diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmane" "b/data_all_eng_slimpj/shuffled/split2/finalzzmane" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmane" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n Supermassive black holes (SMBHs) are thought to inhabit the center of every massive galaxy (e.g., \\citealp{1998AJ....115.2285M}). Throughout its lifetime, a SMBH can accrete significant amounts of matter, creating a very luminous and compact source of photons through the formation of an accretion disk \\citep{1973A&A....24..337S}. When a SMBH is undergoing this accreting phase, it is classified as an active galactic nucleus (AGN). AGN can then be subdivided based on their optical, radio, and X-ray properties. In order to explain the differences in the observational properties of accreting SMBHs, particularly in the optical and X-ray band, the unified model of AGN proposes a large scale dusty torus which sourrounds the broad line region and the accreting SMBH \\citep{1993ARA&A..31..473A,1995PASP..107..803U, 2015ARA&A..53..365N, 2017NatAs...1..679R}. The differences in AGN are then explained by the observers viewing angle with respect to the torus. When viewing an AGN through the torus, the central SMBH and accretion disk are obscured, and narrow emission lines are present. Observing an AGN perpendicular to the plane of the torus leads to an unobscured line of sight of the inner regions of the accreting system, which allows for the detection of both broad and narrow emission lines. \n\n In AGN, X-rays are produced in a small region close to the SMBH (e.g., \\citealp{2009ApJ...693..174C, 2014A&ARv..22...72U}). The accretion disk cannot be responsible for the observed X-ray radiation because the temperature of the disk is not sufficient to create X-ray photons \\citep{1973A&A....24..337S}. Instead, it is now widely accepted that optical\/UV photons from the disk are up-scattered through the process of inverse Compton scattering when interacting with heated electrons in the X--ray corona \\citep{1991ApJ...380L..51H, 1993ApJ...413..507H}. The process in which these electrons become heated is still debated and could be caused by magnetic flares (e.g., \\citealp{1994ApJ...432L..95H,2000MNRAS.318L..15M}), clumpy disks (e.g., \\citealp{1988MNRAS.233..475G}), or aborted jets \\citep{1997A&A...326...87H}. The primary component of the X-ray continuum observed in AGN can be approximated by a power-law with an exponential cutoff at $\\sim 200$ keV (e.g., \\citealp{10.1093\/mnras\/sty1879}). The slope of this power-law is determined by the photon index $\\Gamma$, which typically has a value between 1.8 and 1.9 \\citep{Nandra7469,10.1111\/j.1365-2966.2011.18207.x, 2017ApJS..233...17R}. Absorption in the X-rays is driven by photoelectric absorption and Compton scattering, the latter becoming important at column densities of $~1\/\\sigma_{\\mathrm{T}}\\approx 10^{24}$ cm$^{-2}$, where $\\sigma_{\\text{T}}$ is the Thomson cross section. AGN that are observed through column densities higher than this threshold are dubbed Compton thick (CT; e.g., \\citealp{10.1046\/j.1365-8711.2000.03721.x,10.1111\/j.1365-2966.2010.17902.x,10.1111\/j.1365-2966.2012.20908.x,2015ApJ...815L..13R, 10.1093\/mnras\/stw1764, Hikitani_2018}). Reprocessed radiation is responsible for many characteristic features in the X-ray spectra of AGN, such as the reflection hump at $\\sim 30$\\,keV and the Iron K$\\alpha$ line at 6.4\\,keV. \n \n Until recently, most of the mid-infrared (MIR) emission associated to AGN has been thought to have its origin in the torus \\citep{2008ApJ...685..147N,2012MNRAS.420.2756S}. \\citet{Jaffe2004} reported MIR observations of NGC-1068 showing a warm dusty structure on scales of parsecs which surrounded a smaller hot structure. This was a significant confirmation of the presence of a torus-like structure in an AGN. Now, recent observations from the MID-infrared Interferometric instrument (MIDI) on the Very Large Telescope (VLT) show a large portion of the MIR spectrum of AGN stemming from the polar regions on scales of tens to hundreds of parsecs (e.g., \\citealp{H_nig_2012}) and not only from the torus as previously thought (\\citealp{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220} and references therein). These observations have been made both by MIR interferometric (e.g., \\citealp{2012ApJ...755..149H, 2013A&A...558A.149B, refId03, Leftley_2019}) and single dish observations (e.g., \\citealp{Asmus_2016}). In addition to this, ALMA is also revealing polar outflows on the same spatial scales (e.g., \\citealp{2016ApJ...829L...7G}) which suggests the torus may be an obscuring outflow. This is in contrast with the traditional unified model, which does not predict this elongated polar MIR emission. This polar gas (or component) has a significant effect on the MIR SED, but it is likely optically thin, otherwise the X-ray emission would be obscured for type 1 AGN \\citep{Liu2019XraySO}. The origin of this polar component is still unknown. A plausible explanation is that this polar gas is the result of radiation pressure on dust grains from strong UV emission in the polar region from the accretion disk causing a dusty wind \\citep{Ricci2017, Leftley_2019, 2019ApJ...884..171H, 2020ApJ...900..174V}, a scenario also consistent with hydrodynamical simulations (e.g., \\citealp{2012ApJ...758...66W}). \n\n The two nearest sources in which this polar component has been confirmed by interferometric studies are NGC-1068 \\citep{refId01} and the Circinus galaxy \\citep{refId02}, although such elongated emission has also been observed in several other sources (e.g., \\citealp{refId03, 10.1117\/12.2231077}). There has been extensive literature studying the effect of this polar gas on the MIR spectrum, as well as the effect of different distributions of the polar gas itself (e.g., \\citealp{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220, 10.1093\/mnras\/stz2289}). These studies concluded that the best model for the distribution of this polar gas to fit the MIR spectrum is a hollow cone or a hyperbolic cone on large scales, accompanied by a disk component on smaller scales. The hollow nature of these geometries is naturally consistent with the absence of obscuring material along the line of sight to Type 1 AGNs (e.g., \\citealp{Ricci2017, 2017ApJ...850...74K}). \\cite{Liu2019XraySO} studied the effect of polar gas on the X-ray spectrum of AGN by considering an X-ray source surrounded by an equatorial disk and a hollow cone perpendicular to the plane of the disk. \\cite{Liu2019XraySO} found that the polar gas has a significant effect on the X-ray spectrum of AGN, and argued that the scattered fluorescence line features can act as a potential probe for the kinematics of the polar gas.\n\n So far, most studies of reprocessed X-ray radiation in AGN have been carried out assuming a torus \/ disk and no polar component. Thus, adding this polar component in the study of scattered X-ray emission can provide an important look into the kinematics of the polar gas and thus its origin. Since the polar gas is likely optically thin, it would only affect the soft (0.3--5\\,keV) part of the X-ray spectrum and not the hard portion ($>5$\\,keV). In this paper, we use the ray-tracing simulation software \\textsc{RefleX}\\footnote{\\url{https:\/\/www.astro.unige.ch\/reflex\/}} \\citep{RefleX} to simulate the spectra from an AGN with a torus accompanied by a large scale polar filled and hollow cone corresponding to the best fit model of the MIR data. We study the effect of the polar gas on the X-ray spectrum considering multiple inclination angles and column densities.\n \n \\begin{figure*}\n \\centering\n \n \\includegraphics[scale = 0.14]{TorusD.png}\n \\includegraphics[scale = 0.14]{ConeD.png}\n \\includegraphics[scale = 0.14]{HConeD.png}\n \\caption{Cross-sections for the three geometries considered. (Left panel) Geometry for torus with radius $r = 1.235$\\,pc a distance $R = 1.756$\\,pc from the X-ray source. (Middle panel) Geometry for the torus + filled cone. (Right panel) Geometry for the torus + hollow cone. The cones have an opening angle of 30$^{\\circ}$ with slant lengths of 40\\,pc. The hollow cone has a small angular width of $1^{\\circ}$ corresponding to the best fit MIR data for the galaxy Circinus.}\n \\label{fig1}\n \\end{figure*}\n\n This paper is laid out as follows. In $\\S$2 we give an overview of the parameters and models of our simulations. Then we discuss the results of our simulations in $\\S3$. Next, in $\\S$4 we lay out the equivalent widths of the spectral lines to be compared to current and future observations from X-ray missions as well as consider observational differences in the resulting spectra of our simulations. We also add many components seen from the Narrow Line Region (NLR) such as a scattered power-law and many photoionised lines seen in obscured AGNs. $\\S5$ investigates the difference in the spectra when the opening angle and abundances of the hollow cone is changed. Finally, in $\\S$6 we summarize our findings and present our conclusions. \n\n\\section{Simulation Setup}\n\n \\subsection{\\textsc{RefleX}: a ray-tracing simulation platform}\n \n In this paper we use the ray-tracing simulation platform \\textsc{RefleX} \\citep{RefleX} in order to study the emission from this new polar region. \\textsc{RefleX} is not the first ray-tracing code to be used in studying reprocessed radiation from AGN. Indeed some of the first ray-tracing simulation platforms date to the early 1990s with Monte Carlo methods (e.g., \\citealp{1991MNRAS.249..352G}). Monte Carlo simulations became instrumental in simulating the X-ray spectra of heavily obscured AGN. They were used, for example, to calculate the equivalent width of the Iron K$\\alpha$ complex produced in neutral matter considering various column densities and Iron abundances \\citep{2002MNRAS.337..147M}. Newer models, such as \\textsc{MYTorus} \\citep{10.1111\/j.1365-2966.2009.15025.x}, consider a toroidal geometry with arbitrary column densities, while codes such as \\cite{2011MNRAS.413.1206B} and \\cite{2018ApJ...854...42B} also include varying covering factors. Clumpy torus models have also been developed recently (e.g., \\citealp{2014ApJ...787...52L, 2019A&A...629A..16B}). However, most of these models limit the users input in the allowed geometries. Alternatively, \\textsc{RefleX} allows the user to specify an X-ray source with different geometries and various spectral shapes in the 0.1\\,keV--1\\,MeV energy range. After the geometry and spectral shapes are determined, \\textsc{RefleX} allows the user to build quasi-arbitrary gas distributions around the X-ray source, with user specified densities. When the simulation is run, \\textsc{RefleX} tracks the life of every X-ray photon as it is propagated though the distribution of gas. The photons can undergo several physical processes when propagating through the gas, such as fluorescence, Compton scattering, and Rayleigh scattering. Once a photon escapes the gas and reaches the detector it is collected in energy bins in terms of flux (keV\\,cm$^{-2}$\\,s$^{-1}$\\,keV$^{-1}$), counts (integer number of photons), or photons (cm$^{-2}$\\,s$^{-1}$\\,keV$^{-1}$), depending on the specifications of the user.\n \n \\textsc{RefleX} also allows the user to specify the composition of the material surrounding the X-ray source. This includes setting the gas metallicity, the fraction of Hydrogen in molecular form (H$_{2}$ fraction), as well as the composition of the material which can be set, for example, to follow what was reported by \\cite{1989GeCoA..53..197A} or \\cite{2003ApJ...591.1220L}. All simulations in $\\S$3 are run using the gas composition from \\cite{1989GeCoA..53..197A}, solar metallicity, and a H$_{2}$ fraction of 1. In $\\S$5 we will consider the composition from \\cite{2003ApJ...591.1220L}, as well as an H$_{2}$ fraction of 0.2, showing that no significant difference is expected in the features arising from the polar gas. \n \n \\subsection{Geometries}\n \\subsubsection{Torus model}\n \n In order to study the effect polar gas has on the X-ray spectrum of AGN, we implement three simulation setups. The first is a torus with a Compton thick equatorial column density of $N_{\\rm H,\\, eq} =10^{24.5}$\\,cm$^{-2}$ surrounding an isotropic X-ray source emitting a power-law distribution of photons with an exponential cutoff at 200\\,keV and a photon index of $\\Gamma = 1.9$. This is equivalent to an AGN with no polar component and thus will provide a comparison to our later simulations with the polar component. The distance and size of the torus is characterized by two parameters in \\textsc{RefleX}: $R$ and $r$ as seen on the left panel of Figure\\,\\ref{fig1}. The values of $R$ and $r$ are kept consistent in all the simulations and have values of $R = 1.756$\\,pc and $r = 1.235$\\,pc, corresponding to the best fit model of the Circinus galaxy (Andoine et al. submitted), leading to a covering fraction of $CF = 0.7$. The gas in the torus and all other geometries is smooth. For the torus (as well as all other geometries), we selected the spectra from four inclination angle ranges: $0^{\\circ} \\leq i \\leq 5^{\\circ}$ ($N_{\\text{H}} = 0$\\,cm$^{-2}$), $45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$ ($N_{\\text{H}} = 10^{22.49}$\\,cm$^{-2}$), and $85^{\\circ} \\leq i \\leq 90^{\\circ}$ ($N_{\\text{H}} \\approx 10^{24.50}$\\,cm$^{-2}$) where $i$ is the angle measured from the normal to the plane of the torus. The column densities are calculated from the center of the range. The range $45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$ will be referred to as the intermediate case and the ranges $0^{\\circ} \\leq i \\leq 5^{\\circ}$ and $85^{\\circ} \\leq i \\leq 90^{\\circ}$ will be referred to as the pole-on and edge-on cases respectively. We also ran simulations at an inclination range of $60^{\\circ} \\leq i \\leq 65^{\\circ}$, but with no significant change from the $45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$ case. We note in our simulations we use a linear resolution of 5\\,eV for $E \\leq 10$\\,keV and a logarithmic resolution of 0.001\\,eV for $E > 10$\\,keV \n \n \\begin{figure} \n \\centering\n \\includegraphics[scale = 0.55]{Tor8560450ScattContBRUH.png}\n \\caption{Simulated 0.3$-$200\\,keV spectrum obtained considering a torus with an equatorial column density $N_{\\text{H, eq}} = 10^{24.5}$\\,cm$^{-2}$ at different inclination angles. All the fluxes are normalized as discussed in section 3.1. The edge-on case (black line) has a high photoelectric cutoff due to the strong absorption from the torus. The intermediate case (blue line) displays many fluorescence lines in the soft portion of the spectrum due to the decreased column density. The pole-on inclination angle just shows strong continuum due to the complete lack of line of sight absorbing material along with the Fe K$\\alpha$ complex.}\n \\label{fig2}\n \\end{figure}\n \n \\begin{figure*}\n \\ContinuedFloat*\n \\includegraphics[scale = 0.384]{004.png}\n \\includegraphics[scale = 0.384]{005.png}\n \\includegraphics[scale = 0.384]{006.png}\n \\caption{\\label{fig3a}Simulated spectra for a torus + hollow cone geometry (shown in blue) along with the torus only simulations (shown in black). Moving from the left panel to the right panel, inclination angle decreases as the slant column density of the hollow cone remains constant ($\\log{N_{\\text{H}}}\\big\/\\text{cm}^{-2} = 21$). The torus only simulations shown are linearly shifted to higher energies by 0.02\\,keV to better view the differences between spectral lines in the soft spectrum. The largest difference between the torus + hollow cone simulations and the previous torus simulations comes at the edge-on case where the previous torus simulations contain a photoelectric cutoff (left panel). The intermediate case (middle panel) shows little deviation from the torus only case. This is also the case for the pole-on case (right panel) due to the continuum domination.}\n \\end{figure*}\n \\begin{figure*}\n \\ContinuedFloat\n \\includegraphics[scale = 0.384]{003.png}\n \\includegraphics[scale = 0.384]{002.png}\n \\includegraphics[scale = 0.384]{001.png}\n \\caption{\\label{fig3b}Same as top, however, moving from the left panel to the right panel represents simulations with constant inclination angle ($85^{\\circ} \\leq i \\leq 90^{\\circ}$) and increasing slant column densities for the polar component. The black line again represents the previous torus only simulation for the edge-on case. As the density of the polar component increases, the flux when compared with the torus only simulations increase as there is now more gas for photons to interact with.} \n \\end{figure*}\n \n \\subsubsection{Torus + Hollow\/Filled Cone Model}\n \n The geometry for the torus along with the hollow cone can be seen on the right panel of Figure\\,\\ref{fig1}. The slant length of the cone was chosen to be 40\\,pc with an opening angle of $\\alpha = 30^{\\circ}$ and small angular width of 1$^{\\circ}$, consistent with the best MIR model for the Circinus galaxy \\citep{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220}. The same three inclination angles are considered as in the torus-only model. However, to investigate the effect of the density of the polar gas on the X-ray spectrum, four slant column densities\\footnote{the column density as seen looking along the slant length of the cone} are considered for the hollow cone at each inclination angle. These are $\\log{(N_{\\rm H}\\big \/\\text{\\,cm}^{-2})} = 21, 21.5, 22,$ and $22.5$. The cone starts at a height of 0.1\\,pc above the X-ray source, corresponding to the sublimation radius of dust in AGN (e.g., \\citealp{2007A&A...476..713K}). The setup for the simulations including the torus along with the filled cone can be seen in the middle panel of Figure\\,\\ref{fig1}. The filled cone is equivalent to the hollow cone except for the fact that the outer wall of the hollow cone is removed and the inner portion of the cone is filled with gas. The same three inclination angles and column densities are considered for both the hollow and filled cone. It should be noted that it is unlikely that the polar gas has such a filled geometry in AGN. Otherwise, we would expect a large fraction of type 1 AGN spectra to show obscured X-ray spectral features, which is not the case \\citep{Ricci2017, 2017ApJ...850...74K}. The material in these cones is also thought to be dusty (i.e., \\citealp{10.1093\/mnras\/stx2227, 10.1093\/mnras\/stz220}) and thus must have a low level of ionisation. Therefore, we assume neutral material for the cones. \n\n \\begin{figure*}\n \\centering\n \\includegraphics[scale = 0.35]{images.png}\n \\caption{Images of the torus + hollow cone simulations for the edge-on case at the highest polar gas column density ($85^{\\circ} \\leq i \\leq 90^{\\circ}$, $\\log{N_{\\text{H}}}\\big\/\\text{cm}^{-2} = 22.5$). The colour bars represent the number of total interactions (scattering + fluorescence) undergone in that particular pixel. (Left panel) Image showing the most interactions in the 0.3--4\\,keV range takes place is in the polar component. (Middle panel) 5--6\\,keV range which is mostly devoid of interactions due to the fact that this range is dominated by continuum. (Right panel) 6.3--6.5\\,keV range which is dominated by the torus due to the production of the Iron K$\\alpha$ line at $\\sim6.4$\\,keV.}\n \\label{fig4}\n \\end{figure*}\n \n \\begin{figure}\n \\centering\n \\includegraphics[scale = 0.55]{nophotons.png}\n \\caption{Simulation showing the result of increasing the equatorial column density of the torus to $N_{\\text{H, eq}} = 10^{25}$\\,cm$^{-2}$ for the edge-on inclination range and the highest polar gas column density. The torus only component (black) shows very poor photon transmission due to the increased column density allowing most of the emission to stem from the polar component in the total spectrum (purple).}\n \\label{fig4.5}\n \\end{figure}\n \n\\section{Results}\n\n \\subsection{Torus}\n \n First we present the results for the torus only geometry as to provide a comparison for our later simulations with the polar component. The results for the torus simulations at each inclination angle are shown in Figure\\,\\ref{fig2}. The the edge-on case ($85^{\\circ} \\leq i \\leq 90^{\\circ}$) is characterized by a high photoelectric absorption due to the large column density in the torus and a more prominent Iron K$\\alpha$ line at $\\sim$6.4\\,keV which is also seen at all other inclination ranges. The Compton shoulder is also visible at $\\sim$ 6.2--6.4\\,keV. The simulation run at the intermediate inclination angle ($45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$) has a much lower column density ($N_{\\rm{H}} = 10^{22.49}$\\,cm$^{-2}$) which allows for the detection of more fluorescence lines in the soft portion of the spectrum, due to the decreased importance of photoelectric absorption. The fluorescence lines implemented in \\textsc{RefleX} can be found in Appendix A of \\cite{RefleX}. The hard portion of the spectrum in all simulations shows a reflection hump peaking $\\sim$ 20--30 keV. The pole-on case shows no fluorescence lines in the soft band as the line of sight is no longer obscured by the torus and the spectral lines are washed out by the dominating X-ray continuum.\n \n \\subsection{Torus + Hollow Cone}\n \n The results for all the torus + hollow cone simulations for the different inclination angles and polar gas column densities are shown in Figures\\,\\ref{fig3a} and \\ref{fig3b}. Moving from left to right, Figure\\,\\ref{fig3a} shows how the X-ray spectra are affected by the polar gas as different inclination angles are considered while keeping column density constant (in this case, the column density shown is $\\log{N_{\\mathrm{H}}}\\big\/\\text{cm}^{-2} = 21$). Figure\\,\\ref{fig3b} shows how the X-ray spectra are affected by the polar gas as different slant column densities are considered while keeping inclination angle constant. The inclination angle shown in Figure\\,\\ref{fig3b} is $85^{\\circ} \\leq i \\leq 90^{\\circ}$. The black spectra shown in each panel shows the torus only simulation at the same inclination angle. This gives the ability to clearly distinguish which parts of the X-ray spectrum are most affected by adding a polar component to the simulations. The simulations with a hollow cone clearly show the most deviations from the torus at the edge-on case in the 0.3--4\\,keV portion of the spectrum at the edge-on inclination range. The appearance of the polar gas allows many strong fluorescence line features to dominate the 0.3--4\\,keV range due to its optically thin nature. The strongest of these lines being O, Ne, Mg, and Si, with the O line dominating in equivalent width when the polar gas is at its highest column density and the largest inclination angle ($\\log{N_{\\mathrm{H}}}\\big\/\\text{cm}^{-2} = 22.5$, $85^{\\circ} \\leq i \\leq 90^{\\circ}$; see Table \\ref{table1}). The flux of these fluorescence lines increase with the density of the polar gas. However, for the intermediate and pole-on cases, little deviation from the torus model occurs. The deviations that do occur in the intermediate case only appears in the very soft (0.3--1.2\\,keV) portion of the spectrum.\n \n In addition to these simulations, we also considered how the new polar component would be effected if the torus considered had a much higher equatorial column density, such as $N_{\\text{H, eq}} = 10^{25}$\\,cm$^{-2}$. The results of this for the edge-on case can be seen in Figure\\,\\ref{fig4.5} with the torus component shown in black and the torus + hollow cone shown in purple. In this case the X-ray emission is much weaker, due to the higher column density. This allows us to attribute most of the scattered radiation below $\\sim$ 6\\,keV to the polar component.\n\n \\subsubsection{Imaging the system}\n \n To better understand the morphology of the torus + hollow cone simulations, we took advantage of a feature of \\textsc{RefleX} which allows the user to create simulated images. The user inputs the location of a detector, its aperture size, field of view, number of desired pixels, and creates an image. The user can also select the energy range of the photons which are collected. Using this feature, we created three images of the simulations of a torus + hollow cone at the largest inclination angle and highest polar gas density [$85^{\\circ} \\leq i \\leq 90^{\\circ}$, $\\log{(N_{\\text{H}}\\big \/\\text{\\,cm$^{-2}$}}) = 22.5$]. We selected three energy ranges to create the images: 0.3--4\\,keV, 5--6\\,keV, and 6.3--6.5\\,keV. In addition to this, we isolated only photons which undergo at least one interaction (scattering or fluorescence) in order to remove the direct emission from the central source. These three images are shown in Figure\\,\\ref{fig4}. The left panel of Figure\\,\\ref{fig4} shows a large amount of interactions due to the fluorescent line features in the soft X-rays occurring in the polar region as expected due to the low density nature of the polar gas. The middle panel shows the range from 5--6\\,keV which contains less interactions than the other ranges due to the fact that this range is dominated by the scattered continuum and no strong fluorescent line is present. In this range there is emission from both regions, however, more interactions occur in the torus due to its higher density. Finally, the right panel shows the 6.3--6.5\\,keV range. This range contains fluorescent emission such as the Iron K$\\alpha$ line, and a dominating torus component due to the fact that the hollow cone is not dense enough to scatter those high energy photons. \n \n \n \\begin{figure}\n \\centering\n \\includegraphics[scale = 0.55]{Cone.png}\n \\caption{This figure shows the pole-on spectra for the torus + filled cone for the four slant column densities considered. All the column densities show absorption in the soft portion of the spectrum which shows that this geometry for the polar component is unlikely, as Type 1 AGN spectra would be obscured if this was the true geometry of the polar component.}\n \\label{figCone}\n \\end{figure}\n \n \\subsection{Torus + Filled Cone}\n \n When considering the geometry of a torus + filled cone, the spectra show many similarities to the simulations with the torus + hollow cone. These similarities are a large increase in flux and fluorescence line features in the soft portion of the spectrum while the hard portion of the spectrum remains the same as the simulations for a torus-only geometry, with a large reflection hump, Fe K$\\alpha$ line, and continuum domination. In addition to this, the intermediate case shows little deviation in flux from the torus-only case. However, the filled cone deviates from the hollow cone in a few important ways. First, the flux increase in the soft portion of the spectrum is not as large as in the hollow cone case. This is due to the fact that in this geometry it is more likely for the photons to be absorbed as there is more gas to interact with. Also, Figure\\,\\ref{figCone} shows at the pole-on case, the filled cone does not show only continuum, as the X-ray source is obscured by the polar gas, unlike the pole-on case for the hollow cone. This allows for some line features to appear in the soft spectrum for the pole-on case. It was noted earlier that this geometry is unlikely for the polar component as many Type 1 AGN spectra would be obscured. Simulations were also performed considering the same total number of atoms in the filled cone as in the hollow cone and showed that the spectra still showed a clear photoelectric cutoff at low energies, giving further evidence that this geometry is unlikely for the polar component. \n \n \\begin{figure}\n \\centering\n \\includegraphics[scale = 0.55]{EWvNH85HIGH.png}\n \\includegraphics[scale = 0.55]{EWvNH85LOW.png}\n \\caption{Equivalent widths of spectral lines for a torus + hollow cone as a function of polar gas column density for the edge-on simulations ($85^{\\circ} \\leq i \\leq 90^{\\circ}$). (Top panel) EWs for the Si, S, Ar, and Ca lines as a function of polar gas column density. (Bottom panel) Same as top but for the O, F, Ne, and Mg lines. These plots show a clear energy dependence with the EWs of lines of lower energy (O, F, Ne, Mg, Si) increasing while lines with higher energy (S, Ar, Mg) decrease in EW due to the increasing continuum at higher energies. The equivalent widths for all simulations are reported in Tables \\ref{table1}-\\ref{table6} in Appendix A.}\n \\label{fig6}\n \\end{figure} \n \n \\section{Detecting Polar Gas in Obscured AGN}\n \n \\subsection{Equivalent Width of Spectral Lines}\n \n We discuss here the equivalent widths of the strongest spectral lines in the 0.3$-$4\\,keV range for the torus + hollow cone simulations (see Figures\\,\\ref{fig3a} and \\ref{fig3b}). The simulations for the pole-on case ($0^{\\circ} \\leq i \\leq 5^{\\circ}$) were ignored since the continuum is not suppressed and thus contrast between the lines and the continuum is not enough for detection. Therefore, these equivalent width measurements are achievable only in an absorbed AGN. The equivalent widths for these simulations are reported in Tables \\ref{table1}-\\ref{table6} in Appendix A. We also note that, when considering a galactic absorption of 10$^{21}$\\,cm$^{-2}$, no significant difference in the lines occurred. However, significant deviations in ISM abundances may cause challenges in attributing these EWs to any particular geometry. \n \n A plot of the equivalent widths (EWs) as a function of polar gas density of the chosen spectral lines for a torus + hollow cone for the edge-on case is shown in Figure\\,\\ref{fig6}. The O, F, Ne, Mg, and Si lines all show an increase in equivalent widths as the column density of the polar gas increases, with the O line increasing by the largest amount ($\\Delta\\text{EW}=$ 437\\,eV) followed by the Ne ($\\Delta\\text{EW}=$ 158\\,eV) and Florine ($\\Delta\\text{EW}=$ 82\\,eV) lines. This increase is due to the fact that, as the density of the polar gas increases, there is more material for which to scatter the soft X-rays and create larger fluorescence lines. An interesting feature seen in this plot is that the equivalent widths of the S, Ar and Ca lines decrease with the polar gas column density. To investigate this further, we noted that the flux of the continuum around these lines increases at a larger rate than the line fluxes themselves as we increase the column density. This explains the behaviour of those two lines, since the equivalent width is proportional to the ratio between the flux of the line and that of the continuum. \n\n \\subsection{Observational Differences in Simulations}\n \n \\begin{figure}\n \\centering\n \\includegraphics[scale = 0.55]{Torus60HCone85.png}\n \\caption{Flux in the soft X-ray band for the edge-on case of a torus + hollow cone (purple) with the torus-only (black) intermediate case ($45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$). This shows the comparison of the slopes of the power-laws in the soft X-ray bands for these two geometries and inclinations with the intermediate torus only case having a power-law slope of $\\Gamma = -1.63$ and the edge-on torus + hollow cone having a slope of $\\Gamma = -0.8$.}\n \\label{fig8.5}\n \\end{figure}\n \n \\begin{figure}\n \\centering\n \\includegraphics[scale = 0.365]{MyModel4.pdf}\n \\caption{Plot showing the additional components which will most likely be observed stemming from the NLR added to the edge-on spectrum for a torus + hollow cone at the highest polar gas column density considered (original simulation shown as the dash-dotted black line). This includes a power-law model with 0.1\\% of the flux of the primary continuum (solid black line), as well as many photoionized lines seen in the NLR such as the ones from the soft X-ray spectrum of NGC 3393 (see Table 4 in \\citealp{2006A&A...448..499B}), as well as the Circinus galaxy (see Tables 1, A1 of \\citealp{2001ApJ...546L..13S, 2014ApJ...791...81A}, Andonie et al. submitted, respectively). These photoionized lines are plotted as dashed lines in the figure and are labeled with their transitions.}\n \n \\label{fig10.5}\n \\end{figure}\n \n \\begin{figure}\n \\centering\n \\includegraphics[scale = 0.42]{res5.pdf}\n \\caption{(Top panel) Spectrum with a torus + hollow cone for the edge-on case ($85^{\\circ} \\leq i \\leq 90^{\\circ}$) with the highest polar gas column density ($\\log{N_{\\mathrm{H}}}\\big\/\\text{cm}^{-2}$ = 22.5) with \\textit{XRISM} \\citep{xrismscienceteam2020science} response and background files. (Bottom panel) Same as top but with \\textit{Athena} \\citep{2016SPIE.9905E..2FB} response and background files (Barret et al. 2021 in prep.). For both simulations, we used an exposure time of 200\\,ks and considered a $2-10$\\,keV flux of $F_{2-10} = 12.7\\times10^{-12}$\\,erg\\,s$^{-1}$\\,cm$^{-2}$, as in the Circinus galaxy.}\n \\label{figres}\n \\end{figure}\n \n \\begin{figure*}\n \\centering\n \\includegraphics[scale = 0.55]{10v30.png}\n \\includegraphics[scale = 0.55]{LvAG85.png}\n \\caption{(Left panel) Resulting spectra when considering an opening angle of $\\alpha = 10^{\\circ}$ for an edge-on torus + hollow cone at the highest polar gas column density (shown in black). The soft spectra (0.3$-$4\\,keV) for this new opening angle is best modeled with a power-law with a photon index of $\\Gamma = -0.71$ while the $\\alpha = 30^{\\circ}$ case (shown in red) is best modeled with a photon index of $\\Gamma = -0.80$. The EWs for the $\\alpha = 10^{\\circ}$ case are shown in Table \\ref{table7}. (Right panel) Comparison of simulations with a torus + hollow cone at the edge-on inclination with polar gas column density of $N_{\\text{H}} = 10^{22.5}$\\,cm$^{-2}$ using matter compositions from \\citeauthor{1989GeCoA..53..197A} (\\citeyear{1989GeCoA..53..197A}; red) and \\citeauthor{2003ApJ...591.1220L} (\\citeyear{2003ApJ...591.1220L}; black) with H$_{2}$ fractions of 1 and 0.2 respectively. The simulations obtained by considering an opening angle of $\\alpha = 30^{\\circ}$, as well as a matter composition from \\protect\\cite{2003ApJ...591.1220L} are shifted to the right by 4$\\%$ of the energy in order to clearly see the difference in the spectral lines. There is little spectral difference either in the lines or continuum between the two compositions.}\n \\label{fig10}\n \\end{figure*}\n \n Looking at the simulations in the soft X-ray band for the torus with a hollow cone gives rise to a question: would an observer be able to distinguish between the edge-on case with a torus + hollow cone at the highest polar gas column density ($N_{\\text{H}} = 10^{22.5}$\\,cm$^{-2}$) against the intermediate case ($45.6^{\\circ} \\leq i \\leq 50.6^{\\circ}$) with a torus alone (blue line in Figure\\,\\ref{fig2})? Both cases show strong fluorescence lines in the soft X-ray band along with very similar features in the hard X-ray band.\n \n In order to distinguish between the spectra obtained by these two geometries, we fit the slope of the continuum in the soft X-ray band (0.3$-$3\\,keV) with simple power-laws. When fitting the spectra, fluorescence lines were not masked when computing the slope, since most instruments cannot mask these lines. The torus alone was best fit with a photon index of $\\Gamma = -1.63$. However, the power-law fit for the edge-on case with a torus + hollow cone gave a photon index of $\\Gamma = -0.80$. A plot of the soft spectrum for both simulations considered is shown in Figure\\,\\ref{fig8.5}. Another approach to distinguish between these two spectra is compare the difference in equivalent widths of the strongest spectral lines in the soft X-ray band for intermediate case with a torus alone and the edge-on case with a torus + hollow cone (the soft spectra for both these cases are shown in Figure\\,\\ref{fig8.5}). The equivalent widths of the O, Ne, and Si lines were compared for the two geometries. The equivalent widths of these lines are 569, 171, and 173\\,eV respectively for the torus alone and 944, 321, and 203\\,eV respectively for the torus + hollow cone. All these lines are stronger in the torus + hollow cone scenario with the O line showing the strongest difference, increasing by 375\\,eV, compared to the torus-only case.\n \n However, It should be noted that this X-ray emission from the polar gas might be combined with other features, such as those stemming from the NLR on scales of hundreds of parsecs (e,g., \\citealp{2006A&A...448..499B, 2007MNRAS.374.1290G}). These contributions include an extra scattered power-law component, as well as the addition of many narrow photoionized lines. The contribution from the NLR to the scattered flux is typically $\\sim$0.1--1\\% \\citep{Ueda_2007,2017ApJS..233...17R, 2021MNRAS.504..428G}, with the lowest values typically associated to the most obscured objects \\citep{2021MNRAS.504..428G}. In Figure\\,\\ref{fig10.5} we add, to the edge-on simulation with a torus + hollow cone at the highest polar gas column density, a power-law component to reproduce a 0.1\\% scattered fraction of the intrinsic flux of Circinus, as well as photoionized lines from the obscured AGN NGC 3393 \\citep{2006A&A...448..499B} and the Circinus galaxy (\\citealp{2001ApJ...546L..13S, 2014ApJ...791...81A}, Andonie et al. submitted), in order to verify how visible the lines in the soft X-ray band will be. The normalisation of these lines were set as to keep the ratio of the flux of line to the power-law the same as in their original source. The value of the extra scattered fraction was selected in order to be consistent with an AGN obscured by CT column densities \\citep{2021MNRAS.504..428G}. The resulting spectrum is shown in blue, with the original simulation in dash-dotted black, the scattered power-law as solid black, and the extra photoionized lines dashed and labeled with their transitions. These photoionized lines were chosen from Tables 1, 4, A1 of \\citealp{2001ApJ...546L..13S, 2014ApJ...791...81A}, Andonie et al. submitted, as well as Table 4 in \\cite{2006A&A...448..499B} based on their proximity to our simulated lines. These new photoionized lines can be distinguished from the O, Ne, and Mg lines. However, the Mg XII - Si II-VI, S II-X, and Ar II-XI transitions have the potential to interfere with the detection of the Si, S, and Ar lines from the polar component. But if the bulk motion of the polar component is blue-shifted enough, these lines could still be distinguished. With this in mind, we calculated the EWs of the O, Ne, Mg and Si (assuming no interference from the photoionized lines) lines which had values of 7\\,eV, 12\\,eV, 33\\,eV, and 87\\,eV respectively. \n \n Additionally, X-ray emission in obscured AGN below $\\sim$ 2\\,keV caused by populations of X-ray binaries and collisionally ionised plazma in star forming regions (e.g., \\citealp{2008MNRAS.386.1464R}) could also pose a challenge in detecting the true emission from the polar component. To consider this additional contribution, we added to the model shown in Figure\\,\\ref{fig10.5} a thermal plasma model with a temperature of 0.5\\,keV which corresponds to the median temperature in nearby obscured AGN \\citep{2017ApJS..233...17R}. We found the addition of this component should not add additional difficulty in detecting the spectral lines from the polar component in obscured AGN.\n \n \\subsection{Observations with \\textit{XRISM} and \\textit{Athena}}\n \n Here we consider how our previous simulations of the polar component would be observed with instruments on board \\textit{XRISM} (Resolve; \\citealp{xrismscienceteam2020science}) and \\textit{Athena} (X-IFU; \\citealp{2016SPIE.9905E..2FB}). \\textit{XRISM}\/Resolve boasts a $\\sim 5-7$\\,eV FWHM spectral resolution over the entire bandpass, while \\textit{Athena}\/X-IFU in expected to operate under a $\\sim 2.5$\\,eV spectral resolution up to 7\\,keV. Based on results of our spectral simulations, these instruments could be extremely well suited to identify the X-ray signature of the polar component. We use the X-ray spectral-fitting program software \\textsc{XSPEC} \\citep{1996ASPC..101...17A} along with the latest \\textit{XRISM}\/Resolve and \\textit{Athena}\/X-IFU (Barret et al. 2021 in prep.) response and background files to simulate the spectra that would be expected from these two instruments. The spectral simulations are illustrated in Figure\\,\\ref{figres}, where we consider the edge-on torus + hollow cone geometry, assuming the highest polar gas column density ($\\log{N_{\\mathrm{H}}\\big\/\\text{cm}^{-2}} = 22.5$; left panel of Figure\\,\\ref{fig3b}), along with the components discussed in section 4.2 (power-law with 0.1\\% of the scattered flux with various photoionized lines from the NLR). We assumed an exposure time of 200\\,ks with an observed flux in the $2-10$\\,keV range of $F_{2-10} = 12.7\\times10^{-12}$\\,erg s$^{-1}$ cm$^{-2}$, as expected for the Circinus galaxy \\citep{2017ApJS..233...17R}. The top panel displays the \\textit{XRISM}\/Resolve response while the bottom shows the \\textit{Athena}\/X-IFU response. Based on this figure, \\textit{Athena}\/X-IFU will be better suited to detect lines from the polar component, since \\textit{XRISM}\/Resolve is unable to distinguish between some of the lines from the polar component and and photoionized lines from the NLR. However, \\textit{XRISM}\/Resolve is still able to resolve the Ca line at $\\sim$ 3.69\\,keV. \\textit{Athena}\/X-IFU is able to clearly distinguish the O, Ne, Mg, and Ca lines from the NLR lines. It should be stressed that, if this polar component in caused by out-flowing gas \\citep{Ricci2017, Leftley_2019, 2019ApJ...884..171H, 2020ApJ...900..174V}, it is likely that these lines will be shifted to different energies. We expect, given the resolutions of \\textit{XRISM}\/Resolve and \\textit{Athena}\/X-IFU, to be able to recover velocity shifts of $\\sim$ 1000\\,km\/s and 375\\,km\/s for the two instruments respectively for the O, F, Ne, Mg, and Si lines. \n \n \\section{Testing Different Cone Geometries and Abundances}\n \n \\subsection{Changing the Opening Angle}\n \n It is possible that the opening angle of the polar component could be controlled through collimation by the absorbing material surrounding the BLR and acccretion disk (i.e., the torus), as is the case for the NLR \\citep{2013ApJS..209....1F}. With this in mind, we considered a new opening angle of $\\alpha = 10^{\\circ}$ to simulate a nucleus with higher collimation of ionizing radiation which corresponds to a torus with a larger covering factor. The resulting spectra compared with the previous opening angle of $\\alpha = 30^{\\circ}$ for an edge-on torus + hollow cone at the highest column density can be seen on the left panel of in Figure\\,\\ref{fig10}. Fitting the soft spectra with a simple power-law for the $\\alpha = 10^{\\circ}$ case gives a photon index of $\\Gamma = -0.71$ compared to an index of $\\Gamma = -0.80$ for the $\\alpha = 30^{\\circ}$ case. Table \\ref{table7} shows all the EWs of the lines in the soft spectra for this new opening angle for different densities of the polar gas as well. We note that, while the O line has a higher EW for the $\\alpha = 30^{\\circ}$ case (an average ratio of 1.29 when averaging with all four column densities), the Ne and Si lines tend to have larger EWs in the $\\alpha = 10^{\\circ}$ case with average ratios of 0.79 and 0.99 respectively.\n \n \\subsection{Changing Abundances}\n \n The simulations carried out so far considered a matter composition from \\cite{1989GeCoA..53..197A} as well as a molecular hydrogen fraction of one. In this section, we consider the more recent gas composition from \\cite{2003ApJ...591.1220L} as well as a Hydrogen fraction of 0.2 (H$_{2} = 0.2$). We ran simulations using this new matter composition for the edge-on case at the highest polar gas column density in order to compare with the previous simulations. The results are seen on the right panel of Figure\\,\\ref{fig10} with the black plot representing the composition from \\cite{2003ApJ...591.1220L} (H$_{2}$ = 0.2) and the red plot being from the previous simulations using \\cite{1989GeCoA..53..197A} (H$_{2}$ = 1). As can be seen, there is no significant difference in spectral or continuum features. \n \n \\section{Summary and Conclusion}\n \n In this paper we have investigated the effect polar gas has on the X-ray spectra of accreting supermassive black holes. Three geometries (torus alone, torus + hollow cone, torus + filled cone), along with three inclination angles and four slant length column densities for the filled and hollow cones were considered. We summarize our main results below:\n \n \\begin{itemize}\n \\item The polar gas was found to have a significant impact on the soft X-ray spectrum producing several fluorescence lines in the 0.3--5\\,keV band such as the O ($524.9$\\,eV), Ne ($848.6$\\,eV), Mg ($1253.6$\\,eV), and Si ($1739.38$\\,eV) lines. The most significant impact on the X-ray spectra from the polar gas occurs at the edge-on inclination angle where photons which would normally be absorbed in the torus are allowed to scatter from the less dense polar component (see Figures\\,\\ref{fig2}, \\ref{fig3a} and \\ref{fig3b}, as well as Figure\\,\\ref{fig6} and Tables \\ref{table1}-\\ref{table6} for the EWs of selected spectral lines). \n \\item The most significant impact the polar component has is when it is in the form of a hollow cone, as the filled cone causes more self-absorption as there is more gas to interact with. Simulations were also run using the same number of atoms in the filled as in the hollow cone. These showed that the soft spectrum of the pole-on case was still obscured and, therefore, is likely not the geometry for the polar component (see Figure \\,\\ref{figCone}). \n \\item When considering 0.1\\% of extra scattered continuum stemming from the NLR in addition to the new polar gas emission, as well as extra photoionized lines from NGC 3393 and the Circinus galaxy, we find that the spectral lines from the polar component would still be detectable, particularly by future X-ray calorimeters such as those onboard \\textit{XRISM} (Resolve; \\citealp{xrismscienceteam2020science}) and \\textit{Athena} (X-IFU; \\citealp{2016SPIE.9905E..2FB}) (see Figure\\,\\ref{fig10.5}). We also simulate spectra using \\textit{XRISM} and \\textit{Athena} (Barret et al. 2021 in prep.) to see how our simulated data would be observed (see Figure\\,\\ref{figres}). This figure shows that \\textit{Athena}\/X-IFU is better suited to observe more spectral features from the polar component than \\textit{XRISM}\/Resolve, given the extra features stemming from the NLR. \n \n \n \\item When considering a smaller opening angle for the polar component we find that, while the EWs of many spectral lines change, the general trends as seen in Figure\\,\\ref{fig6} remain the same with the EWs of lower energy lines (O, F, Ne, Mg, Si) increasing with increasing column density of the polar component and the EWs of the higher energy lines (Ar, Ca) decreasing with column density. This is with the exception of the S line which increases in EW, contrary to what is seen in Figure\\,\\ref{fig6} (see Table \\ref{table7}). In addition to this, when considering an abundance from \\cite{2003ApJ...591.1220L}, we find little difference in spectral or continuum features (see the right panel of Figure\\,\\ref{fig10}). \n \\end{itemize}\n \n These simulations show that low-energy fluorescent lines could be an important tracer of the polar gas in AGN. Observations of nearby AGN with future high-resolution X-ray instruments, such as those on-board {\\it XRISM} \\citep{xrismscienceteam2020science} and {\\it Athena} X-IFU \\citep{2016SPIE.9905E..2FB}, will allow to search for signatures of polar gas. Comparison with simulations, such as those reported in this paper, will allow further constrain the kinematics, geometry, and origins of the polar gas. \n \n\n\n\n\n\\newpage\n\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\\section{Limitations}\n\n\nSocial media does not necessarily represent offline events, \nand unlike the surveys we use for comparison, we cannot control for demographics.\nWe study texts written in English and platforms that attract a U.S. audience, as the authors are all most familiar with this setting.\nWe focus on one specific community out of many, and we do not expect that the findings in this paper will necessarily generalize outside of the U.S. or to other online spaces; \nindeed, one of our results indicates that different platforms display different patterns.\nFinally, averaging over posts allows us to track patterns and make comparisons but can also reduce nuance.\nOur work is best read in conjunction with ethnographic studies like \\citet{homewood2017turned} and \\citet{daley2014influences}, which highlight individual voices of birth control users.\n\\section{Introduction}\n\n\nBirth control users include nearly two thirds of women in the U.S. between the ages of 15-49---almost 50 million women \\citep{daniels2020current}.\nIn addition to providing a means of family planning, birth control methods can also be used to treat and manage many medical conditions, including acne, endometriosis, cancer risks, gender dysphoria, and irregular and painful menstruation \\citep{schindler2013non, nahata2017gender}.\nHowever, birth control methods are not one-size fits all, and the choices of whether to use birth control and which method to select are complicated by personal beliefs, cost and accessibility, and a wide array of side effects that are difficult to predict and identify \\citep{manzer2022did,yoost2014understanding,polis2016typical}.\n\n\nWhen navigating this decision, birth control users face a sensemaking challenge that involves dealing with social stigma, understanding contraceptive methods and potential side effects, and seeking normalcy after upsetting experiences seeking and using contraception.\nThis challenge leads many birth control users to the internet \\citep{yee2010role,russo2013women}, where they can engage in community sensemaking activities, e.g., seeking and sharing information, reviews, advice, hypotheses, and stories \\citep{carron2015help,yang2019seekers}.\n\n\nPrior work has shown that these various sensemaking activities can vary in frequency across different online platforms \\citep{choudhury2014seeking,rivas2020classification,zhang2021distress}.\nDifferent healthcare conditions \\citep{sannon2019really} and side effects \\citep{choudhury2014seeking} can result in different behavior,\nand recent work has demonstrated differences in the online sensemaking practices surrounding reproductive healthcare topics like pregnancy and vulvodynia \\citep{young2019girl,andalibi2021sensemaking,chopra2021living}.\nHowever, while birth control researchers have examined the frequencies of different method discussions over time \\citep{nobles2019repeal,merz2020population}, a comparison across platforms of the difficult decision-making of birth control users and their sensemaking practices has so far been unexplored. \n\n\nWe study birth control users' unique combination of sensemaking activities---deciding between methods, interpreting side effects, calculating risks, etc.---and we use computational text analysis methods to discover the intersections between these activities and online platforms.\nBecause different methods and side effects could lead to different activities, we first\nexplore \\textit{which birth control methods and side effects are more likely to be discussed on different online platforms} (\\textbf{RQ1}).\nUsing lexicons crafted for our online settings, we compare these discussion frequencies to the distribution of experienced side effects as reported via a nationally representative survey by \\citet{nelson2018women}, indicating increased or decreased interest in discussing these concerns online.\nNext, using topic modeling and hand-annotations, we discover and characterize \\textit{the kinds of sensemaking activities that birth control users engage in online and how these differ by platform} (\\textbf{RQ2}).\nWe contrast these activities with prior work on related online healthcare communities, and we highlight the particular strategies of birth control users.\n\n\n\\paragraph{Contributions.}\nWe identify and characterize how birth control users make sense of their experiences and options via three large English-language online platforms (Reddit, Twitter, and WebMD).\nTheir unique combination of sensemaking strategies included \\textit{storytelling}, \\textit{risk analysis}, \\textit{timing and calculations}, \\textit{causal reasoning}, \\textit{method and hormone comparison}, and \\textit{information and explanations}; in particular, \\textit{storytelling} is used to prepare for and overcome painful insertion experiences.\nWe examine these patterns across birth control methods, side effects, and platforms, unlike prior work, and we compare our observations of side effect discussions to survey data, finding large variations and highlighting the importance of integrating different methods.\nThroughout this study, we employ a mixture of lexicons (which we provide as a community resource\\footnote{\\url{https:\/\/github.com\/maria-antoniak\/birth-control-across-platforms}}), topic modeling, and hand annotation, demonstrating how a detailed and diverse analysis across online platforms can expand our understandings of social sensemaking, web and social media activity, and vital healthcare issues.\n\n\n\\section{Related Work}\n\\label{section-related-work}\n\n\n\n\n\n\\smallskip\n\\noindent\\textbf{Sensemaking in online communities.}\nAs explained by \\citet{weick2005organizing}, sensemaking necessarily involves communication and is an ``\\textit{activity that talks events and organizations into existence},'' in part through retrospective storytelling.\nIt is a process that \nrelies on collaborative problem solving \\citep{pirolli2005sensemaking}, and more recently, it has been defined in closely related work by \\citet{andalibi2021sensemaking} \nas ``\\textit{how individuals make sense of complex phenomena by constructing mental models that draw on new or existing experiences, information, emotions, ideas, and memories}.''\nIn online healthcare communities, the individual works to make sense of their healthcare experience, and the community collectively gathers information, compares stories, and makes sense of a shared experience \\citep{mamykina2015collective}. \nSharing experiences can help narrators make sense of their stories \\citep{tangherlini2000heroes} and can transfer important information to others without firsthand experience \\citep{Bietti2019StorytellingAA}, leading users through a transformative process via the gathering and organizing of information \\citep{genuis2017looking,patel2019feel}.\nThis process is contextual; each community can rely on different strategies \\citep{young2019girl}.\n\n\n\n\\smallskip\n\\noindent\\textbf{Healthcare activity across platforms.}\nSpecific platform affordances can facilitate different types of healthcare-related disclosure and interactions.\n\\citet{zhang2021distress} formulate a \\textit{social media disclosure ecology} in which platform affordances like anonymity, persistence, and visibility control can predict, e.g., pandemic-related disclosures.\nHealth information seeking behavior can differ across platforms, with search engines more frequently used for serious and stigmatized conditions and Twitter more frequently used for symptoms \\citep{choudhury2014seeking}, and community affordances such as threaded conversations and hashtags can influence participation decisions of those with invisible chronic health conditions \\citep{sannon2019really}.\nThe distribution of content type can also differ by platform; e.g., prior work found that \\textit{sharing experiences} is less frequent on social networks than forums, although these rates can depend on healthcare topic \\citep{rivas2020classification}.\n\n\n\n\n\\smallskip\n\\noindent\\textbf{Reproductive healthcare and online sensemaking.}\nStudies of reproductive healthcare communities have emphasized some unique sensemaking practices.\nFor example, in a study of sensemaking practices after pregnancy loss, \\citet{andalibi2021sensemaking} highlighted the importance of seeking emotional validation, rather than just information, when trying to re-discover normalcy.\n\\citet{young2019girl} similarly supported these dual information management and emotional needs in a study of a vulvodynia Facebook group,\nand \\citet{chopra2021living} discussed polycystic ovary syndrome and why personal tracking is an important communal activity that involves comparison to others' experiences.\nWhether these themes hold for online birth control communities is not currently known.\n\n\nTwo recent studies have focused specifically on online discussions of birth control.\nFirst, \\citet{nobles2019repeal} used search queries to track interest in different birth control methods, finding that U.S. political events correlate strongly with increased information seeking behavior online.\nSecond, \\citet{merz2020population} examined the prevalence and sentiment of tweets mentioning different birth control methods. \nThe authors found that long-acting methods like the IUD were mentioned more often on Twitter, and the proportion of these tweets increased over time.\nWhile most tweets expressing sentiment about contraception were negative, tweets about long-acting methods were more likely to express positive sentiment.\n\n\n\\section{Discussion}\n\n\n\n \\paragraph{Methods and side effects across platforms.}\nIn response to our first research question, we find that birth control discussions on Twitter, WebMD, and Reddit substantially differ in their distributions of methods and side effect mentions.\nWe cannot claim that using a method or experiencing a side effect \\textit{cause} people to choose a specific platform, but our observations add detail and sometimes contradict prior findings.\nFor example, in a study of general healthcare information seeking, \\citet{choudhury2014seeking} found that people more often use search engines for serious and stigmatized conditions and more often use Twitter to discuss symptoms.\nWe do not include search engines in our study, but in our comparison across platforms, we find that Twitter has a higher frequency of severe side effect discussions for birth control, while WebMD and Reddit have higher frequencies of general side effect discussions.\nOur focus on birth control might explain these variations, as many of the birth control side effects are themselves highly stigmatized (e.g., menstrual bleeding, vaginal discharge). \nThe discussion of severe side effects on Twitter is likely related to Twitter's tendency toward sensational content, and could also explain the relative frequency of IUD discussion on Twitter; the IUD has been associated with both severe side effects and potential legal bans\n\\citep{nobles2019repeal}.\nThese findings emphasize the importance of considering platform setting when studying online patterns related to specific health conditions, medications, and side effects.\n\n\n\n\\paragraph{Online sensemaking practices of birth control users.}\nIn response to our second research question, our identification and comparison of topics that align with sensemaking themes reveals important activities specific to birth control and to the different platforms and methods.\nPrior work has found that online communities focused on reproductive healthcare (e.g., pregnancy, vulvodynia) employ strategies related to validation \\citep{andalibi2021sensemaking}, information management \\citep{patel2019feel,young2019girl,andalibi2021sensemaking}, personal tracking \\citep{chopra2021living}, and identifying causation \\citep{patel2019feel}.\nWe found these themes again in our birth control communities, with variations; for example, personal tracking is also a prominent theme in birth control discussions, but it is focused on pill timings and calculations and on self-observations of side effects. \nIdentifying causation is a common concern for those experiencing infertility \\citep{patel2019feel}, and we find this theme again in our datasets but focused on side effects like weight gain.\nThis combination of themes reflects the unique problems facing birth control users, as they choose between ``least bad'' options, struggle to identify and treat side effects, determine risk of pregnancy given their circumstances, and avoid rare but alarming outcomes like stroke and heart attack.\n\n\n\n\n\\paragraph{Storytelling, pain, and the IUD.}\nWe would like to particularly highlight storytelling as an important sensemaking strategy for birth control users.\nStorytelling is known to help communities work through trauma \\citep{tangherlini2000heroes}, but birth control users employ storytelling specifically to address physical pain.\nPain is a major cross-cutting theme across the platforms; it is the most frequently and consistently discussed side effect, and it is most often mentioned alongside the IUD.\nWhile \\citet{barr2010managing} identifies pain as the second most common side effect (after bleeding changes) and \\citet{dickerson2013satisfaction} identifies pain as the most common side effect for the IUD and second most common side effect for the implant, we find a much larger gap between pain and the next most frequent side effects in our analysis.\nPain is mentioned frequently in posts about the IUD and in posts whose probable topics are about seeking and sharing IUD insertion stories.\nPain is inherently a subjective experience that cannot be precisely communicated \\citep{scarry1987body}, but sharing of personal experiences provides one way for a community to build a sense of what is \\textit{normal} or \\textit{to be expected} \\citep{patel2019feel,andalibi2021sensemaking}. \nWe suggest that there is an urgent unmet need for (a) honest education and preparation before IUD insertions and (b) pain treatment options during this procedure.\n\n\n\n\n\\paragraph{Online discussions differ from survey reports.}\nIn comparison to the survey results in \\citet{nelson2018women}, we find not only large differences in the frequency at which different side effects are discussed but also differences across the platforms.\nIt could be that the differences from the survey are due to the demographic distribution opting into the survey versus those opting to post online; future work surveying the demographic distribution of these users would address this question, but our results take a first step at measuring differences between these settings. \nWe find, e,g., while mood changes are discussed at similar frequencies in comparison to the survey data across the platforms, other side effects like strokes, bloating, fatigue, bleeding, and dizziness might be discussed more, less, or at equal frequency ranks to the reports in \\citet{nelson2018women}, depending on the platform.\nBloating is less frequently mentioned on all the platforms and for all the methods in comparison to the survey data, perhaps indicating a general lack of concern about this side effect in contrast to its reported frequency.\nBut bloating is much less frequent on Twitter for the pill and implant, which could also indicate that this particular setting is less suited to discussing this side effect, perhaps because of the embarrassing or intimate nature of this side effect.\n\n\n\n\\paragraph{Stigma, privacy, and contextual disclosures.}\nBirth control can be a controversial, stigmatized, and intimate topic.\nThis can lead birth control users to seek out additional information privately.\nFor example, in a set of interviews of young Black and Hispanic women, \\citet{yee2010role} found that a greater number reported seeking decision-marking support on the internet, citing its privacy, in comparison to other sources of information (e.g., talking to physicians, reading provided information).\nInterpreting side effects, analyzing the risk of pregnancy, or normalizing a painful experience require disclosing personal details and stories.\nBirth control users might analyze the risk and benefit of making these disclosures in certain settings and to certain audiences.\nWhile social penetration theory \\citep{altman1973social} posits that more disclosures are possible as social bonds deepen, prior work has also found that intimate language can be frequent among both close connections and strangers (but not in between) \\citep{pei-jurgens-2020-quantifying}.\nThe variations we observe across methods, side effects, and sensemaking practices could be indicative of platform affordances for privacy, audience size, and anonymity, each of which can affect decisions to self-disclose.\nFor example, we found that side effect discussions are less frequent on Twitter, where users are facing a much larger and non-specialized audience, unlike the other platforms.\nGiving users more platform-specific tools to control their audience size and membership \\citep{,mondal2014understanding} could allow for more productive discussions of this sensitive topic.\n\n\n\n\n\\paragraph{Other factors influencing platform decisions.}\nIt is important to note that decisions to seek information online can correlate with demographic characteristics.\nFor example, in a survey of U.S. young adults, those with a sexual risk history (early sexual activity, involvement in an unintended pregnancy) less frequently reported using the internet as a source and more frequently reported seeking information from a doctor\/nurse, and men more frequently reported using the internet than women \\cite{khurana2015young}.\nIt is also possible that users follow a \\textit{birth control journey}, where different needs at different points in one's journey can lead one to different platforms, as has been reported for other healthcare topics \\citep{sannon2019really,andalibi2018announcing}.\nThese journeys can intersect with methods; for example, while the pill is a popular first method, many people report switching to the IUD as they gain more experience with birth control \\citep{nelson2018women}.\nThis would mirror the journeys of those with invisible chronic illnesses who move from one platform to another as their needs evolve and as they grow more comfortable with self-disclosure \\citep{sannon2019really}.\nThis is mirrored in intra-community research that models user trajectories in online cancer support groups, finding that users often transition from information-seeking to information-sharing roles over time \\citep{yang2019seekers}. \n\n\n\n\n\\paragraph{Recommendations.}\nOur results show that patterns found in one platform are not necessarily replicable on other online platforms, even when grounded in the same health topic.\nWe recommend that social media researchers compare results across multiple platforms to increase confidence in shared patterns.\nIf relying on a single platform, its choice should be guided by an understanding of which platforms might be more less or conducive to discussions of the desired methods, side effects, or sensemaking activities.\nOur work also highlights benefits of mixing computational tools like topic modeling with lexicon-based methods and hand-annotation, and how these methods can be used to characterize difficult-to-identify themes like sensemaking practices; we recommend taking this careful approach, putting unsupervised results in context with fine-grained measurements.\n\n\n\\section{Methods}\n\n\n\\subsection{Birth Control Method Lexicon}\n\\label{section:method-labeling}\n\n\nTo explore the prevalence across platforms of different birth control methods, we develop a lexicon to match each document to the primary method being discussed.\nDue to data scarcity and space limitations, we do not consider methods that are not available on all three platforms (e.g., WebMD does not include the male condom), and of the remaining methods, we limit our analysis to the three most commonly discussed reversible, non-emergency methods, mirroring prior work such as \\citet{merz2020population}.\\footnote{Using the lexicons described above for Reddit, we find that the implant (8,578 posts) is discussed more than twice as often as the shot (3,529 posts) or barrier methods like the male condom (3,029 posts). This contrasts with reported rates of contraception use in the U.S., where the pill is used by 14.0\\%, the IUD by 8.4\\%, the male condom by 8.4\\%, the implant by 2.0\\%, and the shot by 2.0\\% of women aged 15-49 \\citep{daniels2020current}.}\nOur goal is to measure \\textit{prevalence of discussions} rather than \\textit{prevalence of usage} because (1) we are interested in the level of interest and concern of birth control users regardless of actual usage and (2) other study designs (e.g., surveys) are better suited to studying usage rates.\n\n\n\nWe rely on sets of keywords to assign each text a primary birth control method.\nBecause of their different structures, each platform requires its own set of keywords and matching techniques.\nAll of these lexicons are made available for future research.\nTo estimate the lexicon performance, we checked 450 documents balanced across Reddit posts, Twitter posts, and Twitter replies.\\footnote{We omitted Reddit comments because of our method detection strategy, which defaulted to the method in the parent post if no method was found in the comment; and we omitted WebMD reviews because our method detection strategy already used hand-labeling of medication tags.}\nOur precision and recall scores were perfect (1.0).\n\n\n\n\n\\subsubsection{WebMD.}\nThe URLs for WebMD reviews include a \\emph{drugname} parameter.\nThis parameter includes the medication name followed by its class. \nWe were able to classify 99.2\\% of the WebMD reviews by mapping these names to their corresponding method, and the remaining reviews were for methods outside our target three. \n\n\n\\subsubsection{Reddit.}\n\\label{subsubsection:reddit-method-labeling}\nTo assign Reddit posts and comments to the birth control methods, we use a custom set of keywords.\nIn all cases, the text was assigned to the method that had the highest keyword count (of all the methods).\\footnote{We could instead have assigned each Reddit post and comment to as many methods as were mentioned in its text, rather than assigning each post and comment only to the method mentioned most frequently. \nWe chose the latter assignment because it better aligns with the WebMD assignments; WebMD reviews are explicitly assigned by the user to a single method (even if other methods are also mentioned).\nWe do not find that these different assignment techniques affect our distributional results.}\nIf the highest keyword count was not for one of our target three methods, or if multiple methods were mentioned with equal frequency, we discarded the text.\nWe follow a similar procedure for Reddit comments, but in cases where the comments do not contain a birth control keyword, we assign the comment to the birth control method of its parent post.\n\n\nWe developed the Reddit methods keyword set by (a) drawing terms from the WebMD keyword set, (b) drawing terms from the Twitter keyword set,\n(c) supplementing the WebMD and Twitter sets with general terms that are overloaded on Twitter but usable on Reddit (e.g., ``pill''), and (d) iteratively running the assignments and examining the unassigned posts and comments.\nWe were able to assign all but 4.6\\% of the Reddit posts and 32.6\\% of the Reddit comments to a birth control method.\nManual examination of the unassigned posts reveals discussions of pregnancy scares, access (e.g., online appointments and prescriptions), and treatment of side effects with non-contraceptive medications.\nThe remainder of these unassigned posts discussed topics that are adjacent to birth control, e.g., insurance, menstruation, but are not explicitly connected to a birth control method.\n\n\n\\subsubsection{Twitter.}\nWe use a more limited keyword set, derived from a similar survey of birth control tweets by \\citet{merz2020population}, to query for tweets about each of our target birth control methods.\nOur focus was on precision rather than recall, as the Twitter API requires a keyword match to retrieve tweets on a particular topic; if we included keywords like ``pill'', our results would contain many false positives, unlike Reddit and WebMD where the topic is already constrained to birth control.\nAfter gathering our initial Twitter dataset, we then apply the full Reddit keyword set, using the same methods described for Reddit above.\nAs with the Reddit data, we find that more texts can be assigned to only the pill, IUD, or implant than to a combination of methods; 12.7\\% of posts and 17.6\\% of the gathered tweets either mentioned multiple methods with equal frequency or most frequently mentioned a method not in our set of three target methods.\n\n\n\n\n\n\\subsection{Side Effects Lexicon}\n\\label{section:side-effects}\n\n\nTo measure the frequency of discussions about side effect birth control, we develop a lexicon of terms and patterns.\nBecause we do not use the side effects lexicon for data collection (unlike the methods lexicon), we can rely on one lexicon across our three datasets.\nAs with the methods lexicon, we focus specifically on measuring the \\textit{prevalence of discussions} rather than the \\textit{prevalence of experiences}, as these are better measured via surveys \\citep{nelson2018women}.\nThe frequency of discussions could be influenced by the prevalence of experiences and the level of \\textit{concern} birth control users on this platforms have about particular side effects, as well as platform setting and a user's individual goals.\nWe select patterns that match any discussion of the side effect, whether or not it is mentioned in the affirmative. \n\n\nWe grounded our development of the side effects lexicon in prior work.\nWe matched the side effect categories from \\citet{nelson2018women} as closely as possible; this study conducted a nationally representative survey of U.S. birth control experiences and reported prevalences of side effect experiences across different birth control methods.\nIn addition to these, we added lexicon categories for pain, skin changes, PMS, appetite changes, sexual partner feeling IUD strings, and heart attack.\nWe identified these additional categories and patterns from topic models trained on our datasets and from other work; for example, we also drew on work by \\citet{barr2010managing} that discusses a variety of side effects and their known frequencies and associations with different birth control methods.\nFor each side effect, we then iteratively queried and made updates to the lexicon when we encountered false positives or false negatives.\n\n\n\n\\input{tables\/lexicon_coverage}\n\n\n\\subsubsection{Lexicon evaluation.}\n\nTo evaluate our lexicon, for each side effect we randomly selected a set of matching and non-matching texts, balanced across platforms and match status, resulting in a set of 1,040 texts.\nWe then manually checked whether each text does or does not contain a discussion of the target side effect.\nAcross the side effects, we found precision of 0.98 and recall of 0.96.\nWe show the lexicon coverage in Table \\ref{table:lexicon-coverage}.\n\n\\subsubsection{Comparing observations with survey responses.}\n\nWe compare the distributions of side effect mentions to prior work surveying people in the U.S. about their experiences with birth control side effects.\nThis allows us to compare the frequency of side effect experiences with the frequency of online discussions.\nDifferences between these distributions can indicate healthcare needs that users are addressing via the internet.\nWe compare to surveys (rather than controlled studies) because surveys better approximate the self-reported anecdotes and personal experiences shared online.\nWe first converted each distribution to a ranking, where lower ranks represent greater percents of discussions $r_{platforms}$ or experiences $r_{survey}$.\nWe then find the difference between the ranks for each side effect, $d_{ranks}$.\n\\begin{equation}\n d_{ranks} = r_{survey} - r_{platforms}\n\\end{equation}\nThese ranks avoid issues in directly comparing percents, which might on average be higher or lower.\nWhen $d_{ranks}$ is positive, it indicates more online discussion than expected given the frequency of reported side effect experiences. \n\n\n\n\n\\begin{figure*}[t]\n \\centering\n \\includegraphics[width=0.49\\textwidth]{figures\/final\/barplot.side_effects_distributions.pdf}\n \\includegraphics[width=0.49\\textwidth]{figures\/final\/barplot.bc_type_x_side_effect.pdf}\n \\caption{Distribution of the side effect mentions across the platforms and methods. Bars represent the percents of documents for the specified platform or method mentioning \\textit{any} side effect that also mention the \\textit{specified} side effect. \n See Table \\ref{table:lexicon-coverage} for the denominator sizes. \n Error bars indicate the standard deviation over 20 bootstrapped samples of the datasets.\n } \n \\label{figure:barplot-side-effects-distribution}\n\\end{figure*}\n\n\n\n\n\\begin{figure}[t]\n \\centering\n\n \\centering\n \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.reddit_posts.normalized_by_year.pdf}\n \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.twitter_posts.normalized_by_year.pdf}\n \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.reddit_comments.normalized_by_year.pdf}\n \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.twitter_replies.normalized_by_year.pdf}\n \\includegraphics[width=0.49\\linewidth]{figures\/final\/barplot.bc_type_dist.webmd_reviews.normalized_by_year.pdf}\n \n \\caption{Document distributions of methods over time.}\n \n \\label{figure:bc-type-distributions}\n\\end{figure}\n\n\n\n\\begin{figure}[!ht]\n\n \\centering\n \n \\includegraphics[width=0.1805\\textwidth]{figures\/final\/barplot.side_effects_distributions.compare_to_nelson.ranks.pill.v.pdf}\n \\includegraphics[width=0.135\\textwidth]{figures\/final\/barplot.side_effects_distributions.compare_to_nelson.ranks.iud.v.pdf}\n \\includegraphics[width=0.135\\textwidth]{figures\/final\/barplot.side_effects_distributions.compare_to_nelson.ranks.implant.v.pdf}\n \\\\\n \\includegraphics[width=0.25\\textwidth]{figures\/final\/legend.png}\n \n \\caption{Distribution of the side effect mentions across the platforms. Bars represent the \\textbf{differences} between the rank reported in the survey results from \\citet{nelson2018women} and the rank on the specified platform. Platform ranks are determined by first finding the frequency of side effect mentions; these are the percent of documents mentioning \\textit{any} side effect that also mention the \\textit{specified} side effect. See Table \\ref{table:lexicon-coverage} for the denominator sizes. Bars to the \\textbf{right} of the x-axis represent side effects that are mentioned \\textbf{more frequently} on the platforms than are reported in the survey.}\n \n \\label{figure:barplot-side-effects-distribution-comparison}\n \n\\end{figure}\n\n\n\n\\begin{figure*}[!ht]\n\n \\centering\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.storytelling.pdf}\n \\caption{Storytelling}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.risk_analysis.pdf}\n \\caption{Risk Analysis}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.timing_and_calculations.pdf}\n \\caption{Timing \\& Calculations}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.24\\textwidth}\n \\includegraphics[width=\\textwidth]{figures\/final\/heatmap.topic_x_methods.percents.method_and_hormone_comparison.pdf} \n \\caption{Method \\& Dose Comparison}\n \\end{subfigure}\n \n\n \\caption{Mean topic probabilities across methods for four of our sensemaking clusters. Rows are normalized to sum to one.}\n \n \\label{figure:heatmaps-topics-methods}\n \n\\end{figure*}\n\n\n\n\n\n\n\\subsection{Sensemaking Topic Model}\n\n\n\nWe compare sensemaking activity distributions through an unsupervised, bottom-up analysis using topic modeling.\nTopic modeling is an automatic method to identify prominent themes and discourses in a dataset \\citep{blei2003latent} and is a popular unsupervised technique for the analysis of online health communities \\citep{nobles2018std,abebe2020quantifying}.\nWhile some sensemaking themes might be shared across settings, other themes might be more prevalent in certain platforms and side effects.\nThis process helps us to avoid biases in our data coding (which might overlook certain themes while over-representing others) by quickly revealing frequent themes in each dataset \\citep{nelson2020computational}.\n\n\n\n\\subsubsection{Model training.}\n\n\nWe trained a latent Dirichlet allocation (LDA) topic model \\citep{blei2003latent} on each of our datasets, by combining the Reddit posts and comments and the Twitter posts and replies for training, resulting in three models, using MALLET\\footnote{\\url{http:\/\/mallet.cs.umass.edu\/}} for training.\nLDA remains a highly consistent and reliable model \\citep{harrando-etal-2021-apples,hoyle2022AreNT}, especially when trained via Gibbs sampling for smaller datasets.\nWe balanced each training set, sampling 8,000 documents for each method for each of the Reddit and Twitter datasets and 800 documents for each method for the WebMD dataset (see Figure \\ref{figure:bc-type-distributions} for method distributions).\nBy balancing the training set, we avoid weighting the topics toward a certain method.\n\n\nWe removed a set of frequent stopwords,\nnumbers, punctuation, and duplicate documents from the training sets.\nRemoving stopwords and duplicate documents has been shown to improve the legibility of the final topics, whereas stemming can be harmful \\citep{schofield-mimno-2016-comparing,schofield-etal-2017-pulling,schofield-etal-2017-quantifying}.\nTo avoid capitalization discrepancies,\nwe lowercase all text.\nWe experimented with different numbers of topics and found 35 to be interpretable across the datasets; at this number of topics, we observed topics that were neither too fine-grained (e.g., splitting a single theme across multiple topics) nor too high level (e.g., combining themes that should be separated), and that produced reasonable evaluation scores (see below).\nHowever, we emphasize that there is no ``correct'' number of topics and that this method is used for exploration and interpretation. \n\n\n\n\n\n\\subsubsection{Model evaluation.}\n\\label{topic-model-evaluation}\n\n\nWhile we do not have space here to list the full sets of topics for each dataset, we provide the topics, most probable words, and example paraphrased documents in our code repository for manual examination.\nWe report human evaluation of our topics following the recommendations in \\citet{hoyle2021automated}.\nUsing the \\textit{word intrusion task} \\citep{chang2009reading}, we show two expert annotators (the first two authors) and one non-expert annotator a set of the four most probable words plus an ``intruder'' word that has low probability for the current topic but high probability for another topic.\nWe report the proportion of topics for which the annotators identified the intruder.\n\n\nWe found that our Reddit and WebMD topics have performance much higher than a random baseline of 0.2 (annotator accuracies for Reddit: 0.71, 0.74, 0.77, WebMD: 0.54, 0.66, 0.77) while the Twitter topics have lower performance but are still substantially above the random baseline (Twitter: 0.46, 0.46, 0.51).\nThe first (non-expert) annotator consistently had lower scores.\nThe lower performance on Twitter is expected, as text processing methods are notoriously challenged by the short text lengths and non-standard language \\citep{gimpel-etal-2011-part}, and the short tweets require contextual knowledge to interpret.\nThis vulnerability of the word intrusion task to esoteric topics is noted by \\citet{hoyle2021automated}.\n\n\nWe also calculate the ``UMass'' Coherence: the log probability that a document containing at least one instance of a higher-ranked word also contains at least one instance of a lower-ranked word \\citep{roder2015exploring}.\nWe find the highest mean coherence scores across topics for Reddit ($-416$), lower scores for WebMD ($-577$), and lowest scores for Twitter ($-712$).\nWe note the criticisms of this and other automatic metrics in \\citet{hoyle2021automated}; in comparison to human evaluation, automatic metrics can exaggerate differences between models.\n\n\n\n\\subsubsection{Identifying sensemaking themes.}\n\n\nAcross the datasets, we find that many of the topics discuss birth control methods (including one method in their 10 most probable words), side effects, pregnancy, and access (costs, appointments).\nWe then identify a series of cross-cutting sensemaking topics.\nThese topics include discussions of information seeking\/sharing, educational links and resources, how-to explanations, experience seeking\/sharing, emotional support, and other sensemaking-related discussions.\nTwo researchers independently coded the topics as more or less related to sensemaking.\nThe researchers then conferred and agreed upon a final set of topics and assigned them to thematic clusters.\nDuring this coding, we relied on the sensemaking definition from \\citet{andalibi2021sensemaking}: ``\\textit{sensemaking describes how individuals make sense of complex phenomena by constructing mental models that draw on new or existing experiences, information, emotions, ideas, and memories}.''\nWe also drew inspiration from prior work studying sensemaking in online healthcare communities, particularly those works also focused on reproductive healthcare (see \\S Related Work).\nWe further explore and validate these sensemaking topics by hand-labeling a small subsection of the data with social support goals \\citep{yang2019seekers}.\nAfter coding 150 documents for each dataset, we measured the agreement between the annotators using Krippendorff $\\alpha$, as each document could receive zero, one, or more labels.\nOur agreement was acceptable, with a score across the labels of 0.74.\\footnote{Lower scores are not surprising for subjective language labeling tasks \\citep{artstein2008inter,godwin2016collecting}, and our scores are substantially higher than the agreement scores for a very similar classification task in \\citet{rivas2020classification}.}\n\n\n\nTo compare the topics across the platforms, we aligned the topics from the different models using Jensen-Shannon divergence (JSD) for the word distributions associated with each pair of topics. \nJSD is frequently used in prior work to compare topic distributions \\citep{hall-etal-2008-studying,fang2012mining}.\nAfter manual examination of the ranked topic pairs, we categorized topics with JSD scores below 0.6 as aligned across the datasets, and those with scores above 0.8 we considered diverging.\n\n\n\\section{Data}\n\\label{section:data}\n\n\nWe collected data related to birth control from three prominent online platforms: posts and comments on \\textit{r\/BirthControl}, birth control reviews on \\textit{WebMD}, and \\textit{Twitter} posts and replies related to birth control.\nThese are all English-language subcommunities of larger websites, and while we have limited demographic information, we observe that the majority of location-specific posts (e.g., politics, insurance) are about the U.S.\nTable \\ref{table:data} summarizes these datasets.\n\n\n\n\\input{tables\/data}\n\n\n\n\\paragraph{Reddit.}\nr\/BirthControl\\footnote{\\url{https:\/\/www.reddit.com\/r\/birthcontrol\/}} is a user-created and user-moderated online forum dedicated to birth control.\nUsers are pseudonymous and range from one-time questioners to experienced question answerers. \nAs of November 20, 2022, r\/BirthControl had 107,537 members.\nAccording to a Pew Research Center survey of U.S. residents, more men (15\\%) than women (8\\%) use Reddit, and more White (12\\%) and Hispanic (14\\%) than Black (4\\%) survey respondents use Reddit \\citep{perrin2019share},\nbut these platform-wide distributions are unlikely to be representative of r\/BirthControl, for which more detailed demographics are unavailable.\nPrior work on a related subreddit found that 81\\% of the users identified themselves as white \\citep{nobles2020examining}.\n\n\nPosts tend to be longer than comments, there are more comments than posts in our dataset, and both posts and comments have increased in frequency over time.\nFor each document, we collected the title (for posts), text, date, and user-assigned tag (for posts).\nWe removed comments written by the parent post's author, and we also removed stickied comments (auto-generated or mod-written comments).\nWe did not include documents deleted by the time of collection.\n\n\n\n\n\\paragraph{WebMD.}\nWebMD\\footnote{\\url{https:\/\/www.webmd.com\/sex\/birth-control\/birth-control-pills}} is a healthcare website that has hosted news, information, prescription discounts, community forums, and user-written medication reviews.\nWe focused on the medication reviews as a contrast to the Reddit and Twitter datasets.\nThe majority of reviews are written by women (71\\%), and unlike Twitter and Reddit, the website has declined in visitors and reviewers since 2009 \\citep{yun2019decline}.\nUnlike Twitter and Reddit, WebMD reviews do not allow users to interact directly, though reviews sometimes mention other reviews or the general trends observed in other reviews.\n\n\nWe gathered reviews by querying \\emph{birth control} as a condition on the drugs and supplements portion of the site and selecting medications that had more than 5 reviews.\\footnote{At the time of data collection, this search term yielded both preventative and emergency contraception, but the terms must be queried separately as of January 2021.}\nFor each WebMD review, we collect the birth control name, the date of the review, and the review text. \n\n\n\n\\paragraph{Twitter.}\nTwitter is a large social network where discussions range widely from personal to global topics.\nCompared to the general public, Twitter users are more likely to be Democrats, and they also skew younger than users of YouTube, Facebook, or Instagram \\citep{perrin2019share}.\nThe gender and racial distribution on Twitter are close to uniform; out of a set of U.S. survey respondents, 24\\% of men and 21\\% women report using Twitter, while 24\\% of Black, 25\\% of Hispanic, and 21\\% of White respondents report using Twitter \\citep{perrin2019share}.\nOn Twitter, birth control discussions take place in the context of many other discussion topics, and while users can group their conversations using hashtags, there are not separated communities like subreddits.\nModeration is organized by Twitter, rather by users as on Reddit.\n\n\nUsing the Twitter Academic API (v2), we collected all the English Twitter posts containing a set of keywords corresponding to our three target birth control methods.\nSeparately, we collected all the Twitter replies containing the same set of keywords.\nRecent work has shown that this API returns reliable representations of the full tweet space \\citep{Pfeffer2022ThisSS}.\nSee \\S Methods below for the design of the Twitter-specific keywords.\nWe removed Twitter handles from tweet texts.\nThe high vocabulary size for Twitter posts (Table \\ref{table:data}) partly reflects many unique URLs shared in these documents, which can be indicative of information-providing behavior.\nThere are many possible design decisions when gathering Twitter data, and we chose not to collect the full conversations (replies and parent posts of the keyword-containing posts and replies) to (a) limit the size of our collection, (b) target texts explicitly discussing birth control, and (c) more closely replicate prior work on birth control tweets \\citep{merz2020population}.\n\n\n\\section{Broader Impacts, Ethics, and Limitations}\n\\label{section:ethics}\n\n\nOur study was considered exempt under our institution's IRB.\nHowever, while Reddit, Twitter, and WebMD posts and replies are ``public,'' they can contain highly personal information,\nrequiring a balance between potential harms and potential benefits to the community, as described in the guiding principles of the Belmont Report: \\textit{respect for persons}, \\textit{beneficence}, and \\textit{justice}.\\footnote{\\url{https:\/\/www.hhs.gov\/ohrp\/regulations-and-policy}}\nConsidering possible harms, e.g., re-identification of those using stigmatized medications, we do not collect any user-specific information, and we do not infer medical conditions for individual users; instead, we rely on patterns averaged across many users.\nWe also anonymize and paraphrase any direct quotations.\nTo protect users' agency to edit or delete their data at its original source, we release our data collection lexicons but not copies of the collected data.\nWe balance these concerns and protective actions against the benefits of this research; among other benefits, our work highlights the unmet pain treatment needs of a population known to be discriminated against by physicians \\citep{Samulowitz2018BraveMA} and examines the kinds of support needed by those facing difficult healthcare choices.\n\\section{Results}\n\n\n\n\n\\subsection{Results: Birth Control Methods Across Platforms}\n\n\nWe compare the rates of discussion of the different birth control methods across platforms and over time.\nWe find that the types of questions that users ask and the type of information they share differs based on birth control method.\nWe measure the rise and fall of discussions of different birth control methods over time, and we compare these distributions across platforms.\nImportantly, these patterns do not necessarily indicate real-world increases or decreases in use of different methods, but they do show the concerns and interests of users on these platforms. \n\n\n\n\nFigure \\ref{figure:bc-type-distributions} shows the frequency of documents for each medication method by platform over time. \nThe pill is the most popular method in both \\textit{Reddit Posts} and \\textit{WebMD Reviews}, while the pill and IUD are tied in \\textit{Reddit Comments}.\nDiscussions of the pill on WebMD are much more frequent that discussions of other birth control methods and do not show a rise in interest for the IUD, counter to both the other datasets and our original hypothesis.\nOn Twitter, the pill begins as the most popular but is replaced by the IUD in posts, while replies always center on the IUD.\nTwitter replies also increase sharply after 2016, differing from the Twitter posts (so is unlikely to be a symptom of our keywords).\nThis suggests that the IUD generates more discussion on Twitter, especially post-2016, compared to the other birth control methods.\nOn Reddit, the number of posts discussing the pill and IUD are similar, with slightly more posts discussing the pill, though this difference is erased in the comments.\nThe implant is always the least discussed of the three methods.\n\n\n\n\\subsection{Results: Side Effects Across Platforms}\n\n\nFigure \\ref{figure:bc-type-distributions} shows the distributions of the side effects across the different platforms, where frequency proportions are calculated by dividing the number of texts mentioning the specified side effect by the total number of texts mentioning any side effect (i.e., $p(s_i|a)$ where $s_i$ is the specific side effect and $a$ is any side effect).\nDiscussions of different methods have changed dramatically over time on Twitter, with discussions of the IUD rising and the pill falling.\nThe pill remains most most mentioned in Reddit posts.\n\n\nFigure \\ref{figure:barplot-side-effects-distribution} shows the distribution of side effects across platforms.\nWe find that pain, cramps, menstrual bleeding irregularities, mood changes, and skin conditions are the most commonly discussed side effects.\nIn particular, \\textit{pain is consistently and frequently discussed across the platforms} and is an outlier among the side effects.\nDiscussions of stroke are more frequent in Twitter posts than in the other datasets, but for all other side effects, Twitter has the lowest discussion frequencies (perhaps because of stigma around publicly sharing such information compared to the more frequently pseudonymous settings on Reddit and WebMD).\nExcept for pain, menstrual bleeding, and severe effects, WebMD has the highest frequency of side effects discussions compared to the other platforms.\nIn particular, WebMD has higher frequencies for discussions of mood changes, skin conditions, and headache discussions.\nReddit has has the highest frequency of discussions for menstrual bleeding irregularities.\n\n\n\n\n\nFigure \\ref{figure:barplot-side-effects-distribution} also compares the side effects by birth control method.\nAcross the birth control methods, we again find that pain is a frequently mentioned side effect, but it is most frequently mentioned in discussions that also mention the IUD rather than the pill or implant.\nThe implant is the only method shown to cause weight gain \\citep{barr2010managing}, so this association is expected---but the discussions of weight gain for the other methods are less expected and could indicate that this potential side effect is a concern across methods.\nMenstrual bleeding, mood changes, headache, and libido are least often discussed in with the IUD, as are general mentions of side effects, while nausea, stroke, and skin conditions are most often discussed in discussions of the pill.\n\n\n\n\n\n\n\\subsection{Results: Comparison to Observed Distributions}\n\n\nFigure \\ref{figure:barplot-side-effects-distribution-comparison} shows the differences between our observed distributions of side effect mentions and the reported distributions of side effect experiences in \\citet{nelson2018women} (described above).\nWe find large differences between how frequently side effects are discussed online compared with how frequently they are reported in the survey from \\citet{nelson2018women}.\nFor example, strokes are rarely experienced according to \\citet{nelson2018women}, but when mentioned with the pill, they are discussed more frequently on Twitter in comparison to other side effects; this is unsurprising, since sensational topics are frequently discussed on Twitter.\nMood changes are more likely to be discussed across the platforms for the implant in comparison to the survey data.\nDizziness is more likely to be discussed for the IUD than expected, while weight gain is universally discussed less frequently than expected.\nThese patterns indicate cases in which users turn to the internet more or less frequently than expected based on their self-reported experiences in the survey.\n\n\n\n\n\\subsection{Results: Sensemaking Across Platforms}\n\n\nOur final set of sensemaking themes included: \\textbf{storytelling} (e.g., \\textit{implant insertion and removal} on Twitter), \\textbf{risk analysis} (e.g., \\textit{risks of serious side effects} on WebMD), \\textbf{timing and calculations} (e.g., \\textit{daily timing of pill} on WebMD), \\textbf{method and dose comparison} (e.g., \\textit{hormone dosages and comparison} on Reddit), \\textbf{causal reasoning} (e.g., \\textit{weight changes and causes} on Reddit), and \\textbf{information and explanations} (e.g., \\textit{research on the male pill} on Twitter).\nWe show four of these themes (others omitted for space) in Figure \\ref{figure:heatmaps-topics-methods}, with the topics from each platform included in that theme and their probabilities for each birth control method.\nWe observe a greater number of storytelling topics for the IUD and implant than for the pill, and examination of these topics shows that insertion and removal experiences fuel this pattern.\nEach method is associated with risks and is included in method comparisons, but the pill in particular is included in discussions of timing and calculations.\n\n\n\nUsing Jensen-Shannon divergence to compare topics across the platforms, we find that the most \\textbf{aligned} ($JSD < 0.6$) topic categories were: \\textit{weight changes}, \\textit{general side effects}, \\textit{IUD insertion experiences}, \\textit{implant insertion experiences}, \\textit{menstrual timing and cycles}, and \\textit{bleeding changes}.\nEach of our three datasets included at least one representative topic from these categories.\n\n\nThe most \\textbf{diverging} ($JSD > 0.8$) topics were: \\textit{causes and side effects of vaginal infections} and \\textit{explanations of how birth control works} on Reddit; \\textit{IUD jokes and random}, \\textit{viral folk stories}, and \\textit{implant news about unexpected experiences} on Twitter; and \\textit{treating or causing other medical issues} and \\textit{secondhand advice and experiences} on WebMD.\nThese topics were the most unique to their training dataset, without directly comparable topics in the other datasets.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nTo control the occuracy of perturbative QCD predictions for physical\nobservables it becomes important to have a regular way of \nestimating corrections suppressed by powers of large momentum scale,\n$\\Lambda_p\/Q^p$, and if not calculate explicitly the nonperturbative \nscale $\\Lambda_p$ then find the leading exponent $p$ at least. \nAt present, the understanding of power corrections has a solid\ntheoretical status only for the special class of processes like\n$\\mbox{e}^+ \\mbox{e}^-\\to\\mbox{hadrons}$ and deeply inelastic\nscattering (DIS) for which the analysis based on the operator product\nexpansion (OPE) is applicable. The OPE fixes the structure of power\ncorrections (as $1\/Q^4$ for the total cross-section of $\\mbox{e}^+\\mbox{e}^-$ \nannihilation and $1\/Q^2$ for the structure functions of DIS) and \nallows to identify the corresponding scales $\\Lambda_p$ as matrix elements \nof higher twist composite operators in QCD. The understanding of power \ncorrections in the processes which do not admit the OPE is the subject \nof intensive discussions (see review~\\ci{B} and references therein). \n\nAn important example of the process, which does not admit the OPE is the \nDrell-Yan (DY) process $h_A+h_B\\to \\mu^+\\mu^- +X$. Here, the lepton pair\nwith invariant mass $Q^2$ is created in the partonic subprocess\n$q + \\bar q\\to\\mu^+\\mu^-(Q^2)+X$ of annihilation of two quarks with \ninvariant energy $\\hat s$ and we choose the kinematics such that \n$\\tau=Q^2\/\\hat s \\sim 1$. One of the reasons for this is that as $\\tau\\to\n1$, the final state in the partonic subprocess consists of the lepton\npair with the energy $Q$ accompanied by a soft gluon radiation with the \ntotal energy $(1-\\tau)Q\/2$. Due to the presence of two different scales\nthe partonic cross-section gets large perturbative (Sudakov logs in \n$\\mbox{ln}(1-\\tau)$) and nonperturbative (as inverse powers of $(1-\\tau)Q$) \ncorrections induced by soft gluons \\ci{CS,KS}. \nCalculating the moments $\\sigma_N=\\int_0^1 d\\tau \\tau^N d\\sigma_{{\\rm _{DY}}}\n\/d Q^2$ at large $N\\sim\\frac1{1-\\tau}$ and subtracting collinear \ndivergences in the DIS-scheme one can represent the leading term \nof the expansion of $\\sigma_N$ in powers of $1\/Q$ as \n\\begin{equation}\n\\mbox{ln}\\,\\sigma_N=\\sum_k \\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^k(Q\/N) C_k(\\mbox{ln} N) + \\lr{\\Lambda_{{\\rm _{DY}}} N\/Q}^p,\n\\lab{moment}\n\\end{equation}\nwhere the coupling constant is defined at the characteristic soft \ngluon scale, $C_k$ contains the powers of Sudakov logs plus finite, \n$\\mbox{ln}^0 N$, contributions and the second term represents the leading\npower correction. As $N$ increases, the energy of soft gluons in the\nfinal state decreases as $Q\/N$ toward the infrared region in which\nperturbative expressions should fail. Indeed, it is well-known\nthat the perturbative expansion in \\re{moment} is not well-defined due to\nfactorial growth of coefficients $C_k \\sim \\beta_0^k k! p^{-k}$\nat higher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$. This makes the perturbative series non Borel \nsummable and induces an IR renormalon ambiguity into perturbative \ncontribution to \\re{moment} at the level of power corrections,\n$\\delta_{{\\rm _{IR}}} \\sigma_N\\sim \\exp(-\\frac{p}{\\beta_0\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi(Q\/N)})\n=(\\Lambda_{\\rm QCD} N\/Q)^p$. Then, IR renormalon ambiguity of \nperturbative expression is compensated in \\re{moment} by\nambiguity in the definition of the scale $\\Lambda_{{\\rm _{DY}}}$.\nThus, examing the large order behaviour of the perturbative series \none can determine the level $p$ at which the leading power correction\nappear in \\re{moment}. However, since it is impossible to calculate \nthe coefficients $C_k$ exactly to higher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$ one has to \nfind a reasonable approximation to $C_k$ which would allow to identify \nthe $k!$ factorial growth of the coefficients. To this end different \nschemes were proposed~\\ci{BB,DMW,AZ}. Being applied to the Drell-Yan \ncross-section they predict that there are no $N\/Q$ corrections to \n\\re{moment} and the leading renormalon contribution has the following \nform\n\\begin{equation}\n\\delta_{{\\rm _{IR}}}\\sigma_N = 0\\cdot (N\/Q) + \\lr{A_{{\\rm _{DY}}} N\/Q}^2\\,.\n\\lab{zero}\n\\end{equation}\nWe would like to stress that these schemes are {\\it approximate\\\/}\nand although they were tested in the case of $\\mbox{e}^+\\mbox{e}^-$ annihilation \nand DIS, it is not clear yet whether they are applicable in the \nDrell-Yan process and whether they really predict the {\\it leading\\\/}\npower correction to the cross-section. \n\n\\section{Wilson line approach}\n\nA different approach to the analysis of power corrections has been proposed \nin Ref.~\\ci{KS}. It is based on the remarkable property of soft gluons that\ntheir contribution to the Drell-Yan cross-section can be factorized with\na power occuracy into universal factor having the form of a Wilson line.\nSimilar to the OPE approach, the power corrections can be identified\nwith the matrix elements of certain operators built from the Wilson line. \n\nThe Wilson line operator appears in the Drell-Yan partonic subprocess as an\neikonal phase of quarks interacting with soft gluon radiation. Combining the\neikonal phases of quark and antiquark together one gets the expression for\npartonic cross-section as\n$$ \n\\frac{d\\sigma_{{\\rm _{DY}}}}{d Q^2}\n=\\sum_n |\\langle n | T U_{{\\rm _{DY}}}(0) |0\\rangle|^2\n\\delta(E_n-(1-\\tau)Q\/2)\n$$\nwhere summation is performed over all possible final states consisting\nof an arbitrary number, $n$, of soft gluons with the total energy\n$E_n=(1-\\tau)Q\/2$ and $T$ stands for time-ordering of gauge fields. \nHere, $U_{{\\rm _{DY}}}(0)=P\\exp(i\\int_{C(0)} dx \\cdot A(x))$ is the eikonal \nphase of quark + antiquark and\n$A_\\mu(x)$ is the gauge field operator describing soft gluons. \nThe contour $C(0)$ in Minkowski space-time corresponds to the classical\ntrajectory of quark and antiquark. It goes from $-\\infty$ to the\nannihilation point $0$ along the quark momentum and then returns \nto $-\\infty$ in the direction opposite to the antiquark momentum.\nPerforming summation over final states in the last relation one can \nobtain the representation for the Drell-Yan cross section for $\\tau\\to\n1$ as a Fourier transformed Wilson line expectation value~\\ci{KM}\n\\begin{eqnarray}\n\\frac{d\\sigma_{{\\rm _{DY}}}}{d Q^2}\n&=&\n\\frac{Q}2\\int_{-\\infty}^\\infty\\frac{dy_0}{2\\pi}\\mbox{e}^{-iy_0(1-\\tau)Q\/2}\n\\langle 0| W(y_0) |0 \\rangle\n\\nonumber\n\\\\\nW(y_0)&\\equiv&\n\\widebar T U^\\dagger_{{\\rm _{DY}}}(0)\\, T U_{{\\rm _{DY}}}(y_0)\\,,\n\\lab{W}\n\\end{eqnarray}\nwhere $U_{{\\rm _{DY}}}(y_0)$ is defined at the point $y=(y_0,\\vec 0)$. \nCalculating the moments of this expression one obtains the following \nrelation \\footnote{This identity is an analog of the relation between \nthe moments of the structure functions of DIS and matrix elements of \ncomposite operators in the OPE.}\n\\begin{equation}\n\\sigma_N= \n\\frac1{N_c} \\mbox{tr\\,}\n\\langle 0 | W(y_0) |0 \\rangle\\,,\\quad\n\\mbox{for $y_0=iN\/Q$}\\,,\n\\lab{rel}\n\\end{equation}\nwhich is valid up to corrections vanishing as $N\\to \\infty$ and which \ntakes into account all Sudakov logs and power corrections in $N\/Q$. \nThe scale $Q\/N$ enters into \\re{rel} just as a parameter of \nthe integration contour in $U_{{\\rm _{DY}}}(y_0)$. Eq.\\re{rel}\nstates that the asymptotic expansion of\nthe moments of the Drell-Yan cross section is related to the behaviour\nof the Wilson line expectation value at small $y_0=\\frac{iN}{Q}$. \n\nPerturbative expansion of \\re{rel} involves standard ``short-distance'' \nlogarithms $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^n\\mbox{ln}^m(|y_0|\\mu)$, which are transformed into Sudakov logs\nfor $y_0=iN\/Q$. The factorization scale $\\mu\\approx Q$ has a meaning of \nthe maximal energy of soft gluons contributing to the partonic\ncross-section and after factorization of soft gluon emission it\nappears in \\re{W} and \\re{rel} as a {\\it ultraviolet\\\/} renormalization \nscale of the Wilson line operators. As a result, all perturbative Sudakov\ncorrections to the Drell-Yan cross-section can be effectively resummed \nusing the remarkable renormalization properties of Wilson lines. They\nare summarized by the following RG equation\n\\begin{equation}\n\\frac{d\\mbox{ln} \\sigma_N}{d\\mbox{ln}\\mu^2}= 2 \\Gamma_{\\rm cusp}(\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi) \\mbox{ln}(N\\mu\/Q)\n+ \\Gamma_{{\\rm _{DY}}}(\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi)\\,, \n\\lab{RG}\n\\end{equation}\nwith $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi=\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi(\\mu)$ and $\\Gamma_{\\rm cusp\/DY}$ being some anomalous \ndimensions. The evolution of $\\sigma_N$ from $\\mu=Q\/N$ up to $\\mu=Q$ \ngenerates Sudakov logs while the initial condition for $\\sigma_N$\nat $\\mu=N\/Q$ takes into account nonperturbative power corrections in\n$N\/Q$. To understand their structure we perform the operator expansion \nof the Wilson line $W(y_0)$ in powers of $y_0$\n\\begin{equation}\n\\vev{W(y_0)}=\\vev{W(0)} + y_0 \\vev{\\partial W(0)} + ... \\ \\,.\n\\lab{ope}\n\\end{equation}\nSubstituting this expansion into \\re{rel} we find that the power \ncorrections to the cross-section are described by matrix elements \nof the operators defined as derivatives of the Wilson line~\\ci{KS}. \nAlthough these operators are nonlocal in QCD, one can represent them as \nlocal operators in an effective nonabelian Bloch-Nordsieck theory in \nwhich Wilson line appears as a result of integration over effective\nquark fields. Solving the RG \nequation \\re{RG} and using \\re{ope} as a boundary condition we obtain \nthat the leading power correction to the Drell-Yan cross-section appears\nas~\\ci{KS}\n\\begin{equation}\n\\sigma_N^{\\rm power} = \\vev{i\\partial W(0)}N\/Q + {\\cal O}(N^2\/Q^2)\\,,\n\\lab{fin}\n\\end{equation}\nprovided that there is no any hidden symmetry which would prohibit the \nappearence of the linear $\\sim y_0$ term in the expansion of the Wilson \nline \\re{ope} and would enforce the matrix element\n$\\vev{i\\partial W(0)}$ to vanish. \n\n\\section{IR renormalon analysis}\n\nOne possibility to test the presence of the linear term in \\re{ope} is to\napply the IR renormalon approach and identify the source of linear\nterms inside perturbative series for the Wilson line. At the leading\norder of renormalon calculus~\\ci{B}, one can perform the explicit one-loop \ncalculation of $\\vev{W(y_0)}$ with a gluon mass $\\lambda$ and identify \nthe power corrections as nonanalytical terms in the small $(y_0\\,\\lambda)$ \nexpansion. One finds~\\ci{BB} that $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi \\mbox{ln} (y_0\\lambda)$ term is absent in \nagreement with the Bloch-Nordsieck cancellation, the linear term \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi (\\lambda y_0)$ also vanishes and the leading term is \n$\\sim\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi (\\lambda y_0)^2$. This means, that at the leading order the\ncontribution of IR renormalons has the form \\re{zero}.\nThe same result can be obtained in the \nlimit of large number of light quark flavours in which one performs 1-loop \ncalculation of $\\vev{W(y_0)}$ with the gluon propagator dressed by a \nchain of quark loops~\\ci{BB}. We notice that in both cases one calculates \nessentially abelian diagrams containing the color factors $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi C_F$ and \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi C_F (\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi N_f)^{n}$, respectively. It is well-known that the \ncontribution of abelian diagrams to the Wilson line exponentiates to \nhigher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$ and one can generalize the above statement \nabout the absence of linear $\\sim y_0$ term to a much larger class of \nabelian diagrams beyond the leading order of the renormalon \ncalculus. However, the natural question rises whether the abelian\ndiagrams provide a meaningful approximation to the exact expression \nfor the Wilson line or may be there is the leading contribution\nto $W(y_0)$ coming only from nonabelian diagrams. To give an example\nwhere the second possibility is realized we mention the breakdown of\nthe Bloch-Nordsieck cancellation of IR logs in the Drell-Yan\ncross-section at twist $1\/Q^4$. The IR divergent part of \nthe cross-section has the form~\\ci{IR} \n$\\sim\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2 C_A C_F Q^{-4}\\mbox{ln} \\lambda^2$ with $C_A (C_F)$ \nbeing gluon (quark) Casimir operators. It does not appear at order \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$ and at order $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2$ it comes only from diagrams with nonabelian \ncolor structure. One might expect that similar phenomenon may happen when \none considers linear in $y_0$ contributions to the Wilson line beyond \nthe leading order. \n\nThe straighforward way to check this possibility would be to perform \n2-loop calculation of the Wilson line with a power occuracy in\n$\\lambda$. However, even without going through this complicated\ncalculation one can employ the operator methods of the QCD asymptotic \ndynamics~\\ci{CCM} in order to analyse the properties of the matrix element\n\\begin{equation}\n\\vev{\\partial W(0)}=\\frac1{N_c}\\mbox{tr\\,}\n\\vev{\n\\widebar T U_{{\\rm _{DY}}}^\\dagger(0|A)\\,\\partial\\, T U_{{\\rm _{DY}}}(0|A)\n}\\,,\n\\lab{dW}\n\\end{equation}\nparameterizing the leading $N\/Q$ power correction to the Drell-Yan\ncross-section. The operator $U_{{\\rm _{DY}}}(y_0|A)$ describes the eikonal\nphase of fast quark and antiquark annihilating at the space-time \npoint $y_0$ and $T-$ordering of gauge field operators $A_\\mu^a(x)$\nis needed in \\re{W} and \\re{dW} to describe properly the possibility \nfor quark and antiquark to exchange by a virtual soft gluon in the\ninitial state. As a result, due to effects of the initial state \ninteraction~\\ci{CCM}\n$$\nTU_{{\\rm _{DY}}}(y_0|A)\\neq U_{{\\rm _{DY}}}(y_0|A)\\,.\n$$\nThis does not happen however for single-parton initiated process like \nDIS and it is this phenomenon that makes different the properties of \npower corrections in DY and DIS. Indeed, if we omit the $T-$ordering \nin \\re{dW} then it is easy to show by direct calculation that the derivative\n$U_{{\\rm _{DY}}}^\\dagger\\partial U_{{\\rm _{DY}}}$ is proportional to the \ngenerators of the $SU(N_c)$ group and it does not contribute to \n\\re{dW}. To work out the effects of quark-antiquark correlations in\n$\\vev{W(y_0)}$ one defines two gauge field operators\n$A_\\mu^\\pm(x)=\\frac12(A_\\mu(x)\\pm A_\\mu(-x))$ and notices~\\ci{CCM}\nthat the field $A_\\mu^+$ describes the Coulomb gluons while the field \n$A_\\mu^-$ is associated with radiative gluons forming quark and\nantiquark coherent states. Since $[A^{a,\\pm}_\\mu(x),A^{b,\\pm}_\\nu(y)]=0$ and\n$[A^{a,\\pm}_\\mu(x),A^{b,\\mp}_\\nu(y)]\\neq 0$, the expansion of the matrix\nelement $\\vev{n|TU_{{\\rm _{DY}}}(y_0|A^++A^-)|0}$ takes the following form~\\ci{CCM}\n\\begin{eqnarray}\n&&\\hspace*{-5mm}\n\\vev{n|TU_{{\\rm _{DY}}}^{ij}\n(y_0|A^++A^-)|0}\n=[\\mbox{e}^{i\\Phi_C}]^{ij}_{i'j'}\n\\lab{V}\n\\\\\n&&\\hspace*{5mm}\n\\times\n\\vev{n|\nU_{{\\rm _{DY}}}^{i''j''}(y_0|A^-)\nV_{i''j''}^{i'j'}(y_0|A^+,A^-)\n|0} \\,,\n\\nonumber\n\\end{eqnarray}\nwhere the flow of quark color is indicated explicitly.\nHere, in the r.h.s.\\ the first factor is the nonabelian unitary\nCoulomb phase matrix and the eikonal phase $U_{{\\rm _{DY}}}$ depends\nonly on commutative operators $A^-$. The residual\nfactor $V$ is unitary, $V^\\dagger V=\\hbox{{1}\\kern-.25em\\hbox{l}}$, and it\ndescribes the correlations between Coulomb and radiative\ngluons. The perturbative expansion of $V$ looks like~\\ci{CCM}\n$$\nV=\\hbox{{1}\\kern-.25em\\hbox{l}}+ig^3 f^{abc}t^a\\otimes t^b\\!\\int\\!\\! d^4x\\, \nG(x,y_0) A_\\mu^{c,+}(x)+{\\cal O}(g^4)\n$$\nwhere effective nonlocal coupling $G(x,y_0)$ resembles the \nFadin-Kuraev-Lipatov vertex. In abelian theory $V=\\hbox{{1}\\kern-.25em\\hbox{l}}$ to all orders, \nbut in QCD it gets {\\it nonabelian} corrections starting from order \n$g^3$ in the strong coupling constant. \n\nSubstituting \\re{V} into \\re{dW} one observes that \nnonabelian Coulomb phase is canceled. Moreover, if we neglect \n$g^3-$corrections to $V$, then due to unitarity and commutativity of \n$U_{{\\rm _{DY}}}(y_0|A^-)$, the operator $U_{{\\rm _{DY}}}^\\dagger\\partial U_{{\\rm _{DY}}}$\nis again proportional to the $SU(N_c)$ generators and its matrix element\nvanishes identically in Eq.\\re{dW}. Thus, a nonzero contribution to \n\\re{dW} comes only from higher order terms in $V$. It immediately \nfollows from the explicit form of $V$ that the corresponding Feynman \ndiagrams should have a nonabelian color structure and they arise \nstarting from $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2$ order. The set of relevant 2-loop Feynman \ndiagrams was identified in Ref.~\\ci{CCM}. This also explains why \nlinear term did not appear in 1-loop calculation of the Wilson line \nperformed in Ref.~\\ci{BB}.\n\nThe fact that the contribution of the residue factor $V$ to the Wilson \nline $W(y_0)$ is nonzero was demonstrated by a direct calculation \nin Ref.~\\ci{CCM}, in which a small gluon mass $\\lambda$ was used as an \nIR cutoff and only logarithmic terms, $\\sim\\mbox{ln}^k\\lambda$, were kept. \nIt was shown that the leading term $W(0)$ of the expansion of the\nWilson line in Eq.\\re{ope} gets a higher twist correction \n$\\sim\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2 C_FC_A Q^{-4}\\mbox{ln}\\lambda$, which leads to the breakdown of \nBloch-Nordsieck theorem for the Drell-Yan cross-section but is \nnevertheless consistent with the Kinoshita-Lee-Nauenberg (KLN) theorem. \nTo find the linear term in \\re{ope} one has to perform the\ncalculation of the same diagrams with a power \naccuracy in $\\lambda$, although interpretation of the linear term\n$\\sim\\lambda$ to order $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2$ as a ``true'' power correction is not \nobvious. One may dress instead the gluon propagators by a single \nchain of quark loops, resum all 2-loop diagrams of order \n$\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi^2 C_FC_A(\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi N_f)^k$ in the large $N_f$ limit and identify the \ncontribution of the leading IR renormalon. \n\n\\section{Conclusions}\n\nWe conclude that the effects of the initial state interaction, that is\ncorrelations between Coulomb and radiative gluons, provide a new source \nof $N\/Q$ power corrections in the Drell-Yan process. In the Wilson\nline approach they are described by the matrix element of the \nderivative of the Wilson line operator in Eq.\\re{fin}. In the IR\nrenormalon analysis these effects could manifest themselves only \nin nonabelian Feynman diagrams at higher orders in $\\ifmmode\\alpha_{\\rm s}\\else{$\\alpha_{\\rm s}$}\\fi$.\nThe presence of $N\/Q$ corrections in the Drell-Yan cross-section is\nconsistent with the KLN theorem. Summation over degenerate initial \npartonic states in \\re{W} and \\re{rel} \nleads to the expression for the KLN cross-section, $\\sigma_N^{\\rm _{KLN}}=\n\\sum_k \\vev{k|W(iN\/Q)|k}$, where sum goes over an arbitrary number of\nincoming soft gluons. As was shown in Ref.~\\ci{CCM}, the initial \nstate interaction effects can be analysed in $\\sigma_N^{\\rm _{KLN}}$ on \nthe same footing as the final state effects thus resulting in the \ncancellation of $N\/Q$ corrections in $\\sigma_N^{\\rm _{KLN}}$.\n\n\\section*{Acknowledgements}\n\nThe author is most grateful to George Sterman for useful discussions.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{section.01.introduction}\n{\\it \nWe plan to start from the refined topological vertex and obtain Virasoro $A$-series minimal model \nconformal blocks times Heisenberg factors.\n}\n\n\\medskip\n\n\\subsection{Background and basic concepts}\n\n\\subsubsection{Topological strings} Perturbative string theory, seen from a world-sheet point of view, \nis a 2D conformal field theory, coupled to 2D gravity, on Riemann surfaces. If the 2D conformal field \ntheory is topological, in the sense that the correlation functions are independent of the metrics on \nthe Riemann surfaces, the resulting string theory is {\\it topological}. For an introduction, we refer \nto \\cite{vafa.neitzke}, and references therein.\nThere are two constructions of topological string theories, the A-model and the B-model. For the \npurposes of this note, it suffices to say that we work are in the context of the A-model, and that \nthe target space, that the strings propagate in, is $\\mathbb{R}^{\\, 1, 3} \\times \\mathcal{X}$, where $\\mathcal{X}$ \nis a toric Calabi-Yau complex 3-manifold. For suitable choices of $\\mathcal{X}$, topological strings are \nnon-trivial but tractable, and one can compute their partition functions. For further details, we \nrefer to the reviews \\cite{marino.2005, marino.book}, and references therein.\n\n\\subsubsection{From topological strings to 5D gauge theories} Topological string partition functions \nare interesting in themselves, and for yet another, an {\\it apriori} unexpected reason. Namely, for \nsuitable choices of the Calabi-Yau manifold $\\mathcal{X}$, one can compute the topological string partition \nfunction $\\mathcal{Z}^{\\, top}$, and moreover, identify the result with the instanton partition function \n$\\mathcal{Z}^{\\, 5D}_{instanton}$ of a corresponding 5D $\\mathcal{N} \\! = \\! 2$ supersymmetric quiver gauge theory,\nthereby {\\it \\lq geometrically-engineering\\rq} the latter theory \n\\cite{geometric.engineering.01, geometric.engineering.02}. \n\n\\subsubsection{Refined partition functions} A 5D instanton partition function depends, in general, \non two deformation parameters, $\\epsilon_1$ and $\\epsilon_2$. \nFor $\\epsilon_1 + \\epsilon_2 = 0$, the instanton partition function is unrefined. \nFor $\\epsilon_1 + \\epsilon_2 \\neq 0$, the instanton partition function is refined. \nGiven the identification of 5D \ninstanton partition function and topological strings, the latter also depend, in general on \nthe deformation parameters, $\\epsilon_1$ and $\\epsilon_2$.\nFor $\\epsilon_1 + \\epsilon_2 = 0$, the topological string partition function is unrefined.\nFor $\\epsilon_1 + \\epsilon_2 \\neq 0$, the instanton partition function is refined. \n\n\\subsubsection{The refined topological vertex} To compute topological string partition functions,\none splits the full partition function into basic building blocks, computes the contribution of each \nbuilding block, then combines the contributions to obtain the required result.\nIn the unrefined case, $\\epsilon_1 + \\epsilon_2 = 0$, this was achieved in \\cite{topological.vertex}. \nIn this case, the building blocks are copies of the original, unrefined {\\it \\lq topological vertex\\rq} \nintroduced in \\cite{topological.vertex}.\nIn the refined case, $\\epsilon_1 + \\epsilon_2 \\neq 0$, this was achieved in \n\\cite{awata.kanno.01, awata.kanno.02, ikv}\nIn this case, the building blocks are copies of the {\\it \\lq refined topological vertex\\rq} \n\\cite{awata.kanno.01, awata.kanno.02, ikv}. \n\n\\subsubsection{From 5D to 4D gauge theories} \nA 5D quiver gauge theory instanton partition function, $\\mathcal{Z}^{\\, 5D}_{instanton}$, depends on the radius \n$R$ of a space-like circle. In the limit $R \\rightarrow 0$, $\\mathcal{Z}^{\\, 5D}_{instanton}$ reduces to \na corresponding 4D instanton partition function $\\mathcal{Z}^{\\, 4D}_{instanton}$ \n\\cite{nekrasov, hollowood.iqbal.vafa, ikv, eguchi.kanno, zhou, taki.01}.\nThe deformation parameters $\\epsilon_1$ and $\\epsilon_2$ of the 5D theory are inherited by the 4D theory.\n\n\\subsubsection{From 4D gauge theories to generic 2D conformal field theories} The 4D instanton partition \nfunctions $\\mathcal{Z}^{\\, 4D}_{instanton}$ are identified {\\it via} the AGT correspondence with 2D Virasoro \ngeneric conformal blocks times Heisenberg factors, $\\mathcal{B}^{\\, gen, \\, \\mathcal{H}}$ \\cite{agt}. In its original \nformulation, the AGT correspondence applies only to generic, that is non-minimal conformal field \ntheories. From 4D instanton partition functions with $\\epsilon_1 + \\epsilon_2 = 0$, one obtains\nconformal blocks of the conformal field theory of a Gaussian free field at the free fermion point.\nFrom 4D instanton partition functions with $\\epsilon_1 + \\epsilon_2 \\neq 0$, one obtains\nconformal blocks of non-Gaussian, but also non-minimal conformal field theories.\n\n\\subsubsection{From generic conformal field theories to minimal models} One can choose the parameters \nthat appear in $\\mathcal{Z}^{\\, 4D}_{instanton}$, and restrict the states that are allowed as intermediate states, \nin such a way that one obtains restricted 4D instanton partition functions $\\mathcal{Z}^{\\, 4D, \\, min}_{instanton}$ \nthat can be identified with conformal blocks in Virasoro $A$-series minimal models times Heisenberg factors \n\\cite{alkalaev.belavin, bershtein.foda}. In particular, as we will show in the sequel, choosing \n$\\epsilon_1 = - \\sqrt{ p \/ p^{\\prime}}$, and \n$\\epsilon_2 = \\sqrt{ p^{\\prime} \/ p}$, \nwhere $p$ and $p^{\\prime}$ are two co-prime positive integers, $0 < p < p^{\\prime}$, one obtains conformal blocks in \na minimal model parameterised by $p$ and $p^{\\prime}$. \n\n\\subsection{Plan of this work}\nFrom the above chain of connections, it is expected that one can start from $\\mathcal{Z}^{\\, ref\\, top}$, \nchoose the parameters and restrict the intermediate states to obtain 5D instanton partition functions \n$\\mathcal{Z}^{\\, 5D, \\, min}_{instanton}$, reduce to the corresponding $\\mathcal{Z}^{\\, 4D, \\, min}_{instanton}$, and \ncompute minimal model conformal blocks from the latter. \nIn this note, we work out the above chain of connections, which amounts to extending the result \nof \\cite{alkalaev.belavin, bershtein.foda} by starting from topological strings and topological \nvertices rather than from 4D instanton partition functions. We glue four refined topological \nvertices to obtain the building block of the 5D instanton partition functions, and take the 4D \nlimit of the latter to obtain the building block of the 4D instanton partition functions\n$\\mathcal{Z}^{\\, 4D}_{building.block}$. From that point on, we use the results of \\cite{alkalaev.belavin, \nbershtein.foda} to obtain conformal blocks in minimal models.\n\nSince $\\mathcal{Z}^{\\, 4D}_{building.block}$, which can be regarded as the starting point of the results \nin \\cite{alkalaev.belavin, bershtein.foda}, is constructed here from refined topological vertices, \nthe construction in this note is, in this sense, more basic than that in \\cite{alkalaev.belavin, \nbershtein.foda}. \n\nThe Virasoro $A$-series minimal conformal blocks that we obtain can be computed using other methods, \nbut we view this note as an accessible introduction to one more approach to the minimal conformal \nblocks that, hopefully, can be extended to minimal blocks beyond what can currently be computed, \nincluding the $W_N$ minimal blocks that do not satisfy the conditions of \\cite{fateev.litvinov, wyllard}\n\\footnote{\\,\nExpressions for the 3-point functions of $W_N$ Toda theory were proposed, starting from topological \nstrings, in \\cite{mitev.01}, and checked in \\cite{mitev.02}. These results should be useful in \nsubsequent studies of $W_N$ minimal models.\n}. \n\n\\subsection{Outline of contents}\nTo simplify the presentation, we divide the content into short sections.\nIn {\\bf \\ref{section.02.refined.topological.vertex}}, we recall basic definitions related to Young \ndiagrams and Schur functions, followed by\nthe definition of the refined topological vertex $\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu}[q, t]$ of Iqbal, \nKozcaz and Vafa \\cite{ikv}, where $\\lambda$, $\\mu$, and $\\nu$ are Young diagrams, $q$ and $t$ are \nparameters.\nIn {\\bf \\ref{section.03.5d.web}}, we glue four copies of $\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu} [q, t]$ \nto build a $U(2)$ basic web partition function $\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta} [q, t, R]$, where \neach of ${\\bf V}$ and ${\\bf W}$ is a pair of Young diagrams, and \n${\\bf \\Delta} = \\{\\Delta_1, \\Delta_{ { \\scriptstyle M} }, \\Delta_2 \\}$ are K{\\\"a}hler parameters.\nIn {\\bf \\ref{section.04.4d.web}}, we set $q= e^{\\, R \\epsilon_2}$, $t= e^{\\, - R \\epsilon_1}$, where $\\epsilon_1$ and \n$\\epsilon_2$ are Nekrasov's regularisation parameters, then take the limit $R \\rightarrow 0$, to obtain \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta} [\\epsilon_1, \\epsilon_2, R \\! \\rightarrow \\! 0]$.\nIn {\\bf \\ref{section.05.nekrasov.partition.functions}}, we recall the definition of the normalised \ncontribution of the bifundamental hypermultiplet, \n$\\mathcal{Z}_{building.block}^{\\, 4D}$ to $\\mathcal{Z}^{\\, 4D}_{instanton}$. \nIn {\\bf \\ref{section.06.identification.numerators}}, we compare the numerators of \n$\\mathcal{W}^{\\, norm}_{\\bf V, W, \\Delta}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$, and identify the K{\\\"a}hler \nparameters with gauge theory parameters.\nIn {\\bf \\ref{section.07.identification.denominators}}, we compare the denominators of \n$\\mathcal{W}^{\\, norm}_{\\bf V, W, \\Delta}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$, and show that there is \na normalisation such that the two denominators agree, without changing the results of the computations \nof the instanton partition functions.\nIn {\\bf \\ref{section.08.restricted.instanton.partition.functions}}, we recall the choice of parameters \nthat allows us to use $\\mathcal{Z}^{\\, 4D}_{building.block}$ to build 4D instanton partition functions that can \nbe identified with Virasoro $A$-series minimal model conformal blocks times Heisenberg factors.\nIn {\\bf \\ref{section.09.from.gauge.theory.to.minimal.model}}, we identify the parameters of \n$\\mathcal{W}^{\\, norm}_{\\bf V \\, W \\, \\Delta}$ and of $\\mathcal{Z}^{\\, 4D}_{building.block}$. \nIn {\\bf \\ref{section.10.burge.pairs}}, we outline a proof, following \\cite{bershtein.foda}, of \nthe statement that we need to impose Burge conditions on the partition pairs that appear in our \nconstructions, to make the topological string partition functions free of non-physical singularities, \nonce we choose the parameters to coincide with those of the minimal model.\nSection {\\bf \\ref{section.11.comments}} includes comments and remarks. \n\\subsection*{Abbreviations and notation} \nWe focus on $U(2)$ quiver gauge theories, and simply say \n{\\it \\lq the instanton partition functions\\rq\\,}.\nWe assume that every conformal block, whether Liouville or minimal, \nincludes a factor from a field theory \nof a free boson on a line, and omit {\\it \\lq times Heisenberg factors\\rq}. \nThe normalised contribution of the bifundamental hypermultiplet is understood as the \nbuilding block of the instanton partition function, as all other contributions can be \nobtained from it. We simply say {\\it \\lq the bifundamental partition function\\rq\\,}.\n\n\\subsubsection*{Topological string-related notation}\n$\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu} [q, t]$ is the refined topological vertex.\n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta} [q, $ $t, R]$ is a basic web refined topological \nstring partition function.\n${\\bf V}$ and ${\\bf W}$ are two pairs of Young diagrams. \n${\\bf \\Delta}$ is a set of three K{\\\"a}hler parameters. \n$q$ and $t$ are deformation parameters. \n$R$ is the radius of a space-like circle\n\\footnote{\\,\nThe 5D $U(2)$ quiver theories in this note live on D5-branes in time-like $x^0$, and \nspace-like $x^1, x^2, x^3, x^5$ and $x^6$ dimensions. Following \\cite{bao.01}, we take \n$x^5$ to be a circle of radius $R$, $\\beta$ in \\cite{bao.01}, such that setting \n$R \\rightarrow 0$ is equivalent to taking the 4D limit. For a complete discussion, we \nrefer to \\cite{bao.01}.\n}. \n$\\mathcal{Z}^{\\, ref\\, top}$ is the topological string partition function.\n\n\\subsubsection*{Gauge theory-related notation}\n$\\mathcal{Z}^{\\, 5D}_{instanton}$ is the 5D $U(2)$ quiver gauge theory instanton partition functions.\n$\\mathcal{Z}^{\\, 4D}_{instanton}$ is the 4D $U(2)$ quiver gauge theory instanton partition functions.\n$\\mathcal{Z}_{building.block}^{\\, 4D}$ is the normalised contribution of the bifundamental hypermultiplet.\nThe parameters $\\epsilon_1$ and $\\epsilon_2$ are deformation parameters.\n\n\\subsubsection*{Conformal field theory-related notation}\n$\\mathcal{B}^{\\, gen, \\, \\mathcal{H}}$ are 2D Virasoro generic conformal blocks times Heisenberg factors.\n$\\mathcal{B}^{\\, gen, \\, min}$ are 2D Virasoro A-series minimal model conformal blocks times Heisenberg \nfactors.\nThe parameters $p$ and $p^{\\prime}$ are coprime positive integers that label a Virasoro minimal model. \nThe parameters $r$ and $s$ are integers that satisfy $0 < r < p$, $0 < s < p^{\\prime}$ and label Virasoro \nminimal model highest weight representations.\n\n\\section{The refined topological vertex}\n\\label{section.02.refined.topological.vertex}\n{\\it We recall basic definitions related to Young diagrams and Schur functions, followed by the \ndefinition of the refined topological vertex.}\n\n\\subsection{Partitions and Young diagrams}\n\n\\subsubsection{Partitions}\n\\label{partitions} A partition $\\pi$ of a non-negative integer $|\\pi|$\nis a set of non-negative integers $\\{\\pi_{1},\\pi_{2},$ $\\cdots,\\pi_{p}\\}$,\nwhere $p$ is the number of parts, $\\pi_{i}\\geqslant\\pi_{i+1}$, and $\\sum_{i=1}^{p} \\pi_{i}= | \\pi |$.\n\n\\subsubsection{Young diagrams}\n\\label{young.diagrams} A partition $\\pi$ is represented as a Young diagram $Y$, as in \nFigure {\\bf \\ref{A.Young.diagram}}, which is a set of $p$ rows $\\{y_1, y_2, \\cdots, y_p \\}$, such \nthat row-$i$ has $y_i = \\pi_{i}$ cells\n\\footnote{\\, \nWe use $y_i$ for $i$-th row as well as for the number of cells in that row.\n}, \n$y_i \\geqslant y_{i+1}$, and $\\sum_i y_i = |Y| = |\\pi|$. We use $Y^{\\intercal}$ for the transpose of $Y$.\n\n\\subsubsection{Cells}\n\nWe use $\\square$ for a cell, or a square in the south-east quadrant\nof the plane, and refer to the coordinates of $\\square$ as $\\{ {\\tt \\scriptstyle R} , {\\tt \\scriptstyle C} \\}$.\nIf $\\square$ is inside a Young diagram $Y$, then $ {\\tt \\scriptstyle R} $ is the\n$Y$-row-number, counted from top to bottom, and $ {\\tt \\scriptstyle C} $ is the\n$Y$-column-number, counted from left to right, that $\\square$ lies\nin. If $\\square$ is outside $Y$, we still regard $ {\\tt \\scriptstyle R} $ as a $Y$-row-number,\nalbeit the length of this row is zero, and we still regard $ {\\tt \\scriptstyle C} $\nas a $Y$-column-number, albeit the length of this column is zero.\n\n\\subsubsection{Arms and legs, half-extended and extended}\n\\label{arms.legs} Consider a cell $\\square$ that has coordinates $\\{ {\\tt \\scriptstyle R} , {\\tt \\scriptstyle C} \\}$. \nWe define the lengths of \nthe arm $A^{ }_{\\square, Y}$, \nhalf-extended arm $A^{+ }_{\\square, Y}$, \n extended arm $A^{++}_{\\square, Y}$, \nthe leg $L^{ }_{\\square, Y}$,\nhalf-extended leg $L^{+ }_{\\square, Y}$, \n extended leg $L^{++}_{\\square, Y}$, of $\\square$ \nwith respect to the Young diagram $Y$, to be\n\n\\begin{eqnarray}\nA^{ }_{\\square, Y} & = & y_{ {\\tt \\scriptstyle R} } - \\, {\\tt \\scriptstyle C} , \\quad\nA^{+ }_{\\square, Y} = A^{ }_{\\square, Y} + \\frac12, \\quad \nA^{++}_{\\square, Y} = A^{ }_{\\square, Y} + 1, \n\\\\\nL^{ }_{\\square, Y} & = & y_{ {\\tt \\scriptstyle C} }^{\\intercal} - \\, {\\tt \\scriptstyle R} , \\quad\nL^{+ }_{\\square, Y} = L^{ }_{\\square, Y} + \\frac12, \\quad \nL^{++}_{\\square, Y} = L^{ }_{\\square, Y} + 1,\n\\end{eqnarray}\n\n\\subsubsection{Remark} $A_{\\square,Y}$ and $L_{\\square,Y}$ can be negative when $\\square$ lies \noutside $Y$. \n\n\\begin{figure}\n\\begin{tikzpicture}[scale=.8]\n\\draw [thick] (0, 0) rectangle (1,1);\n\\draw [thick] (1, 0) rectangle (1,1);\n\\draw [thick] (2, 0) rectangle (1,1);\n\\draw [thick] (3, 0) rectangle (1,1);\n\\draw [thick] (4, 0) rectangle (1,1);\n\\draw [thick] (5, 0) rectangle (1,1);\n\\draw [thick] (0,-1) rectangle (1,1);\n\\draw [thick] (1,-1) rectangle (1,1);\n\\draw [thick] (2,-1) rectangle (1,1);\n\\draw [thick] (3,-1) rectangle (1,1);\n\\draw [thick] (4,-1) rectangle (1,1);\n\\draw [thick] (0,-2) rectangle (1,1);\n\\draw [thick] (1,-2) rectangle (1,1);\n\\draw [thick] (2,-2) rectangle (1,1);\n\\node at (1.5,-0.5) {$\\checkmark$};\n\\end{tikzpicture}\n\\caption{\n{\\it\nThe Young diagram $Y$ of $5 + 4 + 2$.\nThe rows are numbered from top to bottom. \nThe columns are numbered from left to right. \nThe checkmarked cell has \n$A^{ } = 2$, \n$A^{+ } = \\frac52$, \n$A^{++} = 3$, \n$L^{ } = 1$, {\\it etc.}\n}\n}\n\\label{A.Young.diagram}\n\\end{figure}\n\n\\subsubsection{Partition pairs}\nA partition pair ${\\bf Y}$ is a set of two Young diagrams, $\\{ Y^{\\, 1}, Y^{\\, 2} \\}$, as in \nFigure {\\bf \\ref{A.partition.pair}}, where $|\\bf Y| = |Y^{\\, 1}| + |Y^{\\, 2}|$ is the total \nnumber of cells in ${\\bf Y}$.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=.8] \n\\draw [thick] (0, 0) rectangle (1, 1); \n\\draw [thick] (1, 0) rectangle (2, 1);\n\\draw [thick] (2, 0) rectangle (3, 1);\n\\draw [thick] (3, 0) rectangle (4, 1);\n\\draw [thick] (4, 0) rectangle (5, 1);\n\\draw [thick] (0,-1) rectangle (1, 0);\n\\draw [thick] (1,-1) rectangle (2, 0);\n\\draw [thick] (2,-1) rectangle (3, 0);\n\\draw [thick] (3,-1) rectangle (4, 0);\n\\draw [thick] (0,-2) rectangle (1,-1);\n\\draw [thick] (1,-2) rectangle (2,-1);\n\\draw [thick] (7, 0) rectangle (8, 1); \n\\draw [thick] (8, 0) rectangle (9, 1); \n\\draw [thick] (9, 0) rectangle (10,1); \n\\draw [thick] (10,0) rectangle (11,1); \n\\draw [thick] (7,-1) rectangle (8, 0); \n\\draw [thick] (7,-2) rectangle (8,-1); \n\\node at (3.5,-0.5) {$\\checkmark$};\n\\node at (10.5,-0.5) {$\\checkmark$};\n\\end{tikzpicture}\n\\caption{\n{\\it \nA partition pair $\\{\\mu, \\nu\\}$. $\\mu$ is on the left, $\\nu$ is on the right. \nThe checkmarked cell at coordinates $(2, 4)$, in the lower right quadrant of \nthe plane, is in $\\mu$, but not in $\\nu$, and has\n$A^{ }_{\\mu} = 0$,\n$A^{+ }_{\\mu} = \\frac12$,\n$A^{++}_{\\mu} = 1$,\n$L^{ }_{\\mu} = 0$,\n$L^{+ }_{\\mu} = \\frac12$,\n$L^{++}_{\\mu} = 1$,\nas well as\n$A^{ }_{\\nu} = -3$, {\\it etc.} and\n$L^{ }_{\\nu} = -1$, {\\it etc.} \n}\n}\n\\label{A.partition.pair}\n\\end{figure}\n\\bigskip\n\n\\subsubsection{Sum of row-lengths and squares of row-lengths} Given \na Young diagram $Y$, with rows of length $y_1 \\geqslant y_2 \\geqslant \\cdots$, \nwe define\n\n\\begin{equation}\n| Y | = \\sum_{i=1} y_i, \n\\quad\n\\parallel Y \\parallel = \\sum_{i=1} y_i^{\\, 2}\n\\end{equation}\n\n\\noindent where the sum is over all parts of $Y$\n\\footnote{\\\nIn \\cite{ikv}, Iqbal {\\it et al.} use \n$\\parallel Y \\parallel^{\\, 2} = \\sum_{i=1} y_i^{\\, 2}$. We define $\\parallel Y \\parallel = \\sum_{i=1} y_i^{\\, 2}$ \nto simplify the notation. This is consistent with other notation and should cause no confusion.\n}.\nFurther, to simplify the equations, we define\n\n\\begin{equation}\n| Y |^{\\prime} = \\frac12 \\sum_{i=1} y_i,\n\\quad\n\\parallel Y \\parallel^{\\prime} = \\frac12 \\sum_{i=1} y_i^{\\, 2}\n\\end{equation}\n\n\\subsubsection{Sequences} Given the sequence of row-lengths \n$ \\lambda = \\{ \\lambda_1, \\lambda_2, \\cdots \\}$, the sequence of half-integers\n$ \\rho = \\{ - \\frac12, - \\frac32, \\cdots \\}$, and two variables $x$ and $y$, \nwe define the \\lq exponentiated\\rq\\, sequences \n\n\\begin{equation}\nx^{-\\lambda}= \\{ x^{-\\lambda_1}, x^{-\\lambda_2}, \\cdots \\}, \n\\quad\ny^{\\, -\\rho}= \\{ y^{\\, \\frac12}, y^{\\, \\frac32}, \\cdots \\},\n\\quad\n\\textit{and}\n\\quad\nx^{\\, -\\lambda} y^{\\, -\\rho} = \n\\{\nx^{\\, -\\lambda_1} \\, y^{\\, \\frac12}, \nx^{\\, -\\lambda_2} \\, y^{\\, \\frac32}, \n\\cdots \\}, \n\\end{equation}\n\n\\subsubsection{A function of the arm-lengths and the leg-lengths} Given a Young diagram \n$\\lambda$, we define the function\n\n\\begin{equation}\n\\label{z.factor}\nZ_{\\lambda} [q, t]\n= \\prod_{\\square \\in \\lambda} \n\\frac{1}{ \\left\\lgroup 1 - \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\, \\frac12} \\, \nq^{\\, A_{\\square, \\lambda}^{+ }} \\, \nt^{\\, L_{\\square, \\lambda}^{+ }} \n\\right\\rgroup}\n= \\prod_{\\square \\in \\lambda} \\frac{1}{ \\left\\lgroup 1- \\, q^{\\, A_{\\square, \\lambda}^{++}}\\, \n t^{\\, L_{\\square, \\lambda}^{ }} \\right\\rgroup}\n\\end{equation}\n\n\\subsubsection{Remark} \nThe expression in the middle of Equation {\\bf \\ref{z.factor}} corresponds to \nsplitting the cell $\\square$ at the corner of a hook in the partition $\\lambda$ into two halves, then\nattaching one half to the arm of that hook to form {\\it a half-extended arm} of length \n$A^{+ }_{\\square, \\lambda} = A^{ }_{\\square, \\lambda} + \\frac12$, and \nattaching the other half to the leg of that hook to form {\\it a half-extended leg} of length \n$L^{+ }_{\\square, \\lambda} = L^{ }_{\\square, \\lambda} + \\frac12$. \nThe expression on the right corresponds to \nattaching the cell $\\square$ at the corner of a hook to the arm of that hook to form \n{\\it an extended arm} of length \n$A^{++}_{\\square, \\lambda} = A^{ }_{\\square, \\lambda} + 1$. The length of the leg of that hook remains \n$L^{ }_{\\square, \\lambda}$. \n\n\\subsection{Schur and skew-Schur functions}\nGiven an $n$-row Young diagram $\\lambda$, with parts $\\lambda_1 \\geqslant \\lambda_2 \\geqslant \\cdots$, \nand a set of $n$ variables $\\{x_1, x_2, \\cdots, x_n \\}$, the Schur function $s_{\\lambda} [{\\bf x}]$\nis defined as\n\\begin{equation}\ns_{\\lambda}[{\\bf x}] =\n\\frac{\ndet \\left\\lgroup x_i^{\\, \\lambda_j+n-j} \\right\\rgroup_{1 \\leqslant i, j \\leqslant n}\n}\n{\n\\prod_{1 \\leqslant i < j \\leqslant n} \\left\\lgroup x_i - x_j \\right\\rgroup\n}\n\\end{equation}\nThe skew-Schur function $s_{\\lambda\/\\mu}[{\\bf x}]$ is defined as\n\\begin{equation}\ns_{\\lambda\/\\mu}[{\\bf x}]=\\sum_{\\nu}c_{\\mu \\nu}^{\\, \\lambda}\\ \ns_{\\lambda}[{\\bf x}]\n\\end{equation}\n\\noindent where $c_{\\mu \\nu}^{\\, \\lambda}$ are Littlewood-Richardson coefficients defined by\n\\begin{equation}\ns_{\\mu}\\ \ns_{\\nu}=\\sum_{\\lambda}\\ c_{\\mu \\nu}^{\\, \\lambda}\\ \ns_{\\lambda}\n\\end{equation}\n\n\\subsection{Topological vertices}\nOur story starts from A-model closed topological string theory on non-compact Calabi-Yau \nthreefolds.\nWe cannot afford to review this vast subject and refer the reader to excellent available \nintroduction, including \\cite{marino.2005, marino.book}.\n\n\\subsubsection{The topological vertex of Aganagic {\\it et al}} \nIn \\cite{topological.vertex}, Aganagic, Klemm, Marino and Vafa introduced a systematic procedure \nto calculate A-model topological string partition functions on resolved conifolds. The main \ningredient of this procedure is the topological vertex, which has a combinatorial representation \nin terms of plane partitions with three boundaries specified by three Young diagrams, and can be \nschematically represented as a trivalent vertex, with bonds labelled by Young diagrams, as in \nFigure {\\bf \\ref{A.topological.vertex}}. For details, we refer to \\cite{topological.vertex}.\n\n\\subsubsection{The refined topological vertex}\nIn \\cite{ikv}, Iqbal, Kozcaz and Vafa defined the refined topological vertex, up to simple \nre-arrangements\n\\footnote{\\,\nWe use \n$\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu} [q, t]$, and\n$Z_Y [q, t]$, while Iqbal {\\it et al.} use\n$\\mathcal{C}_{\\, \\lambda \\, \\mu \\, \\nu} [t, q]$, and\n$\\tilde{Z}_{\\mu} [t, q]$.\n}, as \n\n\\begin{multline}\n\\label{refined.topological.vertex}\nC_{\\lambda \\, \\mu \\, \\nu} [q, t] =\n\\\\\nq^{\\, \\left\\lgroup \n | \\lambda |^{\\prime} \n - | \\mu |^{\\prime} \n + \\parallel \\mu \\parallel^{\\prime} \n + \\parallel \\nu \\parallel^{\\prime} \n \\right\\rgroup} \\, \nt^{\\, \\left\\lgroup - | {\\lambda}^{ } |^{\\prime} \n + | {\\mu }^{ } |^{\\prime} \n - \\parallel {\\mu }^{\\intercal} \\parallel^{\\prime}\n\\right\\rgroup} \\, \nZ_{\\nu}[q, t]\n \\left\\lgroup\n\\sum_{\\eta} \n \\left\\lgroup \n\\frac{q}{t} \n\\right\\rgroup^{\\,| \\eta |^{\\prime}}\n\\ s_{ {\\lambda}^{\\intercal} \/ \\eta}[ q^{\\,- \\nu } \\ t^{\\,- {\\rho}^{ }}]\n\\ s_{ {\\mu }^{ } \/ \\eta}[ q^{\\,- \\rho} \\ t^{\\,- {\\nu }^{\\intercal}}]\n\\right\\rgroup\n\\end{multline}\n\n\\noindent Note that the refined vertex is not manifestly cyclically-symmetric, in the sense\nthat the partitions that label the external legs do not appear on equal footing. In particular, \nthe partition $\\nu$ is distinguished from the other two. The external leg labelled by $\\nu$ is\nreferred to as {\\it \\lq preferred\\rq\\,}. The original, unrefined topological vertex of Aganagic \n{\\it et al.} is recovered by setting $q=t$.\n\n\\begin{figure}\n\\begin{tikzpicture}\n\\draw [ultra thick] (0.0,2.0)--(2.0,2.0);\n\\draw [ultra thick] (2.0,2.0)--(2.0,4.0);\n\\draw [ultra thick] (2.0,2.0)--(3.5,0.5);\n\\node [left] at (2.0,3.0) {$\\lambda$};\n\\node [left] at (2.8,1.0) {$\\mu$};\n\\node [above] at (1.0,2.0) {$\\nu$};\n\\end{tikzpicture}\n\\caption{\n{\\it \nThe topological vertex $C_{\\lambda \\, \\mu \\, \\nu} [q, t]$ is trivalent and depends \non two parameters $q$ and $t$. \nThe segments are labelled by three partitions $\\lambda, \\, \\mu$ and $\\nu$, such that \n$\\lambda$ is assigned to the vertical segment, $\\mu$ is assigned to the segment that \nfollows in a clockwise direction, and $\\nu$ to the segment that follows. \nThe \\lq preferred leg\\rq\\, is labelled by $\\nu$. \n} \n}\n\\label{A.topological.vertex}\n\\end{figure} \n\n\\subsubsection{On framing vectors and framing factors} \nIn defining the refined topological vertex, one labels each of the three boundaries of a vertex \nby {\\it a framing vector} that indicates a possible twisting of the boundary. On gluing two \nvertices along a common boundary, there is in general {\\it a framing factor} that accounts for \na possible mismatch in the orientations of the relevant framing vectors. In this note, we glue \nvertices such that we do not require framing factors. \n\n\\section{A 5D $U(2)$ basic web partition function}\n\\label{section.03.5d.web}\n{\\it We glue four copies of the refined topological vertex to obtain a 5D basic web that can be \nused as a building block of $U(2)$ topological string partition functions.}\n\n\\subsection{Geometric engineering} Following \\cite{geometric.engineering.01, geometric.engineering.02}, \nwe write the normalised $U(2)$ basic web partition function as\n\n\\begin{equation}\n\\label{W.norm}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\, norm} [q, t, R] = \n\\frac{\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{ } [q, t, R]\n}\n{\n\\mathcal{W}_{\\, \\bf \\emptyset\\, \\emptyset\\, \\Delta}[q, t, R]\n}\n\\end{equation}\n\n\\noindent where $\\emptyset$ is the trivial, or empty partition with no cells. The numerator is\n\n\\begin{multline}\n\\label{the.numerator}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]\n=\n\\sum_{\\xi_1, \\, \\xi_{ { \\scriptstyle M} }, \\, \\xi_2}\n \\left\\lgroup -Q_1 \\right\\rgroup^{\\, |\\xi_1 |} \\,\n \\left\\lgroup -Q_{ { \\scriptstyle M} } \\right\\rgroup^{\\, |\\xi_{ { \\scriptstyle M} } |} \\,\n \\left\\lgroup -Q_2 \\right\\rgroup^{\\, |\\xi_2 |} \n\\\\\n\\times\nC_{\\emptyset \\, \\xi_1 \\, V^{\\, 1} } [q, t] \\ \nC_{\\xi_{ { \\scriptstyle M} } \\, \\xi_1^{\\intercal} \\, W^{\\, 1 \\, \\intercal}} [t, q] \\ \nC_{\\xi_{ { \\scriptstyle M} }^{\\intercal} \\, \\xi_2 \\, V^{\\, 2} } [q, t] \\ \nC_{\\emptyset \\, \\xi_2^{\\intercal} \\, W^{\\, 2 \\, \\intercal}} [t, q]\n\\end{multline}\n\n\\noindent where we use\n\n\\begin{equation}\n\\boxed{\nQ_i = e^{\\, - R \\Delta_i}, \\quad i = 1, { { \\scriptstyle M} }, 2\n}\n\\end{equation}\n\n\\noindent and the denominator \n$\\mathcal{W}_{\\, \\bf \\emptyset \\, \\emptyset \\, \\Delta}[q, t, R]$ \nis identical to the numerator \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]$\nbut with all external partition pairs empty.\n\n\\begin{figure}\n\\begin{tikzpicture}[scale=.7]\n\\draw [ultra thick] (0,4)--(2,4);\n\\draw [ultra thick] (2,6)--(2,4);\n\\draw [ultra thick] (2,4)--(4,2);\n\\node [left, red] at (0,4) {$V^{\\, 1}$};\n\\node [right] at (3,3) {$\\xi_1, \\, \\Delta_1$};\n\\node [above] at (2,6) {$\\emptyset$};\n\\draw [ultra thick] (4,2)--(8,2);\n\\draw [ultra thick] (4,0)--(4,2);\n\\node [right, blue] at (8,2) {$W^{\\, 1}$};\n\\node [right] at (4,1) {$\\xi_{ { \\scriptstyle M} }, \\, \\Delta_{ { \\scriptstyle M} }$};\n\\draw [ultra thick] (0,0)--(4,0);\n\\node [left, red] at (0,0) {$V^{\\, 2}$};\n\\draw [ultra thick] (4,0)--(6,-2);\n\\node [right] at (5,-1) {$\\xi_2, \\, \\Delta_2$};\n\\draw [ultra thick] (6,-2)--(6,-4);\n\\draw [ultra thick] (6,-2)--(8,-2);\n\\node [right,blue] at (8,-2) {$W^{\\, 2}$};\n\\node [below] at (6,-4) {$\\emptyset$};\n\\end{tikzpicture}\n\\caption{\n{\\it \nThe $U(2)$ basic web diagram. Each external line is labeled by a partition. \nEach internal line is labeled by a partition and a K{\\\"a}hler parameter. This basic \nweb can be glued to form topological partition functions. The preferred legs are all\nexternal, and labelled by the partitions $V^1, V^2, W^1$ and $W^2$.\n}\n}\n\\label{basic.web}\n\\end{figure}\n\nAs shown in Figure {\\bf \\ref{basic.web}}, the basic web has two external horizontal legs \ncoming in from the left, two external horizontal legs going out to the right, and a pair \nof vertical legs, one going up and one down.\nThe horizontal external legs on the left are assigned partitions $\\{V^{\\, 1}, V^{\\, 2}\\}$, \nthe horizontal external legs on the right are assigned partitions $\\{W^{\\, 1}, W^{\\, 2}\\}$. \nThe internal lines are assigned parameters $ Q_1$, $ Q_{ { \\scriptstyle M} }$ and $ Q_2$, and partitions \n $\\xi_1$, $\\xi_{ { \\scriptstyle M} }$ and $\\xi_2$,\nfrom top to bottom. \nThe vertical external legs are assigned empty partitions. \n\nEach trivalent vertex corresponds to a refined topological vertex $\\mathcal{C}_{\\lambda \\, \\mu \\, \\nu}$. \nOur convention is such that each vertex has one vertical leg, we associate $\\lambda$ to that \nvertical leg, regardless of whether it is internal or external, pointing upwards or downwards, \nthen $\\mu$ and $\\nu$ to the remaining two legs, encountered sequentially as we start from the \nvertical leg and move around the vertex clockwise. \nUsing Equation {\\bf \\ref{refined.topological.vertex}}, we re-write the numerator in Equation \n{\\bf \\ref{the.numerator}} as\n\n\\begin{multline}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{ } [q, t, R]\n\\\\\nq^{\\, \\left\\lgroup \\parallel V^{\\, 1} \\parallel^{\\prime} + \\parallel V^{\\, 2} \\parallel^{\\prime} \\right\\rgroup}\nt^{\\, \\left\\lgroup \\parallel W^{\\, 1 \\, \\intercal} \\parallel^{\\prime} + \\parallel W^{\\, 2 \\, \\intercal} \\parallel^{\\prime} \\right\\rgroup}\nZ_{ V^{\\, 1} } [q, t]\\\nZ_{ W^{\\, 1 \\, \\intercal}} [t, q]\\\nZ_{ V^{\\, 2} } [q, t]\\\nZ_{ W^{\\, 2 \\, \\intercal}} [t, q]\n\\\\\n\\times \\,\n\\sum_{\\, \\xi_1, \\, \\xi_{ { \\scriptstyle M} }, \\, \\xi_2, \\, \\eta_1, \\, \\eta_2}\n \\left\\lgroup - Q_1 \\right\\rgroup^{\\, |\\xi_1 |} \n \\left\\lgroup - Q_{ { \\scriptstyle M} } \\right\\rgroup^{\\, |\\xi_{ { \\scriptstyle M} } |}\n \\left\\lgroup - Q_2 \\right\\rgroup^{\\, |\\xi_2 |}\n \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\, \\left\\lgroup | \\eta_2 |^{\\prime} - |\\eta_1 |^{\\prime} \\right\\rgroup}\n\\\\\n\\times\ns_{\\xi_1 } \\, [q^{\\, -\\rho} \\, t^{\\, -V^{\\, 1 \\, \\intercal}}] \\ \ns_{\\xi_1^{\\intercal} \/ \\eta_1} \\, [q^{\\, - W^{\\, 1}} \\, t^{\\, -\\rho }]\\\ns_{\\xi_{ { \\scriptstyle M} }^{\\intercal} \/ \\eta_1} \\, [q^{\\, -\\rho } \\, t^{\\, -W^{\\, 1 \\, \\intercal}}]\\ \n\\\\\n\\times\ns_{\\xi_{ { \\scriptstyle M} } \/ \\eta_2} \\, [q^{\\, - V^{\\, 2}} \\, t^{\\, -\\rho }] \\\ns_{\\xi_2 \/ \\eta_2} \\, [q^{\\, -\\rho} \\, t^{\\, -V^{\\, 2 \\, \\intercal}}]\\ \ns_{\\xi_2^{\\intercal} } \\, [q^{\\, - W^{\\, 2}} \\, t^{\\, -\\rho }] \\ \n\\label{web.02}\n\\end{multline}\n\n\\noindent where we used the fact that for an empty partition $\\emptyset$, the skew \npartition $\\emptyset \/ \\eta$ exists only for $\\eta = \\emptyset$, the sum over $\\eta$ \ntrivialises, and the skew Schur function $s_{\\emptyset \/ \\eta} = s_{\\emptyset} = 1$.\n\n\\subsubsection{Two skew Schur function identities}\nTo evaluate the sums in Equation {\\bf \\ref{web.02}} for $w^{\\, num}$, we need the two identities\n\\footnote{\\,\nExercise {\\bf 26}, page 93 of \\cite{macdonald.book}.}\n\n\\begin{eqnarray}\n\\sum_{\\lambda} \\, s_{\\lambda \/ \\eta_1}[x] \\, s_{\\lambda \/ \\eta_2}[y] \n& = & \n\\prod_{i, j} \\left\\lgroup 1-x_i y_j \\right\\rgroup^{-1} \n\\sum_{\\tau} \\, s_{\\eta_1 \/ \\tau}[y] \\, s_{\\eta_2 \/ \\tau} [x]\n\\label{schur.01}\n\\\\\n\\sum_{\\lambda} \\, s_{\\lambda \/ \\eta_1}[x] \\, s_{\\lambda^{\\intercal} \/ \\eta_2}[y] & = & \n\\prod_{i, j} \\left\\lgroup 1+x_i y_j \\right\\rgroup \n\\sum_{\\tau} \\, s_{\\eta_1^{\\intercal} \/ \\tau^{\\intercal}}[y] \\, s_{\\eta_2^{\\intercal}\/ \\tau} [x]\n\\label{schur.02}\n\\end{eqnarray}\n\n\\noindent as well as the property that\n\n\\begin{equation}\nQ^{\\, |\\lambda| - |\\eta| } \\, s_{\\lambda\/\\eta}[x]= s_{\\lambda \/ \\eta} \\, [Q x]\n\\end{equation}\n\n\\noindent which follows from the definition of the skew Schur function. \n\n\\subsubsection{The basic web in product form}\nWe evaluate the sums over the right hand side of Equation {\\bf \\ref{web.02}}, using \n$\\rho_i = - i + \\frac12$.\n\n\\subsubsection{The sums over $\\xi_1$ and $\\xi_2$}\n\n\\begin{multline}\n\\sum_{\\xi_1} \\left\\lgroup - Q_1 \\right\\rgroup^{\\,|\\xi_1|} \\, \ns_{\\xi_1 } [q^{\\, -\\rho} \\, t^{\\, -V^{\\, 1 \\, \\intercal}}]\\, \ns_{\\xi_1^{\\intercal} \/ \\eta_1} [q^{\\, - W^{\\, 1}} \\, t^{\\, -\\rho }]\n\\\\\n= \\prod_{i, j = 1}^{\\infty}\n \\left\\lgroup\n 1 - Q_1 \\, q^{\\, -W_i^{\\, 1} + j - \\frac12} \\, t^{\\, -V_j^{\\, 1 \\, \\intercal} + i - \\frac12}\n\\right\\rgroup \\, \ns_{\\eta_1^{\\intercal}} [- Q_1 \\, q^{\\, -\\rho} \\, t^{\\, -V^{\\, 1 \\, \\intercal}}],\n\\label{4.3}\n\\end{multline}\n\n\\begin{multline}\n\\sum_{\\xi_2} \\,\n \\left\\lgroup -Q_2 \\right\\rgroup^{\\, |\\xi_2|} \\, \ns_{\\xi_2 \/ \\eta_{2}} [ q^{\\, -\\rho} \\, t^{\\, - V^{\\, 2 \\, \\intercal}}] \\, \ns_{\\xi_2^{\\intercal}}[ q^{\\, - W^{\\, 2}} \\, t^{\\, - \\rho }]\n\\\\\n=\\prod_{i,j=1}^{\\infty} \\,\n \\left\\lgroup\n1-Q_2 \\, q^{\\, -W_i^{\\, 2 }+j-\\frac12} \n \\, t^{\\, -V_j^{\\, 2 \\, \\intercal}+i-\\frac12}\n\\right\\rgroup \\ \ns_{\\eta_2^{\\intercal} }[-Q_2 q^{\\, -W^{\\, 2}} \\, t^{\\, -\\rho} ]\n\\label{4.4}\n\\end{multline}\n\n\\subsubsection{The sum over $\\xi_{ { \\scriptstyle M} }$} We re-write this in terms of a sum over \na new set of partition $\\tau$,\n\n\\begin{multline}\n\\sum_{\\xi_{ { \\scriptstyle M} } } \\, \n \\left\\lgroup -Q_{ { \\scriptstyle M} } \\right\\rgroup^{\\, | \\xi_{ { \\scriptstyle M} } | - | \\eta_1 | } \\,\ns_{\\xi_{ { \\scriptstyle M} }^{\\intercal} \/ \\eta_1} [ \\, q^{\\, -\\rho} \\, t^{\\, -W^{\\, 1, \\, \\intercal}} ] \\, \ns_{\\xi_{ { \\scriptstyle M} } \/ \\eta_2} [ \\, q^{\\, -V^{\\, 2}} \\, t^{\\, -\\rho} ]\n\\\\\n=\\prod_{i, j}\n \\left\\lgroup\n1 - Q_{ { \\scriptstyle M} } \\, \nq^{\\, -V_i^{\\, 2 } + j - \\frac12} \\,\nt^{\\, -W_j^{\\, 1 \\, \\intercal} + i - \\frac12}\n\\right\\rgroup\n\\sum_{\\tau}\n\\ \ns_{\\eta_1^{\\intercal}\/ \\tau^{\\intercal}}[ q^{\\, -V^{\\, 2} } \\, t^{\\, -\\rho} ] \\, \ns_{\\eta_2^{\\intercal}\/ \\tau }[-Q_{ { \\scriptstyle M} } \\, q^{\\, -\\rho} \\, t^{\\, -W^{\\, 1 \\, \\intercal}} ]\n\\label{4.5}\n\\end{multline}\n\n\\subsubsection{The sums over $\\eta_1$ and $\\eta_2$} \n\n\\begin{multline}\n\\sum_{\\eta_1}\n \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\, - |\\eta_1|^{\\prime}} \\, \n \\left\\lgroup - Q_{ { \\scriptstyle M} }\\right\\rgroup^{ |\\eta_1| } \\,\ns_{\\eta_1^{\\intercal}} [ -Q_1 \\, q^{\\, -\\rho} \\, t^{\\, -V^{\\, 1 \\, \\intercal}}]\\ \ns_{\\eta_1^{\\intercal} \/ \\tau^{\\intercal}} \\, [ q^{\\, - V^{\\, 2}} \\, t^{\\, -\\rho }]\n\\\\\n=\\prod_{i,j}\n \\left\\lgroup\n1-Q_1 \\, Q_{ { \\scriptstyle M} }\\, q^{\\, -V_i^{\\, 2} + j - 1} \\, t^{\\, -V_j^{\\, 1 \\, \\intercal}+i} \n\\right\\rgroup^{-1}\n\\ \ns_{\\tau^\\intercal}[Q_1 \\, Q_{ { \\scriptstyle M} } \\, q^{\\, j - \\frac12} \\, t^{\\, -V_j^{\\, 1 \\, \\intercal}}]\n\\label{4.6}\n\\end{multline}\n\n\\begin{multline}\n\\sum_{\\eta_2} \n \\left\\lgroup \\frac{q}{t} \\right\\rgroup^{\\, |\\eta_2|^{\\prime}} \\,\ns_{\\eta_2^\\intercal }[-Q_2 \\, q^{\\, - W^{\\, 2}} \\, t^{\\, -\\rho } ] \\, \ns_{\\eta_2^\\intercal \/ \\tau}[-Q_{ { \\scriptstyle M} } \\, q^{\\, -\\rho} \\, t^{\\, -W^{\\, 1 \\, \\intercal}} ]\n\\\\\n=\\prod_{i,j}\n \\left\\lgroup 1 - Q_{ { \\scriptstyle M} } \\, Q_2 \\, q^{\\, -W_i^{\\, 2} + j} \\, t^{\\, -W_j^{\\, 1 \\, \\intercal} + i - 1} \\right\\rgroup^{-1} \\, \ns_{\\tau}[ - Q_2 \\, q^{\\, -W_i^{\\, 2} } \\, t^{\\, i - \\frac12} ]\n\\label{4.7}\n\\end{multline}\n\n\\subsubsection{The sum over $\\tau$} We finally evaluate the sum over the partitions \nthat were introduced in an intermediate step above, \n\n\\begin{equation}\n\\sum_{\\tau}\\ \ns_{\\tau}[-Q_2 \\, q^{\\, -W_i^{\\, 2}} \\, t^{\\, i - \\frac12 }] \\, \ns_{\\tau^{\\intercal}}[Q_1 \\, Q_{ { \\scriptstyle M} } \\, q^{\\, j - \\frac12} \\, t^{\\, - V_j^{\\, 1 \\, \\intercal} }]\n=\\prod_{i, j}\n \\left\\lgroup 1 - Q_1 \\, Q_{ { \\scriptstyle M} } \\, Q_2 \\, q^{\\, - W_i^{\\, 2} + j - \\frac12} t^{\\, -V_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup\n\\label{4.8}\n\\end{equation}\n\n\\noindent to obtain \n\n\\begin{multline}\n\\label{w.num}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{ } [q, t, R]\n= \n\\\\\nq^{\\, \\left\\lgroup \\parallel V^{\\, 1 } \\parallel^{\\prime} + \\parallel V^{\\, 2 } \\parallel^{\\prime} \\right\\rgroup}\nt^{\\, \\left\\lgroup \\parallel W^{\\, 1 \\, \\intercal} \\parallel^{\\prime} + \\parallel W^{\\, 2 \\, \\intercal} \\parallel^{\\prime} \\right\\rgroup} \\, \nZ_{ V^{\\, 1} }[q, t]\\\nZ_{ W^{\\, 1 \\, \\intercal}}[t, q]\\\nZ_{ V^{\\, 2} }[q, t]\\\nZ_{ W^{\\, 2 \\, \\intercal}}[t, q]\n\\\\\n\\times \\,\n\\prod_{i, j = 1}^{\\infty} \\, \\left\\lgroup 1 - Q_1 \\, \nq^{\\, -W_i^{\\, 1} + j - \\frac12} \\, \nt^{\\, -V_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup \\, \n\\prod_{i, j = 1}^{\\infty} \\, \\left\\lgroup 1 - Q_2 \\, \nq^{\\, -W_i^{\\, 2} + j - \\frac12} \\, \nt^{\\, -V_j^{\\, 2 \\, \\intercal} + i - \\frac12} \\right\\rgroup \\,\n\\\\\n\\times \\,\n\\prod_{i, j = 1}^{\\infty} \\, \\left\\lgroup 1 - Q_{ { \\scriptstyle M} } \\, \nq^{\\, -V_i^{\\, 2} + j - \\frac12} \\, t^{\\, - W_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup \\,\n\\\n\\prod_{i, j = 1}^{\\infty} \\, \\left\\lgroup 1 - Q_1 \\, Q_{ { \\scriptstyle M} } \\, Q_2 \\, \nq^{\\, -W_i^{\\, 2} + j - \\frac12} \\, t^{\\, - V_j^{\\, 1 \\, \\intercal} + i - \\frac12} \\right\\rgroup\n\\\\\n\\times \\,\n\\prod_{i, j = 1}^{\\infty} \\, \\left\\lgroup 1 - Q_1 \\, Q_{ { \\scriptstyle M} } \\, \nq^{\\, - V_i^{\\, 2} + j - 1} \\, \nt^{\\, - V_j^{\\, 1 \\, \\intercal} + i } \\right\\rgroup^{-1} \n\\\n\\prod_{i, j = 1}^{\\infty} \\, \\left\\lgroup 1 - Q_{ { \\scriptstyle M} } \\, Q_2 \\, \nq^{\\, - W_i^{\\, 2} + j } \\, \nt^{\\, - W_j^{\\, 1 \\, \\intercal} + i - 1} \\right\\rgroup^{-1} \n\\end{multline}\n\n\\noindent Note that regarding the right hand side of Equation {\\bf \\ref{w.num}} as a rational function, \nthe initial four products are in the numerator, while the latter two are in the denominator. \n\n\\subsubsection{Normalised products}\nGiven two partitions, $V$ with parts $v_i$, $i = 1, 2, \\cdots$, and \n $W$ with parts $w_i$, $i = 1, 2, \\cdots$, \nand two sequences of integers $\\alpha_k$ and $\\beta_k$, $k = 1, 2, \\cdots$, we define \n\n\\begin{equation}\n\\label{normalised.product}\n{\\prod_{i, j=1}^{\\infty}}^{^\\prime} \n \\left\\lgroup 1-Q \\, q^{\\, - v_i + \\alpha_j} \\, t^{\\, - w_j + \\beta_i} \\right\\rgroup\n=\n\\prod_{i, j=1}^{\\infty}\n \\left\\lgroup\n\\frac{\n1-Q \\, q^{\\, - v_i + \\alpha_j} \\, t^{\\, - w_j + \\beta_i}\n}{\n1-Q \\, q^{\\, \\alpha_i} \\, t^{\\, \\beta_i}\n}\n\\right\\rgroup\n\\end{equation}\n\n\\noindent In this notation, the expression for \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\, norm} [q, t, R]$ \nin Equation {\\bf \\ref{W.norm}}, is identical to that for \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]$, \nin Equation {\\bf \\ref{web.02}}, up to replacing each product $\\prod_{i, j}$ \nby a normalised product $\\prod_{i, j}^{^\\prime}$.\n\n\\subsubsection{From infinite to finite products}\n\n\\noindent Using Equation {\\bf \\ref{normalised.product}}, we have the following identities \n\\cite{nakajima.yoshioka, awata.kanno.02}\n\\footnote{\\\nFor an excellent reference and compendium of relevant combinatorial identities, including \nproofs, see \\cite{awata.kanno.02}. Equation {\\bf \\ref{nakajima.identity}} in this note \nfollows from Equations {\\bf 2.8--2.11} in \\cite{awata.kanno.02}.\n}. Firstly,\n\n\\begin{eqnarray}\n\\label{nakajima.identity}\n{\\prod_{i, j=1}^{\\infty}}^{^\\prime} \n \\left\\lgroup 1-Q \\, q^{\\, -w_i + j - 1} \\, t^{\\, -v_j^{\\intercal} + i} \\right\\rgroup\n& = & \n\\prod_{\\square \\in V}\n \\left\\lgroup\n1-Q\\, q^{\\, -A_{\\square, V}^{++}} \\, t^{\\, -L_{\\square, W}^{ }}\n\\right\\rgroup\n\\prod_{\\blacksquare \\in W}\n \\left\\lgroup\n1-Q\\, q^{\\, A_{\\blacksquare, W}^{ }} \\, t^{\\, L_{\\blacksquare, V}^{++}}\n\\right\\rgroup\n\\nonumber\n\\\\\n& = & \n\\prod_{\\blacksquare \\in W}\n \\left\\lgroup\n1-Q\\, q^{\\, -A_{\\blacksquare, V}^{++}} \\, t^{\\, -L_{\\blacksquare, W}^{ }}\n\\right\\rgroup\n\\prod_{\\square \\in V}\n \\left\\lgroup\n1-Q\\, q^{\\, A_{\\square, W}^{ }} \\, t^{\\, L_{\\square, V}^{++}}\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent Note that while the product on the left of Equation {\\bf \\ref{nakajima.identity}} is \nnormalised in the sense of Equation {\\bf \\ref{normalised.product}}, the remaining products are \nnot. \n\n\\subsubsection{The 5D basic web in product form}\n\nUsing the identities {\\bf \\ref{nakajima.identity}} in Equation {\\bf \\ref{web.02}} for the numerator, \nwe write the normalised $U(2)$ basic web partition function in Equation {\\bf \\ref{W.norm}} as\n\n\\begin{equation}\n\\label{W.5D.norm}\n\\mathcal{W}^{\\, norm}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]\n=\nq^{\\, \\left\\lgroup \\parallel V^{\\, 1} \\parallel^{\\prime} + \\parallel V^{\\, 2} \\parallel^{\\prime} \\right\\rgroup}\nt^{\\, \\left\\lgroup \\parallel W^{\\, 1 \\, \\intercal} \\parallel^{\\prime} + \\parallel W^{\\, 2 \\, \\intercal} \\parallel^{\\prime} \\right\\rgroup}\n\\\n\\frac{\nw^{\\, num} [{\\bf V, \\, W, \\, \\Delta}, q, t, R]\n}{\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, q, t, R]\n}\n\\end{equation}\n\n\\noindent where, using $Q_3 = Q_1 Q_{ { \\scriptstyle M} } Q_2$, $w^{num}$ is \n\n\\begin{eqnarray}\n\\label{w.numerator.5D}\nw^{\\, num} [{\\bf V, \\, W, \\, \\Delta}, q, t, R] \n& = & \n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup 1-Q_1 \\, q^{\\, -A_{\\square, W^{\\, 1}}^+} \\, t^{\\, -L_{\\square, V^{\\, 1}}^+} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup 1-Q_1 \\, q^{\\, A_{\\blacksquare, V^{\\, 1}}^+} \\, t^{\\, L_{\\blacksquare, W^{\\, 1}}^+} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup 1-Q_2 \\, q^{\\, -A_{\\square, W^{\\, 2}}^+} \\, t^{\\, -L_{\\square, V^{\\, 2}}^+} \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup 1-Q_2 \\, q^{\\, A_{\\blacksquare, V^{\\, 2}}^+} \\, t^{\\, L_{\\blacksquare, W^{\\, 2}}^+} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup 1-Q_{ { \\scriptstyle M} } \\, q^{\\, -A_{\\blacksquare, V^{\\, 2}}^+} \\, t^{\\, -L_{\\blacksquare, W^{\\, 1}}^+} \\right\\rgroup\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup 1-Q_{ { \\scriptstyle M} } \\, q^{\\, A_{\\square, W^{\\, 1}}^+} \\, t^{\\, L_{\\square, V^{\\, 2}}^+} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup 1- Q_3 \\, q^{\\, -A_{\\square, W^{\\, 2}}^+} \\, t^{\\, -L_{\\square, V^{\\, 1}}^+} \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup 1- Q_3 \\, q^{\\, A_{\\blacksquare, V^{\\, 1}}^+} \\, t^{\\, L_{\\blacksquare, W^{\\, 2}}^+} \\right\\rgroup, \n\\end{eqnarray}\n\n\\noindent where we have used the second equality in Equation {\\bf \\ref{nakajima.identity}} \nto put the products in the above uniform form. The denominator $w_{den}$ is \n\n\\begin{eqnarray}\n\\label{w.denominator.5D}\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, q, t, R] \n& = &\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup 1- q^{\\, A_{\\square, V^{\\, 1}}^{++}} \\, t^{\\, L_{\\square, V^{\\, 1}}^{ }} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup 1- q^{\\, A_{\\blacksquare, W^{\\, 1}} } \\, t^{\\, L_{\\blacksquare, W^{\\, 1}}^{++}} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup 1- q^{\\, A_{\\square, V^{\\, 2}}^{++}} \\, t^{\\, L_{\\square, V^{\\, 2}}^{ }} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup 1- q^{\\, A_{\\blacksquare, W^{\\, 2}} } \\, t^{\\, L_{\\blacksquare, W^{\\, 2}}^{++}} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup 1-Q_1 \\, Q_{ { \\scriptstyle M} } \\, q^{\\, -A_{\\square, V^{\\, 2}}^{++}} \\, t^{\\, -L_{\\square, V^{\\, 1}} } \\right\\rgroup\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup 1-Q_1 \\, Q_{ { \\scriptstyle M} } \\, q^{\\, A_{\\square, V^{\\, 1}} } \\, t^{\\, L_{\\square, V^{\\, 2}}^{++}} \\right\\rgroup \n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup 1-Q_{ { \\scriptstyle M} } \\, Q_2 \\, q^{\\, -A_{\\blacksquare, W^{\\, 2}} } \\, t^{\\, -L_{\\blacksquare, W^{\\, 1}}^{++}} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup 1-Q_{ { \\scriptstyle M} } \\, Q_2 \\, q^{\\, A_{\\blacksquare, W^{\\, 1}}^{++}} \\, t^{\\, L_{\\blacksquare, W^{\\, 2}} } \\right\\rgroup\n\\end{eqnarray}\n\n\\noindent where the first four products on the right hand side of Equation {\\bf \\ref{w.denominator.5D}} \nare due to the product \n$ Z_{ V^{\\, 1} }[t, q]\\,\n Z_{ W^{\\, 1 \\, \\intercal} }[q, t]\\, \n Z_{ V^{\\, 2} }[t, q]\\, \n Z_{ W^{\\, 2 \\, \\intercal} }[q, t]$ in Equation {\\bf \\ref{web.02}}.\n$w_{den}$ is equal to the denominator $\\mathcal{W}_{\\, \\bf \\emptyset \\, \\emptyset \\, \\Delta} [q, t, R]$ on the right hand \nside of Equation {\\bf \\ref{W.norm}}.\n\n\\subsubsection{Remark} One can glue copies of the basic web partition function \n$\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}[q, t, R]$ in several ways. In this note, we \nrestrict our attention to gluing linearly or cyclically, to form linear or \ncyclic $U(2)$ quiver gauge theories, as described in paragraphs \n{\\bf \\ref{linear.conformal.blocks}} and {\\bf \\ref{cyclic.conformal.blocks}}.\nWe do not, for example, glue basic webs to form a Hirzbruch surface.\n\n\\section{A 4D $U(2)$ basic web partition function}\n\\label{section.04.4d.web}\n{\\it \nWe take the $R \\! \\rightarrow \\! 0$ limit of the 5D basic web partition function to obtain \nits 4D analogue. \n}\n\n\\subsection{Two parameters} We take the relationship between the parameters $q$ and $t$ of \nthe refined vertex and the parameters $\\epsilon_1$ and $\\epsilon_2$ of the instanton \npartition function to be,\n\\begin{equation}\n\\boxed{\nq=e^{\\, R \\epsilon_2},\n\\quad\nt=e^{\\, - R \\epsilon_1}\n}\n\\end{equation} \n\n\\noindent where $R$, the radius of a space-like circle, plays the role of a deformation parameter. \nWe write $\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\, norm} [\\epsilon_1, \\epsilon_2, R]$, then take the limit \n$R \\! \\rightarrow \\! 0$. The prefactor on the left hand side of Equation {\\bf \\ref{W.5D.norm}} tends \nto 1 in the limit $R \\rightarrow 0$, and we obtain\n\n\\begin{equation}\n\\mathcal{W}_{\\, \\bf V \\, W \\, \\Delta}^{\\, norm} [\\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n=\n\\frac{\nw^{\\, num} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n}{\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n}\n\\label{4.15}\n\\end{equation}\n\n\\noindent where, using $\\Delta_3 = \\Delta_1 + \\Delta_{ { \\scriptstyle M} } + \\Delta_2$, we have\n\n\\begin{multline}\n\\label{w.numerator.4D}\nw^{\\, num} [{\\bf V}, {\\bf W}, {\\bf \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n= \n\\\\\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup \\Delta_1 + A^+_{\\square, W^{\\, 1}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup \\Delta_1 - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup \\Delta_2 + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup \\Delta_2 - A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup \\Delta_{ { \\scriptstyle M} } - A^+_{\\square, W^{\\, 1}} \\epsilon_2 + L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup \\Delta_{ { \\scriptstyle M} } + A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 - L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup \\Delta_3 + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup \\Delta_3 - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\end{multline}\n\n\\noindent and\n\n\\begin{multline}\n\\label{w.denominator.4D}\nw_{\\, den} [{\\bf V}, {\\bf W}, {\\bf \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] \n= \n\\\\\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup - A^{++}_{\\square, V^{\\, 1}} \\epsilon_2 + L^{ }_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup - A^{++}_{\\square, V^{\\, 2}} \\epsilon_2 + L^{ }_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times \n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup - A^{ }_{\\blacksquare, W^{\\, 1}} \\epsilon_2 + L^{++}_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup - A^{ }_{\\blacksquare, W^{\\, 2}} \\epsilon_2 + L^{++}_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup \\Delta_1 + \\Delta_{ { \\scriptstyle M} } + A^{++}_{\\square, V^{\\, 2}} \\epsilon_2 - L^{ }_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup \\Delta_1 + \\Delta_{ { \\scriptstyle M} } - A^{ }_{\\square, V^{\\, 1}} \\epsilon_2 + L^{++}_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\\\\n\\times \n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup \\Delta_2 + \\Delta_{ { \\scriptstyle M} } + A^{ }_{\\blacksquare, W^{\\, 2}} \\epsilon_2 - L^{++}_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup \n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup \\Delta_2 + \\Delta_{ { \\scriptstyle M} } - A^{++}_{\\blacksquare, W^{\\, 1}} \\epsilon_2 + L^{ }_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup \n\\end{multline}\n\n\\section{The building block of the 4D $U(2)$ quiver instanton partition function}\n\\label{section.05.nekrasov.partition.functions}\n{\\it \nWe recall the normalised contribution of the bifundamental hypermultiplet which acts as a building block \nof the instanton partition function.\n}\n\nIn the notation of Equation {\\bf 6} and Section {\\bf 2} of \\cite{bershtein.foda}, the normalised bifundamental \npartition function \n$\\mathcal{Z}^{\\, 4D}_{building.block}$ is \n\\begin{equation}\n\\label{z.bb}\n\\mathcal{Z}^{\\, 4D}_{building.block} \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ \\mu \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup = \n\\frac{\nz^{\\, num} \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ \\mu \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup\n}\n{\nz_{\\, den} \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup\n} \n\\end{equation}\n\n\\noindent where ${\\bf a} = \\{a, -a\\}$, ${\\bf b} = \\{b, -b\\}$, $a, b$ and $\\mu$ are linear combinations \nof $\\epsilon_1$ and $\\epsilon_2$, as will be explained in \nSection {\\bf \\ref{section.08.restricted.instanton.partition.functions}} below, and we use \n${\\bf V^{\\prime}}$ and \n${\\bf W^{\\prime}}$ for partition pairs that we will relate in \nSection {\\bf \\ref{section.07.identification.denominators}} to the pairs that appear in the 4D basic web. \nWe refer to \\cite{bershtein.foda} for brief explanations of the parameters that appear in \n$\\mathcal{Z}^{\\, 4D}_{building.block}$. Defining\n\n\\begin{equation}\n\\alpha_0 = \\frac12 \\left\\lgroup \\epsilon_1 + \\epsilon_2 \\right\\rgroup\n\\end{equation}\n\n\\noindent the numerator $z^{\\, num}$, as given in Equation {\\bf 9} of \\cite{bershtein.foda}, is\n\n\\begin{multline}\n\\label{z.numerator}\nz^{\\, num} \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}}\\ | \\ \\mu \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup = \n\\\\\n\\prod_{\\square \\in V^{\\, 1 \\, \\prime}} \\left\\lgroup [ a - b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 1 \\, \\prime}} \\epsilon_1 \n - L^+_{\\square, W^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1 \\, \\prime}} \\left\\lgroup [ a - b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 1 \\, \\prime}} \\epsilon_1 \n + L^+_{\\blacksquare, V^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 2 \\, \\prime}} \\left\\lgroup [- a + b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 2 \\, \\prime}} \\epsilon_1 \n - L^+_{\\square, W^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2 \\, \\prime}} \\left\\lgroup [- a + b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 2 \\, \\prime}} \\epsilon_1 \n + L^+_{\\blacksquare, V^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\\\\n\\times \n\\prod_{\\square \\in V^{\\, 2 \\, \\prime}} \\left\\lgroup [- a - b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 2 \\, \\prime}} \\epsilon_1 \n - L^+_{\\square, W^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1 \\, \\prime}} \\left\\lgroup [- a - b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 1 \\, \\prime}} \\epsilon_1 \n + L^+_{\\blacksquare, V^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\\\\n\\times\n\\prod_{\\square \\in V^{\\, 1 \\, \\prime}} \\left\\lgroup [ a + b - \\mu + \\alpha_0] + A^+_{\\square, V^{\\, 1 \\, \\prime}} \\epsilon_1 \n - L^+_{\\square, W^{\\, 2 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2 \\, \\prime}} \\left\\lgroup [ a + b - \\mu + \\alpha_0] - A^+_{\\blacksquare, W^{\\, 2 \\, \\prime}} \\epsilon_1 \n + L^+_{\\blacksquare, V^{\\, 1 \\, \\prime}} \\epsilon_2 \\right\\rgroup\n\\end{multline}\n\nThe denominator $z_{\\, den}$, as given in Equation {\\bf 7} of \\cite{bershtein.foda}, is\n\n\\begin{equation}\n\\label{z.denominator}\nz_{\\, den} \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup\n= \n \\left\\lgroup \nz^{\\, num} \\left\\lgroup {\\bf a}, {\\bf V^{\\prime}} \\ | \\ 0 \\ | \\ {\\bf a}, {\\bf V^{\\prime}} \\right\\rgroup \\ \nz^{\\, num} \\left\\lgroup {\\bf b}, {\\bf W^{\\prime}} \\ | \\ 0 \\ | \\ {\\bf b}, {\\bf W^{\\prime}} \\right\\rgroup \\ \n\\right\\rgroup^{\\frac{1}{2}}\n\\end{equation}\n\n\\section{Identification of $\\mathcal{W}_{\\bf \\, V \\, W \\, \\Delta}^{\\, norm}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$. The numerators}\n\\label{section.06.identification.numerators}\n\nComparing Equation {\\bf \\ref{w.numerator.4D}} and Equation {\\bf \\ref{z.numerator}}, we find that if \nwe set \n$V^{\\, i \\, \\prime} = V^{\\, i \\, \\intercal}$, \n$W^{\\, i \\, \\prime} = W^{\\, i \\, \\intercal}$, and multiply each factor by $-1$, which is possible \nsince the number of factors is even by construction, we obtain\n\n\\begin{eqnarray}\n\\label{numerator.nekrasov.02}\n& & \n\\\\\n& & z^{\\, num} \\left\\lgroup {\\bf a}, {\\bf V}\\, | \\ \\mu \\ | \\, {\\bf b}, {\\bf W} \\right\\rgroup =\n\\nonumber\n\\\\\n& &\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup [- a + b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 1}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup [- a + b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup [ a - b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup [ a - b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup [ a + b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 1}} \\epsilon_2 - L^+_{\\square, V^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup [ a + b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 2}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup [- a - b + \\mu - \\alpha_0] + A^+_{\\square, W^{\\, 2}} \\epsilon_2 - L^+_{\\square, V^{\\, 1}} \\epsilon_1 \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup [- a - b + \\mu - \\alpha_0] - A^+_{\\blacksquare, V^{\\, 1}} \\epsilon_2 + L^+_{\\blacksquare, W^{\\, 2}} \\epsilon_1 \\right\\rgroup\n\\nonumber\n\\end{eqnarray}\n\n\\noindent which leads to the identification\n\n\\begin{equation}\n\\label{identification.12}\n\\boxed{\n\\Delta_1 = - a + b + \\mu - \\alpha_0, \\quad\n\\Delta_2 = \\phantom{-} a - b + \\mu - \\alpha_0, \\quad\n\\Delta_{ { \\scriptstyle M} } = - a - b - \\mu + \\alpha_0\n}\n\\end{equation}\n\n\\section{Identification of $\\mathcal{W}_{\\bf \\, V \\, W \\, \\Delta}^{\\, norm}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$. The denominators}\n\\label{section.07.identification.denominators}\n\nUsing the identification of parameters obtained in Equation {\\bf \\ref{identification.12}} in\n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}]$ \nand $z_{\\, den}$ as given in Equations {\\bf \\ref{w.denominator.4D}} and {\\bf \\ref{z.denominator}}, \nit is clear that these two functions are not the same. However, what matters is not the denominator \nof s single factor, but the product of all denominators, as we explain below.\n\nThe denominator $w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}]$ is a natural object, as we can see in the \nderivation in Section {\\bf \\ref{section.03.5d.web}}. \nOn the other hand, the denominator $z_{\\, den}$ was obtained in \\cite{bershtein.foda} by taking the \nfull denominator that appears in expressions for the 4D $U(1)$ linear and cyclic quiver instanton \npartition functions and factoring that into denominators for the contributions of the bifundamental \nhypermultiplets. Such a factorisation is not unique and any factorisation is allowed for as long as \nthe product of all factors is equal to the full denominator of the original expression. \n\nIn this work, to identify $\\mathcal{W}^{\\, 4D}$ and $\\mathcal{Z}^{\\, 4D}_{building.block}$, we need \na factor $\\mathcal{F}$, such that the product of all normalisation factors that appear in a conformal block \nis equal to 1. Consider the abbreviations\n\\begin{eqnarray}\n\\mathcal{A}_{V^{ij}} [x] = \\mathcal{A}_{V^i, V^j} [x] & = & \\prod_{\\square \\in V^i} \\left\\lgroup x + A^{++}_{\\square, V^i} \\epsilon_2\n- L^{ }_{\\square, V^j} \\epsilon_1 \\right\\rgroup,\n\\quad\n\\mathcal{A}_{\\emptyset, \\emptyset} [x] = 1,\n\\\\\n\\mathcal{L}_{W^{ij}} [x] = \\mathcal{L}_{W^i, W^j} [x] & = & \\prod_{\\blacksquare \\in W^j} \\left\\lgroup x - A^{ }_{\\blacksquare, W^j} \\epsilon_2\n+ L^{++}_{\\blacksquare, W^i} \\epsilon_1 \\right\\rgroup,\n\\quad\n\\mathcal{L}_{\\emptyset, \\emptyset} [x] = 1,\n\\nonumber\n\\\\\n\\mathcal{H}_{Y^{ij}} [x] & = & \\mathcal{A}_{Y^{ij}} [x] \\, \\mathcal{L}_{Y^{ij}} [x]\n\\nonumber\n\\end{eqnarray}\nIn this notation, $z_{\\, den}$ and $w_{\\, den}$ are \n\\begin{multline}\nz_{\\, den} \\left\\lgroup {\\bf a}, {\\bf V} \\ | \\ 0 \\ | \\ {\\bf b}, {\\bf W} \\right\\rgroup =\n\\\\\n \\left\\lgroup\n\\mathcal{H}_{V^{11}} [0] \\, \\mathcal{H}_{V^{12}} [2a] \\, \\mathcal{H}_{V^{12}} [-2a] \\, \\mathcal{H}_{V^{22}} [0]\n\\mathcal{H}_{W^{11}} [0] \\, \\mathcal{H}_{W^{12}} [2b] \\, \\mathcal{H}_{W^{12}} [-2b] \\, \\mathcal{H}_{W^{22}} [0]\n\\right\\rgroup^{\\frac12}\n\\end{multline}\n\n\\noindent and \n\\begin{multline}\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0] = \nw_{\\, den} \\left\\lgroup {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right\\rgroup =\n\\\\\n \\left\\lgroup - \\right\\rgroup^{\\, | V^{\\, 1} | + | V^{\\, 2} | + | W^{\\, 1} | + | W^{\\, 2} |}\n\\mathcal{L}_{V^{11}} [ 0] \\,\n\\mathcal{L}_{V^{22}} [ 0] \\,\n\\mathcal{H}_{V^{12}} [-2a] \\,\n\\mathcal{H}_{W^{12}} [ 2b] \\,\n\\mathcal{A}_{W^{11}} [ 0] \\,\n\\mathcal{A}_{W^{22}} [ 0]\n\\end{multline}\n\nNow consider the factor\n\n\\begin{equation}\n\\mathcal{F} \\left\\lgroup {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right\\rgroup\n=\n\\frac{\nz_{\\, den} \\left[ {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right]\n}{\nw_{\\, den} \\left[ {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right]\n}\n\\end{equation}\n\n\\noindent and define \n\n\\begin{equation}\nz_{\\, den}^{\\prime} = {\\mathcal{F}}^{-1} \\, z_{\\, den} = w_{\\, den}, \\quad\n\\mathcal{Z}_{building.block }^{\\prime} = \\mathcal{F} \\, \\mathcal{Z}_{building.block}\n\\end{equation}\n\n\\noindent $\\mathcal{Z}^{\\, \\prime}_{building.block}$ is constructed such that \n{\\bf 1.} It has the same numerator as $\\mathcal{Z}^{\\, 4D}_{building.block}$, which \nis the same as that of $\\mathcal{W}^{\\, norm}_{\\bf V W \\Delta}$, \nwhen we choose the parameters as in Equation {\\bf \\ref{identification.12}} and\n{\\bf 2.} It has the same denominator as $\\mathcal{W}^{\\, norm}_{\\bf V W \\Delta}$, also \nwhen we choose the parameters as in Equation {\\bf \\ref{identification.12}}.\n\nSince the denominator of $\\mathcal{Z}^{\\, \\prime}_{building.block}$ is not manifestly \nthe same as that of $\\mathcal{Z}^{\\, 4D}_{building.block}$, we need to show that \ngluing copies of $\\mathcal{Z}^{\\, \\prime}_{building.block}$ to build a topological \npartition function, leads to the same result obtained by gluing copies of the \noriginal $\\mathcal{Z}^{ }_{building.block}$. $\\mathcal{F}$ can be written \nin a simpler form as follows,\n\n\\begin{equation}\n \\left\\lgroup \\mathcal{F} \\left\\lgroup {\\bf a}, {\\bf V} \\, | \\, {\\bf b}, {\\bf W} \\right\\rgroup \\right\\rgroup^{\\, 2} \n\\\\\n=\nF^{\\, left}_{\\bf V} [a] \\,\nF^{\\, right}_{\\bf W} [b]\n\\end{equation}\n\n\\noindent where \n\n\\begin{eqnarray}\nF^{\\, left}_{\\bf V} [a] & = &\n \\left\\lgroup - \\right\\rgroup^{\\, | V^{\\, 1} | + | V^{\\, 2} | }\n \\left\\lgroup\n\\frac{\n\\mathcal{A}_{V^{11}} [0] \\,\n\\mathcal{A}_{V^{22}} [0]\n}{\n\\mathcal{L}_{V^{11}} [0] \\,\n\\mathcal{L}_{V^{22}} [0]\n}\n\\right\\rgroup \\, \n \\left\\lgroup\n\\frac{\n\\mathcal{H}_{V^{12}} [ 2a]\n}{\n\\mathcal{H}_{V^{12}} [-2a]\n}\n\\right\\rgroup \n\\\\\nF^{\\, right}_{\\bf W} [b] & = &\n \\left\\lgroup - \\right\\rgroup^{\\, | W^{\\, 1} | + | W^{\\, 2} |}\n \\left\\lgroup\n\\frac{\n\\mathcal{H}_{W^{12}} [-2b]\n}{\n\\mathcal{H}_{W^{12}} [ 2b]\n}\n\\right\\rgroup \\, \n \\left\\lgroup\n\\frac{\n\\mathcal{L}_{W^{11}} [0] \\,\n\\mathcal{L}_{W^{22}} [0]\n}{\n\\mathcal{A}_{W^{11}} [0] \\,\n\\mathcal{A}_{W^{22}} [0]\n}\n\\right\\rgroup\n\\end{eqnarray}\n\n\\noindent $F^{\\, right}$ and $F^{\\, left }$ satisfy the obvious properties \n\n\\begin{equation}\n\\label{property.00}\nF^{\\, right}_{\\bf \\emptyset} [x] =\nF^{\\, left }_{\\bf \\emptyset} [x] = 1\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\n\\label{property.01}\nF^{\\, right}_{\\bf Y} [x] \\, F^{\\, left }_{\\bf Y} [x] = 1\n\\end{equation}\n\nThe physical objects that we are interested in are the conformal blocks which are constructed \nby gluing copies of $\\mathcal{Z}_{building.block}$ \\cite{bershtein.foda}. We need to show that gluing \ncopies of $\\mathcal{Z}_{building.block}^{\\prime}$ leads to the same result, which will be the case if \nproducts of the normalisation factors trivialise. This will follow directly from Equations \n{\\bf \\ref{property.00}} and {\\bf \\ref{property.01}}. There are two cases to consider, the \nlinear conformal block case and the cyclic conformal block case.\n\n\\subsubsection{Linear conformal blocks}\n\\label{linear.conformal.blocks}\nConsider the linear conformal block obtained by gluing $n$ copies of $\\mathcal{Z}_{building.block}$, \nthat is $\\mathcal{Z}_{building.block.1}$, $\\mathcal{Z}_{building.block.2}$, $\\cdots$, $\\mathcal{Z}_{building.block.n}$, \nsequentially. Using copies of $\\mathcal{Z}_{building.block}^{\\prime}$, we obtain the same result as \nusing copies of $\\mathcal{Z}_{building.block}$ up to a factor \n\n\\begin{equation}\nF^{\\, left}_{\\bf \\emptyset} [x_0 ] \\, F^{\\, right}_{\\bf Y^{\\, 1}} [x_1] \nF^{\\, left}_{\\bf Y^{\\, 1}} [x_1 ] \\, F^{\\, right}_{\\bf Y^{\\, 2}} [x_2] \n\\cdots\nF^{\\, left}_{\\bf Y^{n-1}} [x_{n-1}] \\, F^{\\, right}_{\\bf \\emptyset} [x_n] \n= 1\n\\end{equation}\n\n\\subsubsection{Cyclic conformal blocks}\n\\label{cyclic.conformal.blocks}\n\n\\begin{equation}\nF^{\\, left}_{\\bf Y^0} [x_0] \\, F^{\\, right}_{\\bf Y^{\\, 1}} [x_1]\nF^{\\, left}_{\\bf Y^{\\, 1}} [x_1] \\, F^{\\, right}_{\\bf Y^{\\, 2}} [x_2]\n\\cdots\nF^{\\, left}_{\\bf Y^{n-1}} [x_{n-1}] \\, F^{\\, right}_{\\bf Y^0} [x_0]\n= 1\n\\end{equation}\n\nWe conclude that $\\mathcal{Z}_{building.block}^{\\prime}$ leads to the same conformal blocks as $\\mathcal{Z}_{building.block}$. \n\n\\subsubsection{The denominator $z_{\\, den}^{\\prime}$ and the Burge conditions}\nIn \\cite{alkalaev.belavin, bershtein.foda}, it was shown that for the choice of parameters that \nleads to Virasoro $A$-series minimal conformal blocks, the denominator $z_{\\, den}$ will contain non-physical \nzeros, unless we restrict the partition pairs, that $\\mathcal{Z}^{\\, 4D, \\, min}_{building.block}$ depends on, to obey \nBurge conditions. These conditions were derived in \\cite{bershtein.foda} using $z_{\\, den}$ rather than \n$z_{\\, den}^{\\prime}$. Using $z_{\\, den}^{\\prime}$ leads to the same conditions, since the product of all \n$z_{\\, den}^{\\prime}$ is the same as the product of all $z_{\\, den}$ that show up in the conformal block\n\\footnote{\\,\nIn Section {\\bf \\ref{section.10.burge.pairs}}, we outline the derivation of the Burge conditions from \n$z_{\\, den}^{\\prime}$.}.\n\n\\section{Restricted instanton partition functions for $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$. The parameters}\n\\label{section.08.restricted.instanton.partition.functions}\n\n{\\it \nWe recall the choice of parameters such that $\\mathcal{Z}^{\\, 4D}_{building.block}$ reduces to \n$\\mathcal{Z}^{\\, 4D, \\, min}_{building.block}$ which is the building block of Virasoro $A$-series minimal \nmodel conformal blocks times Heisenberg factors.\n}\n\n\\subsection{AGT parameterisation of minimal models}\n\\label{parameterisation.minimal.models}\n\nA minimal model $\\mathcal{M}^{\\, p, \\, p^{\\prime}}$, based on a Virasoro algebra $\\mathcal{V}^{\\, p, \\, p^{\\prime}}$, characterised \nby a central charge $c_{p, \\, p^{\\prime}} < 1$, that we parameterise as\n\n\\begin{equation}\n\\label{central.charge.minimal}\nc_{p, \\, p^{\\prime}} = 1 - 6 \\left\\lgroup a_{p, \\, p^{\\prime}} - \\frac{1}{a_{p, \\, p^{\\prime}}} \\right\\rgroup^{\\, 2},\n\\quad\na_{p, \\, p^{\\prime}} = \\left\\lgroup \\frac{p^{\\prime}}{p} \\right\\rgroup^{\\frac12},\n\\end{equation}\n\n\\noindent where $\\{p, \\, p^{\\prime}\\}$ are co-prime integers that satisfy $0 < p < p^{\\prime}$. In the Coulomb \ngas approach to computing conformal blocks in minimal models with $c_{p, \\, p^{\\prime}} < 1$ \n\\cite{nienhuis, dotsenko.fateev}, the screening charges $\\{\\alpha_{+}, \\alpha_{-}\\}$, and the background \ncharge $\\alpha_{back.ground}$, satisfy \n\n\\begin{equation}\n\\label{minimal.model.charges}\n\\boxed{\n\\alpha_{+} = a_{p, \\, p^{\\prime}}, \n\\quad\n\\alpha_{-} = - \\frac{1}{a_{p, \\, p^{\\prime}}},\n\\quad\n\\alpha_{back.ground} = - 2 \\alpha_0, \n\\quad \n\\alpha_0 = \\frac12 \\left\\lgroup \\alpha_{+} + \\alpha_{-} \\right\\rgroup \n}\n\\end{equation}\n\n\\noindent The AGT parameterisation of $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$ is obtained by choosing \n\n\\begin{equation}\n\\label{neg.0.pos}\n\\boxed{\n\\epsilon_1 < 0 < \\epsilon_2, \n\\quad\n\\epsilon_1 = \\alpha_{-}, \n\\quad\n\\epsilon_2 = \\alpha_{+} \n}\n\\end{equation}\n\n\\noindent so that $\\alpha_{-} < 0 < \\alpha_{+}$. Since we focus on $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$, \nwe work in terms of $\\{\\alpha_{-}, \\alpha_{+}\\}$ instead of $\\{ \\epsilon_1, \\epsilon_2\\}$.\n\n\\subsection{Two sets of charges in minimal models}\n\\label{charge.content}\nWe consider two types of charges that, in Coulomb gas terms, are expressed in terms of \nthe screening charges $\\{\\alpha_{+}, \\alpha_{-}\\}$. \n{\\bf 1.} The charge $a_{r, s}$ of the highest weight $|a_{r, s} \\rangle$ of the irreducible\nhighest weight representation $\\mathcal{H}^{\\, p, \\, p^{\\prime}}_{r, s}$, and\n{\\bf 2.} The charge $\\mu_{r, s}$ of the vertex operator ${\\mathcal{O}}_{\\mu}$ that intertwines \ntwo highest weight ireducible representations \n$\\mathcal{H}^{\\, p, \\, p^{\\prime}}_{r_1, s_1}$ and \n$\\mathcal{H}^{\\, p, \\, p^{\\prime}}_{r_2, s_2}$.\nThese charges are parameterised in terms of $\\alpha_{+}$ and $\\alpha_{-}$ as follows \n\n\\begin{equation}\n\\label{parameters.03}\n\\boxed{\n a_{r, s} = - \\frac{r}{2} \\, \\alpha_{+} - \\frac{s}{2} \\, \\alpha_{-}, \\quad \n\\mu_{r, s} = - \\frac{r}{2} \\, \\alpha_{+} - \\frac{s}{2} \\, \\alpha_{-} + \\alpha_0, \\quad \n1 \\leqslant r \\leqslant p - 1,\n1 \\leqslant s \\leqslant p^{\\prime} - 1\n}\n\\end{equation}\n\n\\section{From gauge theory parameters to minimal model parameters}\n\\label{section.09.from.gauge.theory.to.minimal.model}\n{\\it We compare the parameters of $\\mathcal{W}^{\\, 4D}$ and the parameters of $\\mathcal{Z}^{\\, 4D}_{building.block}$.}\n\nWe set\n\n\\begin{equation}\n\\label{notation.02.01}\na = - \\left\\lgroup \\frac{r_a}{2} \\right\\rgroup \\, \\alpha_{+} - \\left\\lgroup \\frac{s_a}{2} \\right\\rgroup \\, \\alpha_{-}, \n\\quad\nr_a \\in \\{1, 2, \\cdots, p - 1\\}, \n\\quad\ns_a \\in \\{1, 2, \\cdots, p^{\\prime} - 1\\}, \n\\end{equation}\n\n\\begin{equation}\n\\label{notation.02.02}\nb = - \\left\\lgroup \\frac{r_b}{2} \\right\\rgroup \\, \\alpha_{+} - \\left\\lgroup \\frac{s_b}{2} \\right\\rgroup \\, \\alpha_{-}, \n\\quad\nr_b \\in \\{1, 2, \\cdots, p - 1\\},\n\\quad\ns_b \\in \\{1, 2, \\cdots, p^{\\prime} - 1\\},\n\\end{equation}\n\n\\begin{equation}\n\\label{notation.02.03}\n\\mu = - \\left\\lgroup \\frac{r_{\\mu}}{2} \\right\\rgroup \\, \\alpha_{+} - \\left\\lgroup \\frac{s_{\\mu}}{2} \\right\\rgroup \\, \\alpha_{-} + \\alpha_0,\n\\quad\nr_{\\mu} \\in \\{1, 2, \\cdots, p - 1\\}, \n\\quad\ns_{\\mu} \\in \\{1, 2, \\cdots, p^{\\prime} - 1\\}\n\\end{equation}\n\n\\subsection{The fusion rules}\nFor completeness, let us mention the fusion rules. In the notation\n\n\\begin{equation}\nm_i = r_i - 1, \n\\quad \nn_i = s_i - 1\n\\end{equation}\n\n\\noindent the fusion rules take the simple form\n\n\\begin{equation}\n\\label{fusion.rules}\nm_a + m_b + m_{\\mu} = 0 \\ \\textit{mod} \\ 2, \\quad\nn_a + n_b + n_{\\mu} = 0 \\ \\textit{mod} \\ 2,\n\\end{equation}\n\n\\noindent where the triple $\\{m_a, m_b, m_{\\mu}\\}$ satisfies the triangular conditions\n\n\\begin{equation}\n\\label{triangular.conditions}\nm_a + m_b \\geqslant m_{\\mu}, \\quad \nm_b + m_{\\mu} \\geqslant m_a, \\quad \nm_{\\mu} + m_a \\geqslant m_b\n\\end{equation}\n\n\\noindent with analogous conditions for the triple $\\{n_a, n_b, n_{\\mu}\\}$. \n\n\\section{Restricted instanton partition functions for $\\mathcal{M}^{\\, p, \\, p^{\\prime}, \\, \\mathcal{H}}$. \nThe partition pairs}\n\\label{section.10.burge.pairs}\n{\\it We write the denominator \n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, g \\rightarrow 0]$ as\n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-} ]$, and, following \n\\cite{bershtein.foda}, we check the conditions required so that that it has no zeros.}\n\nUsing $\\alpha_{+}$ and $\\alpha_{-}$, let us write \n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\epsilon_1, \\epsilon_2, R \\rightarrow 0]$ as\n$w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-}]$, that is \n\n\\begin{eqnarray}\n\\label{denominator.4D.02}\nw_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-}]\n& = &\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup A^{++}_{\\square, V^{\\, 1}} \\, \\alpha_{+} - L^{ }_{\\square, V^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup A^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{+} - L^{ }_{\\square, V^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup A^{ }_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{+} - L^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup A^{ }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{+} - L^{++}_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times & \n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-} + A^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{+} - L^{ }_{\\square, V^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-} - A^{ }_{\\square, V^{\\, 1}} \\, \\alpha_{+} + L^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times & \n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-} + A^{ }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{+} - L^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\nonumber\n\\\\\n& \\times &\n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-} - A^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{+} + L^{ }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\end{eqnarray}\n\n\\noindent where we have used \n\n\\begin{eqnarray}\n-2a & = & r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-}, \\quad r = 1, 2, \\cdots, \n\\\\\n-2b & = & r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-}, \\quad s = 1, 2, \\cdots\n\\end{eqnarray}\n\nConsider the denominator $w_{\\, den} [{\\bf V, \\, W, \\, \\Delta}, \\alpha_{+}, \\alpha_{-}]$ in \nEquation {\\bf \\ref{denominator.4D.02}}, \non a product by product basis. We need to check the conditions under which any of these products has a zero, then\nfind the restriction that are necessary and sufficient to remove these zeros. The reasoning that we use to obtain\nthese conditions is the same as that in \\cite{bershtein.foda}. There are eight products to consider.\n\n\\subsection{The initial four products} In each of the initial four products, the product is over the cells inside \na single diagram, thus the arm length $A$ and the leg length $L$ in each of these factors is non-negative. Since \n$\\alpha_{-} < 0 < \\alpha_{+}$, and there is a term $\\alpha_0 > 0$ in each factor, the minimal value of each of these factors \nis greater than zero. Thus there can be no zeros from these factors. \nTo consider the remaining four factors, we require some preparation.\n\n\\subsection{Two zero-conditions}\n\\label{to.vanish}\nFollowing \\cite{bershtein.foda}, we note that, since $\\alpha_{-} < 0 < \\alpha_{+}$, any factor of the type that \nappears in Equation {\\bf \\ref{denominator.4D.02}} has a zero when an equation of type\n\n\\begin{equation}\n\\label{zeros.example.01}\nC_{+} \\, \\alpha_{+} + C_{-} \\, \\alpha_{-} = 0,\n\\end{equation}\n\n\\noindent where $C_+$, $C_-$ $\\in \\mathbbm{Z}$, is satisfied. Since $p$ and $p^{\\prime}$ are coprime, $\\alpha_{-}$ \nand $\\alpha_{+}$ are $\\not \\in \\mathbbm{Q}$, the condition in Equation {\\bf \\ref{zeros.example.01}} is \nequivalent to the two conditions\n\n\\begin{equation}\n\\label{zeros.example.02}\nC_+ = c \\ p, \n\\quad\nC_- = c \\ p^{\\prime}\n\\end{equation}\n\n\\noindent are satisfied, where $c$ is a proportionality constant that needs to be determined. \n\n\\subsection{From two zero-conditions to one zero-condition} \n\\label{from.2.to.1}\n\nConsider the two conditions\n\n\\begin{equation}\n\\label{conditions.example.01}\n\\boxed{\n- A^{ }_{\\square, i} = A^{\\prime} \\geqslant 0,\n\\quad \n L^{ }_{\\square, j} = L^{\\prime} \\geqslant 0\n}\n\\end{equation}\n\n\\noindent which are satisfied if $i \\neq j$, $\\square \\not \\in Y^i$, and $\\square \\in Y^j$.\nIf $\\square$ is in row-$ {\\tt \\scriptstyle R} $ and column-$ {\\tt \\scriptstyle C} $ in $Y^j$, then the second condition in \n(\\ref{conditions.example.01}) implies that there is a cell $\\boxplus \\in Y^{\\, 1}$, strictly \nbelow $\\square$, with coordinates $\\{ {\\tt \\scriptstyle R} + L^{\\prime}, {\\tt \\scriptstyle C} \\}$, such that there are \nno cells strictly below $\\boxplus$. Since there may, or may not, be cells to the right of \n$\\boxplus$, row-$( {\\tt \\scriptstyle R} + L^{\\prime})$ in $Y^j$ has length {\\it at least} $ {\\tt \\scriptstyle C} $,\n\n\\begin{equation}\n\\label{conditions.example.02}\ny^{\\, j}_{\\, {\\tt \\scriptstyle R} + L^{\\prime}} \\geqslant {\\tt \\scriptstyle C} \n\\end{equation}\n\n\\noindent From the definition of $A_{\\square, \\, i}$, we write the first condition in \n(\\ref{conditions.example.01}) as \n$- A_{\\square, \\, i} = A^{\\prime} = {\\tt \\scriptstyle C} - y^{\\, i}_{\\, {\\tt \\scriptstyle R} }$,\nthat is, \n$ {\\tt \\scriptstyle C} = A^{\\prime} + y^{\\, i}_{\\, {\\tt \\scriptstyle R} }$, \nand using (\\ref{conditions.example.02}), we obtain \n$y^{\\, j}_{\\, {\\tt \\scriptstyle R} + L^{\\prime}} \\geqslant A^{\\prime} + y^{\\, i}_{\\, {\\tt \\scriptstyle R} }$,\nwhich we choose to write as\n\n\\begin{equation}\n\\label{conditions.example.04}\ny^{\\, j}_{\\, {\\tt \\scriptstyle R} + L^{\\prime}} - y^{\\, i}_{\\, {\\tt \\scriptstyle R} } \\geqslant A^{\\prime}\n\\end{equation}\n\n\\noindent The condition in Equation {\\bf \\ref{conditions.example.04}} is equivalent to \nthe two conditions in Equation {\\bf \\ref{conditions.example.01}}. \n\n\\subsection{One non-zero condition}\n\\label{1.not.to.vanish}\nConsider a function $f_{Y^{\\, i}, Y^{\\, j}}$, of a pair of Young diagrams $Y^{\\, i}$ and $Y^{\\, j}$, \n$i \\neq j$, such that \n$f_{Y^{\\, i}, Y^{\\, j}} = 0$, {\\it if and only if} (\\ref{conditions.example.04}) is satisfied. This \nimplies that $f_{Y^{\\, i}, Y^{\\, j}} \\neq 0$, {\\it if and only if} $Y^{\\, i}$ and $Y^{\\, j}$ satisfies \nthe complementary condition\n\n\\begin{equation}\n\\label{conditions.example.05}\ny^{\\, j}_{\\, {\\tt \\scriptstyle R} + L^{\\prime}} - y^{\\, i}_{\\, {\\tt \\scriptstyle R} } < A^{\\prime}\n\\end{equation}\n\n\\noindent which we choose to write as \n\n\\begin{equation}\n\\label{conditions.example.06}\n\\boxed{\ny^{\\, i}_{\\, {\\tt \\scriptstyle R} } - y^{\\, j}_{\\, {\\tt \\scriptstyle R} + L^{\\prime}} \\geqslant 1 - A^{\\prime}\n}\n\\end{equation}\n\n\\subsubsection{Remark} \nIn the sequel, we refer \nto Equation {\\bf \\ref{conditions.example.01}} as {\\it \\lq zero-conditions\\rq\\,}, and \nto Equation {\\bf \\ref{conditions.example.06}} as {\\it \\lq a non-zero-condition\\rq\\,}. \n\nNext we consider the latter four products on the right hand side of Equation \n{\\bf \\ref{denominator.4D.02}}.\n\n\\subsection{The first product} \n\\label{den.12}\n\n\\begin{equation}\n\\prod_{\\square \\in V^{\\, 1}} \\left\\lgroup r_a \\, \\alpha_{+} + \n s_a \\, \\alpha_{-} + A^{++}_{\\square, \\, V^{\\, 2}} \\, \\alpha_{+} \n - L^{ }_{\\square, \\, V^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies \n\n\\begin{equation}\n\\label{zeros.12}\n \\left\\lgroup r_a + A^{++}_{\\square, \\, V^{\\, 2}} \\right\\rgroup \\, \\alpha_{+} +\n \\left\\lgroup s_a - L^{ }_{\\square, \\, V^{\\, 1}} \\right\\rgroup \\, \\alpha_{-} = 0\n\\end{equation}\n\\noindent which leads to the conditions\n\\begin{equation}\n\\label{two.zero.conditions.12}\n- A^{ }_{\\square, \\, V^{\\, 2}} = r_a + 1 + c \\, p, \\quad \n L^{ }_{\\square, \\, V^{\\, 1}} = s_a + c \\, p^{\\prime} \n\\end{equation}\n\n\\noindent Since $\\square \\in V^{\\, 1}$, $L_{\\square, V^{\\, 1}} \\geqslant 0$. Given that $s$ and $p^{\\prime}$ are \nnon-zero positive integers, \nthe second equation in Equation {\\bf \\ref{two.zero.conditions.12}} admits a solution \nonly if $c = 0, 1, \\cdots$ \nThe first equation in Equation {\\bf \\ref{two.zero.conditions.12}} admits a solution \nif $\\square \\not \\in V^{\\, 2}$.\n\n\\subsubsection{From two zero-conditions to one non-zero-condition} \n\\label{translating.01}\nFollowing paragraphs {\\bf \\ref{from.2.to.1}} and {\\bf \\ref{1.not.to.vanish}}, the two \nzero-conditions in (\\ref{two.zero.conditions.12}) are equivalent to one non-zero-condition, \n\n\\begin{equation}\n\\label{one.non.zero.condition.12}\nV^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R} + s + c \\, p^{\\prime}} \\geqslant - r - c \\, p \n\\end{equation}\n\n\\subsubsection{The stronger condition} \n\\label{the.stronger.condition.01}\nEquation {\\bf \\ref{one.non.zero.condition.12}} is the statement that to eliminate the zeros, \nwe want \n$V^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R} + s + c \\, p^{\\prime}} \\geqslant - r - c \\, p$, \nwhere $c = \\{0, 1, \\cdots\\}$\nSince the row-lengths of a partition are by definition weakly decreasing, and \n$c = \\{0, 1, \\cdots\\}$, this is the case if \n$V^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R} + s} \\geqslant - r - c \\, p$, \nwhich is the case if \n$V^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R} + s} \\geqslant - r$. \nThus, we should set $c=0$, and obtain \n\n\\begin{equation}\n\\label{weak.Burge.condition.02}\nV^{\\, 2}_{ {\\tt \\scriptstyle R} } - V^{\\, 1}_{ {\\tt \\scriptstyle R} + s} \\geqslant - r \n\\end{equation}\n\n\\subsection{The second product} \n\\label{den.21}\n\n\\begin{equation}\n\\prod_{\\square \\in V^{\\, 2}} \\left\\lgroup r_a \\, \\alpha_{+} + s_a \\, \\alpha_{-} - A^{ }_{\\square, V^{\\, 1}} \\, \\alpha_{+} \n + L^{++}_{\\square, V^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies \n\\begin{equation}\n\\label{zeros.21}\n \\left\\lgroup r_a - A^{ }_{\\square, V^{\\, 1}} \\right\\rgroup \\, \\alpha_{+} \n+ \n \\left\\lgroup s_a + L^{++}_{\\square, V^{\\, 2}} \\right\\rgroup \\, \\alpha_{-} = 0\n\\end{equation}\n\\noindent which leads to the conditions\n\\begin{equation}\n\\label{two.zero.conditions.21}\n- A^{ }_{\\square, V^{\\, 1}} = - r_a + c \\, p, \n\\quad\n L^{ }_{\\square, V^{\\, 2}} = - s_a - 1 + c \\, p^{\\prime},\n\\end{equation}\n\n\\noindent Since $\\square \\in V^{\\, 2}$, $L_{\\square, V^{\\, 2}} \\geqslant 0$. Given that $s_a$ and $p^{\\prime}$ are \nnon-zero positive integers,\nthe second equation in Equation {\\bf \\ref{two.zero.conditions.21}} admits a solution \nonly if $c = 1, \\cdots$ \nThe first equation in Equation {\\bf \\ref{two.zero.conditions.21}} admits a solution \nif $\\square \\not \\in V^{\\, 1}$.\n\n\n\\subsubsection{From two zero-conditions to one non-zero-condition} \n\\label{translating.02}\nFollowing paragraphs {\\bf \\ref{from.2.to.1}} and {\\bf \\ref{1.not.to.vanish}}, the two \nzero-conditions in Equation {\\bf \\ref{two.zero.conditions.21}} are equivalent to one non-zero-condition, \n\n\\begin{equation}\n\\label{condition.21.04}\n V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R} - 1 - s_a + c \\, p^{\\prime}} \\geqslant 1 + r_a - c \\, p \n\\end{equation}\n\n\\subsubsection{The stronger condition} \n\\label{the.stronger.condition.02}\nEquation ({\\bf \\ref{condition.21.04}}) is the statement that to eliminate the zeros, \nwe want \n$V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R} - 1 - s_a + c \\, p^{\\prime}} \\geqslant 1 + r_a - c \\, p$, \nwhere $c = \\{1, 2, \\cdots\\}$.\nSince the row-lengths of a partition are by definition weakly decreasing, and \n$c = \\{1, 2, \\cdots\\}$, this is the case if \n$V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{2, {\\tt \\scriptstyle R} + p^{\\prime} - s_a - 1} \\geqslant 1 + r_a - c \\, p$.\nIn turn, is the case if \n$V^{\\, 1}_{ {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R} + p^{\\prime} - s_a - 1} \\geqslant 1 + r_a - p$. \nThus, we should set $c=1$, to obtain \n\\begin{equation}\n\\label{Burge.condition.01}\nV^{\\, 1}_ { {\\tt \\scriptstyle R} } - V^{\\, 2}_{ {\\tt \\scriptstyle R} + [p^{\\prime} - s_a] - 1} \\geqslant 1 - \\left\\lgroup p - r_a \\right\\rgroup\n\\end{equation}\n\n\\subsection{The third product} \n\\label{den.conjugate.12}\n\n\\begin{equation}\n\\prod_{\\blacksquare \\in W^{\\, 1}} \\left\\lgroup r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-} + A^{ }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{+} \n - L^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies\n\n\\begin{equation}\n\\label{zeros.conjugate.12}\n \\left\\lgroup r_b + A^{ }_{\\blacksquare, W^{\\, 2}} \\right\\rgroup \\, \\alpha_{+} \n+\n \\left\\lgroup s_b - L^{++}_{\\blacksquare, W^{\\, 1}} \\right\\rgroup \\, \\alpha_{-} = 0\n\\end{equation}\n\n\\noindent which leads to the conditions\n\n\\begin{equation}\n\\label{conditions.conjugate.12.01}\n- A_{\\blacksquare, W^{\\, 2}} = r_b + c \\, p, \n\\quad\n L_{\\blacksquare, W^{\\, 1}} = - 1 + s_b + c \\, p^{\\prime},\n\\end{equation}\n\n\\subsubsection{The stronger condition} Using the same arguments as in subsections {\\bf \\ref{den.12}} \nand {\\bf \\ref{den.21}}, are possible for $c = 0, 1, \\dots$, $\\blacksquare \\in W^{\\, 1}$, \n$\\blacksquare \\not \\in W^{\\, 2}$, and we should choose $c = 0$ to obtain \n\n\\begin{equation}\n\\label{Burge.condition.02}\nW^{\\, 2}_{ {\\tt \\scriptstyle R} } - W^{\\, 1}_{ {\\tt \\scriptstyle R} + s - 1} \\geqslant 1 - r \n\\end{equation}\n\n\\subsection{The fourth product} \n\\label{den.conjugate.21}\n\n\\begin{equation}\n\\prod_{\\blacksquare \\in W^{\\, 2}} \\left\\lgroup r_b \\, \\alpha_{+} + s_b \\, \\alpha_{-} - A^{++}_{\\blacksquare, W^{\\, 1}} \\, \\alpha_{+} \n + L^{ }_{\\blacksquare, W^{\\, 2}} \\, \\alpha_{-} \\right\\rgroup\n\\end{equation}\n\n\\noindent vanishes if any factor satisfies \n\n\\begin{equation}\n\\label{zeros.conjugate.21}\n \\left\\lgroup r_b - A^{++}_{\\blacksquare, W^{\\, 1}} \\right\\rgroup \\, \\alpha_{+} \n+ \\left\\lgroup s_b + L^{ }_{\\blacksquare, W^{\\, 2}} \\right\\rgroup \\, \\alpha_{-} = 0,\n\\end{equation}\n\n\\noindent which leads to the conditions\n\n\\begin{equation}\n\\label{conditions.conjugate.21.01}\n- A^{ }_{\\blacksquare, W^{\\, 1}} = 1 - r_b + c\\, p, \n\\quad\n L^{ }_{\\blacksquare, W^{\\, 2}} = - s_b + c\\, p^{\\prime} \n\\end{equation}\n\n\\subsubsection{The stronger condition} Using the same arguments as in subsections {\\bf \\ref{den.12}} \nand {\\bf \\ref{den.21}}, are possible for $c = 1, 2, \\cdots$, $\\square \\in W^{\\, 2}$ and $\\square \\not \\in W^{\\, 1}$, \nand we should choose $c=0$ to obtain\n\n\\begin{equation}\n\\label{weak.Burge.condition.01}\nW^{\\, 1}_{ {\\tt \\scriptstyle R} } - W^{\\, 2}_{ {\\tt \\scriptstyle R} + [p^{\\prime} - s_b]} \\geqslant - \\left\\lgroup p - r_b \\right\\rgroup\n\\end{equation}\n\n\\subsection{The Burge conditions}\n\nEquations {\\bf \\ref{weak.Burge.condition.02}} and {\\bf \\ref{Burge.condition.01}} are \nconditions on the partition pair {\\bf V} on one side of $\\mathcal{Z}_{building.block}$, while \nEquations {\\bf \\ref{Burge.condition.02}} and {\\bf \\ref{weak.Burge.condition.01}} are \nconditions on the partition pair {\\bf W} on the other side of $\\mathcal{Z}_{building.block}$.\nWhen copies of $\\mathcal{Z}_{building.block}$ are glued to form conformal blocks, partition pairs \non one side are identified with partition pairs on the other side. Thus each partition \npair must satisfy all conditions. However, these conditions are not independent as two \nof them are satisfied when the other two are satisfied. More specifically, following \nparagraphs {\\bf \\ref{from.2.to.1}} and {\\bf \\ref{1.not.to.vanish}}, we can see that it \nis sufficient to enforce the two conditions in Equations {\\bf \\ref{Burge.condition.01}} \nand {\\bf \\ref{Burge.condition.02}}, \n\n\\begin{equation}\n\\label{Burge.conditions}\n\\boxed{\n\\mu_{ {\\tt \\scriptstyle R} } - \\mu_{ {\\tt \\scriptstyle R} + [p^{\\prime} - s] - 1} \\geqslant 1 - \\left\\lgroup p - r \\right\\rgroup, \\quad\n\\nu_{ {\\tt \\scriptstyle R} } - \\nu_{ {\\tt \\scriptstyle R} + s - 1} \\geqslant 1 - r\n}\n\\end{equation}\n\n\\noindent which we write in terms of a partition pair $\\{\\mu, \\nu\\}$ that could be \non either side of $\\mathcal{Z}_{building.block}$. Partition pairs that satisfy the conditions \nin Equation {\\bf \\ref{Burge.conditions}} first appeared in the work of W H Burge on \nRogers-Ramanujan-type identities \\cite{burge}. The appeared in earlier studies of Virasoro \ncharacters in \\cite{foda.lee.welsh, foda.welsh, welsh} and more recently in the context \nof the AGT correspondence in \\cite{alkalaev.belavin, bershtein.foda}.\n\n\\section{Comments and open questions}\n\\label{section.11.comments}\n\n\\subsubsection*{Outline of result} We can generate conformal blocks of Virasoro $A$-series \nminimal models, labelled by the co-primes $p$ and $p^{\\prime}$, times a Heisenberg factor, as follows. \n{\\bf 1.} Start from the refined topological vertex of \\cite{ikv} defined in Equation \n{\\bf \\ref{refined.topological.vertex}}, \n{\\bf 2.} Glue four copies of the refined topological vertex to produce a 5D $U(2)$ basic web \npartition function, then take the $R \\rightarrow 0$ limit, to obtain its 4D counterpart \n$\\mathcal{W}_{\\bf V W \\Delta}^{\\, 4D}$ as in Equations {\\bf \\ref{W.5D.norm}, \\ref{w.numerator.5D}, \\ref{w.denominator.5D}}. \n{\\bf 3.} Set the K{\\\"a}hler parameters ${\\bf \\Delta}$, and the deformation parameters $q$ and $t$ in \n$\\mathcal{W}_{\\bf V W \\Delta}^{\\, 4D}$ as in Equations {\\bf \\ref{identification.12}}, \n {\\bf \\ref{minimal.model.charges}},\n\t\t\t\t\t {\\bf \\ref{neg.0.pos}}, and\n\t\t\t\t\t {\\bf \\ref{parameters.03}},\nand\n{\\bf 4.} require each of the partition pairs {\\bf V} and {\\bf W} to satisfy the Burge conditions\nin Equation {\\bf \\ref{Burge.conditions}}.\n\n\\subsubsection*{The refined topological vertex of Awata and Kanno} We have used the refined topological \nvertex of Iqbal, Kozcaz and Vafa \\cite{ikv}, but could have equally well used that of Awata and Kanno \n\\cite{awata.kanno.01, awata.kanno.02}. The two vertices are equivalent as explained in \n\\cite{awata.feigin.shiraishi}. \n\n\\subsubsection*{Layers} The topic discussed in this note is vast and consists of many layers. \nWe could have started our discussion from M-theory and used the language of M5 branes, but we \ndecided to stay away from this, in this short note. \nInstead, we started from A-model topological strings, which live in a corner of the M-theory. \nFrom the refined topological vertex and topological strings, we obtained the building block of \nthe instanton partition function of a 5D quiver gauge theory. We could have used the K-theoretic \nversion of the AGT correspondence to obtain the minimal model analogues of the $q$-deformed \nLiouville conformal blocks discussed in \\cite{awata.yamada.01, awata.yamada.02, aganagic.01, \naganagic.02, nieri.01, nieri.02}. \nInstead, we skipped the $q$-deformed blocks, took the 4D limit, and used the 4D version of AGT \nto obtain minimal model conformal blocks. The M-theoretic origins of the minimal conformal blocks \nand their $q$-deformations should be topics of separate studies.\n\n\\subsubsection*{Interpretation} Missing from this note is an interpretation of the minimal model \nparameters in topological string or gauge theory terms. A complete interpretation will require \nworking in M-theory terms which lies outside the scope of this work. In particular, missing is\nan interpretation of the Burge conditions in topological string or gauge theory terms. \nWe conjecture that such interpretations require re-derivations of existing results in ways that \nallow {\\it ab initio} for minimal model parameters. Current derivations do not do that, and for\nthat reason, one obtains results that are not well-defined upon substitution of minimal-model \nparameters and that require {\\it aposteriori} restrictions as in \n\\cite{alkalaev.belavin, bershtein.foda} \n\\footnote{\\,\nOF wishes to thank J-E Bourgine for discussions on this point.\n}. We plan to address this topic in future work. \n\n\n\\subsubsection*{Postscript} Following the completion of work on this note, we became aware of \nthe fact that $U(N)$ versions of Equations {\\bf 2.14} and {\\bf 2.17} in this note, were obtained \nin Equation {\\bf 4.67} in \\cite{bao.01}, and in Equation {\\bf 5.1} and subsequent equations in \n\\cite{mitev.03}\n\\footnote{\\, \nThe {\\it basic web} in this note is related, by {\\it \\lq a flop\\rq\\,} to the {\\it a strip} in \n\\cite{bao.01}. This term was introduced in \\cite{iqbal.kashani.poor.01}, where strip partition \nfunctions were studied in the context of the original, unrefined topological vertex.\n}. \n\n\\section*{Acknowledgements}\nOF would like to thank M Bershtein for collaboration on \\cite{bershtein.foda}, results of which \nwere used in this note, and H Awata, J-E Bourgine, D Krefl, V Mitev, E Pomoni, A Tanzini and Y \nZenkevich for discussions and useful remarks at various stages of work on this note. We thank \nthe anonymous referee for many useful comments that helped us improve the presentationh. OF is \nsupported by the Australian Research Council [ARC].\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction} \n\nPeriodic oscillations abound in the atmosphere of sunspots as revealed by the temporal variations in, say, Doppler velocities and intensities. These oscillations are usually considered as slow magnetoacoustic waves. The magnetic field orientation is different in the umbra from that in the penumbra; however, the propagation of slow magnetoacoustic waves is anisotropic to the orientation of magnetic fields. So, different types of magnetohydrodynamic waves are displayed inside and above sunspots at different solar atmospheric layers, such as oscillations above light bridges \\citep{2013ApJ...767..169L,2013A&A...560A..84S,2014ApJ...792...41Y, 2015MNRAS.452L..16B, 2016ApJ...816...30S, 2016A&A...594A.101Y}, umbral flashes \\citep{2014RAA....14.1001F}, penumbral running waves \\citep{2000A&A...354..305C,2013ApJ...779..168J,2015A&A...580A..53L} and coronal fan structures \\citep{2011A&A...533A.116Y,2012ApJ...757..160J,2016ApJ...823L..16T}. Slow magnetoacoustic waves are assumed to result from the interaction between the photospheric $p$-modes and magnetic fields, and can propagate upward along magnetic field lines, and finally reach coronal heights \\citep{2012ApJ...757..160J,2016NatPh..12..179J, 2015ApJ...812L..15K, 2017ApJ...847....5K,2008ApJ...682L..65Y}. Studies on the multiple oscillations above the sunspot atmosphere prove useful for understanding the physical conditions of oscillations and waves, and their formation and propagation mechanisms \\citep{2015LRSP...12....6K,2016ApJ...830L..17Z,2016NatPh..12..179J}. \n\nThe frequency distributions of sunspot oscillations usually tend to be concentrated in the bandwidth of 3--6 mHz (i.e., 3--5 minutes). It is believed that three-minute umbral oscillations exist in the chromosphere, and five-minute oscillations exist in the photosphere due to a cutoff frequency. Recently, thanks to the high spatial and temporal observations acquired with ground- and space-based instruments, it has been found that the oscillations below the typical acoustic cutoff frequency can also reach chromospheric heights, and even propagate into the corona along the magnetic field lines. For example, \\citet{2014ApJ...792...41Y} detected five-minute oscillations at a light bridge observed at chromospheric heights. Employing a multi-wavelength approach, \\citet{2012ApJ...757..160J} investigated three-minute magnetoacoustic waves, and revealed the propagation from the photosphere, through the chromosphere, and into the corona. \\citet{2012ApJ...746..119R} studied the perturbations in the 5--9 mHz frequency range in different umbra atmospheres, and examined its propagation along inclined magnetic field lines. \\citet{2013A&A...554A.146K} found a high-frequency mode with 8 mHz (2 minutes) located at a peculiar location inside an umbra, and concluded that the frequency probably relates to the oscillation of umbral dots in the photosphere. \\citet{2014A&A...569A..72S} further revealed that the spectra of umbral oscillations contain distinct peaks at 1.9, 2.3, and 2.8~minutes, and the oscillations in the higher atmospheric layers occur later than in lower ones, and claimed that they are upward propagating waves. \n\nSo far, the high-frequency perturbation modes around one minute have been reported at a particular atmosphere height \\citep{2016A&A...594A.101Y,2017ApJ...847....5K}. Utilizing the multiple optical and UV wavelength observations obtained by the Dunn Solar Telescope (DST) and the \\textit{Interface Region Imaging Spectrograph (IRIS)}, \\citet{2016A&A...594A.101Y} studied the oscillations in the emission intensity of the light bridge plasmas at different temperatures, and revealed that the light bridge exhibits a perturbation with some period shorter than one minute in a particular channel at the solar atmosphere. \n\\citet{2017ApJ...847....5K} confirmed that a high-frequency oscillation mode with 13.1~mHz is located near an umbral center in the DST Ca \\textsc{ii} K and \\textit{IRIS} 2796 \\AA\\ channels, and does not seem to be related to the magnetic field inclination angle effects. A similar period is also found in a flare loop observed by the \\textit{Geostationary Operational Environmental Satellites} \\citep{Ning:2017eh}. However, a periodicity around one minute has not been found in imaging observations at other umbral atmospheric layers. \n\nIn this letter, we use ground- and space-based imaging instruments, with high spatial and temporal resolution, to investigate the high-frequency mode with a period around one minute at different heights above a sunspot umbra. For representing, decomposing, and reconstructing high-frequency oscillation modes, we employ a novel time--frequency analysis technique named synchrosqueezing transform \\citep[SST;][]{Daubechies:2011bg,Thakur:2013kc}. This technique has been used \\citep{2013ApJ...771...33S,2017ApJ...845...11F} to represent intrinsic modes of solar cycles. Furthermore, \\citet{2017ApJ...845...11F} demonstrated that it can decompose and reconstruct intrinsic modes with a high spectral resolution without mode mixing. In Section~\\ref{sec_valid}, we first conduct a simulation experiment to evaluate the performance of SST. We then describe, in Section~\\ref{sec_data}, our observations and data processing procedures as well as an error analysis. Finally, Section~\\ref{sec_disc} concludes this study. \n\n\\begin{figure*}\n\t\\centering\n \\includegraphics[width=18cm]{fig1.eps}\n\t\\caption{Simulation experiment for evaluating the performance of SST. (a) The synthetic signal without noise (black curve) and with noise (blue curve). The noise follows a Poisson distribution. Its standard error is 6.3, and the S\/N ratio is about 3. The synthetic signal consists of three frequency components within 3--4, 6--7, and 9--14 mHz, and their amplitudes are approximately 40, 10, and 4. These are generated by a random process. (b) Power spectrum of Fourier transform. Gray regions denote the frequency ranges of the three components. (c) Power spectrum of the SST. (d) Original components (black curves) and the ones reconstructed from the noisy signal (blue curves).}\n\t\t\\label{fig1}\n\\end{figure*}\n\n\\section{Performance evaluation of the synchrosqueezing transform}\n\\label{sec_valid}\n\nSST is a time--frequency analysis technique based on the continuous wavelet (CWT) and the spectral reassignment method. The reassignment method compensates for the spreading effects inherent to CWT. SST concentrates the frequency content only along the frequency direction, and preserves the time resolution of a quasi-periodic signal. Moreover, the inverse synchrosqueezing algorithm can reconstruct instantaneous frequencies of the signal. More detailed descriptions on SST can be found in \\citet{Daubechies:2011bg} and \\citet{Thakur:2013kc}. \n\nFor evaluating the performance of SST, we synthesize a signal $s(t)$ with three frequency components, and each component varies around the central values of 4, 6, and 12 mHz along the time direction, respectively. Moreover, their amplitudes also randomly fluctuate around specific values. The amplitudes of the 4, 6, and 12 mHz components are approximately 40, 10, and 4, respectively. It should be noted that all the frequency and amplitude fluctuations are generated by a random process around the given value for a near ``real'' signal. \n\nFor an observational image, its counting noise is mainly generated by Poisson processes. So, a Poisson noise with a signal-to-noise ratio (S\/N) being 3 and standard error being 6.3 is considered. Figure 1a shows the signal we actually analyze (blue) and its noise-free counterpart (black). In the noise-free signal, the power is located in the ranges with 3--4 mHz, 6--7 mHz, and 9--14 mHz, respectively. The sampling interval and the total number of data points are 12 s and 300, respectively. The amplitude of the 12 mHz component is only 10 \\% of the 4 mHz component. In absolute values, this amplitude ($\\sim 4$) is less than the standard error ($\\sim 6.3$) of the Poisson noise. \n\n\\begin{figure*}\n\t\\centering\n \\includegraphics[width=8cm]{fig2a.eps}\n \\includegraphics[width=8cm]{fig2b.eps}\n\t\\caption{(a) An AIA 171 \\AA\\ image. (b) The co-temporal and co-spatial NVST TiO image, which is also superimposed on the 171\\AA\\ image. The pixel size of TiO images is 0\\arcsec.041. The umbral region is outlined by a red curve. A region marked ``B\" is used to compare the spectra to the umbral region in Section 4. (c) A co-spatial stacked image. From bottom to top, the images represent observations from a variety of ground- and space-based instruments: NVST TiO, AIA 1700 \\AA, AIA 1600 \\AA, NVST H$\\alpha$, AIA 304 \\AA, and AIA 171 \\AA. The white curves mark the umbral region that is the region of interest in this work.}\n\t\\label{fig2}\n\\end{figure*}\n\nTo evaluate the spectrum representation performance based on SST, the power spectra of the noisy signal (the blue curve in Figure 1a) based on SST and the Fourier transform are shown in Figures 1c and 1b, respectively. The gray regions in Figure 1b denote the ranges of the three frequency components. One can see that the noise floor appears at about 8 mHz, and the component between 9 and 14 mHz is heavily contaminated by the high-frequency noise. In Figure 1c, there are some spectral contents, although weak, in the range 9--15 mHz, demonstrating that the SST has better ability to suppress random noise. Figure 1d shows the original frequency components (black) and the corresponding reconstructed modes (blue). Compared with the original components, the amplitudes of the reconstructed modes are slightly different. However, both the frequency and the phase are almost the same as in the original ones, which is particularly true for the weakest component (12 mHz). To sum up at this point, SST performs well for resolving the physical signals that are heavily contaminated by a Poison noise. \n\n\n\\begin{figure*}\n\t\\centering\n \\includegraphics[width=18cm]{fig3.eps}\n\\caption{Left column: intensity curves averaged over the entire umbra region in different channels. Right column: the corresponding Fourier spectra of the de-trended curves. Red dashed curves denote the 95 \\% confidence levels. The shaded regions denotes the frequency range of interest in this work.}\n\t\t\\label{fig3}\n\\end{figure*}\n\n\\section{Observations and Data Reduction}\n\\label{sec_data}\n\n\\begin{figure*}\n\t\\centering\n \\includegraphics[width=18cm]{fig4.eps}\n\\caption{Left column: SST power spectra in each channel. Black dashed curves denote the influence of cone, and the contours present 95 \\% confidence levels in the 12 mHz components. Right column: the corresponding reconstructed components. }\n\t\t\\label{fig4}\n\\end{figure*}\n\nThis study focuses on active region NOAA 11081, located in the northern hemisphere on 2013 August 1. The New Vacuum Solar Telescope \\citep[NVST;][]{2014RAA....14..705L,2014IAUS..300..117X}, located at the Fuxian Solar Observatory of the Yunnan Observatories, China, was employed to obtain the high-resolution ground-based imaging observations. The two optical channels, centered at the H$\\alpha$ line core (6562.8$\\pm$0.25 \\AA) and TiO (7058$\\pm$10~\\AA), were used to capture the chromospheric and photospheric images between 03:38 and 04:35 UT. The pixel scale is 0\\arcsec.162 for the H$\\alpha$ images, and 0\\arcsec.041 for the TiO ones. The sampling interval is 12 s for the H$\\alpha$ sequences, and 20 s for the TiO ones. \n\n\\begin{figure}\n\t\\centering\n \\includegraphics[width=8cm]{fig5.eps}\n\t\\caption{(a) A reconstructed running difference based on every pixel in the AIA 171 \\AA\\, whose spectra only includes the frequency contents from 10 to 14 mHz. Its original image is observed on 2013 August 1 at 04:04:59 UT. The red and yellow curves outline the umbra and penumbra edge, and the dashed blue lines highlight expanding annulus. A cut taken to make the time--distance diagram is indicated with a green line. (b) The time--distance diagram started at 04:00:35 UT and finished at 04:08:59 UT covers about six cycles of the propagating features. Its propagating velocity is about 49 $\\pm$ 4 km s$^{-1}$. (c) The power spectrum of region ``B\" marked in Figures 2a and b. Two horizontal dashed lines indicate the range between 10 and 14 mHz. }\n\t\t\\label{fig5}\n\\end{figure}\n\nThe co-temporal and co-spatial observations were also obtained from the Atmospheric Imaging Assembly \\citep[AIA;][]{2012SoPh..275...17L} on board the \\textit{Solar Dynamics Observatory} \\citep[SDO;][]{2012SoPh..275....3P}. Our analysis concentrates on the 171 \\AA, 304 \\AA, 1600 \\AA\\, and 1700 \\AA\\ channels. The formation height of the 1700 \\AA\\ channel is located at the temperature minimum, 1600 \\AA\\ at the lower chromosphere, 304 \\AA\\ at the transition region, and 171 \\AA\\ at the corona. To get the level 1.5 data, the level 1.0 data were processed using the routine $aia\\_prep.pro$. The pre-process brings all the data to a common center and plate scale with $\\approx$ 0\\arcsec.6, and the cadence is 12 s for the AIA 171\\AA\\ and 304 \\AA\\ channels, and 24 s for the 1600 \\AA\\ and 1700 \\AA. Furthermore, the images were co-aligned by the sub-pixel registration algorithm \\citep{Feng:2012hk}. Similarly, all the NVST images were aligned to the AIA data with a scale reduction and a rotation transform. Finally, we truncated the size of the data sets into a set of 54\\arcsec\\ $\\times$ 54\\arcsec, which contains the entire sunspot. \n\nFigure 2a shows an AIA 171 \\AA\\ image and its co-temporal and co-spatial TiO thumbnail. The original TiO image is shown in Figure 2b. The umbral edge is marked with a red curve. The entire umbra is employed to analyze high-frequency oscillation modes. The images stacked in Figure 2c represent the observations at different formation heights. \n\n\nFigure~3 presents the average intensity variations of the entire umbra in every channel (left column). We removed long-term trend greater than 2.5 minutes in all data with frequency filtering for concentrating on high-frequency component analysis, and the power spectra of the Fourier transform of the de-trended curves are presented in the right column. Red dashed lines indicate the 95\\% confidence levels, and the shaded regions denote a frequency range between 10 and 14 mHz. We can find that the spectral contents between 10 and 14 mHz exist in each channel, and are higher than the 95 \\% confidence level. There are significant peaks within the shaded regions. In what follows, we use ``the 12~mHz component'' to label the oscillations in this frequency range for brevity. Subsequently, the 12~mHz components are represented and reconstructed by the SST, respectively. The SST spectra are represented in the left column of Figure 4, in which the black dashed curves denote the cone of influence and those contours present 95 \\% of the confidence level. The shaded regions are decomposed and reconstructed by the SST and are shown in the right column of Figure 4.\n\n\nFor estimating the amplitude error of the reconstructed components, we analyze their noise level. The error estimation proposed by \\citet[see Section 3 by][]{2012A&A...543A...9Y} is used to calculate the image noise in each AIA channel. For a pixel with an intensity value $F$ (in units of DN), the standard error of the noise can be approximated by $\\sqrt{2.3+0.06F}$. Because the light curve in the AIA 171 \\AA\\ channel incorporates the entire umbra region that amounts to 190 pixels in total, the estimated data noise should be multiplied by a factor of 1\/$\\sqrt{190}$. Thus, the error of the AIA 171 \\AA\\ channel is about 0.6~DN for the selected data. The error can be estimated similarly for the rest of the EUV\/UV channels. \n\nFor the NVST data, we assume that the noise is primarily due to photon counting, which follows a Poisson distribution. Therefore, its amplitude is the square root of the signal intensity. Because the pixel resolution of the original image of the TiO data is reduced from 0\\arcsec .041 to 0\\arcsec .6, and a 190-pixel region is selected to generate the light curve, the noise is about 1\/202 (1\/50 for the H$\\alpha$ channel) of the square root of the original intensity. The end result is that the amplitude error of the TiO channel is about 0.6, and that of the H$\\alpha$ is about 1.5. So, for AIA 1700, 1600, and 304 \\AA, the amplitudes of the reconstructed mode are two times larger than the estimated noise. Likewise, for AIA 171 \\AA, the NVST TiO, and H$\\alpha$, the amplitudes are at least three times larger. From this error analysis, together with the confidence levels in the Fourier and SST spectra, we conclude that the reconstructed components of 12 mHz are physical rather than high-frequency noise. \n\n\\begin{figure*}\n\t\\centering\n \\includegraphics[width=0.9\\textwidth]{fig6anim.eps}\n\t\\caption{Online animation of the the AIA 171 \\AA\\ images (left) and the 12~mHz modes reconstructed by every pixel in the AIA 171 \\AA\\ channel (right).}\n\t\t\\label{fig6anim}\n\\end{figure*}\n\nWe reconstructed the 12~mHz modes for every pixel in the AIA 171 \\AA\\ channel, and then generated a running-difference animation (Figure 6). This animation is restricted to the time interval between 03:59:59 and 04:09:47 UT, during which the 12~mHz signal is the most evident (see the uppermost panel in Figure~4). In the animation, the footpoint of the coronal fan structures is marked with a white cross. The umbral and penumbral borders as determined with the NVST TiO images are marked with the red and yellow curves, respectively. The dashed blue lines highlight a series of expanding annuli. The movie exhibits an intermittent and outward propagating wave near the footpoint of the coronal fan structures. It emerges first near the footpoint and then propagates outward along the fan structures before disappearing. A running-difference image, extracted from the animation, is shown in Figure 5a, where we also display an artificial slit extending from the footpoint and following the coronal fan structure (the thick green line). A time--distance plot along this slit, 24 pixels long and 3 pixels wide, is shown in Figure 5b. The propagating perturbations are featured by the repeating diagonal ridges, which we outline with a series of red dashed lines to help guide the eye. Standard practice yields that the disturbances propagate at a speed of 49 $\\pm$ 4 km s$^{-1}$. \n\n\nTo further understand where the 12~mHz signal in the TiO image sequences comes from, we also examined the power spectrum in the TiO observations for the region in the quiet photosphere that surrounds the sunspot. Shown in Figure 5c is a spectrum at an arbitrarily chosen location (marked ``B'' in the left column of Figure~2). Evidently, this spectrum does not exhibit any power in the range of 10--14 mHz, delineated by the horizontal dashed lines. This means that the 12~mHz signal originates from within the umbra rather than coming from the surrounding quiet photosphere. \n\n\\section{discussion and conclusion}\n\\label{sec_disc}\n\nA high-frequency oscillation with a frequency range 10--14 mHz (about one minute) above a sunspot umbra atmosphere was found using simultaneous observations captured by the NVST and SDO\/AIA instruments. A novel time--frequency analysis method, called SST, was used to decompose and reconstruct the high-frequency components. To ensure that the reconstructed components are physical rather than high-frequency noise, a synthetic signal (the blue curve in Figure 1a), the noise floor appears at even lower frequencies (the spectral break now occurs at 8 mHz) was analyzed. The results decomposed and reconstructed demonstrate that the SST has no problem in resolving the physical signal even if the contamination from high-frequency noise is even stronger in the synthetic time series. Analyzing Fourier and SST spectra of the intensity curves at six channels: the AIA 171, 304, 1600, 1700 \\AA\\, the NVST H$\\alpha$, and TiO, the spectral contents between 10 and 14 mHz not only have significant peaks, but also higher than 95 \\% confidence levels. Furthermore, error analysis demonstrates that all the reconstructed amplitudes of the 12 mHz signals are at least twice larger than the corresponding error. The time--distance diagram, coupled with a subsonic speed ($\\sim$ 49 km s$^{-1}$), highlights the fact that these coronal perturbations are best described as upwardly propagating magnetoacoustic slow-mode waves.\n\nCompared with the spectra from the region surrounding the sunspot, the 12~mHz periodicity exists only inside the umbra, meaning that the modes at the coronal height are probably related to the perturbations from the photospheric umbra. We note that \\citet{2017ApJ...847....5K} have already found a similar periodicity at chromospheric heights above an umbra. However, to our knowledge, the present study is the first to reveal the existence of such a periodicity at different heights above an umbra in imaging observations. \n\nBefore closing, we would like to point out that our findings benefit from the capability of SST for suppressing spectral smearing. As such, SST is likely to find some wider use for representing and decomposing intrinsic modes in non-stationary signals.\n\n\n\\acknowledgments\nThe authors thank the referee for useful comments. We thank the NVST team for their high-resolution observations and the use of \\textit{SDO\/AIA} image obtained courtesy of NASA\/SDO and the AIA science teams. This work is supported by the National Natural Science Foundation of China (Numbers: 11463003, 11503080, 11761141002, 41474149, 41674172) and the Joint Research Fund in Astronomy (U1531132, U1531140, U1631129) under coperative agreement between the NSFC, the National Key Research and Development Program of China (Number: 2016YFE0100300), and the CAS Key Laboratory of Solar Activity of National Astronomical Observatories (KLSA201715).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}