\alpha(\\tau)=0.\n$$\nUsing Euler's identity, $e^{i\\theta} = \\cos\\theta + i\\sin\\theta$, and\nequating the real and imaginary parts,   this is equivalent to\n\\begin{eqnarray*}\n       c(\\tau)\\cos(\\tau\\omega) + q \\omega\\sin(\\tau\\omega)=&\\omega^2\n       -\\alpha(\\tau),\n       \\\\\n       c(\\tau)\\sin(\\tau\\omega) - q \\omega\\cos(\\tau\\omega)=&p(\\tau)\\omega.\n       \n\\end{eqnarray*}\nSolving for $\\cos(\\tau\\omega)$ and $\\sin(\\tau\\omega)$ gives\n\\begin{subequations}\\label{chpt1sincos}\n\\begin{align}\n&\\sin(\\tau\\omega)=\\frac{c(\\tau)(p(\\tau)\\omega)+q\\omega(\\omega^2-\\alpha(\\tau))}{c(\\tau)^2+q^2\\omega^2\n, \\label{chpt1sincos:sin}\\\\\n&\\cos(\\tau\\omega)=\\frac{c(\\tau)(\\omega^2-\\alpha(\\tau))+q\\omega(-p(\\tau)\\omega)}{c(\\tau)^2+q^2\\omega^2}.\n\\label{chpt1sincos:cos\n\\end{align}\n\\end{subequations}\nSquaring\nboth sides of the equations in (\\ref{chpt1sincos}), adding, and\nrearranging gives\n \\begin{equation}\n\\omega^4+(p(\\tau)^2-q^2-2\\alpha(\\tau))\\omega^2+\\alpha(\\tau)^2-c(\\tau)^2=0.\\label{omegapower4}\n\\end{equation}\nNoting that \\eqref{omegapower4} is a quadratic function of\n$\\omega^2$, we use  the quadratic formula to obtain\n\\begin{equation*}\n\t\\omega_{\\pm}^2(\\tau)=\\frac{1}{2}\\bigg(q^2-p^2(\\tau)+2\\alpha(\\tau) \\pm\n\\sqrt{(q^2-p^2(\\tau)+2\\alpha(\\tau))^2\n-4\\left(\\alpha^2(\\tau)-c^2(\\tau)\\right)}\\bigg).\\label{omega+-2}\n\\end{equation*}\nSubstituting using \\eqref{eqn:coeffs}, it follows that \n  \\begin{equation}\\label{chpt1omegapm_squared}\n\\omega_{\\pm}^2(\\tau)=\\frac{1}{2}\\bigg(-\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^2\n              \\pm\\sqrt{\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^4+s^2\\bigg(12\\frac{s^2 e^{2s\\tau}}{Y^2}\n              -16\\frac{se^{s\\tau}}{Y}+4\\bigg)}\\bigg)  \\ .\t\n\\end{equation}\n\nIn order to determine for what values of $\\tau$ there are  positive real\nroots of (\\ref{omegapower4}), and hence candidates for  pure imaginary\nroots, and possibly Hopf bifurcations, we define\n\\begin{equation}\\label{tstardefined}\n\\tau^*=\\frac{1}{s}\\ln\\left(\\frac{Y}{3s}\\right).\n\\end{equation}\nWe will prove that for $\\tau \\geqslant \\tau^*$, there are no postive real\nroots and  for $0 \\leqslant \\tau < \\tau^*$ \n  \\begin{equation}\\label{chpt1omegapm}\n\\omega_{+}(\\tau)=\\sqrt{\\frac{1}{2}\\bigg(-\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^2\n              +\\sqrt{\\bigg(\\frac{se^{s\\tau}}{Y}\\bigg)^4+s^2\\bigg(12\\frac{s^2 e^{2s\\tau}}{Y^2}\n              -16\\frac{se^{s\\tau}}{Y}+4\\bigg)}\\bigg)}\n\\end{equation}\nis the only positive real root.\n\n\n\\begin{remark}\n  Note that $\\tau^*>0$, if, and only if, $\\frac{s}{Y}<\\frac{1}{3}$, and\n  then $x_+(\\tau^*)=\\frac{s e^{s\\tau^*}}{Y}=\\frac{1}{3}$.\n\\end{remark}\n\nIn the following theorem, proved in \\ref{app:omega+posicond}, we give\nnecessary conditions on $\\tau$ for  Hopf bifurcations to occur.\n\n\\begin{theorem}\\label{omega+posicond}\n  Consider (\\ref{dimenless}). Assume that both components of $E_+$ are\n  positive.\n \n  \\begin{enumerate}\n\\item   $\\omega_+(\\tau^*)=0$.\nIf $\\tau\\geqslant\\tau^*$, then (\\ref{omegapower4}) has no positive real root.\nTherefore, there can be no pure imaginary roots of (\\ref{CharE+})  in this case. In\nparticular, if $\\frac{s}{Y}\\geqslant \\frac{1}{3}$, then $\\tau^*\\leqslant 0$, and so\nthere can be no Hopf bifurcation of $E_+$ for any $\\tau\\geqslant 0$.\n\\item Assume that $\\frac{s}{Y}<\\frac{1}{3}$, and hence, $\\tau^*>0$.\n      If $\\tau\\in[0,\\tau^*)$, then\n$x_+(\\tau)\\in[\\frac{s}{Y},\\frac{1}{3})$, and  (\\ref{omegapower4}) has\nexactly one positive real\nroot, $\\omega_+(\\tau)$, given by (\\ref{chpt1omegapm}). If\n(\\ref{CharE+}) has pure imaginary roots at $\\tau$, and hence $\\tau$ is a\ncandidate for Hopf bifurcation of $E_+$, then  $\\tau\\in(0,\\tau^*)$ and $\\omega_+(\\tau)$ must\nsatisfy (\\ref{chpt1omegapm}).\n  \\end{enumerate}\n\\end{theorem}\n\n\n\n\\begin{remark} \\label{rem:candidates}\n\t\\begin{enumerate}\n\t\t\\item  Note  that even though $\\omega_+(0)>0$ when\n  $\\frac{s}{Y}<\\frac{1}{3}$, $\\tau=0$ is not a candidate for a Hopf bifurcation,\n  since  in this case all roots of (\\ref{CharE+}) have negative real\n  parts. Also, $\\tau^*$ is not a candidate, since $\\omega_+(\\tau^*)=0$.\n\t\t\\item   Haque \\cite{Haque2012} finds an expression for\n\t\t\t$\\omega_+(\\tau^*)$ for a different scaling of\n\t\t\tthe model.  However, he does not go on as we do\n\t\t\tin what follows, to determine when\n\t\t\tthe equations given in  (\\ref{chpt1sincos})\nare both simultaneously  satisfied for \t$\\omega_+(\\tau^*)$, and to\n\t\t\tshow that the Hopf bifurcations are nested and\n\t\t\thow the number of Hopf bifurcations increases as\n\t\t\tthe death rate of the predator decreases.\n\t\\end{enumerate}\n\\end{remark}\n\nSubstituting   the values of the coefficients given by\n(\\ref{eqn:coeffs}), in the right-hand side of  (\\ref{chpt1sincos}), and recalling\nthat $x_+(\\tau)=se^{s\\tau}\/Y$,  we define\n\\begin{subequations} \\label{h1_h2_defined}\n  \\begin{align}\nh_1(\\omega,\\tau)&=\\frac{\\omega}{s}\\left(\\frac{s+x_+(\\tau)-s\nx_+(\\tau)-2x_+^2(\\tau)-\\omega^2}{(1-2\nx_+(\\tau))^2+\\omega^2}\\right),\\label{h1defined} \\\\\nh_2(\\omega,\\tau)&=\\frac{\\omega^2(1+ s -x_+(\\tau))-(1-2 x_+(\\tau)) s\nx_+(\\tau)}{s\\left((1-2x_+(\\tau))^2+\\omega^2\\right)}.\\label{h2defined}\n\\end{align}\n\\end{subequations}\n\nProperties of the functions $h_1$ and $h_2$ are summarized\nin \\ref{app:h1_h2}.\n\n\\begin{remark}\\label{rem:char_pure_imag}\nBy Theorem~\\ref{omega+posicond} and Lemma~\\ref{lem:h1_pos}, $\\tau$ satisfies\n(\\ref{chpt1sincos}) for $\\omega>0$, (and hence (\\ref{CharE+}) has a pair of pure\nimaginary eigenvalues) if and only if $\\tau\\in(0,\\tau^*)$, and\n\\begin{subequations} \\label{eqn:sin_h1_cos_h2}\n   \\begin{align}\n&\\sin(\\tau\\omega_+(\\tau)) \\, \\, =h_1(\\omega_+(\\tau),\\tau)\n,  \\label{eqn:sin_h1_cos_h2:sin}\n\\\\\n&\\cos(\\tau\\omega_+(\\tau)) \\, \\,  =h_2(\\omega_+(\\tau),\\tau).\n\\label{eqn:sin_h1_cos_h2:cos}\n\\end{align}\n\\end{subequations}\n\\end{remark}\n\n\n\n\n\n\nDefine the function\n\\begin{equation}\\label{eq:theta}\n\\theta:[0,\\tau^*]\\rightarrow [0,\\pi]\n\\end{equation}\n\n$$\\theta(\\tau):=\\arccos(h_2(\\omega_+(\\tau),\\tau)).$$\n By part 3 of Lemma~\\ref{lem:h1_pos}, stated and proved in \\ref{app:h1_h2}, $\\theta(\\tau)$ is a well-defined, continuously\ndifferentiable function.   Replacing  $\\tau\\omega_+(\\tau)$ by\n$\\theta(\\tau)+2 n \\pi$ in the left hand side of\n(\\ref{eqn:sin_h1_cos_h2}),\nwe obtain\n\\begin{subequations} \\label{eqn:sin_h1_cos_h2_theta}\n    \\begin{align}\n&\\sin(\\theta(\\tau)+2 n\\pi) \\, \\, =h_1(\\omega_+(\\tau),\\tau)\n,  \\label{eqn:sin_h1_cos_h2:sin_theta}\n\\\\\n&\\cos(\\theta(\\tau)+2 n\\pi) \\, \\,  =h_2(\\omega_+(\\tau),\\tau).\n\\label{eqn:sin_h1_cos_h2:cos_theta}\n\\end{align}\n\\end{subequations}\n Equation \\eqref{eqn:sin_h1_cos_h2:cos_theta} is satisfied directly by the\n definition of $\\theta(\\tau)$.   Equation\n \\eqref{eqn:sin_h1_cos_h2:sin_theta} is also satisfied,\n from  parts 1 and 2 of Lemma~\\ref{lem:h1_pos}, since $0\\leqslant\n \\theta(\\tau)\\leqslant \\pi .$\n\nBy comparing \\eqref{eqn:sin_h1_cos_h2} and\n\\eqref{eqn:sin_h1_cos_h2_theta}, it follows that solutions of\n\\eqref{eqn:sin_h1_cos_h2} occur at precisely those points\nwhere the curves $\\tau\\omega_+(\\tau)$ and $\\theta(\\tau)+2n\\pi$\nintersect.\n\\begin{remark} \\label{rem:characterization}\n  By Remark~\\ref{rem:char_pure_imag}, (\\ref{CharE+}) has a pair of pure imaginary roots at precisely those points\nwhere the curves  $\\tau\\omega_+(\\tau)$ and $\\theta(\\tau)+2n\\pi$\nintersect  for $\\tau\\in(0,\\tau^*)$,  where $n$ is a\nnonnegative integer.\n\\end{remark}\n\n\n\nFor each integer $n\\geqslant 0$, denote the $j_n$ points of intersection of\nthe  curves $\\tau\\omega_+(\\tau)$ and $\\theta(\\tau)+2n\\pi$\n for $\\tau\\in(0,\\tau^*)$, in increasing order,  by $\\tau_n^j, \\, \\,\n j=1,2\\dots,j_n$,\ni.e., for each $n=0,1,2,\\dots$,\n$$\\tau_n^j\\omega_+(\\tau_n^j)=\\theta(\\tau_n^j)+2n\\pi, \\quad\n\\ j=1,2,\\dots,j_n.$$\n\n\n\\begin{theorem}\\label{chpt3intersections\nConsider system (\\ref{dimenless}).\nThe characteristic equation (\\ref{CharE+}) has a  pair of pure\nimaginary eigenvalues, if and only if, $\\tau=\\tau_n^j\\in(0,\\tau^*)$, a point of\nintersection of the curves $\\tau\\omega_+(\\tau)$ and\n$\\theta(\\tau)+2n\\pi$, for some integer $n\\geqslant0.$\nAt  all  such intersections, the pair of pure imaginary eigenvalues is\n simple\nand no other root of (\\ref{CharE+})\nis an integer multiple of $ i \\omega_+(\\tau_n^j).$\nIf in addition, $\\frac{\\mathrm{d}}{\\mathrm{d}\\tau}(\\tau\\omega_+(\\tau))\n\\Big|_{\\tau=\\tau_n^j} \\neq \\frac{\\mathrm{d}}{\\mathrm{d}\\tau}\\theta(\\tau)\n\\Big|_{\\tau=\\tau_n^j}$,\nthe transversality condition  for Hopf bifurcation,\n$\\frac{\\mathrm{d}}{\\mathrm{d}\\tau}\\mathrm{Re}(\\lambda(\\tau))\n\\Big|_{\\tau=\\tau_{n}^j}\\neq0$,\nholds.\n\\end{theorem}\n\n\\noindent\nThe proof is given in \\ref{app:chpt3intersections}.\n\n\n\n\\begin{remark} \\label{rem:test_transversality}\n   If the slope of the curve  $\\theta(\\tau)+2n\\pi$\n  is less than the slope of   $\\tau\\omega_+(\\tau)$ at an intersection\n  point $\\tau_n^j$, then   a pair of complex  roots of\n  (\\ref{CharE+}) crosses the imaginary axis from left to right as $\\tau$\n  increases through  $\\tau_n^j$.\n  On the other hand, if the slope of the curve  $\\theta(\\tau)+2n\\pi$\n  is greater than the slope of   $\\tau\\omega_+(\\tau)$ at an intersection\n  point $\\tau_n^j$, then   a  pair of complex roots of\n  (\\ref{CharE+}) crosses the imaginary axis from right to left as $\\tau$\n  increases through  $\\tau_n^j$.\n\\end{remark}\n\n\\begin{corollary}\\label{cor:2k+1intersection\nConsider system (\\ref{dimenless}). Assume that $\\tau\\in[0,\\tau^*]$ and\nthat there exists $N\\geqslant 0$ such that\n$(2N+1)\\pi\\leqslant\\max_{\\tau\\in[0,\\tau^*]}\\tau\\omega_+(\\tau)\\leqslant\n2(N+1)\\pi$.\n\\begin{enumerate}\n\\item For $0\\leqslant n\\leqslant N$, $\\theta(\\tau)+2n\\pi$ and\n$\\tau\\omega_+(\\tau)$ have at least two intersections in\n$(0,\\tau^*)$.\n \\item For $n\\geqslant N+1$,  $\\theta(\\tau)+2n\\pi$ and\n$\\tau\\omega_+(\\tau)$  do not intersect in $(0,\\tau^*)$.\n\\item\nIf $\\theta(\\tau)+2n\\pi$ and\n$\\tau\\omega_+(\\tau)$ intersect for any $n\\geqslant 0$, then $\\tau_0^1$ is the\nsmallest  and $\\tau_0^{j_0}$ the largest\nvalue of $\\tau$ for which\n(\\ref{CharE+}) has a pair of pure imaginary eigenvalues.\n\\item\n  The\ncoexistence equilibrium $E_+$ is locally asymptotically stable\nfor $\\tau\\in[0,\\tau_0^1)\\cup(\\tau_0^{j_0},\\tau_c)$.\n\\end{enumerate}\n\\end{corollary}\n\n\n\\noindent\nThe proof is given in \\ref{app:2k+1intersection}.\n\n\n\n \\begin{figure}[bt!]\n  \\begin{center}\n    \\includegraphics[width=5.25cm]{Figures\/theta_omega_s_p02.eps}\n   \n \\includegraphics[width=5.25cm]{Figures\/theta_omega_s_p007.eps}\n   \n\\end{center}\n\\caption[  Intersections of $\\theta(\\tau)+2n\\pi$ and $\\tau\n\\omega(\\tau)$] { Intersections of $\\theta(\\tau)+2n\\pi$\nand $\\tau \\omega_+(\\tau), \\ n=0,1,\\dots, $.\nValues of $\\tau$ at which the characteristic equation has pure imaginary eigenvalues, and hence candidates for critical values\nof $\\tau$ at which there could be Hopf bifurcations.\nIn both graphs, at all such intersections, transversality holds, since\nthe slope of these curves\nat these intersections are different.\nParameters: $ m=1,\\ r=1,\\ K=1, \\ Y=0.6 . $\n{\\scriptsize\\bf (LEFT)}  $s=0.02$.\n For $n=0$ there are two\nintersections (i.e. $j_0=2$), at $\\tau_0^1$ and $\\tau_0^2$, but for $n=1$, and hence\n$n\\geqslant 1$, there are no\nintersections. {\\scriptsize\\bf (RIGHT)}  $s=0.007$.\n There are two\nintersections each (i.e. $j_n=2, \\ n=0,1,2$), at $\\tau_n^1$ and $\\tau_n^2$, for $n=0,1$ and $2$,\nbut for $n=3$, and hence\n$n\\geqslant 3$, there are no\nintersections. In both {\\scriptsize\\bf (LEFT)} and  {\\scriptsize\\bf (RIGHT)},  $E_+$ is\nasymptotically stable for $\\tau\\in[0,\\tau_0^1)\\cup(\\tau_0^2,\\tau_c)$\nand unstable for $\\tau\\in(\\tau_0^1,\\tau_0^2)$.}\n\\label{fig:thetatauomega2}\n\\end{figure}\n\n Our results  differ from those in Gourley and Kuang \\cite{Kuang2004},\nsince  we give explicit formulas   for solutions of \\eqref{omegapower4}\nand define $\\theta(\\tau)$ explicitly. These explicit formulas play an\nimportant role in analysis and make numerical simulations more\nstraightforward.  Although $\\omega_{-}^2(\\tau)$ in\nequation\n\\eqref{chpt1omegapm_squared} is negative in\nthis model and hence its square root is not real, in other models\nequation\n\\eqref{omegapower4} can have two positive real\nsolutions (see \\cite{FanW}, where  both $\\omega_+(\\tau)$ and $\\omega_-(\\tau)$\nare positive). In that case, double Hopf bifurcations are possible.\n\n  In the next section we will demonstrate\n  numerically that in the example shown in\n  Figure~\\ref{fig:thetatauomega2},\n  $E_+$ first loses its stability through a supercritical Hopf\n  bifurcation as $\\tau$  increases through $\\tau_0^1$ and then\n  restabilizes  as a result of a second supercritical Hopf\n  bifurcation as $\\tau$   increases through $\\tau_0^2$.  We will also\n  show that between these two values of $\\tau$ there is a\n  sequence of bifurcations resulting in   interesting\n  dynamics, including a strange attractor.\n\n\n\n\n\n\n\n\\section{An example demonstrating complex dynamics}\\label{sec:example}\nIn this section, unless specified otherwise,  we select the following\nvalues for the parameters in model (\\ref{ppreydelay}):\n\\begin{equation}\n  m=1,\\ r=1,\\ K=1, Y=0.6,\\ s=0.02, \\label{example}\n\\end{equation}\n  and consider $\\tau$ as a bifurcation parameter. Since, with this\n  selection of parameters, the model is already in the form of the scaled\n version of the model (\\ref{dimenless}), it is not necessary to apply the scaling given by (\\ref{rescale}).\nWe first use this example  to  illustrate the analytic\nresults given in section~\\ref{analysis} where we provided necessary and sufficient\nconditions for  a simple  pair of pure imaginary eigenvalues of the\ncharacteristic equation  to occur as $\\tau$ varies. We then provide\nbifurcation diagrams with $\\tau$ as the bifurcation parameter,\nsimulations  including time series and time delay embeddings\nfor various values of $\\tau$, and a return map at a value of $\\tau$\nat which there is a chaotic attractor,\nin order  to illustrate the wide variety of dynamics displayed by the\nmodel,   even in the case when there are only two Hopf\nbifurcations.\nThis includes two supercritical Hopf bifurcations, saddle-node  bifurcations of limit cycles and  sequences of period doublings that\nappear to lead to chaotic dynamics with a strange attractor reminiscent\nof the strange attractor for the well-known Mackey-Glass equation\n\\cite{Mackey:1977,Mackey:2009}\n$$  \\frac{dx}{dt}=\\beta \\frac{x(t-\\tau)}{1+(x(t-\\tau))^n}-\\gamma x(t).$$\n\n\\subsection{Illustration of analytic results}\n\nFor the parameters given by (\\ref{example}), the model has three\nequilibria: $E_0=(0,0)$, $E_1=(1,0)$, that always exist, and\n$E_+=(x_+(\\tau),y_+(\\tau))=( 0.0\\dot{3} e^{0.02\\tau},1-0.0\\dot{3} e^{0.02\\tau})$, given by\n(\\ref{notation1}).\nThe components of $E_+$ are both positive, if, and only if, $\\tau\\in[0,\\tau_c)$,\nwhere $\\tau_c\\approx 170$ (see (\\ref{notation2})), and by part 3(b) of\nTheorem~\\ref{th:gasEi}  the model is uniformly persistent\nfor $\\tau\\in[0,170)$.\n\n   $E_0$ is always a saddle, and hence unstable. $E_1$ is globally\nasymptotically stable for $\\tau>\\tau_c$, and unstable for\n$\\tau\\in[0,\\tau_c)$.  For $\\tau=0$, $E_+$ is asymptotically stable, and\nby Lemma~\\ref{lem:imaginarycross}, can only lose stability by means of a Hopf\nbifurcation.  In order to determine th:e\ncritical values of $\\tau$ at which\n there is a pair of  pure imaginary eigenvalues:\n$\\lambda(\\tau)=\\pm i \\omega(\\tau)$,  we consider the interval\n$(0,115)$, since by Theorem~\\ref{omega+posicond}, the interval on\nwhich $\\omega(\\tau)$ is positive, is bounded above  by $\\tau^*\\approx\n115$ (defined in (\\ref{tstardefined})).\n\nWith the parameters  given in (\\ref{example}),  by \\eqref{chpt1omegapm}  the  function\n\t\\[\t\\omega_+(\\tau)= \n\t \\sqrt{ -.\\dot{5} \\ 10^{-3}(e^{0.02\\tau})^2 + \n\t \\frac{\n\\sqrt{1.235\\  10^{-6}(e^{0.02\\tau})^4 +\n\t.5\\dot{3} \\ 10^{-5}e^{0.04\\tau} - 0.5\\dot{3}e^{0.02\\tau} +\n\t4}}{2} } \\]  \n and recall,  by \\eqref{eq:theta}\n $$\\theta(\\tau)=\\arccos(\\cos(\\tau \\omega_+(\\tau))). $$\nThe intersections of the functions $\\tau\\omega_+(\\tau)$  and\n$\\theta(\\tau)+2n\\pi,$  $n$ a nonnegative integer, in the interval\n$(0,\\tau^*)$,   give  the critical values of\n$\\tau$ for  which there is a pair of pure imaginary eigenvalues.\nThere are only two intersections as can be seen in\nFigure~\\ref{fig:thetatauomega2}~{\\scriptsize\\bf (LEFT)}.\nSince,  $\\pi<\\max_{\\tau\\in[0,\\tau^*]} \\tau\\omega(\\tau)<2\\pi$,\nby Corollary~\\ref{cor:2k+1intersection},\nwe are guaranteed at least\n two  values of $\\tau>0$ at which the characteristic\n equation has a pair of pure imaginary roots.  In fact,\n (see Figure~\\ref{fig:thetatauomega2} ({\\scriptsize (LEFT})),  there are\n precisely two such values, $\\tau_0^1$ and\n $\\tau_0^2$.\nUsing Maple \\cite{maple},  we found that $\\tau^1_0\\approx 1.917$ and\n$\\tau^2_0 \\approx 108.365$. By\nTheorem~\\ref{chpt3intersections},    the slopes of the curves\n$\\theta(\\tau)$ and $\\tau\\omega_+(\\tau)$ are different at these\nintersection points, since these curves cross transversally (see\nFigure~\\ref{fig:thetatauomega2} {\\scriptsize (LEFT)}), and hence the\ntransversality required for Hopf bifurcation holds at each\nroot.  By part 4 of Corollary~\\ref{cor:2k+1intersection} and\nRemark~\\ref{rem:test_transversality}, the stability of $E_+$ changes\nfrom asymptotically stable to unstable as $\\tau$ increases through\n$\\tau_0^1$ and from unstable to asymptotically stable as $\\tau$\nincreases through $\\tau_0^2$, and is unstable for\n$\\tau\\in(\\tau_0^1,\\tau_0^2)$.  But recall, even though $E_+$ is unstable\nhere, by  part 3(b) of Theorem~\\ref{th:gasEi},  the model is still uniformly\npersistent in this interval.\n\n Thus, we have shown that for the parameters chosen, there are exactly\n  two candidates for  Hopf bifurcations (see  \\cite{Smith_delay},\nChapter 6, Theorem~6.1, page 89-90).  That both Hopf bifurcations are\nsupercritical (involving first the birth, and then the disappearance of\norbitally asymptotically stable periodic solutions)\n  will be demonstrated in the next section.  As $\\tau$ increases through\n  $\\tau=1.917$, a family of orbitally asymptotically stable periodic\n  orbits is born.  We will see that these periodic orbits  undergo additional bifurcations\n as $\\tau$ increases, and that they disappear when $\\tau$\n  increases through the critical value $\\tau=108.365$, at the second supercritical  Hopf bifurcation.\n\n By decreasing $s$,  the value of $n$ for which the curves\n\n\n\n   $\\theta(\\tau)+2n\\pi$ and $\\tau\\omega_+(\\tau)$ intersect can increase.\n   See\n  Figure~\\ref{fig:thetatauomega2} {\\scriptsize (RIGHT)} for an example\n  with $s=0.007$ for which the curves $\\theta(\\tau)+2n\\pi$ and\n  $\\tau\\omega_+(\\tau)$ intersect when $n=0,1,2$. In fact, there are $6$\n  points of intersection. We see by\n  Remark~\\ref{rem:test_transversality}, that for this example, $E_+$ is\n  asymptotically stable until $\\tau=\\tau_0^1$, is\n  unstable for $\\tau\\in(\\tau_0^1,\\tau_0^2)$, and finally becomes stable\n  again for $\\tau\\in(\\tau_0^2,\\tau_c)$. Again, even though the model is\n  unstable for $\\tau\\in (\\tau_0^1,\\tau_0^2)$, by part 3(b) of\n  Theorem~\\ref{th:gasEi}, it is uniformly persistent\n  in this interval, since the model is uniformly persistent whenever\n  both components of   $E_+$ are positive, i.e. for $\\tau\\in[0,\\tau_c)$.\n\n\n\\subsection{Saddle-node of limit cycles, period doublings, and chaotic dynamics}\n\\label{numerics}\n\nThe computations and figures in this section\nwere done using Maple \\cite{maple},\nMATLAB  \\cite{MATLAB:2018}, and\nXPPAUT \\cite{Ermentrout2002}\n\n\nIn Figure~\\ref{bif_diag_full}~{\\scriptsize (TOP)}, for each value of\n$\\tau\\in[0,120]$, starting with initial data $x(t)=y(t)=0.1$ for\n$t\\in[-\\tau,0]$, we  integrate long enough for the solution to converge\nto an attractor (e.g., equilibrium, periodic orbit, or strange\nattractor), and then\n plot the local minima and maxima of the $y(t)$ coordinate\n on the attractor.\n Because we are interested in period doubling bifurcations, we then\n eliminate certain local maxima and minima that are due to kinks\n in the\n solutions  (see\n Figure~\\ref{kinks}) rather than actual bifurcations, to obtain the graph in\n Figure~\\ref{bif_diag_full} {\\scriptsize (BOTTOM)}).\n Our solutions have\nkinks for values of $\\tau\\in[55,98]$.  Kinks were also observed in\nthe Mackey-Glass equation  \\cite{Mackey:2009}.\n\n\\begin{figure}[tbhp!]\n  \\begin{center}\n  \n \\includegraphics[width=10cm]{Figures\/orbit_diag_y.eps}\\\\\n \\includegraphics[width=10cm]{Figures\/bif_diag_0_120_steps_p2_bold.eps}\n\n  \\end{center}\n\\caption[Parameter range for\n\tinteresting dynamics]{Orbit diagrams.  Initial\n\tdata was taken to be $x(t)=y(t)=0.1$ for\n\t$t\\in[-\\tau,0]$.  However, we found bistability in the portion\n\tof the diagram beween the vertical dots and varied the initial\n\tdata as explained below.   \n\n\n\n\tExcept for the\n\tportion between the vertical dots, the rest of the diagram was the same \n\tfor all of the initial conditions we tried (not shown).\n{\\scriptsize\\bf (TOP)} All local maxima and minima\nfor the $y(t)$ coordinate of the attractor as $\\tau$ varies,\nincluding kinks.  {\\scriptsize\\bf (BOTTOM)}\nDiagram including local maxes and mins\nfor the $y(t)$ coordinate  as $\\tau$ varies, but with kinks eliminated.\n \tThere are two saddle-node of limit cycle\n\tbifurcations. They occur for\n\t$\\tau$ approximately equal to  76 and 82, where  the curves  in the orbit diagrams stop abruptly and there\n\tappear to be vertical dots. For $\\tau$ between these values, there\n\tis is bistability. Two orbitally asymptotically stable\n\tperiodic orbits (with their maximum and minimum amplitudes\n\tshown) and an unstable periodic orbit with amplitudes between\n\tthem (not shown).  The two stable periodic orbits were found\n\tby producing this part of the orbit diagram varying $\\tau$\n\tforward and then varying  it backwards but startng at the last\n\tpoint of the attractor for the previous value of $\\tau$.}\n \\label{bif_diag_full}\n \\end{figure}\n\n\\begin{figure}[tbhp!]\n  \\begin{center}\n    \\includegraphics[angle=270, width=7cm]{Figures\/ts_kink_70.eps}\n  \\end{center}\n  \\caption[kinks]{The time series for $y(t)$ when $\\tau=70$, depicting\n   kinks. There are two local maxima and two local minima over each period as shown  in  Figure~\\ref{bif_diag_full} {\\scriptsize (TOP)}, but only one local\n  maxima and one local minima in Figure~\\ref{bif_diag_full} {\\scriptsize\n  (BOTTOM)} in which kinks have been removed. }\\label{kinks}\n\\end{figure}\n\n\n\\begin{figure}[tbhp!]\n  \\begin{center}\n\n\t\\includegraphics[width=10cm]{Figures\/orbit_diag_y_zoom.eps}\n \n  \\end{center}\n \\caption{ Zoom-in of orbit diagram shown in  Figure\n\\ref{bif_diag_full} for\n\t$\\tau\\in[75,100]$\n\tincluding kinks.   The vertical dots indicate the boundary of the\n\tregion of bistability, where the  two saddle-node of limit cycle\n\tbifurcations occur.\n}\n\n\t\\label{zoom-in_80_100}\n\\end{figure}\n\n\nFigure \\ref{bif_diag_full} confirms that there are two\nHopf bifurcations at $\\tau\\approx 1.917$ and $\\tau\\approx 108.365$, and\nallows us to conclude that these Hopf bifurcations are both\nsupercritical, since they involve a family of orbitally asymptotically\nstable periodic orbits.\n\nNext we focus on the  more interesting\ndynamics observed for $\\tau\\in[80,100]$ (see  Figure~\\ref{zoom-in_80_100}).\nThere appears to be a discontinuity in the\nbifurcation diagram for $\\tau\\approx 82.225$.  Upon further\ninvestigation we have determined that there is a saddle-node bifurcation\nof limit cycles at this value of $\\tau$ and another saddle-node\nbifurcation of limit cycles for a value of $\\tau$ smaller than\n$\\tau=81$.  For values of $\\tau$ between these two saddle-node bifurcations,\nthere is bistability.  There are two orbitally asymptotically stable\nperiod orbits.  An example of two such orbits is given in\nFigure~\\ref{bistability}, where $\\tau=81$.\n\n\n \\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=270, scale=.25]{Figures\/bistability_81_345_273p7_y_yd.eps}\n\n\\caption{  Time delay embedding  of two orbitally asymptotically stable periodic orbits\ndemonstrating bistability for $\\tau=81$.  The one with larger amplitude (dashed) has initial data\n$x(t)=y(t)=0.1$, for $t \\in [-\\tau,0]$, and period approximately\n$345$.\nThe one with smaller amplitude (solid) has initial data\n$x(t)=y(t)=0.1$, for $t \\in [-\\tau,0)$ and $x(0)=0.3, \\ y(0)=0.83$ and has\nperiod approximately\n$273.7$.}\\label{bistability}\n\\end{center}\n\\end{figure}\n\nFigure~\\ref{zoom-in_80_100}  suggests that\n there are sequences of period doubling bifurcations,\n one initiating from the left at  $\\tau\\approx 83$, $86$,  $86.6$,\n $\\dots,$\n and one initiating from the right  at  $\\tau\\approx  98.3$,  $93.2$, $92.2$,\n and $\\tau$ between $92$ and $91.85$.\n To demonstrate these sequences,  time series ($y(t)$ versus\n $t$) and  time delay embeddings\n ($y(t)$ versus $y(t-\\tau)$) at values of $\\tau$ between these\n bifurcations are  shown in Figures~\\ref{PD_left} and \\ref{PD_right}.\n\n \\begin{figure}[tbhp!]\n   \\begin{center}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_82.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_82_340p16.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_85.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_85_564p6.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_86p3.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_86p3_1144p48.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_86p8.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_86p8_2298p26.eps}\n   \\end{center}\n   \\caption[ ]{ {\\scriptsize\\bf (LEFT)} Time series starting from the initial data $x(t)=y(t)=0.1, t\\in[-\\tau,0]$\n   indicating how quickly the orbit gets close to the periodic attractor and\n   {\\scriptsize\\bf (RIGHT)} time delay embeddings of\n   the periodic attractors, demonstrating the sequence of  period doubling bifurcations\n   initiating from the left at\n   $\\tau\\approx 83,\\ 86$, and $86.6$. Values of $\\tau$ selected between these bifurcations:\n    $\\tau=82,\\ 85,\\\n   86.3$, and $86.8$, with periods of the periodic attractor approximately equal to:\n   $340.2,\\ 564.6,\\ 1144.5$, and $2298.3$, respectively, are\n   shown. } \\label{PD_left}\n\\end{figure}\n\n \\begin{figure}[tbhp!]\n   \\begin{center}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_91p95.eps}\n\\includegraphics[angle=270, width=5.25cm]{Figures\/delay_pp_y_yd_91p95_4794p3.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_92.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ytau92_2398p3.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_93.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_y_yd_93_period_1211p2.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_96.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_y_yd_96_period_557p6.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_ts_100.eps}\n\\includegraphics[angle=270, width=5.cm]{Figures\/delay_pp_y_yd_100_period_295p3.eps}\n   \\end{center}\n   \\caption[ ] { {\\scriptsize\\bf (LEFT)} Time series starting  from the initial data $x(t)=y(t)=0.1, t\\in[-\\tau,0]$\n   indicating how quickly the orbit gets close to the periodic attractor and\n   {\\scriptsize\\bf (RIGHT)} time delay embeddings of\n   the periodic attractors, demonstrating the sequence of  period\n\t halfing bifurcations\n   initiating  from the left for values of $\\tau$ between  $91.5$ and $92$,\n\t and at\n   $\\tau\\approx 92.2, \\  93.2, \\ 98.3$. Graphs shown are for values of $\\tau$  between these bifurcations:\n    $\\tau=91.95,\\ 92,\\\n   93,\\ 96  $, \\ and \\  $100$, \\ with periods approximately equal to:\n   $4794.3, \\  2398.3, \\ 1211.2,\\  557.6, $ and  $295.3$, respectively. } \\label{PD_right}\n\\end{figure}\n\n    Figure~\\ref{zoom-in_80_100} also suggests that between these sequences of period doubling\n    bifurcations there is a window of values of $\\tau$ at which there\n     are  periodic attractors that do not have a\n     period that  results from a bifurcation with period approximately\n     equal to $2^n$  for some integer $n$, as well as chaotic dynamics.\n     An example of the former is illustrated in\n     Figure~\\ref{period_two_times_three}.  For $\\tau=90.7$, the\n     time-series embedding of a periodic orbit with period approximately\n    equal to  1800 time steps involving six loops (2 times 3)  is\n    shown.\n\n     \\begin{figure}\n    \\begin{center}\n\t    \\includegraphics[width=35ex]{Figures\/delay_pp_y_yd_90p7_per1800_not_2n.eps}\n  \\end{center}\n     \\caption [two times three] { Time delay embedding starting at initial\n     data $x(t)=y(t)=0.1, t\\in[\\tau,0]$ for $\\tau=90.7$ showing a\n   periodic attractor with period approximately\n      $1800$, having $6(\\neq2^n)$ loops for some integer $n$.}\n     \\label{period_two_times_three}\n\\end{figure}\n\nThe time series and the time delay embedding of a chaotic attractor for\n$\\tau=90$ is shown in Figure~\\ref{chaotic_attractor}. The projection of\nthis attractor into $(x(t),y(t))$-space is\nalso shown in Figure~\\ref{chaos_xy}. This strange attractor resembles the\nchaotic attractor of the well-known Mackey-Glass equation (\\cite{Mackey:2009}, Figure 2).\nThe return map\nshown in Figure~\\ref{return_map}, for $\\tau=90$, also resembles the return map for the\nMackey-Glass equation (\\cite{Mackey:2009}, Figure 14) in the case of chaotic dynamics.\nSensitivity to initial data is a hallmark of chaotic dynamics.\nFigure~\\ref{sensitivity}{\\scriptsize\\bf (RIGHT)}\ndemonstrates that there is sensitivity to initial data in the case of the solution for\n$\\tau=90$ that\nconverges to the strange attractor,  shown in Figure~\\ref{chaotic_attractor}.\nTo show that this is not just a numerical artifact, in\nFigure~\\ref{sensitivity}{\\scriptsize\\bf (LEFT)} we show that, as\nexpected,  there is no sensitivity for the solution for $\\tau=92$ that\nconverges to the  periodic solution shown in Figure~\\ref{PD_right}.\n\n\\begin{figure}[htbp!]\n   \\begin{center}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_ts_90_240-260000.eps}\n\\includegraphics[angle=270, width=6cm]{Figures\/delay_pp_y_yd_90.eps}\n   \\end{center}\n   \\caption[ ]{ {\\scriptsize\\bf (LEFT)} Time series for $\\tau=90$  starting  from the initial data\n   $x(t)=y(t)=0.1, t\\in[-\\tau,0]$.\n   {\\scriptsize\\bf (RIGHT)} Time delay embedding of\n   the strange attractor for $\\tau=90$.  Only the portion of the orbit from $t=240,000-260,000$\n   is shown.} \\label{chaotic_attractor}\n \\end{figure}\n\n \\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=270, width=6cm]{Figures\/pp_ts_tau_90_x_y.eps}\n\t\\caption[]{\n\tThe strange attractor, for $\\tau=90$,  shown in Figure~\\ref{chaotic_attractor} in\n\t$(x,y)$-space.   Only the\n   portion of the orbit from $t=240,000$ \\,  to \\, $260,000$ is shown.\n   }\\label{chaos_xy}\n\\end{center}\n\\end{figure}\n\n \\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=0, width=7cm,height=5cm]{Figures\/return_map_tau_90.eps}\n\t\\caption[]\n  \n  \n  \n   The return map for $\\tau=90$, computing the minimum value of $y(t)$\n  \n   as a function of\nthe preceding minimum value  of $y(t)$ for $y(t)<0.7$ in both cases, using the data in Figure~\\ref{chaotic_attractor}. }\\label{return_map}\n\\end{center}\n\\end{figure}\n\n\n\\begin{figure}[htbp!]\n\t\\begin{center}\n\t\\includegraphics[angle=0,width=6.75cm]{Figures\/insensitive_tau_92.eps}\n\t\n\t\\includegraphics[angle=0,width=6.75cm]{Figures\/sensitive_tau_90.eps}\n\t\n\n\\caption[]{ Time series {\\scriptsize\\bf (LEFT)} for the solution that converges to a periodic attractor when $\\tau=92$, and {\\scriptsize\\bf (RIGHT)}\nfor the solution that converges to a strange attractor when $\\tau=90$, demonstrating that there is no sensitivity to initial data in the\nformer case, but that there is sensitivity in the latter case. Initial data used for the solid\ncurves:\n$x(t)=y(t)=0.1$ for $t\\in[-\\tau,0]$, and for the dotted curves:  $x(t)=0.11$ and\n$y(t)=0.1$ for $t\\in[\\tau,0]$.} \\label{sensitivity}\n\\end{center}\n\\end{figure}\n\n\nThis example demonstrates that including delay in a simple\npredator-prey model that always has a globally asymptotically stable\nequilibrium point in the absence of delay,\ncannot only destabilize a\nglobally asymptotically stable equilibrium point, but can  even result in\nthe birth of a strange attractor.\n\n\\section{Discussion and Conclusions} \\label{discussion}\n\nWe investigated the effect of the time  required for predators to process\ntheir prey on the possible  dynamics predicted by a mathematical\nmodel of predator-prey interaction.  We incorporated a discrete\ndelay    to model this process in one of the simplest classical\npredator-prey models,    one that only allows convergence to an\nequilibrium  when this delay is ignored.\nWe showed that including the delay results in a model with much\nricher dynamics.  By choosing one of the simplest models when delay is\nignored, one that predicts that no   oscillatory behaviour is possible, the\neffect of the delay on the dynamics is emphasized.\n\n\n\n This model can also be interpreted as a\nmodel of a stage-structured population  with the\ndelay modelling the  maturation time of the juveniles\n(see  Gourley and Kuang \\cite{Kuang2004}).\n\nIn the model we considered,   the prey are assumed to grow logistically in\nthe absence of the predator.   The interaction of the predator and prey\nis described using a linear response function, often referred to as mass\naction or   Holling type I.\nIt is well-known that when delay is ignored this is one of the simplest\npredator-prey models for which all solutions converge\nto a globally asymptotically stable equilibrium point for all choices of the\nparameters.   Therefore, any resulting non-equilibrium dynamics would\nthen be solely attributable to the introduction of the delay in the\ngrowth term of the predator.  We not only found\nnon-trivial periodic solutions, but also bistability, and chaotic dynamics.\nIt is then likely that there is  similar   rich dynamics\nin most predator-prey models with any reasonable response function, when  such a delay is incorporated,  for\nsome selection of the parameters, including the form most used by\necologists, the  Holling type II form.  This form\ngiven mathematically by $f(x)=mx\/(1+bx))$, can be considered a   generalization of the Holling type I form, obtained by simply adding an extra parameter, $b$.\nHowever, we feel that demonstrating  that this wide range of dynamics is\npossible for even for one of the simplest models gives more compelling\nevidence that delay should not be ignored when making policy decisions.\n\nUnderstanding how changes in average\ntemperature   might affect   survivability of endangered populations or\nresult in invasions by undesirable populations is   important.\nSince temperature can affect how quickly predators process the prey\nthat they capture, based on our results there might be important\nimplications   for  populations in the wild.   In most predator\npopulations,   the processing time, $\\tau$, is faster when it is warmer and\nslower when it is colder.  Our results may help us  understand how a change in\naverage temperature might\ninfluence the dynamics of   particular predator-prey systems of interest.\n\n\nOur study\nsuggests that we need to be careful when measuring population sizes\nin the wild and predicting the general health of the population based\nupon whether the population seems to be increasing or decreasing.\nShort term indications  that a population size is changing\n in a system with oscillatory dynamics  may be misleading\nand predicting future population size may be impossible without more\ninformation. It would be necessary to have some idea of the period of the\nintrinsic oscillations  if the population is suspected to\nvary periodically.  If the dynamics are suspected to be chaotic, this  may be\neven  more complicated, due to sensitivity to initial data.\n\n\nIf a predator-prey system has the potential to have  chaotic dynamics,\nbased on our results, is there anything that is predictable?    Can such\nanalyses suggest how to prevent   extinctions or\ninvasions due to a change in average temperature that could result in a\nchange in the processing time of the prey by the predator? We give some observations based on the predictions of our model and the example we considered in Section~\\ref{sec:example}.  However,  more work would need to be done to determine whether  these predictions are consistent for  more realistic\nmodels,  and if so,   long term observations would have to be made by\necologists  to determine if they are relevant for   populations\nin the wild.\n\n\n  First,     at the one extreme,\nif the processing time is too long, the predator population would not\nbe expected to survive,  since it\nis obvious that if the processing time is longer than the life span of\nthe predator,  the predator population has no chance to avoid\nextinction.   If there was\nno such  threshold for extinction predicted by the model, the model should be abandoned.  It is\ntherefore very important to use the term\n$e^{-s\\tau}y(t-\\tau)$ and not just $y(t-\\tau)$ in order to account for the predators that do not survive\nlong enough to affect growth of the population,  in the equation\ndescribing the growth of the predator in\nmodel (\\ref{ppreydelay}). Due to this term,\n in our model there is such a threshold, $\\tau_c$.\n\n From the orbit diagram (see Figure~\\ref{bif_diag_full}),\nfor the example considered in Section~\\ref{sec:example}, we summarize some\nobservations.  The dynamics for both populations are   oscillatory for a wide\nrange of processing times,  and non-oscillatory for only relatively\n(very) short or relatively long processing times  (i.e.  before the first Hopf bifurcation at\n$\\tau\\approx 1.9 $ and after the last one at $\\tau\\approx 108 $).\nFor relatively long processing times, slightly larger than\nthe value of $\\tau$ at the final Hopf\nbifurcation of the coexistence equilibrium,  the population is no longer\noscillatory.  The size of the predator\npopulation decreases relatively slowly as the processing time increases\nfurther.\nAlthough  the size of the predator population gets smaller as the\nprocessing time increases, it does not get  much smaller,\nuntil the processing time gets close to the threshold  for extinction\n$\\tau_c$.\nIf we had shown the diagram extended to the threshold $\\tau_c=170$, one would\nsee that as the processing time gets close to  $\\tau_c$, the size of the\npredator population suddenly decreases\nrelatively quickly to zero.  So our model predicts that for a predator\npopulation with fairly long processing times to\nbegin with,    cooling of the environment could be expected to be\ndetrimental with respect to the survivability of the predator\npopulation. Thus, this effect on the size of the predator population might be minor if the delay is close to its Hopf\nbifurcation value, but  could be  drastic if it  is close to the extinction value.\n\nOur example also suggests that   the\npredator population may not be oscillatory when it becomes\nendangered and hence close to extinction, i.e., for excessively long\nprocessing times. It would  be interesting to\ninvestigate  if this also holds for the  model,\nwith Holling type I response functions replaced by Holling type II.\nIn the   model with Holling type II response functions, if the  carrying capacity of the environment for\nthe prey is\nnot affected by the cooling and it is relatively high, then the model\npredicts that the predator\npopulation could still be   oscillatory, even if  the processing\ntime for the predator is ignored.  However,   cooling might also be expected to\nreduce  the carrying capacity for the prey, moving the parameters to a\nrange where that   model would also predict non-oscillatory dynamics (the paradox of enrichment \\cite{Rosenzweig1971}).  So once again, perhaps alarm\nbells should be sounded   when  there is cooling and the\npredator population has been non-oscillatory and  appears to decline rapidly\nas the average yearly temperature declines.\n\n\nOn the other hand, at the other extreme,  our model predicts a Hopf\nbifurcation at a relatively small value of the delay, resulting in\nthe birth of a family of periodic orbits  with\namplitude  increasing very quickly as the delay\nincreases with\nboth the prey\nand predator populations spending time very close to zero.\nThis remains the case for  small, but\nintermediate values of the delay,  (before the two  saddle node of limit\ncycles bifurcations near $\\tau=80$).\nFor this range of $\\tau$, these populations are therefore   very\nsusceptible to stochastic extinctions.  If the processing time was\noriginally very small (below the critical value for the first Hopf\nbifurcation)  and cooling made it longer, again a stochastic extinction might be\nlikely.  Similarly, if $\\tau$ was close to the first of the\ntwo  saddle node of limit\ncycles bifurcations near $\\tau=80$), warming of the average temperature\ncould result in a stochastic extinction  of one  or both of the populations.\nSince the predator cannot survive without the prey in our model, even if\nit was the  prey population  that experienced the stochastic\nextinction, the predator population would eventually die out as well.\nBetween the smallest value of $\\tau$ at which there is a period doubling\nbifurcation, and the value of $\\tau$ at the largest Hopf bifurcation,\ncooling would probably be advantageous to the predator population size.\n\nIn summary, it seems that whether  cooling is beneficial or detrimental to\n  the size of the predator population depends on where the delay is on\n  the bifurcation diagram, and it is very likely that\n  this is  very difficult to determine. As well,\n  if   values of $\\tau$ were to lie in the chaotic region, then changes\n  in the environment that changed the value of $\\tau$ to obtain regular\n  oscillatory dynamics may or may not be preferable.  As well,\n as can be seen by the various\nattractors  shown in Figures~\\ref{PD_left}-\\ref{chaos_xy}, and the orbit\ndiagrams in Figure~\\ref{zoom-in_80_100},  certain\nproperties of the system in the chaotic region such as\nthe maximum and minimum values of the predator\nsize  were fairly insensitive to the change in the delay.\n This is only a toy model. However, it suggests that ignoring delay\nin a model can result in incorrect predictions. Here, delay could change\nthe dynamics from convergence to \na globally asymptotically stable equilibrium to wild\noscillations.  It is most likely that in nature, it would not be\npossible to distinguish from data, a priori, if the dynamics were chaotic or\nperiodic, but  more importantly it would not necessarily be predictable\nwhat the effect of an increase or a decrease in the delay would be. Hence,\nour analysis indicates that one should be extra cautious if trying to\nmanipulate the delay to achieve a certain result based on model\npredictions.  \n\nFinally it is worth pointing out that\n   the resulting strange attractor in the predator-prey model studied\nhere bares such a close resemblance to the Mackey-Glass attractor, a model\ninvolving  a single delay differential equation\nto model a simple  feedback\nsystem for  respiratory control or hematopoietic diseases. Understanding\nwhether there is a deeper significiance to this relationship may give us\na better understanding of the class of possible strange attractors\nand warrants further investigation.\n\n\n\n\n\\bibliographystyle{siam}\n\n\n\\section*{Funding}\nThe Research of Gail S. K. Wolkowicz was partially supported by\n Natural Sciences and Engineering (NSERC) Discovery Grant \\# 9358 and\n Accelerator supplement.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nProbabilistic graphical models for regression problems, where both the model parameters and the observation noise can have complex spatio-temporal correlation structure, are prevalent in neuroscience, neuroimaging and beyond \\citep{shiffrin2008survey}. Example domains where these models are being applied include Gaussian process inference \\citep{rakitsch2013all}, sparse signal recovery and multi-task learning \\citep{candes2006stable,donoho2006compressed,bertrand2019handling,chevalier2020statistical}, array signal processing \\citep{malioutov2005sparse}, fMRI analysis \\citep{cai2020incorporating}, climate science \\citep{beirle2003weekly}, computer hardware monitoring \\citep{ranieri2015near}, and brain-computer interfaces (e.g.~\\citep{lemm2005spatio,NIPS2014_fa83a11a}). The focus in this paper will be on brain source imaging (BSI) \\citep{wu2016bayesian}, i.e. the reconstruction of brain source activity from non-invasive magneto- or electroetoencephalography (M\/EEG) sensors \\citep{baillet2001electromagnetic}. Although a number of generative models that encompass  spatio-temporal correlations have been proposed within the BSI regression problem, inference solutions have either imposed specific simplifications and model constraints \\citep{cai2016bayesian,shvartsman2018matrix,cai2019representational} or have ignored temporal correlations overall \\citep{bertrand2019handling}, and therefore not fully addressed the inherent spatio-temporal problem structure. \n\n\nWe propose a novel and efficient algorithmic framework using Type-II Bayesian regression that explicitly considers the spatio-temporal covariance structure in both model coefficients and observation noise. To this end, we focus on the M\/EEG inverse problem of brain source reconstruction, for which we adopt a multi-task regression approach and formulate the source reconstruction problem as a probabilistic generative model with Gaussian scale mixture priors for sources with spatio-temporal prior covariance matrices that are expressed as a Kronecker product of separable spatial and temporal covariances. Their solutions are constrained to the Riemannian manifold of positive definite (P.D.) temporal covariance matrices. Exploiting the concept of geodesic convexity on the Riemannian manifold of covariance matrices with Kronecker structure, we are then able to derive robust, fast and efficient \\emph{majorization-minimization} (MM) optimization algorithms for model inference with provable convergence guarantees. In addition to deriving update rules for full-structural P.D. temporal covariance matrices, we also show that assuming a Toeplitz structure for the temporal covariance matrix results in computationally efficient update rules within the proposed MM framework. \n\n\n\\section{Spatio-temporal generative model}\n\\label{sec:gen-model}\n\nLet us consider a multi-task linear regression problem, mathematically formulated as\n\\begin{align}\n    {\\bf Y}_{g} = {\\bf L}{\\bf X}_{g} + {\\bf E}_{g} \\quad \\mathrm{for}\\;g=1,\\hdots,G \\;,\\label{eq:linearmodel}\n\\end{align}\nin which a forward matrix, ${\\bf L} \\in \\mathbb{R}^{M\\times N}$, maps a set of coefficients or source components ${\\bf X}_{g} \\in\\mathbb{R}^{N\\times T}$ to measurements ${\\bf Y}_{g} \\in \\mathbb{R}^{M\\times T}$, with independent Gaussian noise ${\\bf E}_{g} \\in\\mathbb{R}^{M\\times T}$. \\ali{$G$ denotes the number of sample blocks in the multi-task problem, while $N$, $M$, and $T$ denote the number of sources or coefficients, the number of sensors or observations, and the number of time instants, respectively.}\nThe problem of estimating $\\{{\\bf X}_{g}\\}_{g=1}^{G}$ given ${\\bf L}$ and $\\{{\\bf Y}_{g}\\}_{g=1}^{G}$ can represent an inverse problem in physics, a multi-task regression problem in machine learning, or a multiple measurement vector (MMV) recovery problem in signal processing \\citep{cotter2005sparse}. \n\n\n\nIn the context of BSI, $\\{{\\bf Y}_{g}\\}_{g=1}^{G}$ refers to M\/EEG sensor measurement data, and $\\{{\\bf X}_{g}\\}_{g=1}^{G}$ refers to brain source activity contributing to the observed sensor data. Here, $G$ can be defined as the number of epochs, trials or experimental tasks. The goal of BSI is to infer the underlying brain activity from the M\/EEG measurements given the lead field matrix ${\\bf L}$. In practice, ${\\bf L}$ can be computed using discretization methods such as the finite element method for a given head geometry and known electrical conductivities using the quasi-static approximation of Maxwell's equations\n\\citep{hamalainen1993magnetoencephalography}. \nAs the number of locations of potential brain sources is dramatically larger than the number of sensors, the M\/EEG inverse problem is highly ill-posed, which can be dealt with by incorporating prior assumptions. Adopting a Bayesian treatment is useful in this capacity because it allows these assumptions to be made explicitly by specifying prior distributions for the model parameters. Inference can be performed either through Maximum-a-Posteriori (MAP) estimation (\\emph{Type-I Bayesian learning}) \\citep{pascual1994low,haufe2008combining,gramfort2012mixed} or, when the model has unknown hyperparameters, through Type-II Maximum-Likelihood (ML) estimation (\\emph{Type-II Bayesian learning}) \\citep{mika2000mathematical,tipping2001sparse,wipf2009unified}. \n\nHere we focus on full spatio-temporal Type-II Bayesian learning, which assumes a family of prior distributions $p({\\bf X}_{g}|{\\bm \\Theta})$ parameterized by a set of hyperparameters $\\bm \\Theta$. We further assume that the spatio-temporal correlation structure of the brain sources can be modeled by a Gaussian $p({\\bf x}_{g}|{\\bm \\Gamma},{\\bf B})\\sim\\mathcal{N}({\\bm 0},{\\bm \\Sigma}_{0})$, where ${\\bf x}_{g}=\\text{vec}({\\bf X}_{g}^{\\top}) \\in \\mathbb{R}^{NT \\times 1}$ and where the covariance matrix ${\\bm \\Sigma}_{0}={\\bm \\Gamma} \\otimes {\\bf B}$ has Kronecker structure, meaning that the temporal and spatial correlations of the sources are modeled independently through  matrices ${\\bm \\Gamma}$ and ${\\bf B}$, where $\\otimes$ stands for the Kronecker product. Note \nthat the decoupling of spatial and temporal covariance is a neurophysiologically plausible assumption as neuronal sources originating from different parts of the brain can be assumed to share a similar autocorrelation spectrum. In this paper, we set ${\\bm \\Gamma}=\\mathrm{diag}({\\bm \\gamma})$, ${\\bm \\gamma} = [\\gamma_{1},\\dots,\\gamma_{N}]^\\top$, and ${\\bf B} \\in \\mathbb{S}^{T}_{++}$, where $\\mathbb{S}^{T}_{++}$ denotes the set of positive definite matrices with size $T \\times T$. Note that using a diagonal matrix ${\\bm \\Gamma}$ amounts to assuming that sources in different locations are independent. This assumption may be relaxed in our future work. On the other hand, modeling a full temporal covariance matrix ${\\bf B}$ amounts to assuming non-stationary dynamics of the sources, which is appropriate in event-related experimental designs. To deal with the more stationary sources (ongoing brain activity), we will also constrain ${\\bf B}$ to have Toeplitz structure later on. Based on these specifications, the prior distribution of the $i$-th brain source is modeled as $p(({\\bf X}_{g})_{i.}|\\gamma_{i},{\\bf B})\\sim\\mathcal{N}(0,\\gamma_{i}{\\bf B}),\\text{for}\\; i=1,\\dots,N$, where $({\\bf X}_{g})_{i.}$ denotes the $i$-th row of source matrix ${\\bf X}_g$. \n\nAnalogous to the definition of the sources, we here also model the noise to have spatio-temporal structure. The matrix ${\\bf E}_{g} \\in \\mathbb{R}^{M\\times T}$ represents $T$ time instances of zero-mean Gaussian noise with full covariance, ${\\bm \\Sigma}_{\\mathrm{\\bf e}}={\\bm \\Lambda} \\otimes {\\bm \\Upsilon}$, where ${\\bf e}_{g}=\\text{vec}({\\bf E}_{g}^{\\top}) \\in \\mathbb{R}^{MT\\times1}\\sim \\mathcal{N}(0,{\\bm \\Sigma}_{\\mathrm{\\bf e}})$, and where ${\\bm \\Lambda}$ and ${\\bm \\Upsilon}$ denote spatial and temporal noise covariance matrices, respectively. Here, we assume that the noise and sources share the same temporal structure, i.e., ${\\bm \\Upsilon}={\\bf B}$. We later investigate violations of this assumption in the simulation section. Analogous to the sources, we consider spatially independent noise characterized by diagonal spatial covariance ${\\bm \\Lambda} =\\mathrm{diag}({\\bm \\lambda})$, ${\\bm \\lambda} = [\\lambda_{1},\\dots,\\lambda_{M}]^\\top$. We consider both the general, \\emph{heteroscedastic}, case in which the noise level may be different for each sensor as well as the \\emph{homoscedastic} case in which the noise level is uniform. \n\nFor later use, we also define augmented versions of the source and noise covariance matrices as well as the lead field. Specifically, we set ${\\bf H}$ and ${\\bm \\Phi}$ so that ${\\bf H}:= [{\\bm \\Gamma}, {\\bf 0}; {\\bf 0}, {\\bm \\Lambda}]$, and ${\\bm \\Phi}:=[{\\bf L},{\\bf I}]$. These definitions allow us to unify source and noise covariance parameters within the single variable ${\\bf H}$, which facilitates the concurrent estimation of both quantities. ${\\bm \\Phi}$ now plays the same role as the lead field ${\\bf L}$, in the sense that it maps ${\\bm \\eta}_g:=[{\\bf x}_g^{\\top},{\\bf e}_g^{\\top}]$ to the measurements; i.e., ${\\bf y}_g = {\\bm \\Phi}{\\bm \\eta}_g$. \\ali{Figure~\\ref{fig:graphprob-geometricmean}-(a) illustrates a probabilistic graphical model summarizing the spatio-temporal generative model of our multi-task linear regression problem.}\n\nThe MMV model Eq.~\\eqref{eq:linearmodel} can be formulated equivalently in \\emph{single measurement vector (SMV)} form by vectorizing the spatio-temporal data matrices and using the Kronecker product as follows: ${\\bf y}_{g}={\\bf D}{\\bf x}_{g}+{\\bf e}_{g}$, where ${\\bf y}_{g}=\\text{vec}\\left({\\bf Y}_{g}^\\top \\right)\\in\\mathbb{R}^{MT\\times1}$, and ${\\bf D}={\\bf L}\\otimes {\\bf I}_{T}$. In a Type-II Bayesian learning framework, the hyper-parameters of the spatio-temporal source model are optimized jointly with the model parameters $\\{\\bm X_g \\}_{g=1}^G$. In our case, these hyper-parameters comprise the unknown source, noise and temporal covariance matrices, i.e., ${\\bm \\Theta} = \\{{\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}\\} $. \n\\section{Proposed method --- full Dugh}\n\\label{sec:full-dugh}\nThe unknown parameters ${\\bm \\Gamma}$, ${\\bm \\Lambda}$, and ${\\bf B}$ can be optimized in an alternating iterative process. Given initial estimates, the posterior distribution of the sources is a Gaussian of the form $p({\\bf x}_{g}|{\\bf y}_{g},{\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}) \\sim \\mathcal{N}(\\bar{\\bf x}_{g},{\\bm \\Sigma}_{\\bf x})$, whose mean and covariance are defined as follows:\n\\begin{align}\n    \\bar{\\bf x}_{g} &={\\bm \\Sigma}_{0}{\\bf D}^{\\top}({\\bm \\Lambda} \\otimes {\\bf B} + {\\bf D}{\\bm \\Sigma}_{0}{\\bf D}^{\\top})^{-1}{\\bf y}_{g}={\\bm \\Sigma}_{0}{\\bf D}^{\\top}\\Tilde{\\bm \\Sigma}_{\\bf y}^{-1}{\\bf y}_{g}\\;,\n    \\label{eq:MeanValue-ST} \\\\\n    {\\bm \\Sigma}_{\\bf x} &= {\\bm \\Sigma}_{0}-{\\bm \\Sigma}_{0}{\\bf D}^{\\top}\\Tilde{\\bm \\Sigma}_{\\bf y}^{-1}{\\bf D}{\\bm \\Sigma}_{0}\n    \\label{eq:CovarianceValue-ST}\\;,\n\\end{align}\nwhere ${\\bm \\Sigma}_{\\bf y}={\\bf L}{\\bm \\Gamma}{\\bf L}^{\\top}+{\\bm \\Lambda}$, and where $\\Tilde{\\bm \\Sigma}_{\\bf y} = {\\bm \\Sigma}_{\\bf y} \\otimes {\\bf B}$ denotes the spatio-temporal statistical model covariance matrix. The estimated posterior parameters $\\bar{\\bf x}_{g}$ and ${\\bm \\Sigma}_{\\bf x}$ are then in turn used to update ${\\bm \\Gamma}$, ${\\bm \\Lambda}$, and $\\bf B$ as the minimizers of the negative log of the marginal likelihood $p({\\bf Y}|{\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, which is given by:\n\\begin{align}\n    \\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}) &= T\\log|{{\\bm \\Sigma}_{\\bf y}}|+M\\log|{\\bf B}| +\\frac{1}{G}\\sum_{g=1}^{G}\\text{tr}({{\\bm \\Sigma}_{\\bf y}^{-1}}{\\bf Y}_{g}{\\bf B}^{-1}{\\bf Y}_{g}^{\\top}) \n    \\label{eq:MLKronokCost}\\;,\n\\end{align}\nwhere $|\\cdot|$ denotes the determinant of a matrix. A detailed derivation is provided in Appendix~\\ref{appendix:spatio-temp-cost}. Given the solution of the hyperparameters ${\\bm \\Gamma}$, ${\\bm \\Lambda}$, and $\\bf B$, the posterior source distribution is obtained by plugging these estimates into Eqs.~\\eqref{eq:MeanValue-ST}--\\eqref{eq:CovarianceValue-ST}. This process is repeated until convergence.\n\nThe challenge in this high-dimensional inference problem is to find (locally) optimal solutions to  Eq.~\\eqref{eq:MLKronokCost}, which is a non-convex cost function, in adequate time. Here we propose a novel efficient algorithm, which is able to do so. Our algorithm thus learns the full spatio-temporal correlation structure of sources and noise. \\citet{hashemi2021unification} have previously demonstrated that \n\\emph{majorization-minimization} (MM) \\citep{sun2017majorization}\nis a powerful non-linear optimization framework that can be leveraged to solve similar Bayesian Type-II inference problems. Here we extend this work to our spatio-temporal setting. Building on the idea of majorization-minimization, we construct convex surrogate functions that \\emph{majorize} $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$ in each iteration of the proposed algorithm. Then, we show the minimization equivalence between the constructed majorizing functions and Eq.~\\eqref{eq:MLKronokCost}. These results are presented in theorems~\\ref{Theo:Time-Surrogate} and \\ref{Theo:Space-Surrogate}. Theorems~\\ref{Theo:Time-Sol} and \\ref{Theo:Space-Sol} propose an efficient alternating optimization algorithm for solving $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, which leads to update rules for the spatial and temporal covariance matrices ${\\bm \\Gamma}$ and ${\\bf B}$ as well as the noise covariance matrix ${\\bm \\Lambda}$. \n\nStarting with the estimation of the temporal covariance based on the current source estimate, we can state the following theorem: \n\\begin{figure}\n\\begin{center}\n\\begin{minipage}[b]{0.49\\textwidth}\n    \\centerline{\\includegraphics[width=7.0cm,keepaspectratio]{figures\/Paper-graphical-probabilistic-models_final.pdf} } \n    \\begin{center}\n        \\centerline{(a)}\n    \\end{center}\n\\end{minipage}\n\\begin{minipage}[b]{0.49\\textwidth}\n    \\centerline{\\includegraphics[width=7.0cm,keepaspectratio]{figures\/PSD-NeurIPS-Temporal_2.pdf}} \n    \\begin{center}\n        \\centerline{(b)}\n    \\end{center}\n\\end{minipage}\n\\end{center}\n\\caption{\\ali{(a) Probabilistic graphical model for the multi-task linear regression problem. (b) Geometric representation of the geodesic path between the pair of matrices \n$\\{{\\bf B}^{k},{\\bf M}_{\\mathrm{time}}^{k}\\}$ on the P.D. manifold and the geometric mean between them, which is used to update ${\\bf B}^{k+1}$.}}\n\\label{fig:graphprob-geometricmean}\n\\vspace{-3mm}\n\\end{figure}\n\n\\begin{theo} \n\\label{Theo:Time-Surrogate}\nOptimizing the non-convex Type-II cost function in Eq.~\\eqref{eq:MLKronokCost},  $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, with respect to the temporal covariance matrix ${\\bf B}$ is equivalent to optimizing the following convex surrogate function, which \\emph{majorizes} Eq.~\\eqref{eq:MLKronokCost}:  \n\\begin{align}\n    \\mathcal{L}^{\\mathrm{time}}_{\\mathrm{conv}}({\\bm \\Gamma}^{k},{\\bm \\Lambda}^{k},{\\bf B})=\\tr\\left(({\\bf B}^{k})^{-1}{\\bf B}\\right)+\\tr({\\bf M}_{\\mathrm{time}}^{k}{\\bf B}^{-1}),\n    \\label{eq:TimeSurrogateFunction}    \n\\end{align}\nwhere ${\\bf M}_{\\mathrm{time}}^{k}$ is defined as\n\\begin{equation}\n    {\\bf M}_{\\mathrm{time}}^{k}:=\\frac{1}{MG}\\sum_{g=1}^{G}{\\bf Y}_{g}^{\\top}\\left({\\bm \\Sigma}_{\\bf y}^{k}\\right)^{-1}{\\bf Y}_{g}\n    \\label{eq:TempSampleCov}\\;,\n\\end{equation}\nand where ${\\bm \\Gamma}^{k}$, ${\\bm \\Lambda}^{k}$, and ${\\bm \\Sigma}_{\\bf y}^{k}$ denote the source, noise and statistical model covariance matrices at the $k$-th iteration, which are treated as constants when optimizing over ${\\bf B}$. \n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix \\ref{appendix:Time-Surrogate}.\n\\end{proof}\nThe solution of Eq.~\\eqref{eq:TimeSurrogateFunction} can be obtained by taking advantage of the Riemannian geometry of the search space. The following theorem holds. \n\\begin{theo}\n\\label{Theo:Time-Sol}\nThe cost function $\\mathcal{L}^{\\mathrm{time}}_{\\mathrm{conv}}({\\bm \\Gamma}^{k},{\\bm \\Lambda}^{k},{\\bf B})$ is strictly geodesically convex with respect to the P.D. manifold and its minimum with respect to ${\\bf B}$ can be attained by iterating the following update rule until convergence:\n\\begin{align}\n    {\\bf B}^{k+1} \\leftarrow ({\\bf B}^{k})^{\\nicefrac{1}{2}}\\left(({\\bf B}^{k})^{\\nicefrac{-1}{2}}{\\bf M}_{\\mathrm{time}}^{k}({\\bf B}^{k})^{\\nicefrac{-1}{2}}\\right)^{\\nicefrac{1}{2}}({\\bf B}^{k})^{\\nicefrac{1}{2}}\\;\\label{eq:BGeodesicUpdate}.\n\\end{align}\n\\end{theo}\n\\begin{proof}\n\\ali{A detailed proof is provided in Appendix \\ref{appendix:Time-Sol}.}\n\\end{proof}\n\n\\begin{rem}\n    A geometric interpretation of the update rule \\eqref{eq:BGeodesicUpdate} is that it finds the geometric mean between a spatially whitened version of the empirical temporal covariance matrix, ${\\bf M}_{\\mathrm{time}}^{k}$, and the model temporal covariance matrix from the previous iteration, ${\\bf B}^{k}$. A geometric representation of the geodesic path between the pair of matrices $\\{{\\bf B}^{k},{\\bf M}_{\\mathrm{time}}^{k}\\}$ on the P.D. manifold and the geometric mean between them, representing the update for ${\\bf B}^{k+1}$, is provided in Figure~\\ref{fig:graphprob-geometricmean}-(b).\n\\end{rem}\nWe now derive an update rule for the spatial source and noise covariance matrices using an analogous approach. To this end, we first construct a convex surrogate function that \\emph{majorizes} $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$ in each iteration of the optimization algorithm by considering augmented variables comprising both the source and noise covariances, i.e., ${\\bf H}:= [{\\bm \\Gamma}, \\bf 0; \\bf 0, {\\bm \\Lambda}]$, and ${\\bm \\Phi}:=[{\\bf L},{\\bf I}]$. The following theorem holds. \n\\begin{theo} \n\\label{Theo:Space-Surrogate}\nMinimizing the non-convex Type-II cost function in Eq.~\\eqref{eq:MLKronokCost}, $\\mathcal{L}_{\\text{kron}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B})$, with respect to the spatial covariance matrix ${\\bf H}$ is equivalent to minimizing the following convex surrogate function, which \\emph{majorizes} Eq.~\\eqref{eq:MLKronokCost}:\n\\begin{align}\n    \\mathcal{L}^{\\mathrm{space}}_{\\mathrm{conv}}({\\bm \\Gamma},{\\bm \\Lambda},{\\bf B}^{k}) = \\mathcal{L}^{\\mathrm{space}}_{\\mathrm{conv}}({\\bf H},{\\bf B}^{k})=\\tr\\left({\\bm \\Phi}^{\\top}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bm \\Phi}{\\bf H}\\right)+\\tr\\left({\\bf M}_{\\mathrm{SN}}^{k} {\\bf H}^{-1} \\right) \\;,\n    \\label{eq:SpaceSurrogateFunction}    \n\\end{align}\nwhere ${\\bf M}_{\\mathrm{SN}}^{k}$ is defined as\n\\begin{align}\n        {\\bf M}_{\\mathrm{SN}}^{k} &:= {\\bf H}^{k}{\\bm \\Phi}^{\\top}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bf M}_{\\mathrm{space}}^{k}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bm \\Phi} {\\bf H}^{k} \\;, \\mathrm{with} \\nonumber \\\\ \n        {\\bf M}_{\\mathrm{space}}^{k} &:= \\frac{1}{TG}\\sum_{g=1}^{G}{\\bf Y}_{g}({\\bf B}^{k})^{-1}{\\bf Y}_{g}^{\\top} \\;,\n        \\label{eq:SpaceSampleCov}\n\\end{align}\nand where ${\\bf B}^k$ denotes the temporal model covariance matrix estimated in the $k$-th iteration.\n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix \\ref{appendix:Space-Surrogate}.\n\\end{proof}\n\n\nWhile, in principle, update rules for full-structured spatial source and noise covariances may be conceived, in analogy to the ones presented for the temporal covariance, we here restrict ourselves to the discussion of diagonal spatial covariances. Note, though, that such a choice of prior does not prohibit the reconstruction of parameters with more complex correlation structure. We have ${\\bf H}=\\mathrm{diag}({\\bf h})$, ${\\bf h} = [\\gamma_{1},\\dots,\\gamma_{N}, \\sigma^2_{1},\\dots,\\sigma^2_{M}]^\\top$, and ${\\bm \\Sigma}_{\\bf y}={\\bm \\Phi}{\\bf H}{\\bm \\Phi}^{\\top}$. The update rule for ${\\bf H}$ then takes a simple form, as stated in the following theorem:\n\\begin{theo}\n\\label{Theo:Space-Sol}\nThe cost function $\\mathcal{L}^{\\mathrm{space}}_{\\mathrm{conv}}({\\bf H},{\\bf B}^{k})$ is convex in $\\bf h$, and its minimum with respect to $\\bf h$ can be obtained according to the following closed-form update rule, which concurrently estimates the scalar source variances and heteroscedastic noise variances:\n\\begin{align}\n    {\\bf H}^{k+1}=\\mathrm{diag}({\\bf h}^{k+1}),\\;   h_{i}^{k+1}&\\leftarrow\\sqrt{\\frac{g_{i}^{k}}{z_{i}^{k}}}\\quad \\text{for}\\;i=1,\\hdots,N+M \\;\\label{eq:HDiagonalUpdate}, \\text{where} \\\\ \n    {\\bf g} &:=\\mathrm{diag}({\\bf M}_{\\mathrm{SN}}^{k}) \\label{eq:diagISpace}  \\\\\n    {\\bf z} &:=\\mathrm{diag}({\\bm \\Phi}^{\\top}({\\bm \\Sigma}_{\\bf y}^{k})^{-1}{\\bm \\Phi}) \\label{eq:diagIISpace} \n\\end{align} \n\\end{theo}\n\\begin{proof}\nA detailed proof can be found in Appendix \\ref{appendix:Space-Solution}.\n\\end{proof}\nThe final estimate of the posterior distribution \\eqref{eq:MeanValue-ST} can be obtained by starting from any initial guess for the model parameters, ${\\bf H}^{0}=\\{{\\bm \\Gamma}^{0},{\\bm \\Lambda}^{0}\\}$ and ${\\bf B}^{0}$, and iterating between update rules \\eqref{eq:BGeodesicUpdate}, \\eqref{eq:HDiagonalUpdate}, and \\eqref{eq:MeanValue-ST} until convergence. We call the resulting algorithm \\emph{full Dugh}. Convergence is shown in the following theorem:\n\\begin{theo}\n\\label{theo:MM-Convergence-Guarantees}\nOptimizing the non-convex ML cost function  Eq.~\\eqref{eq:MLKronokCost} with alternating update rules for ${\\bf B}$ and ${\\bf H}=\\{{\\bm \\Gamma},{\\bm \\Lambda}\\}$ in Eq.~\\eqref{eq:BGeodesicUpdate} and Eq.~\\eqref{eq:HDiagonalUpdate} defines an MM algorithm, which is guaranteed to converge to a stationary point. \n\\end{theo}\n\\begin{proof}\nA detailed proof can be found in Appendix~\\ref{appendix:MM-Convergence-Guarantees}.\n\\end{proof}\n\\section{Efficient inference for stationary temporal dynamics --- thin Dugh}\n\\label{sec:thin-Dugh}\nWith full Dugh, we could present a general inference method for data with full temporal covariance structure. This algorithm may be used for non-stationary data, for which the covariance between two time samples depends on the absolute position of the two samples within the studied time window. An example is the reconstruction of brain responses to repeated external stimuli from M\/EEG (event-related electrical potential or magnetic field) data, where data blocks are aligned to the stimulus onset. In other realistic settings, however, stationarity can be assumed in the sense that the covariance between two samples only depends on their relative position (distance) to each other but not to any external trigger. An example is the reconstruction of ongoing brain dynamics in absence of known external triggers. This situation can be adequately modeled using temporal covariance matrices with Toeplitz structure. In the following, we devise an efficient extension of full Dugh for that setting. \n\nTemporal correlation has been incorporated in various brain source imaging models through different priors \\citep[see][and references therein]{pirondini2017computationally,wu2016bayesian}. According to  \\citep{lamus2012spatiotemporal}, the temporal dynamics of neuronal populations can be approximated by a first order auto-regressive (AR(1)) model of the form ${x}_n(t+1)=\\beta {x}_n(t)+\\sqrt{1-\\beta^{2}}\\xi_n(t),~ n=1,\\hdots,N; t=1,\\hdots,T$ with AR coefficient $\\beta \\in(-1,1)$ and innovation noise $\\xi_n(t)$. It can be shown that the temporal correlation matrix corresponding to this AR model has Toeplitz structure, ${\\bf B}_{i,j}=\\beta^{|{i-j}|}$ \\citep{zhang2011sparse}. Consequently,  we now constrain the cost function Eq.~\\eqref{eq:TimeSurrogateFunction} to the set of Toeplitz matrices:\n\\begin{align}\n    {\\bf B}^{k+1} &=\\argmin_{{\\bf B} \\in \\mathcal{B},\\;{\\bf H}= {\\bf H}^{k} } \\tr(({\\bf B}^{k})^{-1}{\\bf B})+\\tr({\\bf M}_{\\mathrm{time}}^{k}{\\bf B}^{-1})\\;,\n    \\label{eq:MinComstrainedTempo}\n\\end{align}\nwhere set $\\mathcal{B}$ denotes the set of real-valued positive-definite Toeplitz matrices of size $T\\times T$. \n\nIn order to be able to derive efficient update rules for this setting, we bring ${\\bf B}$ into a diagonal form. The following proposition holds.\n\\begin{prop} \n\\label{prop:toeplitz}\nLet ${\\bf P} \\in \\mathbb{R}^{L \\times L}$ with $L>T$, be the circulant embedding of matrix ${\\bf B} \\in \\mathcal{B}^L$, where $\\mathcal{B}^L$ is assumed to be the subset of $\\mathcal{B}$ that guarantees that ${\\bf P}$ is a real-valued circulant P.D. matrix.  \nThen the Toeplitz matrix ${\\bf B}$ can be diagonalized as follows:\n\\begin{align}\n    {\\bf B} &= {\\bf Q}{\\bf P}{\\bf Q}^{H} \\quad \\text{with} \\quad \\mathbf{Q} = \n    [{\\bf I}_{M},{\\bf 0}]{\\bf F}_{L}   \\label{eq:FourierDiagonalization},\\;\\text{and}  \\\\\n    [\\mathbf{F}_{L}]_{m,l} &= \\frac{1}{\\sqrt{L}}e^{\\frac{i2\\pi(l-1)}{L}(m-1)} \\label{eq:DFT}\\;,\n\\end{align}\nwhere ${\\bf P}=\\mathrm{diag}({\\bf p}) =\\mathrm{diag}(p_{0},p_{1},\\dots,p_{L-1})$ is a diagonal matrix. The main diagonal coefficients of ${\\bf P}$ are given as the normalized discrete Fourier transform (DFT), represented by the linear operator $\\mathbf{F}_{L}$, of the first row of ${\\bf B}$: ${\\bf p}=\\mathbf{F}_{L}{\\bf B}_{1.}$  \n(Note that a Toeplitz matrix can be represented by its first row or column, ${\\bf B}_{1.}$). \n\\end{prop}\n\\begin{proof}\nThis is a direct implication of the fact that Toeplitz matrices can be embedded into circulant matrices of larger size \\citep{dembo1989embedding} and the result that circulant matrices can be approximately diagonalized using the Fourier transform \\citep{grenander1958toeplitz}. Further details can be found in Appendix~\\ref{appendix:toeplitz}.\n\\end{proof}\nThe solution of Eq.~\\eqref{eq:MinComstrainedTempo} can be obtained by direct application of Proposition~\\ref{prop:toeplitz}. The results is summarized in the following theorem: \n\\begin{theo}\n\\label{Theo:Time-Toeplitz}\nThe cost function Eq.~\\eqref{eq:MinComstrainedTempo} is convex in $\\bf p$, and its minimum with respect to $\\bf p$ can be obtained by iterating the following closed-form update rule until convergence:\n\\begin{align}\n    p_{l}^{k+1} &\\leftarrow\\sqrt{\\frac{\\hat{g}_{l}^{k}}{\\hat{z}_{l}^{k}}}\\;\\text{for}\\;l=1,\\hdots,L \\;\\label{eq:BToeplitzUpdate}, \\text{where} \\\\ \n    \\hat{\\bf g} &:=\\mathrm{diag}({\\bf P}^{k}{\\bf Q}^{H}({\\bf B}^{k})^{-1}{\\bf M}_{\\mathrm{time}}^{k}({\\bf B}^{k})^{-1}{\\bf Q}{\\bf P}^{k}) \\label{eq:diagI}  \\\\\n    \\hat{\\bf z} &:=\\mathrm{diag}({\\bf Q}^{H}({\\bf B}^{k})^{-1}{\\bf Q})\\;. \n    \\label{eq:diagII} \n\\end{align}\n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix~\\ref{appendix:Time-Toeplitz}.\n\\end{proof}\n\n\n\\subsection{Efficient computation of the posterior}\nGiven estimates of the spatial and temporal covariance matrices, we can efficiently compute the posterior mean by exploiting their intrinsic diagonal structure. \n\\begin{theo}\n\\label{theo:eff-post}\nGiven the diagonalization ${\\bf B}={\\bf Q}{\\bf P}{\\bf Q}^{H}$ of the temporal correlation matrix and the eigenvalue decomposition ${\\bf L}\\bm{\\Gamma}{\\bf L}^{\\top}= {\\bf U}_{\\bf x}{\\bf D}_{\\bf x}{\\bf U}_{\\bf x}^{\\top}$ of ${\\bf L}\\bm{\\Gamma}{\\bf L}^{\\top}$, where ${\\bf D}_{\\bf x}=\\mathrm{diag}(d_1,\\hdots,d_M)$, the posterior mean is efficiently computed as \n\\begin{align}\n    \\bar{\\bf x}_{g} & =\\left(\\bm{\\Gamma}\\otimes{\\bf B}\\right){\\bf D}^{\\top}\\Tilde{\\bm \\Sigma}_{\\bf y}^{-1}{\\bf y}_{g}= \\tr\\left({\\bf Q}{\\bf P} \\left({\\bm \\Pi} \\odot{\\bf Q}^{H}{\\bf Y}_{g}^{\\top}{\\bf U}_{\\bf x}\\right)\\left({\\bf U}_{{\\bf x}}^{\\top}{\\bf L}\\bm{\\Gamma}^{\\top}\\right) \\right) \n    \\label{eq:efficientPosterior}\\;,\n\\end{align}\nwhere $\\odot$ denotes the Hadamard product between the corresponding elements of two matrices of equal size. In addition, ${\\bm \\Pi}$ is defined as follows: $[{\\bm \\Pi}]_{l,m}= \\frac{1}{{\\sigma}_m^2 + p_l d_m}\\;\\text{for}\\; l=1,\\hdots,L\\;\\text{and}\\;\\;m=1,\\hdots,M$.\n\\end{theo}\n\\begin{proof}\nA detailed proof is provided in Appendix \\ref{appendix:eff-post}. \n\\end{proof}\n\nThe resulting algorithm, obtained by iterating between update rules \\eqref{eq:BToeplitzUpdate}, \\eqref{eq:HDiagonalUpdate} and \\eqref{eq:efficientPosterior}, is called \\emph{thin Dugh} (as opposed to \\emph{full Dugh} introduced above).\n\n\\section{Simulations}\n\\label{sec:simulation}\n\\vspace{-3mm}\nWe present two sets of experiments to assess the performance of our proposed methods. In the first experiment, we compare the reconstruction performance of the proposed Dugh algorithm variants to that of Champagne \\citep{wipf2010robust} and two other competitive methods -- eLORETA \\citep{pascual2007discrete} and S-FLEX \\citep{haufe2011large} -- for a range of SNR levels, numbers of time samples, and orders of AR coefficients. In the second experiment, we test the impact of model violations on the temporal covariance estimation.\nAll experiments are performed using \\texttt{Matlab} on a machine with a 2.50 GHz Intel(R) Xeon(R) Platinum 8160 CPU. \\ali{The computational complexity of each method in terms of the average running time in units of seconds for 1000 iterations is as follows: \\emph{Full Dugh}: 67.094s, \\emph{Thin Dugh}: 62.289s, \\emph{Champagne}: 1.533s, \\emph{eLORETA}: 2.653s, and  \\emph{S-FLEX}: 20.963s.}\nThe codes are publicly available at \\href{https:\/\/github.com\/AliHashemi-ai\/Dugh-NeurIPS-2021}{\\textcolor{blue}{https:\/\/github.com\/AliHashemi-ai\/Dugh-NeurIPS-2021}}.\n\n\\vspace{-1mm}\n\\subsection{Pseudo-EEG signal generation and benchmark comparison}\n\\vspace{-1mm}\n\nWe simulate a sparse set of $N_0 =3$ active sources placed at random locations on the cortex. To simulate the electrical neural activity, we sample time series of length $T \\in \\{10,20,50,100\\}$ from a univariate linear autoregressive AR(P) process.\nWe use stable AR systems of order $P \\in \\{1,2,5,7\\}$. \nThe resulting source distribution is then projected to the EEG sensors, denoted by ${{\\bf Y}^{\\mathrm{signal}}}$, using a realistic lead field matrix, ${\\bf L} \\in \\mathbb{R}^{58 \\times 2004}$. We generate ${\\bf L}$ using the New York Head model \\citep{huang2016new} taking into account the realistic anatomy and electrical tissue conductivities of an average human head. Finally, we add Gaussian white noise to the sensor space signal.\nNote that the simulated noise and source time courses do not share a similar temporal structure here -- sources are modeled with a univariate autoregressive AR(P) process while a temporally white Gaussian distribution is used for modeling noise. Thus, we could assess the robustness of our proposed method under violation of the model assumption that the temporal structure of sources and noise is similar.\nThe resulting noise matrix ${\\bf E}=[{\\bf e}(1),\\dots,{\\bf e}(T)]$ is first normalized and then added to the signal matrix ${\\bf Y}^{\\mathrm{signal}}$ as follows: ${\\bf Y}={\\bf Y}^{\\mathrm{signal}}+\\frac{(1-\\alpha)\\left\\Vert {{\\bf Y}^{\\mathrm{signal}}}\\right\\Vert_{F}}{\\alpha \\left\\Vert{\\bf E}\\right\\Vert_{F}}{\\bf E}$, where $\\alpha$ determines the signal-to-noise ratio (SNR) in sensor space. Precisely, SNR is defined as $\\mathrm{SNR} = 20\\mathrm{log}_{10}\\left(\\nicefrac{\\alpha}{1-\\alpha}\\right)$. In this experiment the following values of $\\alpha$ are used: $\\alpha \\in $ \\{0.55, 0.65, 0.7, 0.8\\}, which correspond to the following SNRs: SNR $\\in$ \\{1.7, 5.4, 7.4, 12\\}~(dB). {Interested readers can refer to Appendix~\\ref{appendix-Simulation} and \\citep{haufe2016simulation} for further details on the simulation framework.}\nWe quantify the performance of all algorithms using the \\emph{earth mover's distance} (EMD) \\citep{rubner2000emd,haufe2008combining} and the maximal correlation between the time courses of the simulated and the reconstructed sources (TCE). Each simulation is carried out 100 times using different instances of $\\bf{X}$ and $\\bf E$, and the mean and standard error of the mean (SEM) of each performance measure across repetitions is calculated. \n\n\nIn order to investigate the impact of model violations on the temporal covariance estimation, we generate a random Gaussian source matrix, ${\\bf X} \\in \\mathbb{R}^{2004 \\times 30 \\times G}$ representing the brain activity of $2004$ brain sources at $30$ time instances for different numbers of trials $G \\in 10,20,30,40,50$. In all trials, sources are randomly sampled from a zero-mean normal distribution with spatio-temporal covariance matrix ${\\bm \\Gamma} \\otimes {\\bf B}$, where ${\\bf B} \\in \\mathbb{R}^{30 \\times 30}$ is either a full-structural random PSD matrix or a Toeplitz matrix with $\\beta = 0.8$. Gaussian noise $\\bf E$ sharing the same temporal covariance with the sources is added to the measurements ${\\bf Y}={\\bf L}{\\bf X}$, so that the overall SNR is $0$~dB. We evaluate the accuracy of the temporal covariance reconstruction using Thin and full Dugh.\nThe performance is evaluated using two measures: Pearson correlation between the original and reconstructed temporal covariance matrices, $\\bf B$ and $\\hat{{\\bf B}}$, denoted by $r({\\bf B},\\hat{\\bf B})$, and the normalized mean squared error (NMSE) defined as: $\\text{NMSE} = ||\\hat{{\\bf B}}-{\\bf B}||_{F}^{2} \/ ||{\\bf B}||_{F}^{2}$. The similarity error is defined as: $1- r({\\bf B},\\hat{\\bf B})$. Note that NMSE measures the reconstruction at the true scale of the temporal covariance; while $r({\\bf B},\\hat{\\bf B})$ is scale-invariant and hence only quantifies the overall structural similarity between simulated and estimated noise covariance matrices. \n\n\\begin{figure}\n\\begin{minipage}[b]{0.85\\linewidth}\n  \\centering\n    \\centerline{\\includegraphics[width=8cm]{figures\/legend-final-revised.PNG}}\n\\end{minipage}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth,scale=0.5,keepaspectratio]{figures\/Source-eval-metrics-new-final-revised.pdf}}\n\\caption{Source reconstruction performance (mean $\\pm$ SEM) of the four different brain source imaging schemes, namely Thin Dugh, Champagne, eLORETA, and S-FLEX, for data generated by a realistic lead field matrix. Performance is assessed for different settings including a wide range of SNRs, different numbers of time samples, and different AR model orders. Performance is evaluated in terms of the earth mover's distance (EMD) and time-course correlation error (TCE) between each simulated source and the reconstructed source with highest maximum absolute correlation.}\n\\label{fig:metrics-source}\n\\vspace{-3mm}\n\\end{figure}\n\n\n\n\nIn Figure \\ref{fig:metrics-source}, we show the source reconstruction performance (mean $\\pm$ SEM) of the four different brain source imaging schemes, namely thin Dugh, Champagne, eLORETA and S-FLEX. We notice that Dugh achieves superior performance in terms of EMD metric, whereas it is competitive in terms of TCE. Note that since thin Dugh incorporates the temporal structure of the sources into the inference scheme, its performance with respect to EMD and TCE can be significantly improved by increasing the number of time samples.\nFigure~\\ref{fig:metrics-temporal} demonstrates the estimated temporal covariance matrix obtained from our two proposed spatio-temporal learning schemes, namely thin and full Dugh, indicated by cyan and magenta colors, respectively. The upper panel illustrates the reconstruction results for a setting where the ground-truth temporal covariance matrix has full structure, while the lower panel shows a case with Toeplitz temporal covariance matrix structure. It can be seen that \\emph{full Dugh} can better capture the overall structure of ground truth full-structure temporal covariance as evidenced by lower NMSE and similarity errors compared to \\emph{thin Dugh} that is only able to recover a Toeplitz matrix. As can be expected, the behavior is reversed in the lower-panel of Figure~\\ref{fig:metrics-temporal}, where the ground truth temporal covariance matrix is indeed Toeplitz. Full Dugh, however, still provides reasonable performance in terms of NMSE and similarity error, even though it estimates a full-structure temporal covariance matrix. Finally, the performance of both learning schemes are significantly improved by increasing the number of trials. \n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth,trim=0 5cm 0 0,clip]{figures\/NeurIPS-B-reconstrcut-Both.pdf}}\n\\caption{Accuracy of the temporal covariance matrix reconstruction incurred by two different temporal learning approaches, namely thin (cyan) and full (magenta) Dugh, assuming Toeplitz and full temporal covariance structure, respectively. The ground-truth temporal covariance matrix (first column) has full covariance structure in the first row, and Toeplitz structure in the second row. Performance is assessed in terms of the Pearson correlation between the entries of the original and reconstructed temporal covariance matrices, ${\\bf B}$ and $\\hat{{\\bf B}}$, denoted by $r({\\bf B},\\hat{\\bf B})$. Shown is the similarity error $1- r({\\bf B},\\hat{\\bf B})$ (forth column). Further, the normalized mean squared error (NMSE) between ${\\bf B}$ and $\\hat{{\\bf B}}$, defined as $\\text{NMSE} = ||\\hat{{\\bf B}}-{\\bf B}||_{F}^{2} \/||{\\bf B}||_{F}^{2}$ is reported (fifth column).}\n\\label{fig:metrics-temporal}\n\\vspace{-2mm}\n\\end{figure}\n\\vspace{-2mm}\n\\section{Analysis of real MEG and EEG data recordings}\n\\label{sec:real-data-analysis}\n\\vspace{-2mm}\n\\ali{We now demonstrate the performance of the novel algorithms on two MEG datasets (see Figure~\\ref{fig:AEF} and \\ref{fig:VEF}) and one EEG dataset (see Appendix \\ref{appendix:real-data}). All participants provided informed written consent prior to study participation and received monetary compensation for their participation. The studies were approved by the University of California, San Francisco Committee on Human Research. The MEG datasets from two female subjects comprised five trials of visual or auditory stimulus presentation, where the goal was to reconstruct cortical activity reflecting auditory and visual processing. No prior work has shown success in reconstruction in such extreme low SNR data.} \n\n\\ali{Figure~\\ref{fig:AEF} shows the reconstructed sources for auditory evoked fields (AEF) from a representative subject using eLORETA, MCE, thin and full Dugh. In this case, we tested the reconstruction performance of all algorithms with the number of trials limited to 5. As Figure~\\ref{fig:AEF} demonstrates, the reconstructions of thin and full Dugh both show focal sources at the expected locations of the auditory cortex. Limiting the number of trials to as few as 5 does not negatively influence the reconstruction result of Dugh methods, while it severely affects the reconstruction performance of competing methods. For the visual evoked field (VEF) data in Figure~\\ref{fig:VEF}, thin Dugh was able to reconstruct two sources in the visual cortex with their corresponding time-courses demonstrating an early peak at~100 ms and a later peak around~200 ms (similar performance was observed for full Dugh).}\n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth,trim=0 17cm 0 0,clip]{figures\/AEF-final.pdf}}\n\\caption{Analysis of auditory evoked fields (AEF) of one representative subject using eLORETA, MCE, thin and full Dugh. As it can be seen, thin and full Dugh can correctly localize bilateral auditory activity to Heschl's gyrus, which is the characteristic location of the primary auditory cortex, with as few as 5 trials. In this challenging setting, all competing methods show inferior performance.}\n\\label{fig:AEF}\n\\vspace{-2mm}\n\\end{figure}\n\\begin{figure}\n  \\centering\n  \\centerline{\\includegraphics[width=\\textwidth]{figures\/Figure4.png}}\n\\caption{Spatial maps and corresponding time-courses reconstructed from five trials of visual stimulus presentation using Dugh and MCE. Distinct activity in visual cortex was reconstructed.}\n\\label{fig:VEF}\n\\vspace{-6mm}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec:discussion}\nInverse modeling is a challenging problem that can be cast as a multi-task regression problem. Machine learning-based methods have contributed by systematically incorporating prior information into such models (e.g.~\\citep{wipf2009unified,wu2016bayesian,mccann2017convolutional,arridge2019solving,bubba2019learning}). While, so far, explicit models of spatio-temporal dynamics are rare, we contribute in this paper by deriving efficient inference algorithms for regression models with spatio-temporal dynamics in model parameters and noise. Specifically, we employ separable Gaussian distributions using Kronecker products of temporal and spatial covariance matrices. We assume sparsity and independent activations in the spatial domain, while the temporal covariance is modeled to have either full or Toeplitz structure. The proposed Dugh framework is encompassing efficient optimization algorithms for jointly estimating these distributions within a hierarchical Bayesian inference and MM optimization framework. Interestingly, we could theoretically prove convergence for the proposed update rules yielding estimates of the spatial and temporal covariances. In careful simulation studies, we have demonstrated that the inclusion of both spatial and temporal model parameters in the model indeed leads to significantly improved reconstruction performance. Finally, the utility of our algorithms is showcased in challenging real-world applications by reconstructing expected sources from real M\/EEG data based on very few experimental trials.\n\nThe M\/EEG BSI problem has been previously framed as a multi-task regression problem in the context of Type-I learning methods  \\citep{massias2018generalized,bertrand2019handling}. \\citet{bertrand2019handling} proposed a method to extend the group Lasso class of algorithms to the multi-task learning case. Their method shows promising results for large numbers of trials but is not competitive for smaller sample sizes, presumably due to not modeling the temporal structure of the sources. In contrast, Dugh is a Type-II method that learns the full spatio-temporal prior source distribution as part of the model fitting. Note that the assumption of spatially independent noise made here can be readily dropped, as in  \\citep{hashemi2021joint}. A number of alternative approaches have been proposed to estimate the spatio-temporal correlation structure of the \\emph{noise}\n\\citep{huizenga2002spatiotemporal,de2002estimating,bijma2003mathematical,de2004maximum,jun2006spatiotemporal}. These works, however, do not estimate the noise characteristics as part of the source reconstruction problem on the same data but require separate noise recordings. Our proposed algorithm substantially differs in this respect, as it learns the full noise covariance jointly with the brain source distribution. This joint estimation perspective is very much in line with the end-to-end learning philosophy as opposed to a step-wise independent estimation process that can give rise to error accumulation. \n\n\n\n\n \n\n\nWith respect to the limits of our Dugh framework, we would like to note that using the same temporal correlation prior for noise and sources is a potentially restricting assumption in our modeling. It is made here to achieve tractable inference due to the fact that the spatio-temporal statistical model covariance matrix can be formulated in a separable form, i.e., $\\Tilde{\\bm \\Sigma}_{\\bf y} = {\\bm \\Sigma}_{\\bf y} \\otimes {\\bf B}$. Although we have demonstrated empirically that the reconstruction results of our proposed learning schemes are fairly robust against violating this assumption, this constraint may be further relaxed by exploiting eigenvalue decomposition techniques presented in \\citep{rakitsch2013all,wu2018learning}. Another potentially limiting assumption in our model is to assume Gaussian distributions for the sources and noise. Although Gaussian priors are commonly justified, the extension of our framework to heavy-tailed noise distributions, which are more robust to outliers, is a direction of future work. \\ali{Regarding the societal impact of this work, we note that we intend to solve inverse problems that have non-unique solutions, which depend on explicit or implicit assumptions of priors. These have applications in basic, clinical, and translational neuroscience imaging studies. Our use of a hierarchical empirical Bayesian framework allows for explicit specifications of the priors in our model that are learned from data, and a more clear interpretation of the reconstructions with reduced bias. Nevertheless, in our algorithms, we assume sparse spatial priors, and in scenarios where this assumption embodied in the priors is incorrect, the resulting reconstructions will be inaccurate. Users and neuroscientists will need to be cognizant of these issues.} \n\n\n\n\n\\label{sec:conclusion}\nIn conclusion, we could derive novel flexible hierarchical Bayesian algorithms for multi-task regression problems with spatio-temporal covariances. Incorporating prior knowledge, namely, constraining the solutions to Riemannian manifolds, and using ideas of geodesic convexity, circulant embeddings, and majorization-minimization, we derive inference update rules and prove convergence guarantees. The proposed Dugh algorithms show robust and competitive performance both on synthetic and real neural data from M\/EEG recordings and thus contribute to a well-founded solution to complex inverse problems in neuroscience.  \n\n\n\\section*{Broader Impact}\nIn this paper, we focused on sparse multi-task linear regression within the hierarchical Bayesian regression framework and its application in EEG\/MEG brain source imaging. Our algorithm, however, is suitable for a wider range of applications. The same concepts used here for full-structural spatio-temporal covariance learning could be employed in other contexts where hyperparameters like kernel widths in Gaussian process regression \\citep{wu2019dependent} or dictionary elements in the dictionary learning problem \\citep{dikmen2012maximum} need to be inferred from data. Our proposed method, Dugh, may also prove useful for practical scenarios in which model residuals and signals are expected to be correlated, e.g., probabilistic canonical correlation analysis (CCA) \\citep{bach2005probabilistic}, spectral independent component analysis (ICA) \\citep{ablin2020spectral}, direction of arrival (DoA) and channel estimation in massive Multiple Input Multiple Output (MIMO) systems \\citep{prasad2015joint,gerstoft2016multisnapshot,haghighatshoar2017massive}, robust portfolio optimization in finance \\citep{feng2016signal}, covariance matching and estimation \\citep{werner2008estimation,tsiligkaridis2013covariance,zoubir2018robust,benfenati2020proximal,ollila2020shrinking}, graph learning \\citep{kumar2020unified}, thermal field reconstruction \\citep{hashemi2016efficient,flinth2017thermal,flinth2018approximate}, and brain functional imaging \\citep{wei2020bayesian}.\n\n\n\\begin{ack}\nThis result is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 758985).\nAH acknowledges scholarship support from the Machine Learning\/Intelligent Data Analysis research group at Technische Universit{\\\"a}t Berlin. He further wishes to thank the  Charit{\\'e} -- Universit{\\\"a}tsmedizin Berlin, the Berlin Mathematical School (BMS), and the Berlin Mathematics Research Center MATH+ for partial support. CC was supported by the National Natural Science Foundation of China under Grant 62007013. KRM was funded by the German Ministry for Education and Research as BIFOLD -- Berlin Institute for the Foundations of Learning and Data (ref.\\ 01IS18025A and ref.\\ 01IS18037A), and the German Research Foundation (DFG) as Math+: Berlin Mathematics Research Center  (EXC 2046\/1, project-ID: 390685689),\nInstitute of Information \\& Communications Technology Planning \\& Evaluation (IITP) grants funded by the Korea Government (No. 2019-0-00079,  Artificial Intelligence Graduate School Program, Korea University). SSN was funded in part by National Institutes of Health grants (R01DC004855, R01EB022717, R01DC176960, R01DC010145, R01NS100440, R01AG062196, and R01DC013979), University of California MRPI MRP-17\u2013454755, the US Department of Defense grant (W81XWH-13-1-0494).\n\\end{ack}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe gravity duals of gauge theories with running coupling constants have received\nconsiderable attention in the last few\nyears. The original gauge\/gravity duality \\cite{Mal-1}, \\cite{Witt-1} deals exclusively with theories that have no\nrunning of the coupling constants, or with theories that have some running of the coupling constants but eventually\nfall into fixed point surfaces, for example \\cite{klebwitt}. The first kind of dualities that consider the actual\nrunning of the coupling constants leading to, say, confining theories were discussed some time back in\n\\cite{KS}, \\cite{vafa}, \\cite{MN} and its extension to include fundamental flavors in \\cite{ouyang}. The\ntype IIA brane constructions for theories like \\cite{klebwitt} were first discussed in \\cite{dasmukhi}, and for\ntheories with running couplings were discussed in \\cite{ohtatar}. In fact in the fourth reference of \\cite{ohtatar}\nthe precise distinctions between \\cite{KS} and \\cite{vafa} were pointed out in details.\n\nIn recent times we have seen many new advantages of studying theories like \\cite{KS} and \\cite{vafa} that deal with\nrunning couplings. The confining behavior of these theories in the far IR is of course very powerful in extending them\nto more realistic scenarios like high temperature QCD \\cite{FEP, jpsi}. The cascading nature of these theories allow\nthem to remain strongly coupled throughout the RG flow from UV to IR, and therefore {\\it supergravity} duals can\ndescribe the full dynamics of the corresponding gauge theories. For the Klebanov-Strassler (KS) theory \\cite{KS} even the\nfull UV completion, that allow no Landau poles or UV divergences of the Wilson loops, {\\it can} be achieved by\nattaching an UV cap to the KS geometry\\cite{FEP}. An example \nof the full UV completion of the KS geometry both at\nzero and non-zero temperatures has been recently accomplished in \\cite{jpsi}. The UV cap therein is given by an\nasymptotic AdS space that, in the dual gauge theory, will allow for an asymptotic conformal behavior in the UV and\nlinear confinement in the far IR.\n\nFor the model studied by Vafa \\cite{vafa} the full UV completion would be more non-trivial. We expect the UV to be a\nsix-dimensional theory instead of a four-dimensional one. A six-dimensional UV completion that allows no Landau poles\nin the presence of fundamental flavors has not been constructed so far.\nIn fact a proper study of fundamental flavors\n{\\it a-la} \\cite{ouyang} is yet to be done for this case. The F-theory \\cite{vafaF, DM, senF}\nembedding of this model would be crucial\nin analysing the full UV completion. However some aspects of an intemediate UV behavior, for example cascading dynamics,\nhave been discussed in the past \\cite{katzvafa} where the cascade is likened to an infinite sequence of flop transitions.\nThe IR dynamics of the theory where we expect geometric transition to happen is actually the last stage\nin this sequence of transformations\nwhere the flop is immediately followed by a conifold transition. At this point we should\nexpect the wrapped D5-branes to be completely\nreplaced by fluxes (at least in the absence of fundamental flavors) \\cite{katzvafa}.\nWhat happens in the presence of fundamental flavors is rather subtle, and we will not discuss this here anymore. In fact\nwe will only concentrate on the last stage of the transition, namely, the geometric transition in this paper. The\nintermediate cascading dynamics or the UV completion will be discussed elsewhere \\cite{toappear}.\n\nSince the geometric transition leads to a confining theory, the corresponding gauge dynamics is strongly coupled.\nTherefore the physics of this transition can be captured exclusively by supergravity backgrounds. In some\nof our earlier works \\cite{gtpapers} we managed to study this purely using the supergravity backgrounds in the {\\it local}\nlimit, meaning that the sugra background was studied around a specific chosen point in the internal six-dimensional space.\nThe reason for this was the absence of a known globally defined\nsupergravity solution of the wrapped D5-branes on the two-cycle of the\nresolved conifold. The only known global solution i.e \\cite{pandoz} was unfortunately not supersymmetric (see\n\\cite{cvetic, anke} for details) although it satisfied the type IIB EOMs. In this paper, among other things, we will\nbe able to solve this problem and provide a fully supersymmetric globally defined solution for the wrapped D5-branes on a\ncertain resolved conifold. What we will argue soon is that the resolved conifold should have a non-K\\\"ahler metric to\nallow for supersymmetric solutions. This non-K\\\"ahlerity appears exactly from the back-reactions of the wrapped D5-branes.\n\nDespite the absence of supersymetric solutions, in \\cite{gtpapers} we managed to show, at least locally, the\nfull geometric transitons in type II theories. The gravity duals for the IR confining gauge theories on\nthe wrapped D6-branes in type IIA and wrapped\nD5-branes in type IIB were completely captured by non-K\\\"ahler deformations of the resolved and the deformed conifolds\nrespectively. In this paper we will show that globally under some simplifying assumptions\nthis conclusion remains unchanged, but generically\nthese manifolds would become non-geometric (see \\cite{halmagyi} for a recent discussion on this). In the following\nsub-section we will briefly review the state of geometric transition using local supergravity analysis before we proceed\nto compute the full global picture.\n\n\\subsection{Geometric transition and supersymmetric solution}\n\nLet us begin with a bit of historical notes.\nThe original study of open-closed string duality in type II theory\nstarts with D6 branes wrapping a three cycle of a non-compact\ndeformed conifold. Naively one might expect the deformed conifold\nto be a complex K\\\"ahler manifold with a non-zero three cycle.\nHowever as discussed earlier in \\cite{gtpapers} this is not quite correct,\nand the manifold that actually would solve the string equations of\nmotion is a non-K\\\"ahler deformation of the deformed conifold. It\nalso turns out that the manifold has no integrable complex\nstructure, but only has an almost complex structure. This is\nconsistent with the prediction of \\cite{vafaF}.\n\nHowever, as one may recall, in all our earlier papers we managed to study only the\n{\\it local} behavior of the manifolds. This is because the full global picture was\nhard to construct, and any naive procedure always tend to lead to non-supersymmetric\nsolutions. In deriving the local metric, we took a simpler model where all\nthe spheres were replaced by tori with periodic coordinates ($x,\n\\theta_1$) and ($y,\\theta_2$). The coordinate $z$ formed a\nnon-trivial $U(1)$ fibration over the $T^2$ base. The replacement\nof spheres by two tori was directly motivated from the\ncorresponding brane constructions of \\cite{dasmukhi}, where\nnon-compact NS5 branes required the existence of tori instead of\nspheres in the T-dual picture.\n\nLocally\nthe non-K\\\"ahlerity of\nthe underlying metric can be easily seen from its explicit\nform:\n\\bg\\label{iiamet}\n&&ds_{IIA}^2 = g_1~\\left[(dz -\n{b}_{z\\mu}~dx^\\mu) + \\Delta_1~{\\rm cot}~\\hat\\theta_1~(dx -\nb_{x\\theta_i}~d\\theta_i) + \\Delta_2~{\\rm cot}~ \\hat\\theta_2~(dy -\nb_{y\\theta_j}~d\\theta_j)+ ..\\right]^2\\nonumber\\\\\n&& + g_2~ {[} d\\theta_1^2\n+ (dx - b_{x\\theta_i}~d\\theta_i)^2] + g_3~[ d\\theta_2^2 + (dy -\nb_{y\\theta_j}~d\\theta_j)^2{]} +  g_4~{\\rm sin}~\\psi~{[}(dx -\nb_{x\\theta_i}~d\\theta_i)~d \\theta_2 \\nonumber\\\\\n&& ~~~~~~~~~ + (dy -\nb_{y\\theta_j}~d\\theta_j)~d\\theta_1 {]} + ~g_4~{\\rm\ncos}~\\psi~{[}d\\theta_1 ~d\\theta_2 - (dx - b_{x\\theta_i}~d\\theta_i)\n(dy - b_{y\\theta_j}~d\\theta_j)]\n\\nd\nwhere the coefficients $g_i$\nand the coordinates $\\theta_i, \\hat\\theta_i$ etc. are defined in\n\\cite{gtpapers}. The background has\nnon-trivial gauge fields (that form the sources of the wrapped D6\nbranes) and a non-zero string coupling (which could in principle\nbe small).\n\nExistence of such an exact supergravity background helps us to obtain\nthe corresponding mirror type IIB background. One would expect that\nthis can be easily achieved\nusing the mirror rules of \\cite{syz}. It turns out however that\nthe mirror rules of \\cite{syz}, as discussed in \\cite{gtpapers}, do not\nquite suffice.\nA detailed analysis of this is\npresented in \\cite{gtpapers}.\nAs discussed therein, we have to be careful\nabout various subtle issues while doing the mirror transformations:\n\n\\noindent (a) The mirror rules of \\cite{syz} tells us that {\\it any} Calabi-Yau\nmanifold with a mirror admits, at least {\\it locally}, a $T^3$ fibration over a\nthree dimensional base. This seems to fail for the deformed\nconifold as it does not possess enough isometries to represent it as a $T^3$\nfibration.\nOn the other hand, a resolved conifold does have a well defined $T^3$\ntorus over a three-dimensional base, which can be exploited to get the mirror (see also\n\\cite{agavafa}). It also turns out that the $T^3$ torus is a lagrangian\nsubmanifold, so a mirror transformations will not break any\nsupersymmetry.\n\n\\noindent (b) Viewing the mirror transformation naively as three T-dualities\nalong the $T^3$ torus {\\it does~not} give the right mirror metric. There are\nvarious issues here. The rules of \\cite{syz} tell us that the mirror transformation\nwould only work when the three dimensional base is very large. The configuration that\nwe have is exactly opposite of the case \\cite{syz}. Recall that\nour configuration lies at the end of a much larger cascading theory.\nBy UV\/IR correspondences, this means that the\nbase manifold is very small. Furthermore we are at the {\\it tip} of the\ngeometric transition and therefore we have to be in a situation with very\nsmall base (in fact very small fiber too). In \\cite{gtpapers} we showed\nthat we could still apply the rules of \\cite{syz} if we impose a non-trivial large\ncomplex structure on the underlying $T^3$ torus. The complex structure\ncan be integrable or non-integrable. Using an integrable complex structure, we\nshowed in \\cite{gtpapers} that we can come remarkably close to getting\nthe right mirror metric. Our conjecture there was that if we use a\nnon-integrable complex structure we can get the right mirror manifold.\n\nIt seems therefore natural to start with the manifold that exhibits three isometry\ndirections --- the resolved conifold. We can, however, not use the metric for D5\nbranes wrapping the $S^2$ of a resolved conifold as derived in \\cite{pandoz}, because\nit breaks all supersymmetry \\cite{cvetic}.  The metric that we\nproposed in \\cite{gtpapers} (where we kept the harmonic functions undetermined) is very\nclose to the metric of \\cite{pandoz} but differs in some subtle way:\n\n\\noindent (a) The type IIB resolved conifold metric that we proposed in\n\\cite{gtpapers} is a D5 wrapping a two cycle that {\\it preserves} supersymmetry.\nWe will discuss this issue in more detail below.\n\n\\noindent (b) As explained in \\cite{gtpapers}, our IIB manifold\nalso has seven branes (and possibly orientifold planes) along with\nthe type IIB three-form fluxes. The metric constructed in \\cite{pandoz}\ndoesn't have seven branes but allows three-form fluxes.\n\n\\noindent The {\\it local} behavior of the type IIB metric is expressed in terms\nof non-trivial complex structures $\\tau_1$ and $\\tau_2$ as $dz_1 = dx - \\tau_1 d\\theta_1$ and\n$dz_2 = dy - \\tau_2 d\\theta_2$. The local metric then reads\n\\bg\\label{iibmet}\nds^2 = (dz + \\Delta_1 ~ {\\rm cot}~\\theta_1 ~ dx  +\n\\Delta_2 ~{\\rm cot}~\\theta_2 ~dy)^2 + \\vert dz_1 \\vert^2 + \\vert dz_2\\vert^2\n\\nd\nwhere all the warp factors can locally be absorbed in to the coordinate differentials.\nIn this formalism the metric may naively look similar to the one studied in\n\\cite{pandoz} but the global picture is completely different from the\none proposed by \\cite{pandoz}. Our aim in this paper is therefore two-fold: to determine the full global picture (at\nleast without the inclusion of UV caps), and to follow the duality cycle that will lead us to analyse\ngeometric transitions in type II theories.\n\n\\subsection{Organisation of the paper}\n\nThe paper is organised as follows. In section 2\nwe start with geometric preliminaries about the resolved conifold and the\nblown-up conifold.  We then discuss the mathematical construction of a non-K\\\"ahler resolved\nconifold with an $SU(3)$ structure, following \\cite{llt}.\nThe components of the metric of this $SU(3)$ structure\n(which automatically satisfy the\ntorsional equations) are given in {\\bf Appendix 1}, and we use\nthese components to determine the metric of the wrapped D5-branes on the non-K\\\"ahler resolved conifold in\nsection 4.2. Existence of an $SU(3)$ structure will guarantee that the solution we get is supersymmetric, and we\ndiscuss the issue of supersymmetry further in section 4.1.\n\nSections 4.3 to 4.6 are the main sections where we compute the full geometric transitions in the global framework\nusing duality cycle that were used earlier in \\cite{gtpapers}. In the full global picture the fluxes are very involved\ncompared to the local picture. We managed to work out all the fluxes in the type IIA mirror set-up. These flux\ncomponents are given in {\\bf Appendix 2}. It is necessary to track these fluxes because they would eventually determine\nthe non-K\\\"ahler fibration structure in type IIB theory discussed in section 4.6. We discuss the components of the\nmetric in type IIB after geometric transition in {\\bf Appendix 3}.\n\nIn the above discussion we have briefly alluded to the fundamental flavors. In our model they appear from orientifolding\nthe resolved conifold. The orientifolding is subtle, and we give a brief discussion of this in section 3. This\norientifolding will allow us to add seven-branes in type IIB and six-branes in the mirror type IIA picture. The\nseven-branes should be embedded as in \\cite{pandoz} or \\cite{kuperstein}.\n\nWe end with a conclusion and a short discussion on the topics that we will study in \\cite{toappear}.\n\n\n\\section{Mathematical constructions and $SU(3)$ structures}\n\nThe blown-up conifold describes one of the topologies that we will be using.  We will describe it as well as its relation to the resolved conifold which is more familiar\nin string theory.  These ideas are\nwell known in algebraic geometry.  Rather than starting with the standard algebro-geometric constructions, we instead begin with a description more suited\nto the description of an $SU(3)$ structure.  The algebro-geometric description\nwill follow.\n\n\\subsection{The blown-up conifold}\n\\label{blowup}\n\nLet us begin by explaining how the blown-up conifold arises for us.  The\nconifold is a cone over $S^3\\times S^2$ \\cite{candelas}, arising as a quotient\n$(S^3\\times S^3)\/U(1)$, with the $U(1)$ diagonally embedded and on each\nfactor identified with the $U(1)$ of the Hopf fibration $S^3\\to S^2$.\nIf we take the self-product of the Hopf fibration\n$S^3\\times S^3\\to S^2\\times S^2$ and mod out by the diagonal $U(1)$, we are\nleft with a $U(1)$ fibration $S^3\\times S^2\\to S^2\\times S^2$ which restricts\nto the Hopf fibration over either $S^2$ factor.\n\nIn the geometry we will be using which supports the metric (\\ref{susybg}),\nthe internal space is a 6-manifold, the total space of a complex line bundle\n$L$ with base $S^2\\times S^2$ which is described in spherical coordinates\n$(\\theta_i,\\phi_i)$ for $i=1,2$.  The fiber is described by an angular coordinate $\\psi$ describing the nontrivial $U(1)$ bundle\nover $S^2\\times S^2$ just described.  The $U(1)$ bundle is completed to\na complex line bundle by introducing the radial coordinate $r$.\n\nThe topology of either the line bundle or the $U(1)$ bundle is described completely by its Chern class on $S^2\\times S^2$.\nAs noted above, the\n$U(1)$ bundle restricted to either $S^2$ is the Hopf bundle $S^3$, whose Chern class on $S^2$ is well known to have degree $-1$.  So on\n$S^2\\times S^2$, we learn that $L$ has degrees $(-1,-1)$.\n\nWe can now identify $S^2$ with the complex projective line $\\bfP^1$ and switch to the language of algebraic geometry, whereby we see that the internal manifold $X$ is the total space of the line\nbundle ${\\cal O}(-1,-1)$ on $\\bfP^1\\times\\bfP^1$.\n\nThis manifold appears as\nthe blown-up conifold in algebraic geometry.  Rather than refer to known results, we prefer to directly identify $X$ with the blown-up conifold by describing the map from $X$ to the conifold which shrinks $\\bfP^1\\times\\bfP^1$ (identified with the zero section of $L$) to the conifold point.\n\nWe introduce homogeneous coordinates $(u^1,u^2)$ and $(v^1,v^2)$ on the respective\n$\\bfP^1$'s, and identify sections of $L$ with functions on $\\bfP^1\\times\\bfP^1$ which are homogeneous of degree $-1$ with respect to $(u^1,u^2)$ as well as with respect to $(v^1,v^2)$.  Thus a point of $X$ can be described by homogeneous coordinates\n$(u^1,u^2,v^1,v^2,s)$, where $s$ is thought of as a section of $L$.  The homogeneity is\ndescribed by two $\\bfC^*$ actions whose respective weights are\n$(1,1,0,0,-1)$ and $(0,0,1,1,-1)$.\n\nThe map from $X$ to the conifold is realized by the map from $X$ to $\\bfC^4$ given by\n\n\\bg\\label{bdtoconifold}\n(u^1,u^2,v^1,v^2,s)\\mapsto (u^1v^1s,u^2v^2s,u^1v^2s,u^2v^1s).\n\\nd\n\nIf we introduce coordinates $(x_1,\\ldots,x_4)$ on $\\bfC^4$, we see that the image of\n$X$ satisfies the equation\n\\bg\nx_1x_2-x_3x_4=0\n\\nd\nof the conifold.  Identifying $\\bfP^1\\times\\bfP^1$ with the zero section $s=0$, we see\nthat $\\bfP^1\\times\\bfP^1$ is collapsed to the conifold point $(0,0,0,0)$ as claimed.\n\nWhen described by homogeneous coordinates as above, Calabi-Yau manifolds are characterized by the condition that the sum of the weights is zero for any $\\bfC^*$. Since the sum of the weights is one for either $\\bfC^*$, we conclude that the blown-up conifold is not a Calabi-Yau manifold.  We will also check this\ndirectly in the next section by the adjunction formula.\n\n\\subsection{The resolved conifold}\\label{rescon}\n\nWe can relate the blown-up conifold to the more familiar resolved conifold.\nRather than blow down $\\bfP^1\\times\\bfP^1$ to the conifold point as in (\\ref{bdtoconifold}), we can instead partially blow down $\\bfP^1\\times\\bfP^1$ by\nprojecting to one $\\bfP^1$. This gives the usual resolved\nconifold.\n\n\\smallskip\nUsing the coordinates of the blown-up conifold introduced in Section~\\ref{blowup}, the partial blowdown is described by\n\n\\bg\\label{bdtoresolved}\n(u^1,u^2,v^1,v^2,s)\\mapsto\n(u^1,u^2,v^1s,v^2s),\n\\nd\nso that $\\bfP^1\\times\\bfP^1$, identified with $s=0$ as before, is mapped to\n$(u^1,u^2,0,0,0)$, and $\\bfP^1\\times\\bfP^1$ is projected to the first coordinate, as\nclaimed. We let $(z^1,z^2,z^3,z^4)$ be homogeneous coordinates on the image of (\\ref{bdtoresolved}).\n\nOnly one $\\bfC^*$ remains nontrivial on $(z^1,z^2,z^3,z^4)$, with weights $(1,1,-1,-1)$.  The image is the resolved conifold, the total space of ${\\cal O}(-1)\\oplus{\\cal O}(-1)$ on $\\bfP^1$.  The coordinates $(z^1,z^2)$ can be identified\nwith the homogeneous coordinates of $\\bfP^1$, while $z^3$ and $z^4$ are identified with\nsections on the respective copies of ${\\cal O}(-1)$.   The resolved conifold is of course\nCalabi-Yau, which can be seen since the sum of the weights is 0.\n\n\\bigskip\nWe remark that we have taken a circuitous path to get from\nthe blown-up conifold to the more familiar resolved conifold, but we\nhave reached the usual descriptions of the resolved conifold as either\nthe total space of the bundle ${\\cal O}(-1)\\oplus{\\cal O}(-1)$ on $\\bfP^1$\nor as a toric quotient of $\\bfC^4$ by $\\bfC^*$ with weights $(1,1,-1,-1)$.\n\n\\bigskip\nHowever, it is not the Calabi-Yau structure that is relevant in our model, but rather\na non-K\\\"ahler structure.  To construct this non-K\\\"ahler structure, the holomorphic\nhomogeneous coordinates are not particularly useful.\nAt times it will be useful to describe the resolved conifold as a\nsymplectic quotient, or equivalently as the space of vacua of a $U(1)$ gauge theory with four scalar fields with $U(1)$ charges $(1,1,-1,-1)$, whose\nvevs are identified with $(z^1,z^2,z^3,z^4)$.  There is an FI term $u$, which we take to be positive.  If we take $u<0$, we get the flopped version of the resolved conifold.\n\nSo the resolved conifold can be described by\n\\bg\\label{symprescon}\n|z^1|^2+|z^2|^2-|z^3|^2-|z^4|^2=u\n\\nd\nmodulo the $U(1)$ action.\n\nWe can see directly from this description that the resolved conifold is smooth.  We can for\nexample describe the patch in which $z^1\\ne0$ by the complex coordinates $(z^2,z^3,z^4)$\nand solve (\\ref{symprescon}) by\n\\bg\\label{resconcoords}\nz^1=\\sqrt{u-|z^2|^2+|z^3|^2+|z^4|^2}.\n\\nd\nNote that in (\\ref{resconcoords}) we have fixed the gauge by choosing the positive real\nsolution of (\\ref{symprescon}), so that $(z^2,z^3,z^4)$ are in fact local coordinates.\nHowever, they are in no sense to be considered as holomorphic coordinates since\n(\\ref{resconcoords}) is not holomorphic.\n\n\\bigskip\nWe now turn to the holomorphic description.  Start with\nthe conifold singularity $X$ with\nequation\n$$x_1x_2-x_3x_4=0$$\nwhich we denote by $f=0$.\nThe usual resolved conifold $X'$ can be described as the submanifold of\n$\\bfC^4\\times \\bfP^1$ with equations which we informally write as\n$$\\frac{x_1}{x_3}=\\frac{x_4}{x_2}=\\frac{y_2}{y_1},$$ or more formally as\n$$x_1x_2-x_3x_4=0,\\ x_1y_1=x_3y_2,\\ x_4y_1=x_2y_2.$$  In the above,\n$(y_1,y_2)$ are the homogeneous coordinates on $\\bfP^1$.\n\nIf $x=(x_1,x_2,x_3,x_4)\\ne (0,0,0,0),$ then there is a unique solution for\n$y=(y_1,y_2)$, so that $X$ and $X'$ are isomorphic away from the origin.  If\nhowever, $x=0$, then $y$ is unconstrained and we replace the origin by\na $\\bfP^1$ to form $X'$ from $X$.\n\nThis $\\bfP^1$ can be flopped to produce another resolved conifold $X''$.\nThe flop can be realized directly by the equations\n$$\\frac{x_1}{x_4}=\\frac{x_3}{x_2}=\\frac{y_2}{y_1}.$$\n$X'$ and $X''$ are isomorphic in this local model, but in global models\n$X'$ and $X''$ containing a resolved conifold and its flop respectively, the\nCalabi-Yaus $X'$ and $X''$ need not be isomorphic.\n\n\\bigskip\nIn this paper, we use the model where both $\\bfP^1$'s are introduced\nsimultaneously.  This is accomplished by the algebro-geometric construction\nof blowing up the conifold, which complements our description in Section 2.1. We introduce a $\\bfP^3$ with homogeneous\ncoordinates $(y_1,y_2,y_3,y_4)$ and the blowup $\\tilde{X}$ is constructed\nas the submanifold of $\\bfC^4\\times\\bfP^3$ with equation informally expressed as\n$$(x_1,x_2,x_3,x_4)=(y_1,y_2,y_3,y_4).$$  As before, if $x\\ne0$ then there is\na unique solution for $y$ and so $\\tilde{X}$ is isomorphic to $X$ away from\nthe origin.  If $x=0$, there are more solutions, but now there is a\nconstraint $y_1y_2=y_3y_4$.  This is a quadric surface in $\\bfP^3$, isomorphic\nto $\\bfP^1\\times \\bfP^1$ by the isomorphism\n$$\\left((u_1,u_2),(v_1,v_2)\\right)\\mapsto (u_1v_1,u_2v_2,u_1v_2,u_2v_1).$$\n\nNote that the blown-up conifold\n$\\tilde{X}$ is not Calabi-Yau, but we can realize\nthis within string theory by turning on an appropriate flux.\n\n\\smallskip\nWe consider the holomorphic 3-form $\\Omega$ on $X$ given by the usual\nresidue construction\n$$\\Omega=\\frac{dx_2\\wedge dx_3\\wedge dx_4}\n{\\partial f\/\\partial x_1}=\\frac{dx_2\\wedge dx_3\\wedge dx_4}{x_2}$$ and pull it back to a holomorphic\n3-form $\\tilde{\\Omega}$ on $\\tilde{X}$.\nTo see that $\\tilde{X}$ is not Calabi-Yau, it suffices to show that\n$\\tilde\\Omega$ vanishes somewhere on $\\tilde{X}$. It\nsuffices to compute in one coordinate patch, say where $x_1\\ne0$.  In\nthis patch, $(x_1,y_3,y_4)$ are local coordinates.  To see this, we may\nset $y_1=1$, and then\nwe use $y_i=x_i\/x_1$ for $i=2,3,4$ to compute\n$$x_2=x_1y_3y_4,\\ x_3=x_1y_3,\\ x_4=x_1y_4.$$\nIn these coordinates, we have $\\tilde{\\Omega}=x_1dx_1\\wedge dy_3\\wedge dy_4$,\nwhich clearly vanishes on the surface $x_1=0$.\nThus $\\tilde{X}$ is not Calabi-Yau.\n\nFor later use, note that in this coordinate patch, if $x_1=0$ then necessarily\n$x_2=x_3=x_4=0$ as well.  Thus $x_1=0$ is the local equation of the\nexceptional $\\bfP^1\\times\\bfP^1$  of $\\tilde{X}$, which we denote by $E$.\n\nAlternatively we can see the same result by the adjunction formula \\cite{gh}, which\nsays that for any hypersurface $H$ in a complex manifold $M$, we have\n$K_H=(K_M+[H])\\mid_H$.\n\nWe realize $\\tilde{X}$ as a hypersurface in the blowup of $\\bfC^4$ at a point\nand apply the adjunction formula.\n\nLet $E'$ be the $\\bfP^3$ which is the exceptional divisor of the\nblowup $Z$ of $\\bfC^4$ at the origin; then $K_Z=3E'$ \\cite{gh}.  Then the\nadjunction formula yields\n$$K_{\\tilde{X}}=\\left(K_Z+\\tilde{X}\\right)|_{\\tilde{X}}.$$\n\nNow $\\tilde{X}$ is obtained by subtracting off the exceptional divisor from\nthe pullback of $X$ via the blowup.  Since $X$ has a multiplicity~2 singularity\nat the origin, the pullback of $X$ actually contains the exceptional\ndivisor $E'$ with multiplicity~2.\nSince $2E'$ has to be subtracted off to obtain $\\tilde{X}$, we conclude\nthat $\\tilde{X}$  has divisor class $-2E'$.  We\nconclude that\n$$K_{\\tilde{X}}=\\left(3E'-2E'\\right)|_{\\tilde{X}}=E'|_{\\tilde{X}}=E,$$\nso that $K_{\\tilde{X}}$ is nontrivial and $\\tilde{X}$ is not Calabi-Yau.\n\nThis is consistent with the explicit calculation.  Since $x_1=0$ defines\nthe exceptional divisor $E$, the fact that $\\tilde{\\Omega}$ vanished precisely\non $E$ tells us that $K_{\\tilde{X}}=E$.\n\n\\subsection{$A_1$ fibered geometry}\n\nWe start with the blown up $A_1$ geometry fibered over the complex numbers\nwith parameter $x_4\\in\\bfC$.  The equation is just the $A_1$ equation\n\\bg\\label{a1}\nx_1x_2-x_3^2=0\n\\nd\nand the blowup is performed by introducing a $\\bfP^1$ with homogeneous\ncoordinates $(y_1,y_2)$ and imposing the equations\n$$\\frac{x_3}{x_1}=\\frac{x_2}{x_3}=\\frac{y_2}{y_1},$$\nor more formally\n\\bg\\label{resolveda1}\nx_3y_1=x_1y_2,\\qquad x_2y_1=x_3y_2.\n\\nd\nNote that $x_4$ does not appear explicitly, and can be interpreted as a\nparameter for the\nlocation of $\\bfP^1$.  The $\\bfP^1$ corresponding to $x_4=\\phi$ will be written\nas $C_\\phi$.\n\nFor later use, the normal bundle of $C_\\phi$ is $\\cO_{C_\\phi}\\oplus\n\\cO_{C_\\phi}(-2)$.\n\nWe now deform this geometry with deformation parameter $t$:\n\\bg\\label{a1deformed}\nx_1x_2-x_3^2+t^2x_4^{2n}=0\n\\nd\nand the blowup is performed by introducing a $\\bfP^1$ with homogeneous\ncoordinates $(y_1,y_2)$ and imposing the equations\n$$\\frac{x_3-tx_4^n}{x_1}=\\frac{x_2}{x_3+tx_4^n}=\\frac{y_2}{y_1}.$$\n\nThis geometry corresponds to the superpotential $W(\\phi)=t\\phi^{n+1}\/(n+1)$.\nNote that for $t=0$ we have $W(\\phi)\\equiv0$, and the curve $C_\\phi$ is\nholomorphic for all $\\phi$.  For $t\\ne0$, we have $W'(\\phi)=t\\phi^n$, and\n$C_\\phi$ only  persists holomorphically for $\\phi=0$.  To see this, the\nrequirement is\n$$x_3-t\\phi^n=x_1=x_2=x_3+t\\phi^n=0,$$\nwhich implies that $\\phi^n=0$.\n\nThe superpotential can be obtained by integrating the holomorphic\n3-form $\\Omega=dx_2dx_3dx_4\/x_2$ over a three chain $\\Gamma$\nconnecting $C_0$ to $C_\\phi$.  This can be reinterpreted in terms of\nthe relative homology class of $\\Gamma$.  The same relatively\ncohomology group can be realized after blowing up $C_0$ and $C_\\phi$.\n\nInitially putting $t=0$, the blowup of $C_0$ has exceptional divisor isomorphic\nto the Hirzebruch surface $F_2$.  A similar blowup can be performed on\n$C_\\phi$.\n\nThe Hirzebruch surface $F_2$ deforms if $\\phi=0$ but not otherwise.  If\n$n>1$, the deformed surface is still $F_2$.  If $n=1$, then the deformed\nsurface is $\\bfP^1\\times\\bfP^1$.\n\nIf desired, a toric description of the blowup can be given.  The $A_1$\nsurface singularity is a toric variety whose fan has a single\ntwo-dimensional cone with edges spanned by $(1,0)$ and $(1,2)$.  The\nsingularity gets resolved by inserting an extra edge $(1,1)=(1\/2)\n((1,0)+(1,2)$.  This resolved $A_1$ gets fibered over $\\bfC$ in the\nusual way: by adding another coordinate, appending a zero to the\ncoordinates of the vectors spanning the edges, and adding the new\ncoordinate vector.  Hence the fan has edges spanned by\n\n$$(1,0,0), (1,1,0), (1,2,0), (0,0,1).$$\n\nThe curve $C_0$ corresponds to the 2-dimensional cone spanned by $(1,1,0)$\nand $(0,0,1)$, so $C_0$ gets blown up by inserting a new edge spanned by\n$(1,1,1)=(1,1,0)+(0,0,1)$.  In summary, the toric variety has edges\n\n$$(1,0,0), (1,1,0), (1,2,0), (0,0,1), (1,1,1).$$\n\n\n\\subsection{$SU(3)$ structure}\\label{su3}\nWe follow \\cite{llt} which gives a general procedure for constructing string\ncompactifications on toric varieties via a method for producing\n$SU(3)$ structures.  Any $SU(3)$ structure arises as a string compactification \\cite{torsion}.\n\nWe apply the method to the resolved conifold.  The method\nwas designed to apply to compact toric varieties, but since the method has a\nlocal character, it may be applied to the resolved conifold.  We set ourselves\nto that task.\n\nFor the convenience of the reader, we collect some facts about $SU(3)$\nstructures.\n\n\\bigskip\nAn $SU(3)$ structure on a 6-manifold $M$\nis determined by a complex decomposible 3-form $\\Omega$ and a real 2-form $J$ which are\nrelated by\n\\bg\\label{su3conds}\n\\Omega\\wedge J=0, \\qquad \\Omega\\wedge\\overline{\\Omega}=-\\frac{4i}3J\\wedge J\\wedge J.\n\\nd\n\nAt each point $p\\in M$, we can find complex cotangent vectors $dz^1,dz^2,dz^3$ so that $\\Omega=\ndz^1\\wedge dz^2\\wedge dz^3$ at $p$.  The first condition of (\\ref{su3conds}) and the\nreality of $J$ imply that we can ``diagonalize\" $J$, writing it as\n$$J=\\frac{i}2\\left(a_1dz^1\\wedge dz^{\\bar{1}}+a_2dz^2\\wedge dz^{\\bar{2}}+a_3dz^3\\wedge dz^{\\bar{3}}\\right)$$\nfor some real constants $a_i$, while retaining the form of $\\Omega$.  Then the second condition of (\\ref{su3conds})\nimplies that we can rescale the $dz^i$ so that\n\\bg\nJ=\\frac{i}2\\left(dz^1\\wedge dz^{\\bar{1}}+dz^2\\wedge dz^{\\bar{2}}+dz^3\\wedge dz^{\\bar{3}}\\right)\n\\nd\nwhile $\\Omega=dz^1\\wedge dz^2\\wedge dz^3$ still holds.\n\nThere is still the freedom of\nmultiplying the $dz^i$ by phases whose product is 1.\n\nThese coordinates determine a Euclidean metric $g_{i\\bar{j}}=i\/2$ at $p$, which\nis independent of the phase ambiguity.\n\nThis pointwise analysis extends to all of $M$, showing that the data of $\\Omega$ and\n$J$ satisfying (\\ref{su3conds}) completely determines a metric, the metric associated\nwith an $SU(3)$ structure.\n\nThere is an explicit procedure to calculate the metric directly from $\\Omega$ and $J$.\nThe first step is to calculate the (not necessarily integrable) complex structure $I$.  In pointwise Euclidean coordinates at $p$, the complex structure $I$ is the standard one.  But it can be computed intrinsically following \n\\cite{hitchin} as follows\\footnote{For more details on the following analysis, and also to connect to recent conifold \nliterature the readers may refer to \\cite{papat} and references therein.}.\n\nFirst define an unnormalized complex structure by\n\\bg\n\\tilde{I}_j^k=\n\\epsilon^{klmnop}\\left(\\mathrm{Re}\\Omega\\right)_{jlm}\\left(\\mathrm{Re}\\Omega\\right)_{nop},\n\\nd\nwhere $\\epsilon$ is the completely antisymmetric tensor.  It is shown in \\cite{hitchin}\nthat $\\tilde{I}^2$ is a diagonal matrix with negative real entries.  Then\n\\bg\nI=\\frac{\\tilde{I}}{\\sqrt{-\\frac16\\mathrm{Tr}\\tilde{I}^2}}\n\\nd\nis the desired complex structure.\n\nFrom here, the metric is determined by\n\\bg\ng_{ij}=I^k_jJ_{ki}.\n\\nd\n{}From the definitions, $\\Omega$ has type $(3,0)$ and $J$ has type $(1,1)$\nin the complex structure $I$.\nIf $(g,I)$ determines a K\\\"ahler structure, then $J$ is just the usual\nK\\\"ahler form.\n\n\\bigskip\nWe also quickly review the method of \\cite{llt} while applying it to the resolved\nconifold.  A general method is developed for producing $SU(3)$ structures on\nthree-dimensional complex toric varieties, by producing an $\\Omega$ and $J$ satisfying\n(\\ref{su3conds}).  The novel ingredient is to produce a $(1,0)$ form on complex\nEuclidean space satisfying certain conditions, and then then $SU(3)$ structure is\ndetermined by formulae.\n\nWe recall that the resolved conifold has been described as a quotient of $\\bfC^4$ by\nthe $\\bfC^*$ with weights $Q=(1,1,-1,-1)$.\nSo it suffices to produce a $(1,0)$ form\n$K=K_idz^i$ on $\\bfC^4$ satisfying (3.15), (3.16) and the normalization condition\n(3.18) of \\cite{llt}.  We interpret these conditions in concrete terms.\n\nThe condition (3.15) is equivalent to $Q^iz^iK_i=0$.  The condition (3.16)\nsays that $K$ has half the $U(1)$ charge as the holomorphic volume form\n$\\Omega_{\\bfC}=dz^1\\cdots dz^4$ of $\\bfC^4$, which is zero in this case.  Our normalization condition is $\\sum |K_i|^2 =1$, slightly different from\n(3.18) but we will adjust for it later.\n\nAn obvious solution is\n\n\\bg\\label{kdefn}\nK=\\frac{z^3dz^1+z^1dz^3+z^4dz^2+z^2dz^4}{|z|}.\n\\nd\n\nWe now compute the $SU(3)$ structure, following the formulas in \\cite{llt}.  We first construct\nthe standard $SU(3)$ structure on the resolved conifold before modifying it.  The\n$\\bfC^*$ action on $\\bfC^4$ is generated by\n\\bg\nV=z^1\\partial_{z^1}+z^2\\partial_{z^2}-z^3\\partial_{z^3}-z^4\\partial_{z^4}\n\\nd\nand then the standard Calabi-Yau 3-form is\n\\bg\n\\widetilde{\\Omega}&=&~i_V\\Omega_{\\bf C}\\\\\n&=&~z^1dz^2\\wedge dz^3\\wedge dz^4-z^2dz^1\\wedge dz^3\\wedge\ndz^4-z^3dz^1\\wedge dz^2\\wedge dz^4+z^4dz^1\\wedge dz^2\\wedge dz^3.\\nonumber\n\\nd\nThe K\\\"ahler form $\\tilde{J}$ of the resolved conifold arises by modifying the K\\\"ahler form of\n$\\bfC^4$\n\\bg\nJ_{\\bfC}=\\frac{i}2\\sum_{i=1}^4dz^i\\wedge dz^{\\bar{i}}\n\\nd\nby putting\n\\bg\n\\eta=\\bar{z}^1dz^1+\\bar{z}^2dz^2-\\bar{z}^3dz^3-\\bar{z}^4dz^4,\n\\nd\nwith the coefficients coming from the weights, and then putting\n\\bg\n\\tilde{J}=\\frac1{|z^2|}\\left(J_{\\bfC}-\\frac{i}2\\eta\\wedge\\bar{\\eta}\\right).\n\\nd\nNote that $\\eta$ respects the $U(1)$ action but not the $\\bfC^*$ action.  So from this\npoint forward, we have to understand differential forms as $U(1)$-invariant forms,\nsubject to the D-term constraint (\\ref{symprescon}).  In particular, the K\\'ahler\nform implicitly depends on the FI parameter $u$, as it must.\n\n\n\\bigskip\nWe now use $K$ to modify $\\tilde{\\Omega}$ and $\\tilde{J}$, effectively replacing\n$K$ by $\\bar{K}$ throughout.\n\n\\smallskip\nAn auxiliary $SU(2)$ structure is created, characterized by two-forms $\\omega$ and $j$ satisfying\n\\bg\\label{su2cond}\n\\omega\\wedge j=0,\\qquad \\omega\\wedge\\bar{\\omega}=2j\\wedge j,\n\\nd\nwhere\n\\bg\nj=\\tilde{J}-\\frac{i}2K\\wedge\\bar{K}\n\\nd\nand $\\omega$ is given by\n\\bg\\label{su2}\n\\omega_{ij}=-2\\bar{K_l}\\eta^{m\\bar{l}}\\tilde{\\Omega}_{mij},\n\\nd\nwhere $\\eta$ is the Euclidean metric on $\\bfC^4$. The prefactor of 2 here on the right-hand side of (\\ref{su2}) is not present in \\cite{llt} but is required by our\nnormalization condition for $K$.\n\nFrom here, we get a 2-parameter family of $SU(3)$ structures given by\n\\bg\nJ=aj-\\frac{ib^2}2K\\wedge\\bar{K},\\qquad\n\\Omega=ab\\bar{K}\\wedge\\omega.\n\\nd\nUsing (\\ref{su2cond}), it is immediate to see that $J$ and $\\Omega$ satisfy the conditions (\\ref{su3conds}) for an $SU(3)$ structure.\n\n\\smallskip\nThere is an extra phase parameter for $\\Omega$ in \\cite{llt}, but we supress it here\nsince the metric does not depend on this phase. In {\\bf Appendix 1} we write down all the components of the metric.\n\n\\section{T-duality and Orientifold Projection}\n\nThe  blown-up conifold that we discussed above, has a product structure of\n$\\bfP^1\\times\\bfP^1$ and we would like to discuss the T-duality to a IIA model. Our final aim is to see how\norientifold projection effects the T-duality.\nTo do so we first review the case of \\cite{orientifold} for the resolution of a deformed $A_2$ singularity.\nThis involves a\nnatural way to introduce two $\\bfP^1$ cycles.\n\n\\subsection{Brief review of the deformed $A_2$ case}\n\nLet us consider  the singular space $X_0$ realized as:\n\\begin{eqnarray}\n\\label{X0}\nxy=(u-t_0(z))(u-t_1(z))(u-t_2(z))~~, \\end{eqnarray} where $x,y,u,z$ are the\naffine coordinates of $C^4$ and  $t_j(z)$ are polynomials.\n\nThe affine variety (\\ref{X0}) has $A_1$ singularities at\n$x=y=0$ and $z$ one of the double points of the planar\nalgebraic curve: \\begin{eqnarray}\n\\label{Sigma_0}\n\\Sigma_0:~~(u-t_0(z))(u-t_1(z))(u-t_2(z))=0~~.  \\end{eqnarray}\nThe curve has 3 components $C_j$ given by\n$u=t_j(z)$, each a section of the $A_2$ fibration (\\ref{X0}).\n\nThe resolved space ${\\hat X}$ can be described explicitly as\nfollows. Consider two copies of\n$\\bfP^1$ with homogeneous coordinates $[u_1,u_2]$ and $[v_1,v_2]$, respectively, and local\naffine coordinates $\\xi_1:=u_1\/u_2, \\xi_2:=v_1\/v_2$.  Then\n${\\hat X}$ is realized as :\n \\bea\n\\label{hatX}\nu_2 (u-t_0(z))&=&u_1 x\\nn\\\\\nv_1(u-t_1(z))&=&v_2 y\n\\eea\n\nThe IIA construction is obtained by  performing a T-duality with respect to the\nfollowing $U(1)$ action on ${\\hat X}$, which we denote by\n \\begin{eqnarray}\n\\label{rho_global}\n([u_1,u_2], [v_1,v_2],z,u,x,y)\\stackrel{{\\hat \\rho}(\\theta)}{\\longrightarrow}\n([e^{-i\\theta}u_1,u_2],\n[v_1,e^{i\\theta}v_2],z,u,e^{i\\theta}x,e^{-i\\theta} y)~~.  \\end{eqnarray}\nThis projects as follows on the singular space $X_0$:\n\\begin{eqnarray}\n\\label{U1proj}\n(z,~u,~x,~y)\\stackrel{\\rho_0(\\theta)}{\\longrightarrow} (z,~u,~e^{i\\theta}x,~e^{-i\\theta}y)~~.\n\\end{eqnarray}\n\nIn the type IIB set-up, we consider the case\n\\bea\n\\label{tspec}\nt_0(z)&=&t(z)\\nn\\\\\nt_1(z)&=&t(-z)\\\\\nt_2(z)&=&-t(z)-t(-z)\\nn~~.  \\eea\n\nIn this situation, the resolution ${\\hat X}$ admits a $Z_2$ symmetry ${\\hat \\kappa}$ given by:\n\\begin{eqnarray}\n\\label{or_global}\n([u_1,u_2], [v_1,v_2],z,u,x,y)\\stackrel{\\hat \\kappa}{\\longrightarrow}\n([-v_2,v_1], [-u_2,u_1],-z,u,-y,-x) \\end{eqnarray} which acts\nas follows on the affine coordinates $\\xi_j$ of the\ntwo $\\bfP^1$ factors: \\begin{eqnarray} \\xi_1\\longleftrightarrow -1\/\\xi_2~~ \\end{eqnarray} and\nprojects to the following involution $\\kappa_0$ of $X_0$: \\begin{eqnarray}\n\\label{or_projected}\n(z,~x,~y,~u)\\stackrel{\\kappa_0}{\\longrightarrow} (-z,~-y,~-x,~u)~~.  \\end{eqnarray}\n\n\\subsection{The blown-up conifold case}\n\nWe now use  two $\\bfP^1$ cycles but in the blown-up conifold.\nIn this situation we have\n \\begin{eqnarray}\n\\label{rho_globalblowup}\n([u_1,u_2], [v_1,v_2],x_1,x_2,x_3,x_4)\\stackrel{{\\hat \\rho}(\\theta)}{\\longrightarrow}\n([e^{-i\\theta}u_1,u_2],\n[v_1,e^{i\\theta}v_2],x_1,x_2,e^{i\\theta}x_3,e^{-i\\theta} x_4)~~.  \\end{eqnarray}\nThis projects as follows on the singular space $X_0$:\n\\begin{eqnarray}\n\\label{U1projblowup}\n(x_1,~x_2,~x_3,~x_4)\\stackrel{\\rho_0(\\theta)}{\\longrightarrow} (x_1,~x_2,~e^{i\\theta}x_3,~e^{-i\\theta}x_4)~~.\n\\end{eqnarray}\nThe $Z_2$ symmetry ${\\hat \\kappa}$ is given in the homogeneous coordinates\n$(u^1,u^2,v^1,v^2,s)$ by:\n\\bg\n\\label{or_globalblowup}\n(u^1,u^2,v^1,v^2,s)\\stackrel{\\hat\\kappa}{\\longrightarrow}(-v^2,v^1,-u^2,u^1,s)\n\\nd\nwhich projects on $X_0$ as:\n\\bg\n(x_1,x_2,x_3,x_4) \\rightarrow (x_2,x_1,-x_3,-x_4)\n\\nd\nas is seen from (\\ref{bdtoconifold}).\n\nThe action on the homogeneous coordinates $y_i$ of $\\bfP^3$ is then given by\n\\begin{eqnarray}\n(y_1,y_2,y_3,y_4) \\rightarrow (y_2,y_1,-y_3,-y_4)\n\\end{eqnarray}\nand the action on $\\Omega$ is $\\Omega \\rightarrow - \\Omega$.\n\nWhat is the difference between the orientifold of \\cite{orientifold} and the current one? The orientifold of\n\\cite{orientifold} was an O5 orientifold extended on directions orthogonal to the D5 branes wrapped on\n$\\bfP^1$ cycle. After the T-duality, the involution determined an inversion of the radial direction of the\n$S^1$ which implied that the orientifold  became an O6 plane.\n\nIn the current example, the orientifold extends along the direction $s$ of the complex line bundle $L$ but also needs to\nas an inversion on the ${\\bf P^1} \\times {\\bf P^1}$ fiber which means that it wraps a 2-dimensional surface in the\n${\\bf P^1} \\times {\\bf P^1}$ fiber. Therefore we have O7 planes with the action as before.\n\n\n\n\n\\section{Analysis of the global picture and the cycle of geometric transitions}\n\nWith all the mathematical construction at hand, it is time now to discuss the geometrical aspect of the problem i.e\nthe supergravity metric and the fluxes in type II theories. Our starting point would be the\nissue of supersymmetry in the usual resolved conifold background with fluxes and branes in type IIB background. Once\nwe obtain this, it will prepare us for all the subsequent stages of the duality cycle for the\ngeometric transition \\cite{gtpapers}.\n\n\n\\subsection{Analysis of the global picture in type IIB}\n\n{}From our earlier works we know that there are two ways of extending our local configuration of \\cite{gtpapers} to study\nsupersymmetric cases in the full global picture:\n\n\\noindent (a) The full global geometry is a six-dimensional K\\\"ahler manifold with F-theory\nseven-branes distributed in some particular way. These seven-branes contribute to massive\nfundamental flavors in the gauge theory. Orientation of these seven-branes are the generalised version of the\nOuyang \\cite{ouyang}\n(or the Kuperstein \\cite{kuperstein}) embeddings.\n\n\\noindent (b) The full global geometry is non-K\\\"ahler with or without F-theory seven-branes. The seven-branes could\nbe embedded in this picture via Ouyang or the Kuperstein embedding, which in turn would provide fundamental matters\nin the gauge theory. In fact the possibility of such a global completion was already hinted in the second paper of\n\\cite{gtpapers}.\n\nLet us see how from our local picture studied in \\cite{gtpapers} these two possibilities can be realised. In the\nfirst paper of \\cite{gtpapers}, the local metric was argued to be of the following \nform\\footnote{The local metric \\eqref{localmet}\nthat we consider here is that of\na supergravity background studied around a specific chosen point in the internal six-dimensional space. For \nexample we choose a point \n($r_0, \\langle\\theta_i\\rangle, \\langle\\phi_i\\rangle, \\langle\\psi\\rangle$) in \n\\cite{gtpapers} which is away from the $r=0$ conifold point. \n This is because the full global picture was\nhard to construct, and any naive procedure always lead to non-supersymmetric\nsolutions. In deriving the local metric, we took a simpler model where all\nthe spheres were replaced by tori with periodic coordinates ($x,\n\\theta_1$) and ($y,\\theta_2$). The coordinate $z$ formed a\nnon-trivial $U(1)$ fibration over the $T^2$ base. Here ($r, x, y, z, \\theta_1, \\theta_2$) is the coordinate of a point \naway from \n($r_0, \\langle\\phi_1\\rangle, \\langle\\phi_2\\rangle, \\langle\\psi\\rangle, \\langle\\theta_1\\rangle, \n\\langle\\theta_2\\rangle$). \nThe replacement\nof spheres by two tori was directly motivated from the\ncorresponding brane constructions of \\cite{dasmukhi}, where\nnon-compact NS5 branes required the existence of tori instead of\nspheres in the T-dual picture. On the other hand \nthe term {\\it global} means roughly adding back the curvature, warping, etc., replacing tori by\nspheres, so that at the end of the day, we have a supersymmetric\nsolution to the equations of motion. The purpose of this paper \nis to exactly fill in the long-awaited gap, i.e to provide the full global picture of geometric\ntransition. Note also that the only known global solution, i.e \\cite{pandoz}, before our work\nwas unfortunately not supersymmetric (see\n\\cite{cvetic}, \\cite{anke} for details) although it satisfied the type IIB EOMs.}:\n\\bg\\label{localmet}\nds^2 ~=~ && dr^2 + \\Bigg(dz + \\sqrt{\\gamma' \\over \\gamma}~r_0 ~{\\rm\ncot}~\\langle\\theta_1\\rangle~ dx  + \\sqrt{\\gamma' \\over (\\gamma+4a^2)~}~r_0~ {\\rm\ncot}~\\langle\\theta_2\\rangle~dy\\Bigg)^2~ + \\nonumber\\\\\n&& ~~~~~~~~~~~ + \\Bigg[{\\gamma\\sqrt{h} \\over 4}~d\\theta_1^2 + dx^2\\Bigg] +\n\\Bigg[{(\\gamma + a^2)\\sqrt{h} \\over 4}~d\\theta_2^2 + dy^2\\Bigg] + ....\n\\nd\nwhere all the coefficients are measured at a fixed chosen point ($r_0, \\langle\\psi\\rangle, \\langle\\phi_i\\rangle,\n\\langle\\theta_i\\rangle$). For more details see \\cite{gtpapers}. The local $B_{\\rm NS}$ field was taken to be:\n\\bg\\label{bnslocal}\nB_{\\rm NS} = b_{x\\theta_i} dx \\wedge d\\theta_i + b_{y\\theta_i} dy \\wedge d\\theta_i\n\\nd\nwhere $i= 1,2$. The above background is invariant under the orbifold operation:\n\\bg\\label{orbi} {\\cal I}_{xy}: ~~ x ~ \\to ~-x, ~~~~ y ~\\to ~ -y\n\\nd\nand therefore can support D7\/O7 states at the following orientifold points:\n\\bg\\label{opoitla}\n{{\\bf T}^2 \\over {\\cal I}_{xy}~ \\Omega ~(-1)^{F_L}}\n\\nd\nIt is interesting to note that, at the orientifold point, a component like $b_{xy}$ is projected out. However the\norientifold projection may allow components like $b_{xz}, b_{yz}$ which could in principle make our mirror manifold\nnon-geometric. In the local picture advocated in \\cite{gtpapers} we only see components like \\eqref{bnslocal} so\nthe local mirror is non-K\\\"ahler and geometric.\n\nMore interestingly, the orientifolding operation \\eqref{opoitla} allows, along with the wrapped D5-branes,\nthe D7-branes and O7-planes along the internal directions ($r, z, \\theta_1, \\theta_2$) located at the\nfour fixed points of the torus ${\\bf T^2}$ along ($x, y$) directions. Therefore,\nin the local picture, a possible\nsusy preserving Ouyang-type configuration would be D5-branes wrapped on the two-torus ($\\theta_2, \\phi_2$) and the\nseven branes wrapping ($\\theta_1, \\theta_2, \\psi$) and stretched along the radial direction $r$. On the other hand,\nglobally\nin a resolved conifold background the seven-branes are in a configuration that is the {\\it union} of\nbranch 1 and branch 2 (see \\cite{ouyang, sullyj, FEP, jpsi} for details). Recall that\nin branch 1 the seven-branes wrap the ${\\bf P^1}$ parametrised by ($\\theta_2, \\phi_2$) and are\nembedded along ($r, \\psi$) directions at a point on the other ${\\bf P^1}$ parametrised by ($\\theta_1, \\phi_1$);\nwhereas in branch 2 the\nseven-branes are at a point on the ${\\bf P^1}$\nparametrised by ($\\theta_2, \\phi_2$). Thus globally a susy configuration of\nseven-branes is a {\\it two}-dimensional surface in ${\\bf P^1} \\times {\\bf P^1}$\nand stretched along ($r, \\psi$) directions determined\nby the appropriate embedding equation. Therefore in the local limit the two-dimensional susy preserving surface\nin ${\\bf T^2} \\times {\\bf T^2}$ should be the two-cycle parametrised by ($\\theta_1, \\theta_2$) as\nprescribed in \\cite{gtpapers}.\n\nAway from the orientifold point, the local metric takes the following fibration form:\n\\bg\\label{locfib}\nds^2 = && ~h^{-1\/2}ds^2_{0123} +  \\gamma'\\sqrt{h}~ dr^2 +\n(dz + \\Delta_1~{\\rm cot}~\\theta_1~ dx + \\Delta_2~{\\rm cot}~\\theta_2~\ndy)^2 + \\nonumber\\\\\n&& ~~~+ \\left({\\gamma \\sqrt{h} \\over 4} d\\theta_1^2 + dx^2\\right)\n + \\left({(\\gamma + 4a^2)\\sqrt{h} \\over 4} d\\theta_2^2 + dy^2 \\right)\\nonumber\\\\\nH_3 = && d{\\cal J}_1 \\wedge d\\theta_1 \\wedge dx + d{\\cal J}_2 \\wedge\nd\\theta_2 \\wedge dy \\nonumber\\\\\nF_5 = && K(r)~(1 + \\ast) ~dx \\wedge dy \\wedge\ndz \\wedge d\\theta_1\\wedge d\\theta_2\\\\\nF_3 = && c_1~(dz \\wedge d\\theta_2 \\wedge dy - dz \\wedge d\\theta_1 \\wedge dx)\\nonumber\n\\nd\nwith additional axio-dilaton that appear from the seven-brane sources. The above form of orientifold projection only\nallows a non-trivial fibration structure {\\it away} from the orientifold point. However there exist another orientifold\noperation that may be more well suited at the orientifold point. This can be applied locally via:\n\\bg\\label{orbnow}\n{\\cal I}_{x\\theta_1}: ~~~ x ~\\to ~ -x, ~~~~~~~~~ \\theta_1 ~\\to ~ \\pi - \\theta_1\n\\nd\nThe above action gives rise to the following orientifold action:\n\\bg\\label{oriennnw}\n{{\\bf T}^2 \\over {\\cal I}_{x\\theta_1}~ \\Omega ~(-1)^{F_L}}\n\\nd\nthat will keep the wrapped D5 branes only if they\nare {\\it away} from the orientifold point unless of course\nthey exist as bound states with the seven-branes {\\it at} the orientifold point.\nIn addition there is the $B_{\\rm NS}$ field with the following components at the orientifold point:\n\\bg\\label{hecchu}\nB_{\\rm NS} ~=~&& b_{x\\theta_2}~ dx \\wedge d\\theta_2 + b_{y\\theta_1}~ dy \\wedge d\\theta_1 +\nb_{xy}~ dx \\wedge dy + b_{xz}~ dx \\wedge dz + \\nonumber\\\\\n&& b_{rx}~ dr \\wedge dx + b_{r\\theta_1}~ dr \\wedge d\\theta_1 + b_{\\theta_1\\theta_2}~ d\\theta_1 \\wedge d\\theta_2\n+ b_{z\\theta_1}~ dz \\wedge d\\theta_1\n\\nd\nwhich means that at the orientifold point not only is the IIB metric non-trivial, the mirror can also be\nnon-K\\\"ahler and non-geometric. The seven-branes and the orientifold-planes are parallel to the wrapped D5-brane\nbound states\\footnote{These are in fact the dipole-deformed bound states studied in the last paper of\n\\cite{gtpapers}.}.\nIn the following we will argue how susy is preserved in the global set-up when the seven-branes are moved away from the\nwrapped D5 branes. This is the case where the fundamental hypermultiplets are infinitely massive and therefore susy\nremains unbroken at the scale that we want to study.\n\nThe naive global extension of the above configuration along the lines of \\cite{pandoz} will lead to a non-susy\nconfiguration. This is because we have assumed that the global extension of a configuration like \\eqref{locfib} is\nK\\\"ahler in the presence of a $B_{\\rm NS}$ field like \\eqref{bnslocal} {\\it away} from the orientifold point. The\nsimplest global extension that we will study here as the starting point for the IIB geometric transition is a\nnon-K\\\"ahler manifold with D5-branes wrapping two cycles of this manifold. Of course it may be possible to add other\nbranes and fluxes to make the ambient space conformally K\\\"ahler, but we will not do so here.\nWe will use the following set of duality transformations,\nrecently proposed by \\cite{marmal} (see also figure 1), to get\nour type IIB intial configuration.\n\n\\noindent $\\bullet$ Our starting point would be a non-K\\\"ahler type IIB metric with a background dilaton $\\phi$ and\nNS three-form $H_3$ that satisfies the standard relation $H_3 = e^{2\\phi} \\ast d(e^{-2\\phi}J)$\nwith $J$ being the fundamental\n(1,1) form.\n\n\\noindent $\\bullet$ On this background we perform a S-duality that transforms the NS three-form to RR three-form\n$F_3$, and in the process converts the dilaton to $-\\phi$ without changing the metric in the Einstein frame.\n\n\\noindent $\\bullet$ We now make three T-dualities along the spacetime directions $x^{1, 2, 3}$ that takes us to type\nIIA theory. Observe that this is {\\it not} the mirror construction.\n\n\\noindent $\\bullet$ We lift the type IIA configuration to M-theory and perform a boost along the eleventh direction.\nThis boost is crucial in generating D0-brane {\\it gauge} charges in M-theory.\n\n\\noindent $\\bullet$ A dimensional reduction back to IIA theory does exactly what we wanted: it generates the necessary\nnumber of D0-brane charges from the boost, without breaking the underlying supersymmetry of the system.\n\n\\noindent $\\bullet$ Once we have the IIA configuration, we go back to type IIB by performing the three T-dualities\nalong $x^{1, 2, 3}$ directions. From the D0-branes, we get back our three-brane charges namely the five-form. The\nduality cycle also gives us NS three-form $H_3$ as well as the expected RR three-form $F_3$. Therefore the final\nconfiguration is exactly what we required for IR geometric transition: wrapped D5s with necessary sources on a\nnon-K\\\"ahler globally defined ``resolved'' conifold background. Also as expected, the background preserves\nsupersymmetry and therefore should be our starting point. This background should also be compared with the one\ngiven in \\cite{pandoz} that solves the EOM but does not preserve supersymmetry. \nIn figure 1, we illustrate the above dualities.\n\\begin{figure}[htb]\\label{MMdualities}\n       \\begin{center}\n\\includegraphics[height=12cm]{MMdualities.eps}\n        \\caption{This figure illustrates the series of dualities that we used to generate\nthe full supersymmetric background with non-trivial fluxes.}\n        \\end{center}\n        \\end{figure}\nTo start off, we switch on a non-trivial background dilaton $\\phi$ and a NS three-form $H_{\\rm NS}$ on a\nbackground outlined by the following metric:\n\\bg \\label{startmet}\nds^2 = h^{1\/2} e^\\phi d{\\widetilde s}^2_{0123} + h^{-1\/2} e^\\phi ds^2_6\n\\nd\nwhere note that we can choose the dilaton $\\phi$ appropriately so that in the string frame the spacetime metric may not\nhave a warp factor. This will be consistent with the last reference of \\cite{torsion}.\nWe have defined the other variables in the following way:\n\\bg \\label{varde}\n&&h = {e^{-2\\phi} F_0^{-4} \\over e^{-2\\phi} h^{-2}F_0^{-4}{\\rm cosh}^2\\beta - {\\rm sinh}^2 \\beta},\n~~~~~~ d{\\widetilde s}^2_{0123} = F_0 ds^2_{0123}\\\\\n&& ds^2_6 = F_1~ dr^2 + F_2 (d\\psi + {\\rm cos}~\\theta_1 d\\phi_1 + {\\rm cos}~\\theta_2 d\\phi_2)^2  + \\sum_{i = 1}^2 F_{2+i}\n(d\\theta_i^2 + {\\rm sin}^2\\theta_i d\\phi_i^2)\\nonumber\n\\nd\nwith $\\beta$ being an arbitrary constant, $F_i \\equiv F_i(r), i = 1, ..., 4$ \nare functions of the radial coordinate for simplicity and $F_0 = F_0(r, \\theta_1, \\theta_2)$ since this is unconstrained. \nObserve that in the first equation of \\eqref{varde}, $h$ appears on both sides, and once we put in the value of the\ndilaton we can determine the warp factor $h$. For our case the dilaton will take the following form:\n\\bg\\label{dildil}\n\\phi ~ = ~ -{\\rm log}~F_0 - {1\\over 2} {\\rm log}~h\n\\nd\nso that the starting metric in IIB, that preserves spacetime supersymmetry, becomes:\n\\bg\\label{smetru}\nds^2 ~ = ~ ds^2_{0123} ~+~ \\left({1+F_0^2 {\\rm sinh}^2\\beta\\over F_0 {\\rm cosh}^2\\beta}\\right) ds^2_6\n\\nd\nIn\ngeneral we will continue keeping the dilaton $\\phi$ in the metric components to get the general torsion classes for the\nbackground (see the analysis in \\cite{chen2}). \nWe also\nexpect $F_i$ to be functions of all the internal coordinates. We will give an example of this soon when\nwe derive a more precise initial metric. For the time being we will consider \\eqref{startmet} to be our starting point.\nAlso to preserve supersymmetry\\footnote{Or, equivalently, preserving $SU(3)$ structure.}, we expect:\n\\bg\\label{hns}\nH_{\\rm NS} = e^{2\\phi} \\ast d\\left(e^{-2\\phi} J\\right)\n\\nd\nwith the appropriate dilaton. Here\n$J$ is the fundamental form associated with the metric, and we can choose to\nimpose one of the following two conditions on the\nNS three-form:\n\\bg\\label{condhns}\n&&dH_{\\rm NS} ~\\equiv~ d\\ast dJ - d\\ast (d\\phi \\wedge J) ~=~ {\\rm sources}\\nonumber\\\\\n&&dH_{\\rm NS} ~\\equiv~ d\\ast dJ - d\\ast (d\\phi \\wedge J) ~ = ~\\alpha'({\\rm tr} ~R \\wedge R - {\\rm Tr} ~ F \\wedge F)\n\\nd\nThe first condition is what we require here. This will give rise to the IR wrapped D5 branes theory on non-K\\\"ahler\nresolved conifold set-up (after the chain of dualities mentioned above). The latter case will be for the\nheterotic theory. We can use the non-closure of $H_{\\rm NS}$ to study not only\nthe vector bundles $F$ on the heterotic side, but also the possibility of geometric transition in the heterotic theory!\nWe have alluded to this possibility in our earlier papers \\cite{gtpapers} (see also \\cite{israel}).\nWe have completed that side of the\nstory in our follow-up paper \\cite{chen2}.\n\nNow following the chain of dualities mentioned above, we can get the following type IIB \nbackground\\footnote{Due to an unfortunate choice of notation, the RR three-form and the third warp factor have \nthe same notation of $F_3$ (as this is the standard way to represent them!). \nHowever since we use $F_3$ to mostly denote the third warp factor, we hope that there will be \nno confusion.}:\n\\bg\\label{susybg}\n&& F_3 = h~{\\rm cosh}~\\beta~e^{2\\phi} \\ast d\\left(e^{-2\\phi} J\\right), ~~~~~~~\nH_3 = -hF_0^2{\\rm sinh}~\\beta~e^{2\\phi} d\\left(e^{-2\\phi} J\\right)\\nonumber\\\\\n&& F_5 = -{1\\over 4} (1 + \\ast) dA_0 \\wedge dx^0 \\wedge dx^1 \\wedge  dx^2 \\wedge  dx^3, ~~~~~\n\\phi_{\\rm now} = -\\phi\\\\\n&& ds^2 = F_0 ds^2_{0123} + F_1~ dr^2 + F_2 (d\\psi + {\\rm cos}~\\theta_1 d\\phi_1 + {\\rm cos}~\\theta_2 d\\phi_2)^2\n+ \\sum_{i = 1}^2 F_{2+i}\n(d\\theta_i^2 + {\\rm sin}^2\\theta_i d\\phi_i^2)\\nonumber\n\\nd\nwhich is, by construction, supersymmetric and since the RR three-form\n$F_3$ is not closed it represents precisely the IR configuration\nof wrapped D5-branes on warped non-K\\\"ahler resolved conifold. The above set of equations \\eqref{susybg} is one \nof our main results, and as promised in the introduction, provides the fully supersymmetric globally defined \nsolution for the wrapped D5-branes on a non-K\\\"ahler resolved conifold. \nNote that we have left the warp factors $F_i$ undetermined in \\eqref{susybg}. This means that \nthere is a wide range of choices for $F_i$ related to various gauge theory deformations. This is closely related to \na similar class of solutions illustrated in figure 3 of \\cite{chen2}. Thus the procedure will be to identify \ncertain set of $\\{F_i\\}$ related to deformations in ${\\cal N} = 1$ YM theory, and then our duality chain will reproduce \nthe gravity dual of this YM configuration. One may also refer to a recent class of solutions studied in \\cite{pando2} \nwith a given choice of $\\{F_i\\}$. \n\nFinally, the five-form is switched on to satify the equation of\nmotion with\n\\bg\\label{a0def}\nA_0 ~&=~ &{{\\rm cosh}~\\beta~{\\rm sinh}~\\beta (1-e^{2\\phi}h^{-2}F_0^{-4})\n\\over e^{-2\\phi} h^{-2} F_0^{-4}{\\rm cosh}^2\\beta - {\\rm sinh}^2\\beta}\\nonumber\\\\\n&= ~& (F_0^2 -1 ){\\rm tanh}~\\beta \\left[1 + \\left({1-F_0^2\\over F_0^2}\\right){\\rm sech}^2\\beta +  \n\\left({1-F_0^2\\over F_0^2}\\right)^2 {\\rm sech}^4 \\beta\\right]\n\\nd\nThe above equation \\eqref{susybg} \nis our starting metric, whose local forms\nwe studied in details in \\cite{gtpapers}, and therefore should be taken instead of the metric derived in \\cite{pandoz}.\nThe ISD condition for our case gets modified to the following condition on the fluxes:\n\\bg\\label{isdnow}\n\\cosh\\beta~H_3 ~+~ F_0^2~\\sinh\\beta \\ast F_3 ~ = ~ 0\n\\nd\nwhich may be compared to \\cite{marmal}.\nOur derivation\ncould also solve the long standing problem of finding the supersymmetric configuration of wrapped D5-branes on a\nresolved conifold set-up.\n\n\\subsection{More explicit type IIB background before geometric transition}\n\nIn the above section we saw how one could derive the precise intial\nmetric that not only serves as starting point for geometric\ntransition, but is also supersymmetric. One may make this more specific by\nsolving the $SU(3)$ structure condition specified in section 2.4. This is\nworked out in {\\bf Appendix 1}.\nThe metric derived this way has\nmany non-trivial components compared to our initial ansatze\n\\eqref{susybg}. This is not a problem in itself, because we can\nalways do some coordinate transformations to bring the metric to a form that\ndoesn't have components like $g_{r\\mu}$ where $\\mu = \\theta_i,\n\\phi_i, \\psi$. But the metric will have other non-trivial\ncross-terms. It may be possible to make further coordinate\ntransformations to bring the above metric in a form that closely\nresembles \\eqref{susybg}, but we will not pursue this here as this\ndoesn't change the underlying physics. Instead we will continue\nusing the background \\eqref{susybg} and assume that the values of\nthe coefficients $F_i$ are to be fixed using our above metric\nconfiguration\\footnote{See also \\cite{israel} where a non-K\\\"ahler metric\non the resolved conifold is studied. It would be interesting to compare the\nmetric of \\cite{israel} with the metric components given in {\\bf Appendix 1}.}.\nOther possible cross-terms, not considered in\n\\eqref{susybg}, will only make the IIA background more non-trivial\nwithout revealing new physics\\footnote{We have justified this claim in \\cite{chen2} where we explicitly computed the \ntorsion classes for the background \\eqref{startmet}, \\eqref{dildil} and \\eqref{hns} to argue for supersymmetry. See \nsection 3.1 of \\cite{chen2}.}. \nHenceforth our starting point would\nbe \\eqref{susybg} with the assumption that the coefficients are to\nbe derived from the metric discussed in the above subsection.\n\\begin{figure}[htb]\\label{dualitiesGT}\n        \\begin{center}\n\\includegraphics[height=10cm]{dualitiesGT.eps}\n        \\caption{The duality map to generate the full geometric transitions in the supersymmetric global\nset-up of type IIA and type IIB theories.}\n        \\end{center}\n        \\end{figure}\nOnce we know the metric, we can follow up the steps described\nearlier to compute the three-forms. The NS three-form $H_3$ has the\nform: \\begin{eqnarray}\\label{H3}\n\\frac{H_3}{hF_0^2\\sinh\\beta}=&+&\\Big[k^2(2\\phi_{\\theta_1}\\sqrt{F_1F_2}\\cos\\theta_1+\\sqrt{F_1F_2}\\sin\\theta_1+2\\phi_rF_3\\sin\\theta_1-F_{3r}\\sin\\theta_1)\\nonumber\\\\\n&-&2kF_3k_r\\sin\\theta_1-2k\\sqrt{F_1F_2}k_{\\theta_1}\\cos\\theta_1\\Big]dr\\wedge\nd\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&+&\\Big[k^2(2\\phi_{\\theta_2}\\sqrt{F_1F_2}\\cos\\theta_2+\\sqrt{F_1F_2}\\sin\\theta_2+2\\phi_rF_4\\sin\\theta_2-F_{4r}\\sin\\theta_2)\\nonumber\\\\\n&-&2kF_4k_r\\sin\\theta_2-2k\\sqrt{F_1F_2}k_{\\theta_2}\\cos\\theta_2\\Big]dr\\wedge\nd\\theta_2\\wedge d\\phi_2\\nonumber\\\\\n&-&2k\\sqrt{F_1F_2}\\Big[(k\\phi_{\\theta_1}-k_{\\theta_1})(dr\\wedge\nd\\psi\\wedge d\\theta_1-\\cos\\theta_2dr\\wedge d\\theta_1\\wedge\nd\\phi_2)\\nonumber\\\\\n&+&(k\\phi_{\\theta_2}-k_{\\theta_2} )(dr\\wedge d\\psi\\wedge\nd\\theta_2-\\cos\\theta_1dr\\wedge\nd\\theta_2\\wedge d\\phi_1)\\Big]\\nonumber\\\\\n&-&2kF_3\\sin\\theta_1(k\\phi_{\\theta_2}-k_{\\theta_2})d\\theta_1\\wedge\nd\\theta_2\\wedge\nd\\phi_1\\nonumber\\\\\n&+&2kF_4\\sin\\theta_2(k\\phi_{\\theta_1}-k_{\\theta_1})d\\theta_1\\wedge\nd\\theta_2\\wedge d\\phi_2\n\\end{eqnarray}\nwhere $k^2(r,\\theta_1,\\theta_2)=h^{-1\/2}e^{\\phi}$, and we have\ndefined $\\phi_\\alpha \\equiv \\partial_\\alpha \\phi$ with $\\alpha =\n\\theta_i, r$ as $\\phi \\equiv \\phi (r, \\theta_1, \\theta_2)$ for\nsimplicity. A constant $\\phi$ is not good for us, and also leads to\ncertain issues detailed in \\cite{papad}. Once we have $H_3$, we can\nget $dH_3$ as:\n\\begin{eqnarray}\\label{dH3}\n\\frac{dH_3}{\\sinh\\beta}&=&k\\Big[F_3\\sin\\theta_1(4hk_r\\phi_{\\theta_2}-4hk_{\\theta_2}\\phi_r+2h_{\\theta_2}k_r-2h_rk_{\\theta_2}+2kh_r\\phi_{\\theta_2}-2kh_{\\theta_2}\\phi_r)\\nonumber\\\\\n&&\\sqrt{F_1F_2}\\cos\\theta_1(4hk_{\\theta_1}\\phi_{\\theta_2}-4hk_{\\theta_2}\\phi_{\\theta_1}+2h_{\\theta_2}k_{\\theta_1}-2h_{\\theta_1}k_{\\theta_2}\n+2kh_{\\theta_1}\\phi_{\\theta_2}-2kh_{\\theta_2}\\phi_{\\theta_1})\\nonumber\\\\\n&&-(\\sqrt{F_1F_2}-F_{3r})(2hk\\phi_{\\theta_2}+h_{\\theta_2}k)\\sin\\theta_1\\Big]dr\\wedge\nd\\theta_2\\wedge d\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&&+k\\Big[F_4\\sin\\theta_2(4hk_r\\phi_{\\theta_1}-4hk_{\\theta_1}\\phi_r+2h_{\\theta_1}k_r-2h_rk_{\\theta_1}+2kh_r\\phi_{\\theta_1}-2kh_{\\theta_1}\\phi_r)\\nonumber\\\\\n&&\\sqrt{F_1F_2}\\cos\\theta_2(4hk_{\\theta_2}\\phi_{\\theta_1}-4hk_{\\theta_1}\\phi_{\\theta_2}+2h_{\\theta_1}k_{\\theta_2}-2h_{\\theta_2}k_{\\theta_1}\n+2kh_{\\theta_2}\\phi_{\\theta_1}-2kh_{\\theta_1}\\phi_{\\theta_2})\\nonumber\\\\\n&&-(\\sqrt{F_1F_2}-F_{4r})(2hk\\phi_{\\theta_1}+h_{\\theta_1}k)\\sin\\theta_2\\Big]dr\\wedge\nd\\theta_1\\wedge d\\theta_2\\wedge\nd\\phi_2\\nonumber\\\\\n&&+2k\\sqrt{F_1F_2}\\Big[h_{\\theta_1}k_{\\theta_2}-h_{\\theta_2}k_{\\theta_1}+kh_{\\theta_2}\\phi_{\\theta_1}-kh_{\\theta_1}\\phi_{\\theta_2}+2hk_{\\theta_2}\\phi_{\\theta_1}-2hk_{\\theta_1}\\phi_{\\theta_2}\\Big]\\nonumber\\\\\n&&~~~~~~~~~~~~~~~~\\times dr\\wedge d\\theta_1\\wedge d\\theta_2\\wedge\nd\\psi\n\\end{eqnarray}\nwith $F_{ir} \\equiv \\partial_r F_i$, $k_{i}=\\partial_i k$ and\n$(hF_0^2)_{i}=\\partial_i (hF_0^2)$. From \\eqref{dH3} it means that\nif we make\n\\begin{eqnarray}\n&&h_{\\theta_1}k_{\\theta_2}-h_{\\theta_2}k_{\\theta_1}+kh_{\\theta_2}\\phi_{\\theta_1}-kh_{\\theta_1}\\phi_{\\theta_2}+2hk_{\\theta_2}\\phi_{\\theta_1}-2hk_{\\theta_1}\\phi_{\\theta_2}=0,\\nonumber\\\\\n&&\\frac{4hk_r\\phi_{\\theta_2}-4hk_{\\theta_2}\\phi_r+2h_{\\theta_2}k_r-2h_rk_{\\theta_2}+2kh_r\\phi_{\\theta_2}-2kh_{\\theta_2}\\phi_r}{2hk\\phi_{\\theta_2}+h_{\\theta_2}k}=\\frac{\\sqrt{F_1F_2}-F_{3r}}{F_3},\\nonumber\\\\\n&&\\frac{4hk_r\\phi_{\\theta_1}-4hk_{\\theta_1}\\phi_r+2h_{\\theta_1}k_r-2h_rk_{\\theta_1}+2kh_r\\phi_{\\theta_1}-2kh_{\\theta_1}\\phi_r}{2hk\\phi_{\\theta_1}+h_{\\theta_1}k}=\\frac{\\sqrt{F_1F_2}-F_{4r}}{F_4}.\\nonumber\n\\end{eqnarray}\nthen $H_3$ will be closed. One however may worry that making $H_3$\nclosed implies too much constraints on the $F_i$'s. For the present\ncase this may still be okay because the initial choice of the\nbackground \\eqref{varde} forms a class of solutions parametrised by\nour choice of $F_i$ and $\\phi$, the dilaton\\footnote{See also \\cite{chen2} where a torsion class analysis reveals \n$k$ to be a function of $r$ only. This means we can choose $F_0$ appropriately such that $h$ and $e^\\phi$ are \nfunctions of ($r, \\theta_i$). One possible choice would be $h = {1\\over k^2 F_0}$ and $e^\\phi = {k\\over \\sqrt{F_0}}$.}. \nA specific choice of the\nbackground with a specified complex structure and K\\\"ahler class is\nexemplified in {\\bf Appendix 1}. For this case we can define a\nclosed three-form with appropriate choice of the dilaton, so that\nour choice remains generic enough. Thus the $B_{NS}$ field can be\ngauge transformed to have only the following components:\n\\begin{eqnarray}\\label{bns}\nb_{r\\psi}&=&\\int-2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_1}-k_{\\theta_1})\\sqrt{F_1F_2}~d\\theta_1,\n\\nonumber\\\\\nb_{r\\phi_1}&=&\\int-2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_2}-k_{\\theta_2})\\sqrt{F_1F_2}\\cos\\theta_1~d\\theta_2,\n\\nonumber\\\\\nb_{r\\phi_2}&=&\\int-2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_1}-k_{\\theta_1})\\sqrt{F_1F_2}\\cos\\theta_2~d\\theta_1,\\nonumber\\\\\nb_{\\theta_1\\phi_1}&=&\\int2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_2}-k_{\\theta_2})\nF_3\\sin\\theta_1~d\\theta_2,\\nonumber\\\\\nb_{\\theta_2\\phi_2}&=&\\int2hF_0^2k\\sinh\\beta~(k\\phi_{\\theta_1}-k_{\\theta_1})\nF_4\\sin\\theta_2~d\\theta_1.\n\\end{eqnarray}\nwhere we see that there are three new components of the form $b_{r\\alpha}$ compared to the local case \\cite{gtpapers}.\nThis is expected because we are no longer fixed to $r = r_0$, but have global access. However\nbefore moving ahead we will pause to comment on switching on\nother possible components of the $B_{NS}$ field of the form:\n\\bg\\label{bnsother}\n b_{\\phi_1 \\phi_2}~ d\\phi_1 \\wedge d\\phi_2 ~+~  \\sum_{i = 1}^2 b_{\\phi_i \\psi}~ d\\phi_i \\wedge d\\psi\n\\nd\nSuch choices of $B_{NS}$ fields would make the type IIA background {\\it non-geometric}. So far locally we saw that the\ntype IIA backgrounds remains geometric but does become non-K\\\"ahler. Is there a possibility that the IIA background\nglobally is non-geometric also? We will reflect on this point later, but for the time being we will assume that the\n$B_{NS}$ field is only of the form \\eqref{bns} and doesn't have additional components like \\eqref{bnsother}.\n\nNext comes the RR three-form $F_3$. From our previous set of duality arguments, this is given by:\n\\begin{eqnarray}\\label{F3}\n\\frac{F_3}{h\nF_0^2\\cosh\\beta}=&&2KF_1F_2F_3F_4\\sin\\theta_2\\sin\\theta_1(\\phi_{\\theta_1}\\sin\\theta_1\\cos\\theta_2-\n\\phi_{\\theta_2}\\sin\\theta_2\\cos\\theta_1)dr \\wedge d\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&&+KF_3^2\\sin^2\\theta_1\\sin\\theta_2(2\\phi_{\\theta_2}\\sqrt{F_1F_2}F_4\\sin\\theta_2-F_2\\sqrt{F_1F_2}\n\\cos\\theta_2\\nonumber\\\\\n&&-2\\phi_rF_2F_4\\cos\\theta_2+F_2F_{4r}\\cos\\theta_2)d\\theta_1\\wedge\nd\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&&+KF_4^2\\sin^2\\theta_2\\sin\\theta_1\n(-2\\phi_{\\theta_1}\\sqrt{F_1F_2}\\sin\\theta_1+F_2\\sqrt{F_1F_2}\\cos\\theta_1\\nonumber\\\\\n&&+2\\phi_rF_2F_3\\cos\\theta_1-F_2F_{3r}\\cos\\theta_1)d\\theta_2\\wedge d\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&&-KF_2F_3^2\\sin^2\\theta_1(2\\phi_rF_4\\sin\\theta_2+\\sqrt{F_1F_2}\\sin\\theta_2-F_{4r}\\sin\\theta_2)d\\psi\\wedge\nd\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&&-KF_2F_4^2\\sin^2\\theta_2(2\\phi_rF_3\\sin\\theta_1+\\sqrt{F_1F_2}\\sin\\theta_1-F_{3r}\\sin\\theta_1)d\\psi\\wedge\nd\\theta_2\\wedge d\\phi_2\\nonumber\\\\\n&&-2\\phi_{\\theta_2}KF_1F_2F_3F_4\\sin\\theta_1\\sin^2\\theta_2dr\\wedge\nd\\psi\\wedge d\\phi_2\\nonumber\\\\\n&&-2\\phi_{\\theta_1}KF_1F_2F_3F_4\\sin\\theta_2\\sin^2\\theta_1dr\\wedge\nd\\psi\\wedge d\\phi_1\n\\end{eqnarray}\nwhere as before $\\phi_\\alpha$ should be understood as derivatives on $\\phi$ i.e $\\partial_\\alpha \\phi$,\nand we have defined $K$ as:\n\\bg\\label{kdef}\nK ={{\\rm cosec}~\\theta_1{\\rm cosec}~\\theta_2\\over \\sqrt{F_1F_2}F_3F_4}\n\\nd\nNote that $dF_3$ is no longer\nclosed, and will be related to delta function sources coming from the wrapped \nD5-branes\\footnote{There is a subtlety \nhere: not every non-closed $F_3$ can be interpreted as a source (see for example the criteria presented in \n\\cite{konkal}). Happily,\nour case does fall into one of the required criteria of \\cite{konkal} as should be clear by writing the \nfluxes in the language of G-structure, or in terms of the torsion classes. A more detailed elaboration of this is \ngiven in \\cite{chen2}.}.\n\nOnce we have the explicit forms for the three-forms, to satisy the type IIB EOMs we will now require RR five-form.\nThis is easy to work out, and is given by:\n\\begin{eqnarray}\\label{F5}\nF_5&=&\\frac{1}{4}\\Big[-A_{0r}dr\\wedge dt\\wedge dx\\wedge dy\\wedge\ndz-A_{0\\theta_1}d\\theta_1\\wedge dt\\wedge dx\\wedge dy\\wedge\ndz\\nonumber\\\\\n&&-A_{0\\theta_2}d\\theta_2\\wedge dt\\wedge dx\\wedge dy\\wedge dz\n-PF_2F_3F_4\\sin^2\\theta_1\\sin^2\\theta_2\\nonumber\\\\\n&&\\times (A_{0r}F_3F_4d\\psi\\wedge d\\theta_1\\wedge d\\theta_2\\wedge\nd\\phi_1\\wedge d\\phi_2+A_{0\\theta_1}F_1F_4dr\\wedge d\\psi\\wedge\nd\\theta_2\\wedge\nd\\phi_1\\wedge d\\phi_2\\nonumber\\\\\n&& +A_{0\\theta_2}F_1F_3dr\\wedge d\\psi\\wedge d\\theta_1\\wedge\nd\\phi_1\\wedge d\\phi_2)\\Big]\n\\end{eqnarray}\nwhere $A_{0\\alpha} \\equiv \\partial_\\alpha A_0$ and $A_0$ is given in \\eqref{a0def}. We have also defined $P$ as:\n\\bg\\label{pdef}\nP={{\\rm cosec}~\\theta_1{\\rm cosec}~\\theta_2\\over \\sqrt{F_1F_2}F_0^2F_3 F_4}\n\\nd\nThus with \\eqref{H3}, \\eqref{F3}, \\eqref{F5}\nand \\eqref{susybg} we have the complete susy background in type IIB before geometric transition. A torsion\nclass analysis with susy constraints has been done in \\cite{chen2} (see eq. (3.15) therein). As long as the\nwarp factors satisfy eq. (3.15) of \\cite{chen2} supersymmetry will be preserved.\nIn the following subsection, we will use the above background to\ncompute the type IIA mirror configuration.\n\n\\subsection{The type IIA mirror configuration}\n\nAs it stands, the metric in \\eqref{susybg} has three obvious\nisometries associated with translation along the three angular\ndirections $\\phi_1, \\phi_2$ and $\\psi$. So there is a natural ${\\bf\nT}^3$ embedded in our configuration, and one might naively think\nthat the mirror would be three T-dualities along ${\\bf T}^3$. Such a\nsimple transformation doesn't work for our case because our\nconfiguration represents the IR limit of a cascading gauge theory\nwhere the base of the three torus is {\\it small}. Mirror\ntransformation {\\it a la} SYZ \\cite{syz} works exactly in the\nopposite limit! So naive T-dualities will not give us the mirror\nmetric, and we need to first make the base, paramerised by\n$\\theta_1, \\theta_2$ and $r$, very large\\footnote{This effectively\nmeans that the distances along the $\\theta_i$ directions have to be\nmade very large, as $r$ is non-compact. See also our earlier works \\cite{gtpapers} where this\nhas been explained in more details.}. The simplest way to do\nthis would be to make the following transformation on the background\n\\eqref{susybg}: \\bg\\label{tranbg}\n&& d\\psi ~ \\to ~ d\\psi ~+~f_1~ {\\rm cos}~\\theta_1~d\\theta_1   ~+~f_2~ {\\rm cos}~\\theta_2~d\\theta_2\\nonumber\\\\\n&& d\\phi_1 ~ \\to ~ d\\phi_1 ~ - ~ f_1~d\\theta_1, ~~~~~~~ d\\phi_2 ~\n\\to ~ d\\phi_2 ~ - ~ f_2~d\\theta_2 \\nd with the assumption that $f_i\n= f_i(\\theta_i)$ so that the transformations \\eqref{tranbg} would be\nintegrable\\footnote{Note also that since $f_i = f_i(\\theta_i)$, the transformations \\eqref{tranbg} on the \none-forms ($d\\psi, d\\phi_i$) are just coordinate transformations of ($\\psi, \\phi_i$). Therefore they don't change the \nEOMs.}.  \nRecall that compatibility with the $SU(3)$ structure will fix $f_i$ in the mirror \\cite{gtpapers}.\nNote also that these transformations are similar in form as\nin the first reference of \\cite{gtpapers} and would change the complex structure of the base accordingly.\n\nUnder these transformations the $B_{\\rm NS}$ field generates extra\ncomponents $b_{r\\theta_1}$, $b_{r\\theta_2}$. It is however interesting to note that\nthey vanish as follows:\n\\begin{eqnarray}\nb_{r\\theta_1}=f_1(b_{r\\psi}\\cos\\theta_1-b_{r\\phi_1})=0,\\;\\;b_{r\\theta_2}=f_2(b_{r\\psi}\\cos\\theta_2-b_{r\\phi_2})=0\n\\end{eqnarray}\nimplying that the $B_{\\rm NS}$ field do not change under the transformation \\eqref{tranbg}. This is similar to the\nlocal case also \\cite{gtpapers}.\n\nOn the other hand the RR three-form {\\it does} change under the coordinate transformation \\eqref{tranbg}. The\nextra components of the three-form are the following:\n\\begin{eqnarray}\\label{extra3}\n&&F_{r\\theta_1\\theta_2}=f_1f_2(F_{r\\phi_1\\phi_2}-\\cos\\theta_1F_{r\\psi\\phi_2}+\\cos\\theta_2F_{r\\psi\\phi_1}),\n~~~F_{r\\psi\\theta_1}=-f_1F_{r\\psi\\phi_1}\\nonumber\\\\\n&&F_{r\\theta_1\\phi_2}=-f_1(F_{r\\phi_1\\phi_2}-\\cos\\theta_1F_{r\\psi\\phi_2}),\\quad\\,\nF_{\\theta_1\\theta_2\\phi_1}=f_2(F_{\\theta_1\\phi_1\\phi_2}-\\cos\\theta_2F_{\\psi\\theta_1\\phi_1}),\n\\nonumber\\\\\n&&F_{r\\theta_2\\phi_1}=f_2(F_{r\\phi_1\\phi_2}+\\cos\\theta_2F_{r\\psi\\phi_1}),\\quad\\,\nF_{\\theta_1\\theta_2\\phi_2}=f_1(F_{\\theta_2\\phi_1\\phi_2}+\\cos\\theta_1F_{\\psi\\theta_2\\phi_2}),\n\\nonumber\\\\\n&&F_{r\\theta_2\\phi_2}=f_2\\cos\\theta_2F_{r\\psi\\phi_2},~~~\nF_{r\\psi\\theta_2}=-f_2F_{r\\psi\\phi_2},~~~\nF_{r\\theta_1\\phi_1}=f_1\\cos\\theta_1F_{r\\psi\\phi_1}\n\\end{eqnarray}\nA physical reason for this change can be easily understood: under the coordinate transformation \\eqref{tranbg} the\nbase parametrised by ($\\theta_1, \\theta_2$) become large. This means that the associated RR three-form\nfield strengths increase\nsimultaneously, which is of course what we see in \\eqref{extra3}. Note that the component\n$F_{r\\theta_1\\theta_2}$ dominates over all other extra components in \\eqref{extra3} because this lies exclusively\non the base parametrised by the coordinates ($r, \\theta_1, \\theta_2$) which is made much bigger than the\n${\\bf T}^3$ fibre parametrised by the coordinates ($\\psi, \\phi_1, \\phi_2$).\n\nOnce the three-form $F_3$ changes, the RR five-form also has to change. Its is easy to show that the\nextra components of the five-form are:\n\\begin{eqnarray}\\label{extra5}\n&&F_{r\\theta_1\\theta_2\\phi_1\\phi_2}=f_1\\cos\\theta_1F_{r\\psi\\theta_2\\phi_1\\phi_2}-f_2\\cos\\theta_2\nF_{r\\psi\\theta_1\\phi_1\\phi_2},\\nonumber\\\\\n&&F_{r\\psi\\theta_1\\theta_2\\phi_2}=f_1F_{r\\psi\\theta_2\\phi_1\\phi_2},\\quad\nF_{r\\psi\\theta_1\\theta_2\\phi_1}=f_2F_{r\\psi\\theta_1\\phi_1\\phi_2}\n\\end{eqnarray}\nsatisfying the background EOMs. All these extra components will give rise to RR four-form in Type IIA\nafter mirror transformation, as we will show soon. But before that lets infer how the metric changes.\nUnder the transformation \\eqref{tranbg} the metric \\eqref{susybg} takes the\nfollowing form:\n\\bg\\label{sysymet} ds^2 = && F_0 ds^2_{0123} + F_1~\ndr^2 + F_2 (d\\psi + {\\rm cos}~\\theta_1 d\\phi_1 + {\\rm cos}~\\theta_2\nd\\phi_2)^2\n+ \\sum_{i = 1}^2 F_{2+i} ~{\\rm sin}^2\\theta_i d\\phi_i^2 \\nonumber\\\\\n&& ~~~~~~~~~~~+ \\sum_{i = 1}^2 \\Big[F_{2+i}\\left(1 + f_i^2~{\\rm sin}^2 \\theta_i\\right) d\\theta_i^2 -2 f_i F_{2+i}~\n{\\rm sin}^2 \\theta_i ~d\\phi_i d\\theta_i\\Big]\n\\nd\nwhich in fact does exactly what we wanted\\footnote{The metric \\eqref{sysymet} also solves the supergravity EOM as \nshould be clear from the discussion presented earlier.}: \nit enlarges the $\\theta_i$-cycles, but doesn't change the $B_{\\rm NS}$\nfield. For SYZ to work properly, we require the base size to be very large, and therefore we will require $f_i$ also to be\nlarge. This conclusion fits well with the local picture that we had in \\cite{gtpapers}. Note that we have also\ngenerated cross terms. These cross terms will be useful soon. The eleven metric components are:\n\\bg\\label{metcom}\n&& j_{rr} = F_1, ~~~~ j_{\\phi_1\\theta_1} = -f_1F_3 {\\rm sin}^2\\theta_1, ~~~~\nj_{\\phi_2\\theta_2} = -f_2 F_4 {\\rm sin}^2\\theta_2 \\nonumber\\\\\n&& j_{\\psi\\psi} = F_2(1-\\epsilon), ~~~~ j_{\\phi_1\\psi} = F_2 {\\rm cos}~\\theta_1, ~~~~\nj_{\\phi_2\\psi} = F_2 {\\rm cos}~\\theta_2\\nonumber\\\\\n&& j_{\\phi_1\\phi_1} = F_2 {\\rm cos}^2 \\theta_1 + F_3 {\\rm sin}^2 \\theta_1, ~~~~\nj_{\\phi_2\\phi_2} = F_2 {\\rm cos}^2 \\theta_2 + F_4 {\\rm sin}^2 \\theta_2\\\\\n&& j_{\\phi_1\\phi_2} = F_2 {\\rm cos}~\\theta_1 \\theta_2, ~~~~\nj_{\\theta_1 \\theta_1} = F_3 (1+f_1^2 {\\rm sin}^2 \\theta_1), ~~~\nj_{\\theta_2 \\theta_2} = F_{4} (1+f_2^2 {\\rm sin}^2\n\\theta_2)\\nonumber \\nd where $\\epsilon$ is a finite but small number\\footnote{Of course $\\epsilon < 1$ to preserve the \nsignature. \nIn fact introducing $\\epsilon$ in\n$j_{\\psi\\psi}$ will help us not only to keep $f_i$ large to satisfy SYZ but also satisfy the susy conditions\nin the mirror dual. This will become clear soon.}.\nLet\nus also define another quantity $\\alpha$ in the following way:\n\\bg\\label{alphadef} \\alpha^{-1} ~\\equiv~ {F_3 F_4 {\\rm\nsin}^2\\theta_1 {\\rm sin}^2\\theta_2 + F_2 F_4 {\\rm cos}^2\\theta_1{\\rm\nsin}^2\\theta_2 + F_2 F_3 {\\rm sin}^2\\theta_1 {\\rm cos}^2\\theta_2}\n\\nd away from the point ($\\theta_1, \\theta_2$) $=0$. Now assuming\nthat $f_1, f_2$ are large, we can perform the mirror\ntransformation along ($\\psi, \\phi_1, \\phi_2$) directions. The mirror metric in type IIA takes the following\nform: \\bg\\label{mirrormet} ds^2_{\\rm mirror} = F_0 ds^2_{0123} +\nds^2_6 \\nd where the six-dimensional internal space is a\nnon-K\\\"ahler deformation of the deformed conifold in the following\nway: \\bg\\label{nkdefco} &&ds^2_6 = F_1 dr^2 + {\\alpha F_2 \\over\n\\Delta_1 \\Delta_2} \\Big[d\\psi-b_{\\psi r}dr + \\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1\nr}dr\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+ \\Delta_2\n{\\rm cos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\\Big)\\Big]^2\\nonumber\\\\\n&& ~~~~~~ + \\alpha j_{\\phi_2\\phi_2}\\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)^2 + \\alpha\nj_{\\phi_1\\phi_1}\\Big(d\\phi_2 - b_{\\phi_2\\theta_2}\nd\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\nonumber\\\\\n&& ~~~~~~ -2\\alpha j_{\\phi_1\\phi_2} \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\\\\n&& ~~~~~~ -2\\alpha j_{\\phi_1\\phi_2} \\left({f_1 f_2 \\epsilon\\over\n\\alpha}\\right) d\\theta_1 d\\theta_2 + \\Big(F_3 - \\epsilon ~F_2 f_1^2\n{\\rm cos}^2 \\theta_1\\Big)d\\theta_1^2 + \\Big(F_4 - \\epsilon ~F_2\nf_2^2 {\\rm cos}^2 \\theta_2\\Big)d\\theta_2^2\\nonumber \\nd and we have\ndefined $\\Delta_i$ in the following way: \\bg\\label{deltadef}\n\\Delta_1 ~ = ~ \\alpha F_2 F_4 {\\rm sin}^2 \\theta_2, ~~~ \\Delta_2 ~ =\n~ \\alpha F_2 F_3 {\\rm sin}^2 \\theta_1 \\nd At this stage we can\nextract the consequence of the fact that both $f_1$ and $f_2$ are\nvery large. This fits perfectly well with the mirror metric because\n$f_i^2$ as well as $f_1f_2$ come with the coefficient $\\epsilon$ allowing us to satisfy both SYZ and\nsusy in the mirror.\nThis means that if we impose the following constraint:\n\\bg\\label{consf1f2} f_1 f_2 \\epsilon ~ \\equiv ~ -\\alpha \\nd i.e both\n$f_i$ proportional to $\\epsilon^{-1\/2}$, it will bring the\ncross-terms in the metric to the following suggestive form:\n\\bg\\label{suggest} 2\\alpha j_{\\phi_1\\phi_2} \\Big[d\\theta_1 d\\theta_2\n- \\Big(d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1\nr}dr\\Big)\\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big] \\nd In the limit $b_{\\phi_1\\alpha} =\nb_{\\phi_2\\alpha} =0$ with $\\alpha = r, \\theta_i$, \\eqref{suggest} is in fact a term of the\ndeformed conifold! The above conclusion seems rather encouraging,\nprovided of course \\eqref{consf1f2} is satisfied. In the local\nlimit, similar condition also arose (see the first reference of\n\\cite{gtpapers}) and we argued therein that as long as we can define\n\\bg\\label{fiear} f_i ~\\propto ~ {(-1)^i \\langle\\alpha\\rangle_i \\over\n\\sqrt{\\epsilon}} \\nd where $\\langle\\alpha\\rangle_i$ depend only on\n$\\theta_i$ the constraint \\eqref{consf1f2} is satisfied. Therefore a\ncondition like \\eqref{consf1f2} works perfectly well in the local\ncase. Question is, can we satisfy \\eqref{consf1f2} also for the\nglobal case?\n\nThe answer is now tricky. We demanded that $f_i = f_i(\\theta_i)$, otherwise global coordinate transformation like\n\\eqref{tranbg} {\\it cannot} be defined. This means that $F_i$ appearing in the definition of $\\alpha$ in \\eqref{consf1f2}\nwill have to be highly constrained. Generically this is not possible\\footnote{For the local case $\\alpha$ was\ndefined at $r = r_0$ so this subtlety did not arise and, as we discussed above, we used $\\langle\\alpha\\rangle_i$ to\ndefine $f_i$ so things were perfectly consistent there.} because of the underlying type IIA supersymmetry,\nas constraints on the\ntorsion classes \\cite{torsion} will tend to fix $f_i(\\theta_i)$,\nbut in special case this may happen.\n\nThe special case arises if we allow $F_2$ to depend on the angular coordinates $\\theta_i$\nalso in such a way that the susy constraints on the torsion classes are still satisfied with $F_2$ given by:\n\\bg\\label{f2def}\nF_2(r, \\theta_1, \\theta_2) ~= ~ - {(\\beta_1\\beta_2)^{-1} + F_3 F_4 {\\rm sin}^2\\theta_1 {\\rm sin}^2\\theta_2 \\over\nF_4 {\\rm cos}^2\\theta_1 {\\rm sin}^2\\theta_2 + F_3 {\\rm sin}^2\\theta_1 {\\rm cos}^2\\theta_2}\n\\nd\nwhere $f_i \\equiv {{\\beta}_i\\over \\sqrt{\\epsilon}}$. This tells us that the radial dependence of $F_2$ is fixed by\n$F_3(r)$ and $F_4(r)$, but the angular dependences are pretty much unfixed because $\\beta_i(\\theta_i)$ are arbitrary\nfunctions of $\\theta_i$ respectively\\footnote{Interestingly we can make both $\\beta_i$ and $\\epsilon$ to be\ngeneric functions of the internal coordinates in such a way that $f_i \\equiv {{\\beta}_i\\over \\sqrt{\\epsilon}}$\nremain functions of $\\theta_i$ {\\it only}. The only bound on $\\epsilon$ would be that it never exceeds 1 over any point \nin the internal space.}.\nAll these of course should get fixed once we impose the susy constraints on the\ntorsion classes.\nHowever the above relation \\eqref{f2def} already looks tight, but lets move\non and see how far we can go with these kind of arguments. Our next\nquestion would therefore be: is there a way to fix the angular dependences also?\n\nTo see how to fix the angular dependences, we can go back to the equivalent local limit of\n\\eqref{suggest} where the particular way of writing the\nmetric allows us to make a coordinate rotation to bring the term \\eqref{suggest} into the more familar deformed\nconifold form \\cite{minasian}. This, as we know from \\cite{minasian, gtpapers}, is only possible iff {\\it other}\nterms in the metric remain invariant under the coordinate transformation. If this condition is\nimposed globally, then it would imply the\nfollowing two relations:\n\\bg\\label{2rels}\n&& \\beta_1 ~ = ~ \\pm\\sqrt{F_3 - \\alpha j_{\\phi_2\\phi_2}\\over F_2 {\\rm cos}^2 \\theta_1}\\nonumber\\\\\n&& \\beta_2 ~ = ~ \\mp \\sqrt{F_4 - \\alpha j_{\\phi_1\\phi_1}\\over F_2\n{\\rm cos}^2 \\theta_1} \\nd In the local case, studied in the first\nreference of \\cite{gtpapers}, relations like \\eqref{2rels} are\nconsistent in the sense that \\eqref{consf1f2} is satisfied.\nUnfortunately, this is no longer true for the global case\ngenerically because the above relation along with \\eqref{consf1f2}\nwould lead to inconsistent set of equations, and would probably break susy.\nTherefore in general\nthe mirror metric will take the following form: \\bg\\label{mirmet}\n&&ds^2_6 = F_1 dr^2 + {\\alpha F_2 \\over \\Delta_1 \\Delta_2}\n\\Big[d\\psi-b_{\\psi r}dr + \\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\nonumber\\\\\n&& \\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+ \\Delta_2 {\\rm\ncos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& ~~~~~~ + \\alpha j_{\\phi_2\\phi_2}\\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)^2 + \\alpha\nj_{\\phi_1\\phi_1}\\Big(d\\phi_2 - b_{\\phi_2\\theta_2}\nd\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\nonumber\\\\\n&& ~~~~~~ -2\\alpha j_{\\phi_1\\phi_2} \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\\Big)\\nonumber\\\\ &&\n~~~~~~ -2 j_{\\phi_1\\phi_2} \\beta_1 \\beta_2~ d\\theta_1 d\\theta_2 +\n\\Big(F_3 -  F_2 \\beta_1^2 {\\rm cos}^2 \\theta_1\\Big)d\\theta_1^2 +\n\\Big(F_4 - F_2 \\beta_2^2 {\\rm cos}^2\n\\theta_2\\Big)d\\theta_2^2\\nonumber\\\\ \\nd Only in very special cases,\nwhere \\eqref{2rels} and \\eqref{consf1f2} are both simultaneously\nsatisfied, we expect the mirror to take the following symmetric\nform: \\bg\\label{mirmetspecial} ds^2_6 = && F_1 dr^2 + {\\alpha F_2\n\\over \\Delta_1 \\Delta_2} \\Big[d\\psi -b_{\\psi r}dr+ \\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1\nr}dr\\Big)\n\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\;+ \\Delta_2 {\\rm\ncos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& + \\alpha j_{\\phi_2\\phi_2}\\Big[d\\theta_1^2 + \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)^2\\Big] \\nonumber\\\\\n&&+ \\alpha j_{\\phi_1\\phi_1}\\Big[d\\theta_2^2 + \\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}\\Big[d\\theta_1 d\\theta_2 - \\Big(d\\phi_1\n- b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\\Big)\\Big]\\nonumber\\\\ \\nd\nwhich is strongly reminiscent of the deformed conifold!\nObserve that both the forms of the metrics are finite and well defined.\nThis tells us that our procedure of making the base large before performing\nSYZ \\cite{syz} is logical and correct.\n\nOn the other hand, the cross term that we developed in the metric appears as\nthe $B_{\\rm NS}$ field in type IIA theory. Expectedly, this B-field is large\nand is given by the following form:\n\\begin{eqnarray}\\label{BinIIA}\n\\widetilde{B}&&=~\\alpha\nf_1F_3\\sin^2\\theta_1\\left(F_2\\cos^2\\theta_2+F_4\\sin^2\\theta_2\\right)d\\theta_1\\wedge d\\phi_1\\nonumber\\\\\n&&~+\\alpha\nf_2F_4\\sin^2\\theta_2\\left(F_2\\cos^2\\theta_1+F_3\\sin^2\\theta_1\\right)d\\theta_2\\wedge d\\phi_2\\nonumber\\\\\n&&~+ \\left(1-{\\epsilon \\over \\alpha\nF_2F_4\\sin^2\\theta_1\\sin^2\\theta_2}\\right)\n\\left(f_1\\cos\\theta_1d\\theta_1+f_2\\cos\\theta_2d\\theta_2\\right)\\wedge d\\psi\n\\end{eqnarray}\nIn the limit $\\epsilon \\to 0$, the last two terms are pure gauge. For finite, but small, $\\epsilon < 1$ they cannot be\ngauged away.\nIn the local limit (see the first paper of\n\\cite{gtpapers}) all the $F_i$ were constants, and so $\\widetilde{B}$ became a pure gauge when written in terms\nof $\\langle\\alpha\\rangle_i$. This doesn't seem to be the case globally, unless of course $F_i$'s are of some specific\nforms.\n\nThe wrapped D6 brane two-form charges now come partly from the type IIB three-forms and partly from the five-forms.\nThe three-forms contributions to the IIA two-forms are given by the following\ncomponents:\n\\begin{eqnarray}\\label{2form1}\n&&\\widetilde{F}_{\\psi\\theta_1}=F_{\\phi_1\\phi_2\\theta_1},\\quad\n\\widetilde{F}_{\\psi\\theta_2}=F_{\\phi_1\\phi_2\\theta_2},\\quad\n\\widetilde{F}_{\\psi r}=F_{\\phi_1\\phi_2 r},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_1r}=F_{r\\phi_2\\psi}+\\frac{2j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\nr \\psi}+\\frac{2j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}F_{\nr \\phi_1\\phi_2},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_2 r}=F_{\\phi_1 r\\psi}+2\\alpha\n(j_{\\phi_2\\psi}j_{\\phi_1\\phi_1}-j_{\\phi_1\\phi_2}j_{\\phi_1\\psi})F_{r\\phi_1\\phi_2},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_1\\theta_2}=F_{\\psi\\theta_2\\phi_2}+2\n\\frac{j_{\\phi_1\\psi}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\\phi_2\\theta_2},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_1\\theta_1}=2\\frac{j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\\theta_1\\psi}\n+2\\frac{j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}F_{\\phi_1\\phi_2\\theta_1},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_2\\theta_1}=F_{\\psi\\phi_1\\theta_1}+2\\alpha\n(j_{\\phi_2\\psi}j_{\\phi_1\\phi_1}-j_{\\phi_1\\phi_2}j_{\\phi_1\\psi})F_{\\phi_1\\phi_2\\theta_1},\\nonumber\\\\\n&&\\widetilde{F}_{\\phi_2\\theta_2}=2\\alpha\n(j_{\\phi_2\\psi}j_{\\phi_1\\phi_1}-j_{\\phi_1\\phi_2}j_{\\phi_1\\psi})F_{\\phi_1\\phi_2\\theta_2}\n\\end{eqnarray}\nSimilarly, the five-forms contributions to the type IIA two-forms are given in terms of the following components:\n\\begin{eqnarray}\\label{2form2}\n&&\\widetilde{F}_{\\theta_1\\theta_2}=F_{\\psi\\theta_1\\theta_2\\phi_1\\phi_2}+b_{\\theta_2\\phi_2}F_{\\psi\\theta_1\\phi_1}\n+b_{\\theta_1\\phi_1}F_{\\psi\\theta_2\\phi_2}\\nonumber\\\\\n&&\\widetilde{F}_{r\\theta_1}=F_{r\\theta_1\\phi_1\\phi_2\\psi}+b_{\\theta_1\\phi_1}F_{r\\psi\\phi_2}\n+\\left(b_{\\phi_2r}+\\frac{j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}b_{r\\phi_1}\\right)F_{\\psi\\theta_1\\phi_1}\n+\\frac{j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}b_{\\theta_1\\phi_1}F_{r\\phi_1\\phi_2}\\nonumber\\\\\n&&~~~~~~~~~~~+\\left(b_{r\\psi}-\\frac{j_{\\psi\\phi_1}}{j_{\\phi_1\\phi_1}}b_{r\\phi_1}\\right)F_{\\theta_1\\phi_1\\phi_2}\n+\\frac{j_{\\phi_1\\phi_2}}{j_{\\phi_1\\phi_1}}b_{\\theta_1\\phi_1}F_{r\\psi\\phi_1}\\nonumber\\\\\n&&\\widetilde{F}_{r\\theta_2}=F_{r\\theta_2\\phi_1\\phi_2\\psi}+b_{r\\phi_1}F_{\\psi\\theta_2\\phi_2}\n+b_{r\\psi}F_{\\theta_2\\phi_1\\phi_2}-b_{\\theta_2\\phi_2}F_{r\\psi\\phi_1}\n\\end{eqnarray}\nAll the above components are finite and give rise to the required D6-branes charges. However since the B-field is\nlarge, to compensate this in the EOMs we need large G-fluxes in type IIA. These fluxes come exactly from the\nextra three- and five-form components \\eqref{extra3} and \\eqref{extra5} respectively.\nThese three- and five-form components give rise to twelve components of the four-form\nfluxes in IIA namely:\n\\bg\\label{twelve}\n&&\\widetilde{F}_{r\\psi\\theta_1\\theta_2},~~~\n\\widetilde{F}_{r\\psi\\theta_1\\phi_1}, ~~~\n\\widetilde{F}_{r\\psi\\theta_1\\phi_2}, ~~~\n\\widetilde{F}_{r\\psi\\theta_2\\phi_1}\\nonumber\\\\\n&&\\widetilde{F}_{r\\psi\\theta_2\\phi_2},~~~\n\\widetilde{F}_{r\\theta_1\\theta_2\\phi_1}, ~~~\n\\widetilde{F}_{r\\theta_1\\theta_2\\phi_2}, ~~~\n\\widetilde{F}_{r\\theta_1\\phi_1\\phi_2}\\nonumber\\\\\n&&\\widetilde{F}_{r\\theta_2\\phi_1\\phi_2}, ~~~\n\\widetilde{F}_{\\psi\\theta_1\\theta_2\\phi_1},~~~\n\\widetilde{F}_{\\psi\\theta_1\\theta_2\\phi_2},~~~\n\\widetilde{F}_{\\theta_1\\theta_2\\phi_1\\phi_2}\n\\nd\nThese components are listed in {\\bf Appendix 2} which the readers may refer to for details. Combined with \\eqref{BinIIA},\nthese fluxes lift to M-theory as G-fluxes with components along the spacetime and the eleventh directions respectively.\nInterestingly, both the metric \\eqref{mirmet} or \\eqref{mirmetspecial} along with the two-form flux components\n\\eqref{2form1} and \\eqref{2form2} lift to a geometrical configuration in M-theory, which we expect to have a\n$G_2$ structure. This is of course expected because both the non-K\\\"ahler deformed conifold as well as the wrapped\nD6-branes tend to become geometrical configurations when the type IIA coupling is made very \nlarge\\footnote{The fact that we get \\eqref{mirmet} instead of \\eqref{mirmetspecial} is nothing too surprising. \nOne {\\it does not} expect to get a K\\\"ahler deformed conifold in type IIA. This was already clear from the \npioneering work of \\cite{vafa}. Here we not only seem to confirm the statement of \\cite{vafa} but also determine \nthe precise form of the IIA metric.}.  \nIn the\nfollowing sub-section we will dwell on this in more details.\n\n\n\\subsection{M theory lift, flop transition and type IIA reduction}\n\nThe lift of our type IIA mirror configuration to M-theory is rather straighforward. The eleven-directional fibration\nis given by gauge fluxes derived from the two-form components \\eqref{2form1} and \\eqref{2form2}. It is easy to\nshow that we need only $A_{\\phi_i}, A_{\\theta_i}$ and $A_r$ components. Using these, the M-theory lift of our IIA\nsymmetric mirror metric \\eqref{mirmetspecial} is:\n\\begin{eqnarray}\\label{mlift}\nds^2_{11}=&&e^{-{2\\phi\\over 3}}\\Bigg\\{F_0ds_{0123}^2+F_1 dr^2 + {\\alpha F_2\n\\over \\Delta_1 \\Delta_2} \\Big[d\\psi -b_{\\psi r}dr\\nonumber\\\\\n&&+ \\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)+ \\Delta_2 {\\rm cos}~\\theta_2\n\\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& + \\alpha j_{\\phi_2\\phi_2}\\Big[d\\theta_1^2 + \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)^2\\Big] \\nonumber\\\\\n&&+ \\alpha j_{\\phi_1\\phi_1}\\Big[d\\theta_2^2 + \\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}\\Big[d\\theta_1 d\\theta_2 - \\Big(d\\phi_1\n- b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]\\Bigg\\}\\nonumber\\\\\n&&+e^{4\\phi\\over 3}\\Big[dx_{11}+A_{\\phi_1}d\\phi_1+A_{\\phi_2}d\\phi_2+A_{\\theta_1}d\\theta_1\n+A_{\\theta_2}d\\theta_2+A_rdr\\Big]^2\n\\end{eqnarray}\nIt is easy to see that the non-symmetric mirror metric \\eqref{mirmet} will also lift to M-theory in an\nidentical way. The local limit of the lift of\n\\eqref{mirmet} is precisely the one discussed in the first paper of\n\\cite{gtpapers} and, as discussed therein, we expect the lift of \\eqref{mirmet} to have a $G_2$ structure to\npreserve supersymmetry. To see this for our case, we have to express the lift of the metric \\eqref{mirmet} in terms of\ncertain one-forms similar to the ones given in \\cite{gtpapers} (see also \\cite{brandhuber}). Following the\nfirst paper of \\cite{gtpapers} we first express the B-fields appearing in the fibration \\eqref{mirmet} in terms\nof periodic angular coordinates $\\lambda_i$ in the following way:\n\\begin{eqnarray}\\label{angles}\n&&\\tan\\lambda_1 ~\\equiv ~b_{\\phi_1\\theta_1},\\quad\n\\tan\\lambda_2~\\equiv~b_{\\phi_2\\theta_2},\\quad \\tan\\lambda_3 ~\\equiv~b_{\\psi\nr}\\nonumber\\\\\n&&\\tan\\lambda_4 ~\\equiv ~b_{\\phi_1 r},\\quad \\;\\,\\tan\\lambda_5 ~\\equiv~ b_{\\phi_2 r}\n\\end{eqnarray}\nUsing these we can define two sets of one-forms. The first set, called $\\sigma_i$ with $i = 1, .., 3$, can be expressed\nin terms of $\\lambda_i$ as:\n\\begin{eqnarray}\\label{lift1}\n\\sigma_1&=&\\sin\\psi_1(d\\phi_1-\\tan\\lambda_4dr)+\\sec\\lambda_1\\cos(\\psi_1+\\lambda_1)d\\theta_1,\\nonumber\\\\\n\\sigma_2&=&\\cos\\psi_1(d\\phi_1-\\tan\\lambda_4dr)-\\sec\\lambda_1\\sin(\\psi_1+\\lambda_1)d\\theta_1,\\nonumber\\\\\n\\sigma_3&=&d\\psi_1-\\frac{1}{2}\\tan\\lambda_3dr+\\Delta_1\\cos\\theta_1(d\\phi_1-\\tan\\lambda_1d\\theta_1-\\tan\\lambda_4dr)\n\\end{eqnarray}\nand the second set can be expressed in terms of $\\lambda_i$ as:\n\\begin{eqnarray}\\label{lift2}\n\\Sigma_1&=&-\\sin\\psi_2(d\\phi_2-\\tan\\lambda_5dr)+\\sec\\lambda_2\\cos(\\psi_2+\\lambda_2)d\\theta_2,\\nonumber\\\\\n\\Sigma_2&=&-\\cos\\psi_2(d\\phi_2-\\tan\\lambda_5dr)-\\sec\\lambda_2\\sin(\\psi_2+\\lambda_2)d\\theta_2,\\nonumber\\\\\n\\Sigma_3&=&d\\psi_2+\\frac{1}{2}\\tan\\lambda_3dr-\\Delta_2\\cos\\theta_2(d\\phi_2-\\tan\\lambda_2d\\theta_2-\\tan\\lambda_5dr)\n\\end{eqnarray}\nAt this stage one may compare these two sets of one-forms to the ones given by eq. (6.2) and eq. (6.3) in the first\npaper of \\cite{gtpapers}. The definitions of $\\psi_1$ and $\\psi_2$ follow exactly as in \\cite{gtpapers}, i.e\n\\bg\\label{psi12}\nd\\psi ~ = ~ d\\psi_1 - d\\psi_2, ~~~~~~~~~ dx_{11} ~ = ~ d\\psi_1 + d\\psi_2\n\\nd\nFurthermore we can perform the following rotation of the coordinates:\n\\begin{equation}\\label{rotc}\n\\begin{pmatrix} {\\cal D}\\phi_2 \\\\ d\\theta_2\\end{pmatrix} ~ \\to ~\n\\begin{pmatrix} \\cos~\\psi_0 & -\\sin~\\psi_0 \\\\ \\sin~\\psi_0 & ~~\\cos~\\psi_0 \\end{pmatrix}\n\\begin{pmatrix} {\\cal D}\\phi_2 \\\\ d\\theta_2\\end{pmatrix}\n\\end{equation}\nwith ${\\cal D}\\phi_2 \\equiv d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2r}dr$ and $\\psi_0$ a constant. If we make\nthis transformation to the symmetric mirror metric of type IIA \\eqref{mirmetspecial}, this will lift to M-theory not as\n\\eqref{mlift}, but to a more {\\it suggestive} configuration:\n\\begin{eqnarray}\\label{mliftnow}\nds^2_{11}=&&e^{-{2\\phi\\over 3}}\\Bigg\\{F_0ds_{0123}^2+F_1 dr^2 + {\\alpha F_2\n\\over \\Delta_1 \\Delta_2} \\Big[d\\psi -b_{\\psi r}dr - b_{\\psi\\theta_2} d\\theta_2\\nonumber\\\\\n&&+ \\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)+ \\Delta_2 {\\rm cos}~\\theta_2 {\\rm cos}~\\psi_0\n\\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]^2\\nonumber\\\\\n&& + \\alpha j_{\\phi_2\\phi_2}\\Big[d\\theta_1^2 + \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)^2\\Big] \\nonumber\\\\\n&&+ \\alpha j_{\\phi_1\\phi_1}\\Big[d\\theta_2^2 + \\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)^2\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}{\\rm cos}~\\psi_0\\Big[d\\theta_1 d\\theta_2 - \\Big(d\\phi_1\n- b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)\\Big(d\\phi_2 -\nb_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\\Big]\\nonumber\\\\\n&& + 2\\alpha j_{\\phi_1\\phi_2}{\\rm sin}~\\psi_0\\Big[\\Big(d\\phi_1- b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big) d\\theta_2\n+ \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2r}dr\\Big)d\\theta_1\\Big]\\Bigg\\}\\nonumber\\\\\n&&+e^{4\\phi\\over 3}\\Big[dx_{11}+{\\widetilde A}_{\\phi_1}d\\phi_1+{\\widetilde A}_{\\phi_2}d\\phi_2+\n{\\widetilde A}_{\\theta_1}d\\theta_1\n+{\\widetilde A}_{\\theta_2}d\\theta_2+{\\widetilde A}_rdr\\Big]^2\n\\end{eqnarray}\nwhere we have introduced a $B$-field fibration using\n$b_{\\psi\\theta_2} \\equiv \\Delta_2 ~{\\rm sin}~\\psi_0 {\\rm cos}~\\theta_2$ to modify the $d\\psi$ fibration structure.\nThe eleven-dimensional fibration structure will also change accordingly because we can always express the\n$A_{\\phi_2} d\\phi_2$ term in $dx_{11}$ of \\eqref{mlift} using ${\\cal D}\\phi_2$. Thus the overall eleven-dimensional\nfibration will retain its form but with shifted $A_\\mu$ fields denoted above by the ${\\widetilde A}_\\mu$ fields.\nIn terms of the fibration components of \\eqref{mlift} one can\nshow that ${\\widetilde A}_{\\phi_1} = A_{\\phi_1}, {\\widetilde A}_{\\theta_1} = A_{\\theta_1}$ and the rest of the\ncomponents can be presented in the following matrix form:\n\\begin{equation}\\label{matricre}\n\\begin{pmatrix}{\\widetilde A}_{\\phi_2}\\\\ {}&{}&{} \\\\ {\\widetilde A}_{\\theta_2}\\\\ {}&{}&{}\\\\\n{\\widetilde A}_{r} \\end{pmatrix} ~ = ~\n\\begin{pmatrix} \\cos\\psi_0 + b_{\\phi_2\\theta_2}\\sin\\psi_0 & \\sin\\psi_0 & 0\\\\ {}&{}&{}\\\\\n-(1+b^2_{\\phi_2\\theta_2})\\sin\\psi_0 & \\cos\\psi_0 - b_{\\phi_2\\theta_2}\\sin\\psi_0 & 0\\\\ {}&{}&{}\\\\\nb_{\\phi_2 r}(1-\n\\cos\\psi_0 - b_{\\phi_2\\theta_2}\\sin\\psi_0) & -b_{\\phi_2 r} \\sin\\psi_0 &1 \\end{pmatrix}\n\\begin{pmatrix}{A}_{\\phi_2}\\\\ {}&{}&{}\\\\ {A}_{\\theta_2}\\\\ {}&{}&{}\\\\ {A}_{r} \\end{pmatrix}\n\\end{equation}\nAdditionally with the above modification, the\nabove metric is surprisingly close to the uplift of a non-K\\\"ahler deformed conifold metric with wrapped D6-branes\nto M-theory provided we can make an additional substitution\\footnote{This is compatible with the\nunderlying $G_2$ structure. An analysis of the $G_2$ torsion classes, along the lines of\nthe first paper in \\cite{gtpapers}, will reveal this. We will discuss this more soon.} in \\eqref{mliftnow}:\n\\bg\\label{psis}\n\\psi_0 ~ ~\\to ~~ \\psi\n\\nd\nMaking such a substitution may lead one to think that the $\\psi$ isometry that we have in \\eqref{mlift} is removed.\nThis is {\\it not} the case locally\nwith the non-K\\\"ahler deformed conifold because the extra B-field component $b_{\\psi\\theta_2}$\nin the $d\\psi$ fibration structure of \\eqref{mliftnow} as well as the vector fields ${\\widetilde A}_\\mu$\nin the $dx_{11}$ fibration structure transform non-trivially under\nshift in $\\psi$ to restore the isometry. One may also do a somewhat similar rotation like \\eqref{rotc} to the\nnon-symmetric type IIA metric \\eqref{mirmet} and bring it in a more suggestive format.\n\nThe rotation \\eqref{rotc} should now be captured by\nthe one-forms \\eqref{lift1} and \\eqref{lift2} appropriately. In fact only the\nsecond set of one-forms \\eqref{lift2} is related to the\nchange \\eqref{rotc}.\nThus the one-forms for\nour purposes will be ($\\sigma_i, \\Sigma_i$) with $\\Sigma_i$ to be viewed as the one got from \\eqref{rotc} directly.\nUsing these one-forms\nwe can rewrite the M-theory metric in two possible ways. The first one is the lift of the non-symmetric type IIA\nmetric \\eqref{mirmet} under the rotation \\eqref{rotc} and transformation \\eqref{psis}:\n\\begin{eqnarray}\\label{liftnow}\nds_7^2 && = g_rdr^2+g_1(\\sigma_3+\\Sigma_3)^2+g_2(\\sigma_3-\\Sigma_3)^2\\nonumber\\\\\n&&+~g_3(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2)^2\n+\\widetilde{g}_3(\\cos\\psi_1\\sigma_1-\\sin\\psi_1\\sigma_2)^2\\nonumber\\\\\n&&+~g_4(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2)^2\n+\\widetilde{g}_4(\\cos\\psi_2\\Sigma_1-\\sin\\psi_2\\Sigma_2)^2\\nonumber\\\\\n&&+~g_5(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2)(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2)\\nonumber\\\\\n&&-~\\widetilde{g}_5(\\cos\\psi\\sigma_1-\\sin\\psi_1\\sigma_2)(\\cos\\psi_2\\Sigma_1-\\sin\\psi_2\\Sigma_2)\n\\end{eqnarray}\nwhere we have defined the coefficients $g_i, {\\widetilde g}_i$ as:\n\\begin{eqnarray}\\label{ggdef}\n&&g_r=e^{-2\\phi\/3}F_1,\\quad g_1=e^{4\\phi\/3},\\quad\ng_2=e^{-2\\phi\/3}\\frac{\\alpha F_2}{\\Delta_1\\Delta_2},\\quad g_3=\\alpha\nj_{\\phi_2\\phi_2},\n\\nonumber\\\\\n&&\\widetilde{g}_3=F_3-F_2\\beta_1^2\\cos^2\\theta_1,\\quad g_4=\\alpha\nj_{\\phi_1\\phi_1},\\quad\n\\widetilde{g}_4=F_4-F_2\\beta_2^2\\cos^2\\theta_2\\nonumber\\\\\n&&g_5=2\\alpha j_{\\phi_1\\phi_2},\\quad\n\\widetilde{g}_5=2\\beta_1\\beta_2j_{\\phi_1\\phi_2}\n\\end{eqnarray}\nThe second way to rewrite the metric is a little more suggestive of the way to perform the flop operation on the M-theory\nmanifold and has a nice form for the symmetric case \\eqref{mirmetspecial}, again under \\eqref{rotc} and \\eqref{psis}.\nThe local form of this has already appeared in the first reference of \\cite{gtpapers}, and the readers may\nwant to look at that for more details. In fact we can rewrite \\eqref{liftnow} in the following\nway also.\nHere we will simply quote the generic ansatze using parameters $\\alpha_i$ and $\\zeta$:\n\\bg\\label{sugmet}\nds_7^2 = \\alpha_1^2 \\sum_{a=1}^2 (\\sigma_a + \\zeta \\Sigma_a)^2 + \\alpha_2^2 \\sum_{a=1}^2 (\\sigma_a - \\zeta \\Sigma_a)^2\n+ \\alpha_3^2 (\\sigma_3 + \\Sigma_3)^2 + \\alpha_4^2 (\\sigma_3 - \\Sigma_3)^2 + \\alpha_5^2 dr^2\\nonumber\\\\\n\\nd\nThe above is a familiar form by which any $G_2$ structure metric could be expressed. Once we switch off $\\lambda_i$ the\nmanifolds has a $G_2$ holonomy. The coefficients $\\alpha_i$ and $\\zeta$ are not arbitrary. They are fixed by the EOM\nand, for the case \\eqref{mirmetspecial}, they take the following values:\n\\bg\\label{alphazeta}\n&& \\alpha_1 = {1\\over 2} e^{-{\\phi\\over 3}}\\sqrt{2\\alpha\\left(j_{\\phi_2\\phi_2} + {j_{\\phi_1\\phi_2}\\over \\zeta}\\right)},\n~~~~~ \\alpha_2 = {1\\over 2} e^{-{\\phi\\over 3}}\\sqrt{2\\alpha\\left(j_{\\phi_2\\phi_2}\n- {j_{\\phi_1\\phi_2}\\over \\zeta}\\right)}\\nonumber\\\\\n&& \\alpha_3 = e^{2\\phi\\over 3}, ~~~~ \\alpha_4 = e^{-{\\phi\\over 3}} \\sqrt{\\alpha F_2 \\over \\Delta_1 \\Delta_2}, ~~~~\n\\alpha_5 =  e^{-{\\phi\\over 3}} \\sqrt{F_1}, ~~~~ \\zeta = \\sqrt{j_{\\phi_1\\phi_1}\\over j_{\\phi_2\\phi_2}}\n\\nd\nBefore proceeding further let us pause for a while and ask whether the above set of manipulations would preserve\nsupersymmetry. From type IIA point of view we have done the following:\n\n\\vskip.1in\n\n\\noindent $\\bullet$ Shift of the coordinates ($\\psi, \\phi_i$) using variables $f_i(\\theta_i)$. This shifting of the\ncoordinates mixes non-trivially all the three isometry directions as described in \\eqref{tranbg}.\n\n\\noindent $\\bullet$ Shift the metric along $\\psi$ direction by the variable $\\epsilon$, as given in the second line\nof \\eqref{metcom}. This variable doesn't have to be too small in the global limit. As long as it is smaller than 1 \nit'll suffice.\n\n\\noindent $\\bullet$ Make SYZ transformations along the new shifted directions. Thus the three T-dualities are\n{\\it not} made along the three original isometry directions.\n\n\\noindent $\\bullet$ In the new metric of IIA make a further rotation along the ($\\theta_2, \\phi_2$)\ndirections using a $2\\times 2$\nmatrix given as \\eqref{rotc}. The matrix is described using a constant angular variable $\\psi_0$.\n\n\\noindent $\\bullet$ Finally in the transformed metric convert $\\psi_0$ to $\\psi$ as in \\eqref{psis}.\n\n\\vskip.1in\n\n\\noindent However due to steps 2, 4 and 5 above, it is not guaranteed that the metric will preserve supersymmetry.\nFurthermore one might also question whether the SYZ operation itself could preserve supersymmetry. Therefore to verify\nthis we have evaluated all the torsion classes for this background in sec 3.2 of \\cite{chen2}. The supersymmetry\nconstraints are given by eq. (3.21) and Appendix B of \\cite{chen2}. These set of equations along with the constraint\nequation (3.15) of \\cite{chen2}  are enough to guarantee supersymmetry in the type IIA (or the equivalent\nM-theory) background.\n\nOnce the issue of supersymmetry is resolved, we go to the flop operation.\nThe operation of flop on the above metric \\eqref{sugmet} has already been discussed in details in sec. 7 of the\nfirst reference of \\cite{gtpapers}. Using similar techniques for \\eqref{liftnow},\nafter the flop we expect the metric to look like:\n\\begin{eqnarray}\\label{aflop}\nds_7^2=a_1(\\sigma_1^2+\\sigma_2^2)+a_2(\\Sigma_1^2+\\Sigma_2^2)+a_3(\\sigma_3+\\Sigma_3)^2+a_4(\\sigma_3-\\Sigma_3)^2+a_5dr^2\n\\end{eqnarray}\nwith $a_i$, $i=1, ..., 5$ are some coefficients to be determined. Due to the global nature of our metric,\nthe operation of flop can be performed by\na class of transformations parametrized by the values of\n$a$, $b$ etc. in the following way:\n\\begin{eqnarray}\\label{lbaaz}\n&&\\sigma_1\\mapsto a\\sigma_1+b\\Sigma_1,\\quad \\Sigma_1\\mapsto\ne\\sigma_1+f\\Sigma_1,\\nonumber\\\\\n&&\\sigma_2\\mapsto c\\sigma_2+d\\Sigma_2,\\quad \\Sigma_2\\mapsto\ng\\sigma_2+h\\Sigma_2,\\nonumber\\\\\n&&\\sigma_3+\\Sigma_3\\mapsto \\sigma_3-\\Sigma_3, \\quad\n\\sigma_3-\\Sigma_3\\mapsto \\sigma_3+\\Sigma_3\n\\end{eqnarray}\nNow comparing \\eqref{sugmet} and \\eqref{liftnow} one can pretty much fix the coefficients $c, d$ etc. in terms of\n$a, b$ in the following way:\n\\bg\\label{cdef}\n&& c=a\\sqrt{\\frac{k^2G_2+kG_3+G_1}{\\omega^2G_5+\\omega\nG_6+G_4}},~~~~~~~ d=b\\sqrt{\\frac{\\mu^2G_2+\\mu G_3+G_1}{\\tau^2G_5+\\tau\nG_6+G_4}}\\nonumber\\\\\n&& e=ak,~~~~ g=c\\omega,~~~~ f=b\\mu,~~~~ h=d\\tau\n\\nd\nwhere $a, b$ can in turn be fixed by computing the $G_2$-torsion classes and demanding supersymmetry (see also\n\\cite{chen2}).\nThe other coefficients appearing above, namely, $k$,\n$\\omega$, $\\mu$, $\\tau$ satisfy the following equations:\n\\begin{eqnarray}\n&&2G_1+2k\\mu G_2+(k+\\mu)G_3=0, \\quad~ G_7+k\\tau G_8+\\tau G_9+kG_{10}=0,\\nonumber\\\\\n&&2G_4+2\\tau \\omega G_5+(\\tau+\\omega)G_6=0, \\quad G_7+\\omega\\mu\nG_8+\\omega G_9+\\mu G_{10}=0.\n\\end{eqnarray}\nwhose solutions are fixed by the following values of $G_i$ determined from the $G_2$ structure metric \\eqref{liftnow}\nusing ($g_i, \\widetilde{g}_i$) defined earlier in \\eqref{ggdef}:\n\\begin{eqnarray}\n&&G_1=g_3\\sin^2\\psi_1+\\widetilde{g}_3\\cos^2\\psi_1,\\quad\\quad\nG_2=g_4\\sin^2\\psi_2+\\widetilde{g}_4\\cos^2\\psi_2, \\nonumber\\\\\n&&G_3=g_5\\sin\\psi_1\\sin\\psi_2-\\widetilde{g}_5\\cos\\psi_1\\cos\\psi_2,\\nonumber\\\\\n&&G_4=g_3\\cos^2\\psi_1+\\widetilde{g}_3\\sin^2\\psi_1,\\quad\\quad\nG_5=g_4\\cos^2\\psi_2+\\widetilde{g}_4\\sin^2\\psi_2, \\nonumber\\\\\n&&G_6=g_5\\cos\\psi_1\\cos\\psi_2-\\widetilde{g}_5\\sin\\psi_1\\sin\\psi_2,\\nonumber\\\\\n&&G_7=(g_3-\\widetilde{g}_3)\\sin\\psi_1\\cos\\psi_1,\\quad\\;\nG_9=g_5\\sin\\psi_1\\cos\\psi_2+\\widetilde{g}_5\\cos\\psi_1\\sin\\psi_2,\\nonumber\\\\\n&&G_8=(g_4-\\widetilde{g}_4)\\sin\\psi_2\\cos\\psi_2,\\quad\\;\nG_{10}=g_5\\cos\\psi_1\\sin\\psi_2+\\widetilde{g}_5\\sin\\psi_1\\cos\\psi_2.\\nonumber\\\\\n\\end{eqnarray}\nUsing all the above relations, the $a_i$ coefficients in the M-theory metric after flop transition \\eqref{lbaaz} can\nbe determined in terms of ($a, b$). The final form of the metric therefore is given by:\n\\begin{eqnarray}\\label{flopmet}\nds_{11}^2&&= e^{-{2\\phi\\over 3}}F_0 ds_{0123}^2+g_rdr^2+g_1(\\sigma_3-\\Sigma_3)^2+g_2(\\sigma_3+\\Sigma_3)^2\\nonumber\\\\\n&&+a^2(k^2G_2+kG_3+G_1)(\\sigma_1^2+\\sigma_2^2)+b^2(\\mu^2G_2+\\mu\nG_3+G_1)(\\Sigma_1^2+\\Sigma_2^2)\\nonumber\\\\\n\\end{eqnarray}\nWe are now one step away from getting the type IIA metric from the above metric.\nReducing along $x_{11}$ the metric takes the following form in type IIA theory:\n\\begin{eqnarray}\\label{IIAmetric}\nds_{10}^2&& =F_0ds_{0123}^2+F_1dr^2+e^{2\\phi}\\Big[d\\psi-b_{\\psi\n\\mu}dx^\\mu+\\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 - b_{\\phi_1\\theta_1}\nd\\theta_1-b_{\\phi_1 r}dr\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad +\n\\widetilde{\\Delta}_2 {\\rm cos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2}\nd\\theta_2-b_{\\phi_2 r}dr\\Big)\\Big]^2\\nonumber\\\\\n&&~+e^{2\\phi\\over 3}a^2(k^2G_2+kG_3+G_1)\\Big[d\\theta_1^2+(d\\phi_1^2-b_{\\phi_1\\theta_1}d\\theta_1-b_{\\phi_1\nr}dr)^2\\Big]\\nonumber\\\\\n&&~+e^{2\\phi\\over 3}b^2(\\mu^2G_2+\\mu\nG_3+G_1)\\Big[d\\theta_2^2+(d\\phi_2^2-b_{\\phi_2\\theta_2}d\\theta_2-b_{\\phi_2\nr}dr)^2\\Big]\n\\end{eqnarray}\nwhich has an amazing similarity with the warped resolved conifold! The above metric is completely global and\nsupersymmetric\\footnote{Of course there is a further UV completion that we don't discuss here. The\nUV completion should follow in the same vein as studied recently for the\nKlebanov-Strassler case in \\cite{FEP, jpsi}.}. As before, the full torsion class analysis for this is\ngiven in eq. (3.25) of \\cite{chen2}. Together with equations (3.15), (3.21) and (3.25) of \\cite{chen2} we can\npretty much get most of the warp factor components (plus the other parameters) to demand supersymmetry for all the\nabove backgrounds. Any remaining set of unconstrained parameters would allow us to get a class of gauge\ntheory deformations that span the landscape of solutions in the geometric transition set-up. We will discuss more\non this landscape soon (see also figure 3 of \\cite{chen2}).\n\nTherefore the type IIA background\nshould be viewed as the gravity dual in the IR for the gauge theory on wrapped D6-branes before\ngeometric transition. In this background there are no six-branes. The wrapped D6-branes have {\\it dissolved} in the\ngeometry, and is replaced by the following one-form flux components:\n\\bg\\label{1ffc}\n A=\\Delta_1 {\\rm cos}~\\theta_1 \\Big(d\\phi_1 -\nb_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr\\Big)- \\widetilde{\\Delta}_2 {\\rm\ncos}~\\theta_2 \\Big(d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2\nr}dr\\Big)\n\\nd\nwith $\\widetilde{\\Delta}_2$ is a slight deformation of $\\Delta_2$ appearing from the rotation \\eqref{rotc}\nbefore the flop operation. The type IIA background also supports an effective dilaton, that measures the\nIIA coupling, and is given by:\n\\bg\\label{edil}\n\\phi_{\\rm eff}~ = ~ \\frac{3}{4}\\ln(g_2)\n\\nd\nBefore we end this section, there are a few loose ends that need to be tied up. The first one is related to the M-theory\nG-fluxes. These G-fluxes stem from \\eqref{twelve} and \\eqref{BinIIA} in type IIA,\nand they are in general large\\footnote{As we discussed before $\\epsilon$ in \\eqref{metcom} or \\eqref{consf1f2} is a small\nbut {\\it finite} number less than 1 (otherwise the signature will change), \nthe type IIA flux components \\eqref{BinIIA} and \\eqref{twelve} will be large but finite.\nChoosing a particular \nvalue of $\\epsilon$ and then demanding supersymmetry will consequently fix the resulting fluxes. A range of choices for \n$\\epsilon$ will give a class of backgrounds which are dual to certain continuous deformations parametrised \nby $\\epsilon$ in the gauge theory side. We will discuss more about the class of backgrounds later.}.\nIn the\nlocal picture both \\eqref{BinIIA} as well as \\eqref{twelve} components were all pure gauges, and therefore they\ndid not contribute to the background. Here we expect they would, and therefore we need to see how these fluxes behave\nunder:\n\n\\vskip.1in\n\n\\noindent $\\bullet$ The rotation of coordinates \\eqref{rotc} with shift \\eqref{psis}, and\n\n\\noindent $\\bullet$ The flop transformation \\eqref{lbaaz}.\n\n\\vskip.1in\n\n\\noindent Both these effects can be worked out if we can express our fluxes \\eqref{twelve} and \\eqref{BinIIA}\ncompletely in terms of the one-forms \\eqref{lift1} and \\eqref{lift2}. As we noted before, the rotation\n\\eqref{rotc} and shift \\eqref{psis} is appropriately captured by\nthe one-forms \\eqref{lift2}. Therefore to compensate both the changes, namely rotation\n\\eqref{rotc} (with shift \\eqref{psis})\nand the flop \\eqref{lbaaz}, all we need is to express the M-theory lift of the fluxes in terms of\n\\eqref{lift1} and \\eqref{lift2}. Any {\\it additional} $\\psi$ dependent contributions will come out from\nsusy requirement.\n\nTo achieve all this, we can express the differential coordinates completely in terms of $\\sigma_i$ and $\\Sigma_i$.\nSince there are seven differential coordinates ($dr, d\\theta_1, d\\phi_1, d\\theta_2, d\\phi_2, d\\psi_1, d\\psi_2$) but\nsix one-forms ($\\sigma_i, \\Sigma_i$), we can assume $dr$ goes to itself, and then the rest of the differential\nforms map to the ($\\sigma_i, \\Sigma_i$) in the following way:\n\\begin{eqnarray}\n&&d\\theta_1=\\cos\\psi_1\\sigma_1-\\sin\\psi_1\\sigma_2,\\quad\nd\\theta_2=\\cos\\psi_2\\Sigma_1-\\sin\\psi_2\\Sigma_2,\\nonumber\\\\\n&&d\\phi_1-\\tan\\lambda_4dr=\\sec\\lambda_1\\Big[\\sin(\\psi_1+\\lambda_1)\\sigma_1+\\cos(\\psi_1+\\lambda_1)\n\\sigma_2\\Big],\\nonumber\\\\\n&&d\\phi_2-\\tan\\lambda_5dr=-\\sec\\lambda_2\\Big[\\sin(\\psi_2-\\lambda_2)\\Sigma_1+\\cos(\\psi_2-\\lambda_2)\n\\Sigma_2\\Big],\\nonumber\\\\\n&&d\\psi-\\tan\\lambda_3dr=\\sigma_3-\\Sigma_3-\\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\;+\n\\Delta_2 {\\rm cos}~\\theta_2\n\\Big(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2\\Big)\\nonumber\\\\\n&&dx_{11}=\\sigma_3+\\Sigma_3+\\Delta_1 {\\rm\ncos}~\\theta_1 \\Big(\\sin\\psi_1\\sigma_1+\\cos\\psi_1\\sigma_2\\Big)\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\\quad- \\Delta_2 {\\rm cos}~\\theta_2\n\\Big(\\sin\\psi_2\\Sigma_1+\\cos\\psi_2\\Sigma_2\\Big)\n\\end{eqnarray}\nUnder the rotation, shift and flop the flux components mix in rather\nnon-trivial way.\nWe therefore expect, after the IIA reduction from M-theory, the three-form and four-form flux components of type\nIIA \\eqref{BinIIA} and \\eqref{twelve} respectively before geometric transition go to {\\it new} three- and four-form\nflux components. They can be expressed as:\n\\begin{eqnarray}\\label{BIIA2}\n{B}_{\\rm now}~=~ \\bar{b}_{ij}dx^i\\wedge dx^j, ~~~~~~~~\n{F}_{\\rm now} ~= ~ \\bar{f}_{ijkl}dx^i\\wedge dx^j\\wedge dx^k\\wedge dx^l\n\\end{eqnarray}\nwhere $x^{i,j,k,l}=r, \\theta_i, \\phi_i, \\psi$ and\n$\\bar{b}_{ij}$ and $\\bar{f}_{ijkl}$\nare not only functions of ($r, \\theta_i$) but now also of $\\psi$ because of the shift \\eqref{psis} and rotation\n\\eqref{rotc}.\n\n\n\\subsection{Type IIB after mirror transition}\n\nWith the type IIA picture at hand, we are now at the last chain of the duality transformation that will give us the\nsupergravity dual of the confining gauge theory on the wrapped D5-branes. Our starting points are now:\n\n\\vskip.1in\n\n\\noindent $\\bullet$ The type IIA metric \\eqref{IIAmetric}.\n\n\\noindent $\\bullet$ The remnant of the D6-brane charges, i.e the one-form fluxes \\eqref{1ffc}.\n\n\\noindent $\\bullet$ The type IIA string coupling, or the dilaton \\eqref{edil}, and\n\n\\noindent $\\bullet$ The $B_{\\rm NS}$ and the $F_4$ fluxes \\eqref{BIIA2}\nfrom the remnant of the IIB shift transformations.\n\n\\vskip.1in\n\n\\noindent Before moving ahead, let us make two observations. The first is\nthat now we {\\it do} have components like $\\bar{b}_{\\phi_1\\phi_2}, \\bar{b}_{\\psi\\phi_i}$ in \\eqref{BIIA2}.\nThis would mean that the\nmirror type IIB should become non-geometric! This is what one would have expected generically, and our analysis\ndoes confirm this\\footnote{In fact this tells us that the generic solution spaces we get in type IIB are non-geometric\nmanifolds. For certain choices of parameters (B-fields, and metric components) we can get geometric manifolds like\nKlebanov-Strassler \\cite{KS} or Maldacena-Nunez \\cite{MN}.\nThis is almost like the parameter space of \\cite{butti} but now much bigger,\nand allowing {\\it both} geometric and non-geometric manifolds that cover various\nbranches of the dual gauge theories. One may also note that the susy constraints from the torsion\nclass analysis in equation (3.15), (3.21) and (3.25) of \\cite{chen2} would keep some of the parameters unfixed (one example would be the $\\epsilon$ parameter that we discussed earlier). \nTherefore\nthese parameters could be {\\it varied} to get a class of gauge theory deformations. Typically the duals of\nthese deformations are non-geometric manifolds. This is expressed in figure 3 of \\cite{chen2}.}.\nObserve that locally, as in \\cite{gtpapers}, this aspect of non-geometricity was not visible because\nmost of the extra fluxes were pure gauges. In the global case, the system is rather non-trivial and the dual gravitational\ndescription may become non-geometric. Question now is whether we can look for a special case where we can study the\nsystem as a {\\it geometric} manifold. It turns out at the orientifold point there might be a situation where we can\nswitch off the extra flux components and consider only the standard B-field components. Recall that due to various\nrotations \\eqref{rotc} and shifting \\eqref{tranbg} and \\eqref{psis}\nthe orientifolding is more involved, as all the internal coordinates\nare mixed up in these transformations. However this may not generically remove all the necessary components.\nTherefore to simplify the situation, in the following, we will study the\ntype IIB mirror by first keeping:\n\\bg\\label{nihar} \\bar{b}_{\\phi_1\\phi_2} ~ = ~ \\bar{b}_{\\psi\\phi_i} ~ = ~0\\nd\nso that the mirror could be geometric. Switching on \\eqref{nihar} in the IIA scenario will then make the system\nnon-geometric.\n\n\\noindent Secondly,\nnote that except for the $B_{\\rm NS}$ and the $F_4$ fluxes, rest of the components of the metric or the\none-form fluxes, or even the dilaton are all finite. The $B_{\\rm NS}$ and the $F_4$ fluxes are large, and in the limit\n$\\epsilon$ in \\eqref{metcom} is a small but finite integer, these would also be finite (but large). To proceed further\nlet us define:\n\\bg\\label{defui}\n{\\cal D}\\phi_1 \\equiv d\\phi_1 - b_{\\phi_1\\theta_1} d\\theta_1-b_{\\phi_1 r}dr,~~~~~~\n{\\cal D}\\phi_2 \\equiv d\\phi_2 - b_{\\phi_2\\theta_2} d\\theta_2-b_{\\phi_2 r}dr\n\\nd\nUsing this, the type IIA metric can be rewritten as:\n\\begin{eqnarray}\\label{IIAmetre}\nds_{10}^2&& =F_0ds_{0,1,2,3}^2+F_1dr^2+e^{2\\phi}\\Big(d\\psi-b_{\\psi\n\\mu}dx^\\mu+\\Delta_1 {\\rm cos}~\\theta_1 ~{\\cal D}\\phi_1 +\n\\widetilde{\\Delta}_2 {\\rm cos}~\\theta_2~ {\\cal D}\\phi_2\\Big)^2\\nonumber\\\\\n&&~+ {\\cal F}_1 \\Big(d\\theta_1^2+ {\\cal D}\\phi_1^2\\Big) +\n{\\cal F}_2 \\Big(d\\theta_2^2+ {\\cal D}\\phi_2^2\\Big)\n\\end{eqnarray}\nIn this form the non-K\\\"ahlerity is obvious in terms of the fibrations ${\\cal D}\\phi_i$ and\nthe resolution parameters of the two two-cycles are determined completely in terms of ${\\cal F}_i$ as:\n\\bg\\label{calfo}\n{\\cal F}_1 = e^{2\\phi\\over 3}a^2(k^2G_2+kG_3+G_1), ~~~~\n{\\cal F}_2 = e^{2\\phi\\over 3}b^2(\\mu^2G_2+\\mu G_3+G_1)\n\\nd\nIt is now clear that to determine the type IIB mirror using SYZ \\cite{syz} we have to make the base bigger as before.\nThe manifold \\eqref{IIAmetre} still retains isometries along ($\\phi_i, \\psi$), so after we enlarge the base we can\nperform SYZ in the usual way\\footnote{This is not as simple as it sounds. As we saw above, due to rotation \\eqref{rotc}\nand shift \\eqref{psis}, the fluxes may develop dependences on $\\psi$ coordinate. A way out of this would be to\nexpress the fluxes in terms of an average $\\langle\\psi\\rangle$ and then perform the SYZ transformations. We can then\ndemand supersymmetry in the\nfinal IIB configuration by analysing the torsion classes, exactly as we did for all the above\ncases. This way all intermediate configurations would be supersymmetric.}.\nSince these details are rather straightforward to work out, we will not redo them again\nnow. To put the type IIB metric in some suggestive format, let us define the following quantities:\n\\begin{eqnarray}\\label{newthings}\n&&\\bar{\\alpha}=\\Big(\\bar{j}_{\\phi_1\\phi_1}\\bar{j}_{\\phi_2\\phi_2}-\\bar{j}_{\\phi_1\\phi_2}+\n\\bar{b}_{\\phi_1\\phi_2}^2\\Big)^{-1}, ~~~~~ \\widetilde{\\cal D}\\psi \\equiv d\\psi+g_{r\\psi}dr \\nonumber\\\\\n&& \\widetilde{\\cal D}\\phi_1 \\equiv \\sqrt{g_{\\phi_1\\phi_1}\\over g_{\\theta_1\\theta_1}}\n\\Big(d\\phi_1+g_{\\phi_1\\theta_1}d\\theta_1+g_{r\\phi_1}dr\\Big), ~~~\n\\widetilde{\\cal D}\\phi_2 \\equiv \\sqrt{g_{\\phi_2\\phi_2}\\over g_{\\theta_2\\theta_2}}\n\\Big(d\\phi_2+g_{\\phi_2\\theta_2}d\\theta_2+g_{r\\phi_2}dr\\Big)\\nonumber\\\\\n\\end{eqnarray}\nwhere\n$\\bar{j}_{\\mu\\nu}$ denote the components of the IIA metric \\eqref{IIAmetre}, and $g_{\\mu\\nu}$ are defined in terms of\n$\\bar{j}_{\\mu\\nu}$ in {\\bf Appendix 3}. Using these definitions, and taking $b_{r\\phi_i} = b_{r\\psi} = 0$,\nthe mirror manifold in type IIB theory\ntakes the following form:\n\\begin{eqnarray}\\label{iibmetfinal}\nds^2 & = & F_0^2ds_{0,1,2,3}^2+g_{rr}dr^2  +g_{\\psi\\psi}\\Big(\\widetilde{\\cal D}\\psi\n+ \\widehat{\\Delta}_1 ~\\widetilde{\\cal D}\\phi_1\n+ \\widehat{\\Delta}_2 ~\\widetilde{\\cal D}\\phi_2\\Big)^2\\\\\n&& +g_{\\theta_1\\theta_1}\\Big(d\\theta_1^2 + \\widetilde{\\cal D}\\phi_1^2\\Big)\n+g_{\\theta_2\\theta_2}\\Big(d\\theta_2^2 + \\widetilde{\\cal D}\\phi_2^2\\Big)\n+g_{\\theta_1\\theta_2}\\Big(d\\theta_1d\\theta_2 + \\widehat{\\Delta}_3\n~\\widetilde{\\cal D}\\phi_1 \\widetilde{\\cal D}\\phi_2\\Big)\\nonumber\n\\nd\nwhich looks surprisingly close to the warped resolved-deformed conifold! Clearly the manifold is non-K\\\"ahler and\n$\\hat{\\Delta}_i$ are defined as:\n\\bg\\label{hatdeldef}\n\\widehat{\\Delta}_1 \\equiv \\sqrt{g_{\\theta_1\\theta_1}g^2_{\\psi\\phi_1}\\over g_{\\phi_1\\phi_1}}, ~~~~~\n\\widehat{\\Delta}_2 \\equiv \\sqrt{g_{\\theta_2\\theta_2}g^2_{\\psi\\phi_2}\\over g_{\\phi_2\\phi_2}}, ~~~~~\n\\widehat{\\Delta}_3 \\equiv \\sqrt{g_{\\theta_1\\theta_1} g_{\\theta_2\\theta_2}g^2_{\\phi_1\\phi_2}\\over\ng_{\\phi_1\\phi_1}g_{\\phi_2\\phi_2}g^2_{\\theta_1\\theta_2}}\n\\nd\nThe type IIB fluxes are rather involved, but they could be worked out exactly as in {\\bf Appendix 2}. We will\nnot do so here, but discuss their implications in our follow-up paper \\cite{toappear}. It is interesting that the\nsolutions that we get in type IIA as well as type IIB for the gravity duals look very close to what has been\nadvocated in the literature so far in the limit where we switch off certain\ncomponents of the $\\bar{b}$-fields as well as $b_{r\\phi_i}, b_{r\\psi}$.\nOnce we keep these components then the metric \\eqref{iibmetfinal} {\\it cannot} be the global description. The global\ndescription will have to be a non-geometric manifold. In the present\npaper we will not discuss the non-geometric aspect anymore, and details on this will be presented in our upcoming\npaper \\cite{toappear}. We end this section by noting that\nthe duality cycle that we\nadvocated here (and also in \\cite{gtpapers} earlier) does\nlead to the correct gauge\/gravity dualities for the confining theories.\n\n\\section{Discussions and conclusion}\n\nIn this paper we have made some progress along two related directions. The first one is to find a supersymmetric\nconfiguration of wrapped D5-branes on a two-cycle of a resolved conifold. We find that the {\\it simplest} way to\nachieve supersymmetry here is to allow for a non-K\\\"ahler metric with an $SU(3)$ structure on the resolved conifold.\nIt is possible that once we add extra fluxes and fundamental seven-branes we can also\nget a K\\\"ahler metric on the resolved conifold that is supersymmetric in the presence of the wrapped D5-branes.\n\nThe second progress that we made here is related to the full cycle of geometric transition. It should now be clear\nthat in the limit where the gauge theories on the wrapped D5-branes\nor the D6-branes were\nstrongly coupled, the geometric transition\neffectively  boiled down to a simple series of mirror transformations \\cite{syz} followed\nin-between by a flop transition to an intermediate M-theory configuration with a $G_2$-structure. The fact that\nsuch a complicated set of gauge\/gravity dualities follow from simple sequences of T-dualities is rather \nremarkable\\footnote{An interesting question to ask at this stage is whether the set of operations (mirror symmetry\nand flop) could be described as a transformation in the space of $SU(3)$ structures, much like the one studied in \n\\cite{minabeta}. The fact that this is indeed the case for our set-up can be inferred from the torsion class\nanalysis before and after geometric transition. In the sequel \\cite{chen2} we have worked out the torsion \nclasses both before and after GT in type IIA theory. These torsion classes could be related to each other by \n$SU(3)$ transformations, and therefore our set of operations can be described as transformations in the space of \n$SU(3)$ structure.}.\n\nIn the global case the mirror transformations are more subtle than the local case.\nFor example the fluxes appearing in the mirror\npictures are very involved (we gave one example in {\\bf Appendix 2}). They are not in general pure gauges, and so\nwe need to follow them through every step of the duality cycle because they would eventally influence the gravity\nduals for the confining gauge theories on the wrapped D6- and D5-branes (at least in the far IR). In this\npaper we have managed to work out these details, and showed that despite the complicated underlining flux-structures, the\ngravity duals for the type IIA and the type IIB configurations in certain cases were\n given by non-K\\\"ahler deformations of the resolved and\nthe deformed conifold respectively. In more generic cases, that allow us to keep all components of the B-fields, the\ngravitational descriptions become non-geometric.\n\nThere are many things that need to be done now. One of the issue is to understand the non-geometric aspects of the\nmirror configurations. We have briefly alluded to this by arguing that there are B-field components (even at the\norientifold points) which could in-principle make the IIA, as well as IIB after geometric transition,\npicture non-geometric (see also the recent discussion in\n\\cite{halmagyi}). Such a non-geometric configuration for example\nin type IIA should lift to a non-geometric configuration in\nM-theory. The question then is: how should we perform the flop operation now? Even if there exist a meaningful\nway to perform the flop, how do we understand the SYZ \\cite{syz} mirror map now? How does the non-geometric\nbehavior reflect in the gauge theory side? How should we compute the exact superpotential for the gauge theory in the\nnon-geometric case? Furthermore if we allow the B-fields components $\\bar{b}_{\\phi_1\\phi_2}$ and\n$\\bar{b}_{\\psi\\phi_i}$ in \\eqref{BIIA2}, how would the metric \\eqref{iibmetfinal} change to reflect the non-geometric\naspect?\n\nClearly these are very interesting questions to resolve, and we have barely scratched the surface of the iceberg. In\nour upcoming paper \\cite{toappear} we plan to address some of these issues. Hopefully these questions\nwill have simple tractable\nsolutions. If not, then it will be even more interesting to reveal the underlying structure.\n\n\\vskip.1in\n\n\\noindent {\\bf Note added:}\nWhile we were writing this draft, \\cite{pandozayas} appeared that has some overlap with sections 4.1 and\n4.2 of this paper. See also the recent set of papers \\cite{lilia} where some new properties of non-K\\\"ahler manifolds\nhave been discussed.\n\n\n\\vskip.1in\n\n\\begin{center}\n{\\bf Acknowledgements}\n\\end{center}\n\n\\noindent We would like to thank N.~Halmagyi, M.~Larfors, D.~L\\\"ust, and D.~Tsimpis for helpful\nconversations. The works of FC, KD and PF were supported in parts by the NSERC grants.\nThe work of SK was supported by NSF grants DMS-02-44412 and\nDMS-05-55678. The work of RT was supported by a PPARC grant.\n\n\\newpage\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}