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Localizing AlAdS$_5$ black holes and the SUSY index on $S^1 \times M_3$

Jaeha Park

TL;DR

The paper develops a gravity–field theory bridge for localizing AlAdS$_5$ black holes with $S^1\times M_3$ boundaries by employing equivariant localization in five-dimensional gauged supergravity, tying the SUSY index on twisted backgrounds to topological fixed-point data. It explicitly constructs anti-periodic Killing spinors on complex, non-extremal Euclidean backgrounds and reduces to 3d new minimal supergravity to obtain Cardy-like indices for $\mathcal{N}=4$ SYM on various $M_3$ geometries, including round $S^3$, elliptically/biaxially squashed $S^3$, and Lens spaces; the leading results are of the form $\log \mathcal{I} \sim -\mathrm{i}\pi (N^2-1)\frac{8}{27}\frac{\Delta^3}{\sigma\tau}$ with appropriate refinements. A gravity interpretation via equivariant localization recovers the field theory results in the large $N$ limit, provided one uses a holographic background subtraction that subtracts the supersymmetric Casimir energy, and the boundary data fully determines the fixed-point contributions. The work also outlines a general gluing prescription to compute the index for arbitrary AlAdS$_5$ fillings, suggesting a universal, topological computation of the index from fixed-point data independent of the explicit bulk solution, and points to future extensions to higher dimensions and lower-supersymmetry theories.

Abstract

We consider complex, supersymmetric, non-extremal Euclidean black holes that are asymptotically locally AdS$_5$, with $S^1 \times M_3$ conformal boundary. We study field theory backgrounds consisting of various $M_3$, and explicitly construct Killing spinors that are anti-periodic around the Euclidean time circle. Focussing on elliptically/biaxially squashed three-spheres and Lens spaces, we compute the supersymmetric index of the $\mathcal{N}=4$ SYM in a Cardy-like limit. While such black holes have not been constructed for general $M_3$, we show that our field theory results can be recovered from a gravity computation using equivariant localization, just assuming the solutions exist.

Localizing AlAdS$_5$ black holes and the SUSY index on $S^1 \times M_3$

TL;DR

The paper develops a gravity–field theory bridge for localizing AlAdS black holes with boundaries by employing equivariant localization in five-dimensional gauged supergravity, tying the SUSY index on twisted backgrounds to topological fixed-point data. It explicitly constructs anti-periodic Killing spinors on complex, non-extremal Euclidean backgrounds and reduces to 3d new minimal supergravity to obtain Cardy-like indices for SYM on various geometries, including round , elliptically/biaxially squashed , and Lens spaces; the leading results are of the form with appropriate refinements. A gravity interpretation via equivariant localization recovers the field theory results in the large limit, provided one uses a holographic background subtraction that subtracts the supersymmetric Casimir energy, and the boundary data fully determines the fixed-point contributions. The work also outlines a general gluing prescription to compute the index for arbitrary AlAdS fillings, suggesting a universal, topological computation of the index from fixed-point data independent of the explicit bulk solution, and points to future extensions to higher dimensions and lower-supersymmetry theories.

Abstract

We consider complex, supersymmetric, non-extremal Euclidean black holes that are asymptotically locally AdS, with conformal boundary. We study field theory backgrounds consisting of various , and explicitly construct Killing spinors that are anti-periodic around the Euclidean time circle. Focussing on elliptically/biaxially squashed three-spheres and Lens spaces, we compute the supersymmetric index of the SYM in a Cardy-like limit. While such black holes have not been constructed for general , we show that our field theory results can be recovered from a gravity computation using equivariant localization, just assuming the solutions exist.

Paper Structure

This paper contains 7 sections, 36 equations.

Table of Contents

  1. Introduction
  2. Field theory
  3. Warm up: S1×S3 with twistings Ω1, Ω2
  4. Supersymmetry
  5. Superconformal index in the Cardy-like limit
  6. Lens space index
  7. Elliptical squashing S1×S3b1b2$with twistings Ω1, Ω2 We now consider the following background metric: {{\rm d}} s^2_4= {{\rm d}} t_E^2 + f(\vartheta)^2 {{\rm d}} \vartheta^2 + \frac{\sin^2 \vartheta}{\mathfrak{b}_1^2} \left({{\rm d}} \varphi_1 -\mathrm{i} \Omega_1 {{\rm d}} t_E \right)^2 + \frac{\cos^2 \vartheta}{\mathfrak{b}_2^2} \left({{\rm d}}\varphi_2 - \mathrm{i} \Omega_2 {{\rm d}} t_E \right)^2 \,, with independent coordinate identifications$t_E ∼ t_E + β$,$φ_1 ∼ φ_1 + 2π$,$φ_2 ∼ φ_2 + 2π$, as well as$ϑ ∈ [0, π/2 ]$. The real parameters$b_1, b_2$control the "squashings". We will not specify the function$f(ϑ)$and keep the discussion general, following observations in e.g. Martelli:2011fu that physical observables do not depend on the details of$f(ϑ)$. If we set f(\vartheta) \equiv \sqrt{\frac{\cos^2\vartheta}{\mathfrak{b}_1^2} + \frac{\sin^2\vartheta}{\mathfrak{b}_2^2}} \,, the spatial part of the metric is equivalent to the elliptically squashed three-sphere of Hama:2011ea. As in section \ref{['subsection:S1S3']}, the "twisting" parameters$Ω_1, Ω_2$are \emph{complex}. A similar background was considered in Cassani:2021fyv. The computations are complementary, and we highlight the differences below: The metric considered in Cassani:2021fyv is real. The real parameters $k_{1,2}^{\rm there}$ that appear as "twistings" in their metric correspond to $\mathrm{i} \Omega_{1,2}$ in \ref{['S1S3bmetric']}, i.e. they restrict to pure imaginary $\Omega_1, \Omega_2$.In addition, in Cassani:2021fyv it was assumed that the supercharges do not depend on the Euclidean time direction. Here, we derive Killing spinors that are anti-periodic around the Euclidean time circle.We show that the dependence of the index on the function $f(\vartheta)$ drops out of the index in the Cardy-like limit -- not even its behaviour at the poles. The assumptions made in Cassani:2021fyv, regarding points 1 \& 2, imply that the background considered there cannot arise from the conformal boundary of complex, non-extremal deformations of supersymmetric AdS$_5$black holes Cabo-Bizet:2018ehj. This issue is discussed in pp.~27--28 of Cassani:2021fyv. Though the background metric of Cassani:2021fyv is real, it is possible to complexify the real parameters$k_1,2^ there$and recover the complex background \ref{['S1S3_metric']}, further setting$b_1 = b_2 = 1$. Since the supersymmetric partition functions are metric-independent, and are holomorphic functions of the complex structure parameters Closset:2013vra, the field theory computations performed on real backgrounds Cassani:2021fyvArdehali:2021irqOhmori:2021dzb can be matched with supergravity computations that study complex black hole saddles Bobev:2022bjmCassani:2022lrkCassani:2024tvk via analytic continuations. Our analysis provides a further check for this claim, allowing for arbitrary metrics on$S^1 S^3_b_1, b_2$via the function$f(ϑ)$. We emphasise that it is necessary for our purposes to start with a complex background, in order to explicitly construct Killing spinors that are \emph{anti-periodic} around the Euclidean time circle. This is due to the fact that$S^1_β = ∂ R^2$bounds the$R^2$disc factor that smoothly caps off at the black hole horizon -- it is only the anti-periodic spin structure at the boundary that extends to the bulk. We show that it is possible to work out background fields$A^ nm, V^ nm$with the metric \ref{['S1S3bmetric']}, that realise Killing spinors with respect to this spin structure. We then show that the supersymmetric index computed using Honda's prescription \ref{['HondaIndex']} agrees with (an analytic continuation of) the "second sheet" result, Cassani:2021fyv. Though not highlighted in Cassani:2021fyv, we thus argue that the result can be associated to complex supersymmetric AlAdS$_5$black saddles with elliptically squashed$S^1_β S^3_b_1, b_2$conformal boundary, which we verify from a gravity computation in section \ref{['gravity']}. We use the following frame for the metric \ref{['S1S3bmetric']}: {\rm e}^1= {{\rm d}} t_E \,,{\rm e}^2= \cos(\varphi_1 + \varphi_2) f(\vartheta) {{\rm d}} \vartheta- \frac{1}{2} \sin(2\vartheta) \sin(\varphi_1 + \varphi_2) \left( \frac{1}{\mathfrak{b}_1} ( {{\rm d}}\varphi_1 -\mathrm{i} \Omega_1 {{\rm d}}t_E) - \frac{1}{\mathfrak{b}_2} ({{\rm d}} \varphi_2 - \mathrm{i} \Omega_2 {{\rm d}} t_E) \right) \,,{\rm e}^3= \sin(\varphi_1 + \varphi_2) f(\vartheta) {{\rm d}}\vartheta+ \frac{1}{2} \sin(2\vartheta) \cos(\varphi_1 + \varphi_2) \left( \frac{1}{\mathfrak{b}_1} ( {{\rm d}}\varphi_1 -\mathrm{i} \Omega_1 {{\rm d}}t_E) - \frac{1}{\mathfrak{b}_2} ({{\rm d}} \varphi_2 - \mathrm{i} \Omega_2 {{\rm d}} t_E) \right) \,,{\rm e}^4= \frac{\sin^2 \vartheta}{\mathfrak{b}_1} \left({{\rm d}} \varphi_1 - \mathrm{i} \Omega_1 {{\rm d}} t_E \right) + \frac{\cos^2\vartheta}{\mathfrak{b}_2} \left( {{\rm d}}\varphi_2 -\mathrm{i} \Omega_2 {{\rm d}} t_E \right) \,. We would like to find background field configurations (that is,$A^ nm$and$V^ nm$) that allows for solutions to the Killing spinor equation \ref{['KSE_4d_Dirac']}, which we rewrite here: \left[ \nabla_M - \mathrm{i} A_M^{\rm nm} \gamma_5 + \mathrm{i} V_M^{\rm nm} \gamma_5 - \frac{\mathrm{i}}{2} (V^{\rm nm})^N \gamma_{MN} \gamma_5 \right] \epsilon = 0 \,. By choosing A^{\rm nm}= \frac{\mathrm{i}}{2} \left[ \frac{1}{f(\vartheta)} \left( \frac{\Omega_1}{\mathfrak{b}_1} + \frac{\Omega_2}{\mathfrak{b}_2} - 2 \right) - \left( 3 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi + \Omega_1 + \Omega_2 \right) \right] {{\rm d}} t_E+ \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_1} \right) {{\rm d}}\varphi_1 + \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_2} \right) {{\rm d}}\varphi_2 \,,V^{\rm nm}= - \frac{\mathrm{i}}{f(\vartheta)} {{\rm d}} t_E \,, one can verify that the following is a unit-norm Killing spinor (cf. \ref{['S1S3KSchoice']}): \epsilon = \frac{1}{\sqrt{2}} \begin{pmatrix} \exp \left[ \frac{t_E}{2} \left(3 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi + \Omega_1 + \Omega_2 \right) \right] \\ 0 \\ 0 \\ \exp \left[ - \frac{t_E}{2} \left(3 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi + \Omega_1 + \Omega_2 \right) \right] \\ \end{pmatrix} \,. Some remarks in order: We fixed the ambiguity in $A^{\rm nm}, V^{\rm nm}$ by demanding that $V^{\rm nm}$ only has components along ${{\rm d}} t_E$. The background fields were found starting from the assumption that $A^{\rm nm}_{\varphi_1,\varphi_2}$ are given by \ref{['app:S3bAfield']}. The $\varphi_1$, $\varphi_2$ components of the Killing spinor equation then fixes $V^{\rm nm}_{t_E}$, and the remaining components are solved by choosing $A^{\rm nm}_{t_E}$ as in \ref{['AnmVnm_S1S3b']}.Sanity check: $A^{\rm nm}$ as given in \ref{['AnmVnm_S1S3b']} reduces to that as given in \ref{['S1S3_AnmVnm']} upon taking $\mathfrak{b}_1 = \mathfrak{b}_2 = 1$, given that $f(\vartheta)$ reduces to 1.The gauge field $A^{\rm nm}$ is regular at the two poles of $S^3_{\mathfrak{b}_1,\mathfrak{b}_2}$, provided $f(\vartheta)$ is given by \ref{['f_vartheta']}. As $\vartheta \rightarrow 0$, where ${{\rm d}} \varphi_1$ is not well-defined, we have $A^{\rm nm}_{\varphi_1} \rightarrow 0$, and as $\vartheta \rightarrow \frac{\pi}{2}$, where ${{\rm d}} \varphi_2$ is not well-defined, we have $A^{\rm nm}_{\varphi_2} \rightarrow 0$.The Lie derivatives $\mathcal{L}_{\varphi_1}$, $\mathcal{L}_{\varphi_2}$ acting on individual components of $\epsilon$ are unchanged from the round $S^3$ case, with charges $\pm \frac{1}{2}$. The background admits two supercharges of opposite R-charge.The following is also a solution: $\epsilon = \frac{1}{\sqrt{2}} 0\exp \left[ \frac{t_E}{2} \left(3 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi - \Omega_1 - \Omega_2 \right) \right]\exp \left[ - \frac{t_E}{2} \left(3 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi - \Omega_1 - \Omega_2 \right) \right]0 \,,$ provided that for $A^{\rm nm}$ we choose instead A^{\rm nm}= - \frac{\mathrm{i}}{2} \left[ \frac{1}{f(\vartheta)} \left( \frac{\Omega_1}{\mathfrak{b}_1} + \frac{\Omega_2}{\mathfrak{b}_2} + 2 \right) + \left( 3 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi - \Omega_1 - \Omega_2 \right) \right] {{\rm d}} t_E- \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_1} \right) {{\rm d}}\varphi_1 - \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_2} \right) {{\rm d}}\varphi_2 \,. The boundary supersymmetric Killing vector is then given by (recall$ϵ^* = i C^-1 γ^1 ϵ$): K= \epsilon^* \gamma_1 \gamma^M \epsilon \, \partial_M= \partial_{t_E} + \mathrm{i} (\Omega_1 - \mathfrak{b}_1)\partial_{\varphi_1} + \mathrm{i} (\Omega_2 - \mathfrak{b}_2) \partial_{\varphi_2} \,. The frame \ref{['S1S3bframe']} is such that$L_∂_t_E ϵ = ∂_t_E ϵ$. It follows that anti-periodic boundary condition on the Killing spinor \ref{['S1S3bKS']}, given by$ϵ(t_E+β) = -ϵ(t_E)$, requires (cf. \ref{['KSE_4d_constraint_S3']}): \beta \left( 2\Psi - 3 + \mathfrak{b}_1 + \mathfrak{b}_2 - \Omega_1 - \Omega_2 \right) = 2\pi \mathrm{i} n \,,\qquad n \ \textrm{odd} \,. We can now define "elliptically squashed" supersymmetric chemical potentials as \sigma_{\mathfrak{b}} = \frac{\beta(\Omega_1 - \mathfrak{b}_1)}{2\pi\mathrm{i}} \,,\qquad \tau_{\mathfrak{b}} = \frac{\beta(\Omega_2 - \mathfrak{b}_2)}{2\pi\mathrm{i}} \,,\qquad \Delta_{\mathfrak{b}} = \frac{\beta}{2\pi\mathrm{i}} \left( \Psi - \frac{3}{2} \right) \,, and the constraint \ref{['KSE_4d_constraint_S3b']} is given by 2\Delta_{\mathfrak{b}} - \sigma_{\mathfrak{b}}- \tau_{\mathfrak{b}} = n \,,\qquad n \ \textrm{odd} \,. We now perform the KK reduction along$t_E$, starting from the metric \ref{['S1S3bmetric']}, identify 3d new minimal supergravity fields, and evaluate$A_M_3, L_M_3$given by \ref{['AM3']}, \ref{['LM3']} in the Cardy-like limit. A similar computation was carried out in section 6 of Cassani:2021fyv, though with reality assumptions on the 4d metric, as well as assuming$A^ nm_t_E = V^ nm_t_E$. We are required to lift both assumptions, as$Ω_1, Ω_2$are complex, and the latter assumption does not hold for our background of interest (recall \ref{['AnmVnm_S1S3b']}). The result is as follows. As before, let us write the dimensional reduction ansatz as {{\rm d}}s_4^2 = e^{2w} ( {{\rm d}} t_E + c_\mu (x) {{\rm d}}x^\mu)^2 + h_{\mu\nu}(x){{\rm d}}x^\mu {{\rm d}}x^\nu \,. Comparing with the background metric \ref{['S1S3bmetric']}, we find e^{2w}= 1 - \frac{\Omega_1^2 \sin^2 \vartheta}{\mathfrak{b}_1^2} - \frac{\Omega_2^2 \cos^2 \vartheta}{\mathfrak{b}_2^2} \,,c_\mu {{\rm d}} x^\mu= - \mathrm{i} e^{-2w} \left( \frac{\Omega_1 \sin^2\vartheta}{\mathfrak{b}_1^2} {{\rm d}} \varphi_1 + \frac{\Omega_2 \cos^2 \vartheta}{\mathfrak{b}_2^2} {{\rm d}} \varphi_2 \right) \,, as well as h_{\mu\nu} {{\rm d}}x^\mu {{\rm d}}x^\nu = f(\vartheta)^2 {{\rm d}}\vartheta^2 + \frac{\sin^2 \vartheta}{\mathfrak{b}_1^2} {{\rm d}}\varphi_1^2 + \frac{\cos^2 \vartheta}{\mathfrak{b}_2^2} {{\rm d}}\varphi_2^2 - e^{2w} c^2 \,. From the dimensional reduction Assel:2014paa, 3d new minimal supergravity fields are identified via \ref{['H_AR_id']}-\ref{['vHodge']}. Given \ref{['AnmVnm_S1S3b']}, we have (cf. \ref{['H_AR_S1S3']}) H= - \frac{\mathrm{i}}{f(\vartheta)} e^{-w} \,,A_\mu^{(R)}= \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_1} \right) {{\rm d}} \varphi_1 + \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_2} \right) {{\rm d}} \varphi_2- \mathrm{i} \left[ \frac{1}{2f(\vartheta)} \left( \frac{\Omega_1}{\mathfrak{b}_1} + \frac{\Omega_2}{\mathfrak{b}_2} - 2 \right) + \frac{\pi \mathrm{i} n}{\beta} \right] c_\mu + \frac{1}{2} e^w v_\mu where we have used the constraint \ref{['KSE_4d_constraint_S3b']}. Collecting the above ingredients, we may now compute$A_S^3_b_1,b_2$and$L_S^3_b_1,b_2$given by \ref{['AM3']}, \ref{['LM3']}. We note that in evaluating A_{S^3_{\mathfrak{b}_1,\mathfrak{b}_2}}^{\rm CS} = \frac{1}{\pi^2} \int_{S^3} c \wedge {{\rm d}} c = - \frac{4\Omega_1 \Omega_2}{(\mathfrak{b}_1^2-\Omega_1^2)(\mathfrak{b}_2^2-\Omega_2^2)} \,, the dependence on$f(ϑ)$explicitly cancels out in the integrand and$A_S^3_b_1,b_2^ CS$does not depend on the function at all. With the modified chemical potentials \ref{['susychempot_S3b']}, we now take the "Cardy-like limit" by introducing small parameters$δ_σ_b = βσ_b$and$δ_τ_b = βτ_b$, such that \Omega_1 \simeq \frac{2\pi \mathrm{i}}{\delta_{\sigma_{\mathfrak{b}}}} \,,\qquad \Omega_2 \simeq \frac{2\pi \mathrm{i}}{\delta_{\tau_{\mathfrak{b}}}} \,, and taking$δ_σ_b, δ_τ_b → 0$only at the end of all computations, with fixed$ σ_b τ_b $. As observed in the round$S^3$analysis in section \ref{['subsection:S1S3']},$A^H_S^3_b_1,b_2$is sub-leading to$A^ CS_S^3_b_1,b_2$in the modified Cardy-like limit, and we have A_{S^3_{\mathfrak{b}_1,\mathfrak{b}_2}} \underset{\beta \rightarrow 0}{\simeq} \frac{\beta^2}{\pi^2} \frac{1}{\sigma_{\mathfrak{b}} \tau_{\mathfrak{b}}} \,. Similarly, the leading asymptotic behaviour of$L_S^3_b_1,b_2$in \ref{['LM3']} comes from the$A_μ^(R) v^μ$term, and we have L_{S^3_{\mathfrak{b}_1,\mathfrak{b}_2}} \underset{\beta \rightarrow 0}{\simeq} \frac{\mathrm{i} \beta}{\pi} \frac{\sigma_{\mathfrak{b}} + \tau_{\mathfrak{b}}}{\sigma_{\mathfrak{b}} \tau_{\mathfrak{b}}} \,. Note to evaluate the integrals required to compute$L_S^3_b_1,b_2$, we had to specify the function$f(ϑ)$as in \ref{['f_vartheta']}, but it is likely that the dependence drops out in the Cardy-like limit. Indeed, if one takes the limit first (as opposed to taking the limit after the integration), the terms involving$f(ϑ)$drop out, and we recover the same result, that is, \ref{['LS3b']}. Substituting these into Honda's prescription \ref{['HondaIndex']}, with "elliptically squashed" chemical potentials satisfying the constraint (setting$n=1$in \ref{['SYMconstraint_S3b']}) 2 \Delta_{\mathfrak{b}} - \sigma_{\mathfrak{b}} - \tau_{\mathfrak{b}} = 1 \,, we conclude that the elliptically squashed index in the Cardy-like limit is given by \log \mathcal{I}_{S^1_\beta \times S^3_{\mathfrak{b}_1,\mathfrak{b}_2}} \underset{|\sigma_{\mathfrak{b}}|, |\tau_{\mathfrak{b}}| \rightarrow 0}{\simeq} - \mathrm{i} \pi (N^2-1) \frac{8}{27} \frac{\Delta_{\mathfrak{b}}^3}{\sigma_{\mathfrak{b}} \tau_{\mathfrak{b}}} \,. It is interesting to compare this to the "second sheet" result obtained in Cassani:2021fyv, where the Chern--Simons terms were evaluated on a real background. Recall for this comparison that we computed the index of the$SU(N)$$N=4$SYM, for which$a = c = N^2-14$. Restricting$Ω_1, Ω_2$to be purely imaginary, we may identify \omega_1^{\rm there} \leftrightarrow - 2\pi \mathrm{i} \sigma_{\mathfrak{b}} \,,\qquad \omega_2^{\rm there} \leftrightarrow - 2\pi \mathrm{i} \tau_{\mathfrak{b}} \,, and one finds that \ref{['HondaSquashedIndexb']} matches Cassani:2021fyv, with$n_0^ there = -1$. The fact that the two results agree strongly suggests that the two supersymmetric backgrounds, that is, (6.2), (6.6) there and \ref{['S1S3bmetric']}, \ref{['AnmVnm_S1S3b']} here, are related by a supersymmetry-preserving deformation that leaves the partition function invariant. We note that the "shift" in the chemical potentials considered in Cassani:2021fyv to go to the "second sheet" is precisely such that the constraint \ref{['SYMconstraint_S3b_1']} is satisfied. Here, we showed that the constraint can be realised from the periodicity condition \ref{['KSE_4d_constraint_S3b']} without having to perform such a "shift", extending an analysis in Cabo-Bizet:2018ehj. We return to this point around \ref{['SCIwrittenout']}-\ref{['Zsymv2']}. We now consider the following background metric: {{\rm d}}s^2_4 = {{\rm d}} t_E^2+ {{\rm d}}\vartheta^2 + \sin^2\vartheta \left({{\rm d}} \varphi_1 -\mathrm{i} \Omega_1 {{\rm d}} t_E \right)^2 + \cos^2\vartheta \left({{\rm d}}\varphi_2 - \mathrm{i} \Omega_2 {{\rm d}} t_E \right)^2+ (v^2-1) \left( \sin^2\vartheta ({{\rm d}}\varphi_1 - \mathrm{i} \Omega_1 {{\rm d}}t_E) + \cos^2\vartheta ({{\rm d}}\varphi_2 - \mathrm{i} \Omega_2 {{\rm d}}t_E) \right)^2 \,, with independent identifications$t_E ∼ t_E + β$,$φ_1 ∼ φ_1 + 2π$,$φ_2 ∼ φ_2 + 2π$, with$ϑ ∈ [0,π/2]$. The spatial part of the metric describes a biaxially squashed$S^3_v$, with squashing parameter$v$(cf. \ref{['biaxsquashedS3v']}). Setting$Ω_1 = Ω_2 = 0$, we recover the direct product metric on$S^1_β S^3_v$considered in Cassani:2014zwa (note we rescaled the coordinate$t_E$). We highlight that the "twistings" arise from considering the \emph{Euclidean} black hole solution, demanding absence of conical singularities at the horizon. Supersymmetric black hole solutions with biaxially squashed conformal boundary have been constructed numerically in Cassani:2018mlhBombini:2019jhp (see also Blazquez-Salcedo:2017ghgBlazquez-Salcedo:2017kig) in \emph{Lorentzian} signature, and accordingly their conformal boundary is equivalent to \ref{['S1S3vmetric']} without the "twistings". Supersymmetric non-extremal deformations of these solutions have not been studied previously in the literature. We use the following frame for the metric \ref{['S1S3vmetric']}: {\rm e}^1= {{\rm d}} t_E \,,{\rm e}^2= \cos(\varphi_1 + \varphi_2) {{\rm d}} \vartheta - \frac{1}{2} \sin(2\vartheta) \sin(\varphi_1 + \varphi_2) ( ( {{\rm d}}\varphi_1 -\mathrm{i} \Omega_1 {{\rm d}}t_E) - ({{\rm d}} \varphi_2 - \mathrm{i} \Omega_2 {{\rm d}} t_E) ) \,,{\rm e}^3= \sin(\varphi_1 + \varphi_2) {{\rm d}}\vartheta + \frac{1}{2} \sin(2\vartheta) \cos(\varphi_1 + \varphi_2) ( ( {{\rm d}}\varphi_1 -\mathrm{i} \Omega_1 {{\rm d}}t_E) - ({{\rm d}} \varphi_2 - \mathrm{i} \Omega_2 {{\rm d}} t_E) ) \,,{\rm e}^4= v \left[ \sin^2 \vartheta \left({{\rm d}} \varphi_1 - \mathrm{i} \Omega_1 {{\rm d}} t_E \right) + \cos^2\vartheta \left( {{\rm d}}\varphi_2 -\mathrm{i} \Omega_2 {{\rm d}} t_E \right) \right] \,. We would like to find background field configurations (that is,$A^ nm$and$V^ nm$) that allows for solutions to the Killing spinor equation \ref{['KSE_4d_Dirac']}, which we rewrite here: \left[ \nabla_M - \mathrm{i} A_M^{\rm nm} \gamma_5 + \mathrm{i} V_M^{\rm nm} \gamma_5 - \frac{\mathrm{i}}{2} (V^{\rm nm})^N \gamma_{MN} \gamma_5 \right] \epsilon = 0 \,. By choosing A^{\rm nm}= \frac{\mathrm{i}}{2} \left[ \Omega_1 + \Omega_2 - 2 v - \left( 3 - \frac{2}{v} - 2\Psi + \Omega_1 + \Omega_2 \right) \right] {{\rm d}} t_E+ (v^2 - 1) [ \mathrm{i} (\Omega_1 \sin^2\vartheta + \Omega_2 \cos^2\vartheta) {{\rm d}} t_E - \sin^2\vartheta {{\rm d}}\varphi_1 - \cos^2\vartheta {{\rm d}}\varphi_2 ] \,,V^{\rm nm}= - \mathrm{i} v \, {{\rm d}} t_E \,, one can verify that the following is a unit-norm Killing spinor (cf. \ref{['S1S3KSchoice']}): \epsilon = \frac{1}{\sqrt{2}} \begin{pmatrix} \exp \left[ \frac{t_E}{2} \left(3 - \frac{2}{v} - 2\Psi + \Omega_1 + \Omega_2 \right) \right] \\ 0 \\ 0 \\ \exp \left[ - \frac{t_E}{2} \left(3 - \frac{2}{v} - 2\Psi + \Omega_1 + \Omega_2 \right) \right] \\ \end{pmatrix} \,. Some remarks in order: We fixed the ambiguity in $A^{\rm nm}, V^{\rm nm}$ by demanding that $V^{\rm nm}$ only has components along ${{\rm d}}t_E$. The background fields were found starting from the assumption that $A^{\rm nm}_{\varphi_1,\varphi_2}$ are given by \ref{['app:S3vAfield']}. The $\varphi_1$, $\varphi_2$ components of the Killing spinor equation then fixes $V^{\rm nm}_{t_E}$, and the remaining components are solved by choosing $A^{\rm nm}_{t_E}$ as in \ref{['AnmVnm_S1S3v']}.Sanity check: $A^{\rm nm}$ as given in \ref{['AnmVnm_S1S3v']} reduces to \ref{['S1S3_AnmVnm']} upon taking $v = 1$.The gauge field $A^{\rm nm}$ is regular at the two poles of $S^3_v$. As $\vartheta \rightarrow 0$, where ${{\rm d}} \varphi_1$ is not well-defined, we have $A^{\rm nm}_{\varphi_1} \rightarrow 0$, and as $\vartheta \rightarrow \frac{\pi}{2}$, where ${{\rm d}} \varphi_2$ is not well-defined, we have $A^{\rm nm}_{\varphi_2} \rightarrow 0$.The Lie derivatives $\mathcal{L}_{\varphi_1}$, $\mathcal{L}_{\varphi_2}$ acting on individual components of $\epsilon$ are unchanged from the round $S^3$ case, with charges $\pm \frac{1}{2}$. The background admits two supercharges of opposite R-charge.The following is also a solution: $\epsilon = \frac{1}{\sqrt{2}} 0\exp \left[ \frac{t_E}{2} \left(3 - \frac{2}{v} - 2\Psi - \Omega_1 - \Omega_2 \right) \right]\exp \left[ - \frac{t_E}{2} \left(3 - \frac{2}{v} - 2\Psi - \Omega_1 - \Omega_2 \right) \right]0 \,,$ provided for $A^{\rm nm}$ we choose instead A^{\rm nm}= - \frac{\mathrm{i}}{2} \left[ \Omega_1 + \Omega_2 + 2 v + \left( 3 - \frac{2}{v} - 2\Psi - \Omega_1 - \Omega_2 \right) \right] {{\rm d}} t_E- (v^2 - 1) [ \mathrm{i} (\Omega_1 \sin^2\vartheta + \Omega_2 \cos^2\vartheta) {{\rm d}} t_E - \sin^2\vartheta {{\rm d}}\varphi_1 - \cos^2\vartheta {{\rm d}}\varphi_2 ] \,. The boundary supersymmetric Killing vector is then given by (recall$ϵ^* = i C^-1 γ^1 ϵ$): K= \epsilon^* \gamma_1 \gamma^M \epsilon \, \partial_M= \partial_{t_E} + \mathrm{i} \left( \Omega_1 - \frac{1}{v} \right) \partial_{\varphi_1} + \mathrm{i} \left( \Omega_2 - \frac{1}{v} \right) \partial_{\varphi_2} \,. The frame \ref{['S1S3vframe']} is such that$L_∂_t_E ϵ = ∂_t_E ϵ$. It follows that anti-periodic boundary condition on the Killing spinor \ref{['S1S3vKS']}, given by$ϵ(t_E+β) = -ϵ(t_E)$, requires (cf. \ref{['KSE_4d_constraint_S3']}): \beta \left( 2\Psi - 3 + \frac{2}{v} - \Omega_1 - \Omega_2 \right) = 2\pi \mathrm{i} n \,,\qquad n \ \textrm{odd} \,. We now define "biaxially squashed" supersymmetric chemical potentials as \sigma_v = \frac{\beta}{2\pi\mathrm{i}} \left( \Omega_1 - \frac{1}{v} \right) \,,\qquad \tau_v = \frac{\beta}{2\pi\mathrm{i}} \left( \Omega_2 - \frac{1}{v} \right) \,,\qquad \Delta_v = \frac{\beta}{2\pi\mathrm{i}} \left( \Psi - \frac{3}{2} \right) \,, such that the constraint reads 2 \Delta_v - \sigma_v - \tau_v = n \,,\qquad n \ \textrm{odd} \,. We now perform the KK reduction along$t_E$, starting from the metric \ref{['S1S3vmetric']}, identify 3d new minimal supergravity fields, and evaluate$A_M_3, L_M_3$given by \ref{['AM3']}, \ref{['LM3']} in the Cardy-like limit. As before, let us write the dimensional reduction ansatz as {{\rm d}}s_4^2 = e^{2w} ( {{\rm d}} t_E + c_\mu (x) {{\rm d}}x^\mu)^2 + h_{\mu\nu}(x){{\rm d}}x^\mu {{\rm d}}x^\nu \,. Comparing with the background metric \ref{['S1S3vmetric']}, we find e^{2w} = 1 - \Omega_1^2 \sin^2 \vartheta- \Omega_2^2 \cos^2 \vartheta - (v^2 - 1) ( \Omega_1 \sin^2\vartheta + \Omega_2 \cos^2 \vartheta)^2 \,,c_\mu {{\rm d}} x^\mu = - \mathrm{i} e^{-2w} [ \sin^2\vartheta \left( \Omega_1 + (v^2-1) (\Omega_1 \sin^2\vartheta + \Omega_2 \cos^2\vartheta) \right) {{\rm d}} \varphi_1+ \cos^2\vartheta \left( \Omega_2 + (v^2-1) (\Omega_1 \sin^2\vartheta + \Omega_2 \cos^2\vartheta) \right) {{\rm d}} \varphi_2 ] \,, as well as h_{\mu\nu} {{\rm d}}x^\mu {{\rm d}}x^\nu= {{\rm d}}\vartheta^2 + \sin^2\vartheta {{\rm d}}\varphi_1^2 + \cos^2\vartheta {{\rm d}}\varphi_2^2 + (v^2-1) \left( \sin^2\vartheta {{\rm d}}\varphi_1 + \cos^2\vartheta {{\rm d}}\varphi_2 \right)^2 - e^{2w} c^2 \,. From the dimensional reduction Assel:2014paa, 3d new minimal supergravity fields are identified via \ref{['H_AR_id']}-\ref{['vHodge']}. Given \ref{['AnmVnm_S1S3v']}, we have (cf. \ref{['H_AR_S1S3']}) H= - \mathrm{i} v e^{-w} \,,A_\mu^{(R)}= - (v^2 - 1) \left( \sin^2\vartheta {{\rm d}} \varphi_1 + \cos^2\vartheta {{\rm d}} \varphi_2 \right) + \frac{1}{2} e^w v_\mu- \mathrm{i} \left[ \frac{1}{2} ( \Omega_1+ \Omega_2 ) - v + \frac{\pi\mathrm{i} n}{\beta} + (v^2 - 1)(\Omega_1 \sin^2\vartheta + \Omega_2 \cos^2 \vartheta) \right] c_\mu where we have used the constraint \ref{['KSE_4d_constraint_S3v']}. Collecting the above ingredients, we may now evaluate the local functionals$A_M_3, L_M_3$\ref{['AM3']}, \ref{['LM3']} for the$S^1_β S^3_v$background \ref{['S1S3vmetric']}, \ref{['AnmVnm_S1S3v']}. We note that the integrals were significantly more complicated compared to the round and elliptically squashed cases, and we did not succeed in evaluating them before taking the Cardy-like limit. Nevertheless, it was possible to first take the Cardy-like limit within the integrand and integrate the series expansion. As before, with the biaxially squashed chemical potentials \ref{['susychempot_S3v']}, the Cardy-like limit is taken by introducing small parameters$δ_σ_v = βσ_v$and$δ_τ_v = βτ_v$, such that \Omega_1 \simeq \frac{2\pi \mathrm{i}}{\delta_{\sigma_v}} \,,\qquad \Omega_2 \simeq \frac{2\pi \mathrm{i}}{\delta_{\tau_v}} \,, and taking$δ_σ_v, δ_τ_v → 0$with fixed$σ_vτ_v$. We find A_{S^3_v} \underset{\beta \rightarrow 0}{\simeq} \frac{\beta^2}{\pi^2} \frac{1}{\sigma_v \tau_v} \,, \qquad L_{S^3_v} \underset{\beta \rightarrow 0}{\simeq} \frac{\mathrm{i} \beta}{\pi} \frac{\sigma_v + \tau_v}{\sigma_v \tau_v} \,. Substituting these into Honda's prescription \ref{['HondaIndex']}, with "biaxially squashed" chemical potentials satisfying the constraint (cf. \ref{['SYMconstraint_S3v']}) 2 \Delta_{v} - \sigma_{v} - \tau_{v} = 1 \,, we conclude that the biaxially squashed index in the Cardy-like limit is given by \log \mathcal{I}_{S^1_\beta \times S^3_{v}} \underset{|\sigma_v|, |\tau_v| \rightarrow 0}{\simeq} - \mathrm{i} \pi (N^2-1) \frac{8}{27} \frac{\Delta_{v}^3}{\sigma_v \tau_v} \,. In the previous two subsections, we utilised Honda's prescription \ref{['HondaIndex']} to compute the supersymmetric index on$S^1_β S^3_b_1, b_2$and$S^1_β S^3_v$backgrounds, having worked out a choice of profile for background fields that admit Killing spinors that are anti-periodic around$S^1_β$. As an interlude we briefly discuss complex supercharges preserved by these backgrounds, and the relation between the grand-canonical partition function and the supersymmetric index. The careful reader may worry that the "squashed" backgrounds do not preserve the full$N=1$superconformal algebra. Recall that, even when the$S^3$is \emph{round}, it is only a subalgebra that matters for the superconformal index; one can pick a complex supercharge obeying the following algebra Romelsberger:2005egKinney:2005ej \{ \mathcal{Q}, \overline{\mathcal{Q}} \} = H - J_1 - J_2 - \frac{3}{2} R \,, and define a (refined) Witten index of the specific supercharge$Q$by the trace \mathcal{I}(\sigma_{\rm FT}, \tau_{\rm FT}) = {\rm Tr}'_{\mathcal{H}} (-1)^F e^{-\beta \{ \mathcal{Q}, \overline{\mathcal{Q}} \} + 2\pi\mathrm{i} \sigma_{\rm FT} (J_1 + \frac{1}{2} R) + 2\pi\mathrm{i} \tau_{\rm FT} (J_2 + \frac{1}{2} R) } \,, where the trace$ Tr'$is such that the contribution of the vacuum is 1. Note we introduced the subscript "FT" -- which stands for "field theory" -- to distinguish$(σ_ FT, τ_ FT)$from the "gravitational" chemical potentials$(σ_g, τ_g)$, i.e. the$(σ, τ)$we defined in \ref{['susychempot_S3']}. The partition function on the background \ref{['S1S3_metric']} is given by Z_{S^1 \times S^3}(\sigma, \tau)= {\rm Tr}_{\mathcal{H}} e^{-\beta \{ \mathcal{Q}, \overline{\mathcal{Q}} \} + 2\pi\mathrm{i} \sigma_g J_1 + 2\pi\mathrm{i} \tau_g J_2 + 2\pi\mathrm{i} \Delta R}= {\rm Tr}_{\mathcal{H}} e^{-\beta \{ \mathcal{Q}, \overline{\mathcal{Q}} \} + 2\pi\mathrm{i} \sigma_g (J_1+\frac{1}{2}R) + 2\pi\mathrm{i} \tau_g (J_2 + \frac{1}{2}R) + \pi \mathrm{i} R } \,, where the second line follows from the constraint \ref{['SYMconstraint_S3_1']}. This can be put into the following form Z_{S^1 \times S^3}(\sigma_g, \tau_g)= {\rm Tr}_{\mathcal{H}} e^{2\pi\mathrm{i} J_1 + \pi \mathrm{i} F} e^{-\beta \{ \mathcal{Q}, \overline{\mathcal{Q}} \} + 2\pi\mathrm{i} \sigma_g (J_1+\frac{1}{2}R) + 2\pi\mathrm{i} \tau_g (J_2 + \frac{1}{2}R) + \pi \mathrm{i} R }= {\rm Tr}_{\mathcal{H}} (-1)^F e^{-\beta \{ \mathcal{Q}, \overline{\mathcal{Q}} \} + 2\pi\mathrm{i} (\sigma_g+1) (J_1+\frac{1}{2}R) + 2\pi\mathrm{i} \tau_g (J_2 + \frac{1}{2}R)}= {\rm Tr}_{\mathcal{H}} (-1)^R e^{-\beta \{ \mathcal{Q}, \overline{\mathcal{Q}} \} + 2\pi\mathrm{i} \sigma_g (J_1+\frac{1}{2}R) + 2\pi\mathrm{i} \tau_g (J_2 + \frac{1}{2}R) } \,, where in the last line it is apparent that$Z_S^1 S^3(σ_g, τ_g)$is an "index" graded by the R-charge. This is precisely the "index" we computed using Honda's formula in section \ref{['subsection:S1S3']}, that gives \ref{['roundSCI']} in the Cardy-like limit. Notice that$Z_S^1 S^3(σ_g, τ_g) = I(σ_g+1,τ_g)$, up to the normalisation of the vacuum. Instead of starting from \ref{['Zsym']} on the complex background \ref{['S1S3_metric']}, with anti-periodic Killing spinors around$S^1_β$, one can also start from \ref{['SCIwrittenout']} on a real background with periodic Killing spinors, and shift$σ_ FT → σ_ FT + 1$to land on the same expression, i.e. the "second sheet" of the index Cassani:2021fyv. Given the constraints \ref{['SYMconstraint_S3b']}, \ref{['SYMconstraint_S3v']}, all of the above goes through for the$S^1_β S^3_b_1,b_2$and$S^1_β S^3_v$backgrounds. The chemical potentials$(σ,τ)$are simply replaced by the elliptically/biaxially squashed chemical potentials, \ref{['susychempot_S3b']} and \ref{['susychempot_S3v']}, respectively. The "squashings" -- along with the "twistings" -- enter via geometric fugacities that refine the index Aharony:2013dhaClosset:2013vra. Let us denote the pair of spinors associated with$Q$,$Q$as$ζ_+$,$ζ_-$. The curved space supersymmetry algebra is given by Dumitrescu:2012haKlare:2012gn (see also Cassani:2014zwa, BenettiGenolini:2016tsn) [ \delta_{\zeta_+}, \delta_{\zeta_-} ] \Psi = 2 \mathrm{i} \left( \mathcal{L}_K - \mathrm{i} r K \mathbin{} A^{\rm nm} \right) \Upsilon \,, where$Υ$is a generic field in the theory, and$r$is its R-charge. Given the explicit conformal Killing spinors \ref{['S1S3KSchoice']}, \ref{['S1S3bKS']}, and \ref{['S1S3vKS']}, this can be evaluated explicitly. For$S^1_β S^3$, it is straightforward to see that K \mathbin{} A^{\rm nm} = \mathrm{i} \left( \Psi - \frac{3}{2} \right) \,, from \ref{['S1S3_AnmVnm']} and \ref{['S1S3susyKV']}. Using \ref{['AnmVnm_S1S3b']} and \ref{['S1S3bsusyKV']} for$S^1_β S^3_b_1, b_2$, and similarly \ref{['AnmVnm_S1S3v']}, \ref{['S1S3vsusyKV']} for$S^1_β S^3_v$, one finds that$K A^ nm$is unchaged for both cases. The upshot is that the squashing only affects the coefficients in front of the angular momentum generators in the superalgebra \ref{['QQbar']}. Note the abstract operators$H$,$J_1$,$J_2$that appear in \ref{['QQbar']} are associated with$L_∂_t_L$,$L_∂_ϕ_L$,$L_∂_ψ_L$, where the$(t_L, ϕ_L, ψ_L)$coordinates are related to$(τ, φ_1, φ_2)$by the coordinate transformation \ref{['twistidcoordtransform']}. Taking this into account, we infer that the supercharges preserved by the$S^1_β S^3_b_1, b_2$background obey \{ \mathcal{Q}_{\mathfrak{b}} , \overline{\mathcal{Q}}_{\mathfrak{b}} \} = H - \mathfrak{b}_1 J_1 - \mathfrak{b}_2 J_2 - \frac{3}{2} R \,, whereas those preserved by the$S^1_β S^3_v$background obey \{ \mathcal{Q}_v , \overline{\mathcal{Q}}_v \} = H - \frac{1}{v} J_1 - \frac{1}{v} J_2 - \frac{3}{2} R \,. In Appendix \ref{['app:susyqsr']}, we show that \ref{['QQbarb']}, \ref{['QQbarv']} are indeed consistent with expectations for BPS charge relations of supersymmetric AlAdS$_5$black holes with elliptically/biaxially squashed conformal boundaries. We now consider complex, non-extremal deformations of supersymmetric extremal black holes with topology$R^2 M_3$. While such solutions are known for$M_3 = S^3$Cabo-Bizet:2018ehj, there are no known non-extremal supersymmetric deformations for the numerical black hole solutions of Cassani:2018mlhBombini:2019jhp, with biaxially squashed$S^1_β S^3_v$boundary. Even less is known for the elliptically squashed$S^1_β S^3_b_1, b_2$boundary: to the best of our knowledge, supersymmetric black holes filling this boundary have never been constructed. Remarkably, we find that supersymmetric indices on$S^1_β M_3$backgrounds can be recovered using the equivariant localization technique developed in BenettiGenolini:2025icr, without explicit knowledge of the solutions. The only ingredients are topological data, and the R-symmetry vector,$K$, where the latter is read off from the (conformal) Killing vector$K$formed as a bilinear in the boundary Killing spinors. We start with a brief review of equivariant localization for$D=5$gauged supergravity. In BenettiGenolini:2025icr, the formalism was developed for$D=5$Euclidean gauged supergravity coupled to an arbitrary number of vector multiplets. Since we assumed equal charges in section \ref{['fieldtheory']} for brevity, here we restrict to minimal gauged supergravity, whose bosonic action is given by \begin{split} I_{(5)} & = - \frac{1}{16\pi G_{(5)}} \int_{M_{(5)}} \left[ \left( R_{(5)} + 12 - \frac{2}{3} \mathcal{F} \wedge \ast \mathcal{F} \right) \mathrm{vol}_{(5)} - \frac{8\mathrm{i}}{27} \mathcal{F} \wedge \mathcal{F} \wedge \mathcal{A} \right] \,, \end{split} where the graviphoton$A$has field strength$F = dA$. The Einstein and Maxwell equations read R_{\mu\nu} + \frac{2}{3} \mathcal{F}_{\mu\rho} \mathcal{F}^\rho_{ \ {\nu}} + g_{\mu\nu} \left( 4 + \frac{1}{9} \mathcal{F}_{\rho\sigma} \mathcal{F}^{\rho\sigma} \right) = 0 \,,{{\rm d}} \ast \mathcal{F} + \frac{2}{3} \mathcal{F} \wedge \mathcal{F} = 0 \,. We are interested in Euclidean supersymmetric solutions that admit non-trivial Dirac spinors$ζ, ζ$satisfying 0=\left[\nabla_\mu - \mathrm{i} {\mathcal{A}}_\mu + \frac{1}{2} \gamma_\mu + \frac{\mathrm{i}}{12} {\mathcal{F}}_{\nu\rho} ({\gamma}_{\mu}{}^{\nu\rho}-4\delta_\mu^\nu{\gamma}^\rho)\right]\zeta \,,0=\left[\nabla_\mu+ \mathrm{i} {\mathcal{A}}_\mu - \frac{1}{2} \gamma_\mu + \frac{\mathrm{i}}{12} {\mathcal{F}}_{\nu\rho} ({\gamma}_{\mu}{}^{\nu\rho}-4\delta_\mu^\nu{\gamma}^\rho)\right] \widetilde{\zeta} \,, where$γ_μ$generate Cliff(5). The Killing spinor equations imply that the bilinear$K ≡ ζ γ^μ ζ ∂_μ$is a Killing vector that generates the symmetry of the full solution, i.e.$L_K g = L_K F = 0$. While$K$is a Killing vector for any supersymmetric solution, in the framework of BenettiGenolini:2025icr it is assumed that the solution admits another Killing vector$ℓ$that is no-where vanishing, generating a circle fibration of the$D=5$spacetime$M_(5)$with base$M_(4)$: S^1 \lhook\joinrel\longrightarrow M_{(5)} \overset{\pi}{\longrightarrow} M_{(4)} \,. Introducing local coordinates$x^μ = (x^i, x^5)$, with \ell = \partial_{x^5} \,, \qquad x^5 \sim x^5 + \Delta x^5 \,, we can write the$D=5$metric as {{\rm d}}s^2_{(5)} = e^{-4\lambda} \alpha^2 + e^{2\lambda} {{\rm d}}s^2_{(4)} \,, where$α ≡ e^4λ ℓ^♭$is locally written as$α = dx^5 - A^0$, where$A^0$is a$D=4$gauge field. Performing a Kaluza--Klein (KK) reduction along$ℓ$, the$D=5$action can be written as I_{(5)} = I_{(4)} - \frac{\Delta x^5}{16\pi G_{(5)}} \int_{M_{(4)}} \Lambda_4 \,, where$I_(4)$is given by the action of Euclidean$D=4$,$N=2$gauged supergravity, discussed in BenettiGenolini:2024xeoBenettiGenolini:2024lbj. The additional term is an integral of a closed four-form$Λ_4$, that depends on the choice of gauge for the gauge field$A$. If$A$is a globally defined one-form on$M_(5)$, then$Λ_4$is globally defined, and hence exact on$M_(4)$. Having performed the dimensional reduction, within$D=4$,$N=2$gauged supergravity one can construct a vector$ξ$as a Killing spinor bilinear, and the$D=4$Killing spinor equations imply that$ξ$is a Killing vector BenettiGenolini:2024lbj. In BenettiGenolini:2025icr, it was shown that this vector is precisely the pushforward of$K$under the map \ref{['M5M4fib']}: \xi = \pi_* (\mathcal{K}) \,. Imposing the trace of the Einstein equation on$I_(4)$, we can write the bulk$D=4$on-shell action as BenettiGenolini:2024lbj I_{(4)}^{\rm OS}[M_{(4)}] = \frac{1}{8\pi G_{(4)}} \int_{M_{(4)}} \Phi \,, where$Φ$is a polyform whose top form$Φ_4$is the bulk on-shell action, equivariantly closed with respect to the action of$ξ$: \Phi \equiv \Phi_4 + \Phi_2 + \Phi_0 \,, \qquad {{\rm d}}_\xi \Phi = 0 \,, where$ d_ξ ≡ d - ξ $. Moreover, each$D=4$gauge field strength$F^Λ$($Λ = 0,1$) has an equivariant completion \Phi^\Lambda_{(F)} \equiv F^\Lambda + \Phi_0^\Lambda \,,\qquad {{\rm d}}_\xi \Phi^\Lambda_{(F)} = 0 \,. The total$D=5$action is given by I^{\mathrm{Total}}_{(5)}[M_{(5)}] = I_{(5)}+ I_{\mathrm{GHY}}^{\partial M_{(5)}} + I_{(5)}^{\partial M_{(5)}}\,, where$I_GHY^∂ M_(5)$is the standard Gibbons--Hawking--York term and$I_(5)^∂ M_(5)$are counterterms to remove divergences, as well as finite counterterms, which we discuss further below. Applying the BVAB formula BV:1982Atiyah:1984px to the bulk$D=4$on-shell action \ref{['I4OSbulk']}, one finds that \ref{['Itot']} can be written as BenettiGenolini:2025icr I^{\mathrm{Total}}_{(5)}[M_{(5)}]= I_{(4)}^{\rm FP}[M_{(4)}] - \frac{1}{8\pi G_{(5)}} \int_{\partial M_{(5)}} \alpha \wedge \eta \wedge \left( \Phi_2 + \Phi_0 \, {\rm d} \eta \right)- \frac{\Delta x_5}{16\pi G_{(5)}} \int_{M_{(4)}} \Lambda_4 + I_{\mathrm{GHY}}^{\partial M_{(5)}}+ I_{(5)}^{\partial M_{(5)}}\,, where$I^ FP_(4)[M_(4)]$is a sum over fixed points of$ξ$, given by I^{\rm FP}_{(4)}[M_{(4)}] = \frac{2\pi \Delta x^5}{27 G_{(5)}} \mathrm{i} \sum_{\substack{ {\rm fixed} \\ {\rm points} }} \frac{1}{d} \frac{\left(-\kappa (b_1 - \chi b_2) + 2 Q^{(\ell)}\Phi_0^0\right)^3}{8 b_1b_2 \, \Phi_0^0} \,. Here,$b_1, b_2$are the weights of$ξ$at each of its fixed point, and$κ, χ ∈ { ± 1}$are signs, where$χ$is the chirality of the$D=4$Killing spinor at the fixed point. Both the weights and the signs$κ$,$χ$can be determined in a systematic way, as explained in BenettiGenolini:2024hydBenettiGenolini:2024lbj. We note that$M_(4)$may in general have orbifold singularities, where$d ∈ N$is the order of the orbifold structure group. Finally,$Q^(ℓ)$is defined by the charge of the$D=5$Killing spinor with respect to$ℓ$, \mathcal{L}_\ell \zeta = \mathrm{i} Q^{(\ell)} \zeta \,. We highlight that$Φ_0^0$at the fixed points is determined by the weights of$K$, extending an argument in BenettiGenolini:2024kyy -- see section 2.4 of BenettiGenolini:2025icr. To summarise, the total$D=5$on-shell action$I^Total_(5)[M_(5)]$is expressed in terms of fixed point contributions$I_(4)^ FP[M_(4)]$, along with boundary terms that appear in \ref{['ItotFPbdy']}. The fixed point contributions are determined solely in terms of topological data of the$D=5$spacetime$M_(5)$, and weights of the supersymmetric Killing vector$K$. We will take$M_(5)$to have the black hole topology$R^2 M_3$, where the$R^2$disc factor smoothly caps off at the horizon. Its conformal boundary is given by$S^1_β M_3$, i.e. the field theory backgrounds we constructed in section \ref{['fieldtheory']}. The AdS/CFT correspondence states that, in the appropriate large$N$limit of the field theory, - \log Z_{S^1_\beta \times M_3} = I^{\mathrm{Total}}_{(5)} [M_{(5)}] \,, where the supergravity action$I^Total_(5) [M_(5)]$is regularised in a supersymmetric scheme. However, holographic renormalisation in$D=5$that implements such a scheme involves non-standard, finite counterterms that enter \ref{['Itot']}, which are not known for general supersymmetric solutions BenettiGenolini:2016qwmBenettiGenolini:2016tsn, including the black holes. Verifying the relation \ref{['AdSCFT']} is beyond the scope of the current manuscript. Instead, we may utilise the following prescription to directly compute the supersymmetric index$I_S^1_β M_3$, - \log \mathcal{I}_{S^1_\beta \times M_3} = I^{\mathrm{Total}}_{(5)}[M_{(5)}] - I^{\mathrm{Total}}_{(5)}[N_{(5)}] \,, where the RHS computes a regularised on-shell action of the background$M_(5)$, relative to that of a reference background$N_(5)$. The validity of \ref{['ItotMN']} hinges on the following question: what is the correct choice of$N_(5)$, such that$I^Total_(5)[N_(5)]$precisely accounts for the difference between the partition function and the index? Note the field theory partition function on these backgrounds are related to the supersymmetric index by Assel:2014paaArabiArdehali:2015iowArabiArdehali:2019tdm \log Z_{S^1_\beta \times M_3} = -\beta E_{\rm susy} + \log \mathcal{I}_{S^1_\beta \times M_3} \,, where$E_ susy$is the supersymmetric Casimir energy on$S^1_β M_3$Assel:2015nca. The results of BenettiGenolini:2016tsn show that, for large classes of$S^1_β M_3$backgrounds, there are gravitational fillings with conformal boundary$S^1_β M_3$, whose supergravity on-shell action precisely agrees with$β E_ susy$. Such solutions regularly cap off with no boundary in the interior, and comprise a graviphoton$A$that is a global one-form. By taking$N_(5)$'s to be precisely these backgrounds, we thus expect \ref{['ItotMN']} to hold. The holographic match we show is then between the supersymmetric index on$S^1_β M_3$, and the on-shell action of a \emph{closed} manifold, obtained by gluing$M_(5) = R^2 M_3$and the$N_(5)$'s of BenettiGenolini:2016tsn along their common$S^1_β M_3$conformal boundary. Restricting to$M_3$'s that are squashed three-spheres, the$N_(5)$'s have topology$S^1_β R^4$-- see figure \ref{['figure:MglueN']}. For asymptotically AdS black holes, the prescription \ref{['ItotMN']} is precisely the "background subtraction" method utilised in Colombo:2025ihpBenettiGenolini:2025icr. The black holes of Gutowski:2004ezChong:2005hr studied in these references have$S^1_β S^3$conformal boundary, i.e. the background studied in section \ref{['subsection:S1S3']}.$N_(5)$in this case corresponds to the AdS$_5$vacuum. For asymptotically \emph{locally} AdS black holes,$N_(5)$is no longer global AdS$_5$. We shall refrain from using the term "background subtraction", as not all AlAdS$_5$spacetimes can be embedded in an ambient spacetime. For example, the supersymmetric black holes of Cassani:2018mlhBombini:2019jhp have biaxially squashed$S^1_β S^3_v$conformal boundary, i.e. the$S^1_β S^3_v$background we studied in section \ref{['subsection:S1S3v']}. The results of BenettiGenolini:2016tsn implies that the choice of$N_(5)$, such that \ref{['ItotMN']} is valid, is given by supersymmetric AlAdS$_5$solutions with topology$S^1_β R^4$. Schematic description of gluing $M_{(5)} \cong \mathbb{R}^2 \times S^3$ with $N_{(5)} \cong S^1 \times \mathbb{R}^4$, for Euclidean, squashed, supersymmetric, non-extremal black holes. The gluing is performed along the common $\partial M_{(5)} = \partial N_{(5)} \cong S^1_\beta \times S^3$ conformal boundary as $z \rightarrow 0$, where $z$ is the Fefferman-Graham coordinate. The $S^3$'s in the figure should be interpreted as squashed three-spheres. Generically, the boundary is not conformally flat; both $M_{(5)}$ and $N_{(5)}$ are asymptotically locally AdS$_5$. For the Lens space index on $S^1_\beta \times L(\texttt{p}, \texttt{q})$, we instead take $M_{(5)} \cong \mathbb{R}^2 \times L(\texttt{p},\texttt{q})$ and $N_{(5)} \cong S^1 \times \mathbb{R}^4/\mathbb{Z}_\texttt{p}$. We highlight that numerical solutions in this class exist Cassani:2014zwa, as supersymmetric AlAdS$_5$solutions of$D=5$minimal gauged supergravity. However, this is \emph{not} the solution we will take for$N_(5)$. The non-trivial graviphoton field$A$in Cassani:2014zwa has an intrinsically different asymptotic behaviour to what is needed here, as the gauge choice there is such that the Killing spinors are independent of the Euclidean time direction. As emphasised in section \ref{['fieldtheory']}, our choice of spin structure around$S^1_β$is \emph{anti-periodic}! We demand that the boundary Killing spinors \ref{['S1S3KSchoice']}, \ref{['S1S3bKS']}, and \ref{['S1S3vKS']} on$S^1_β M_3$extend to both$M_(5)$and$N_(5)$, such that$K$is a Killing vector on the \emph{closed} manifold$M_(5) ∪ ( - N_(5))$. We can be more explicit by writing the five-dimensional metric on$M_(5)$and$N_(5)$in Fefferman--Graham (FG) form, {{\rm d}}s^2_{(5)} = g_{\mu\nu}{{\rm d}}y^\mu{{\rm d}}y^\nu = \frac{{{\rm d}}z^2}{z^2} + \frac{1}{z^2} h_{ij}(y,z) {{\rm d}}y^i {{\rm d}}y^j \,,h(y,z) = h^{(0)} + z^2 h^{(2)} + z^4 h^{(4)} + \widetilde{h}^{(4)} z^4 \log z^2 + \dots where the conformal boundary is at$z=0$. The gauge field admits the expansion \mathcal{A}(x,z) = \mathcal{A}^{(0)} + z^2 \mathcal{A}^{(2)} + \widetilde{\mathcal{A}}^{(2)} z^2 \log z^2 + \dots where by definition$A^(0) = A_ bdy$. The boundary metric$h^(0)$can be viewed as a metric on$( M_(5) ∪ (-N_(5)) ) \\ Int ( M_(5) ⊔ N_(5) )$. The name of the game is then to find gravitational fillings$M_(5)$and$N_(5)$, for a given$S^1_β M_3$background, such that$I^Total_(5)[N_(5)]$cancels$β E_ susy$, resulting in the relation \ref{['ItotMN']}. As emphasised in section \ref{['fieldtheory']}, the Killing spinors on$S^1_β M_3$are anti-periodic around$S^1_β$, such that they naturally extend to Killing spinors on$M_(5) ∪ ( - N_(5))$. The field theory backgrounds thus fix a set of boundary conditions, namely the boundary value$A^(0)$of the bulk gauge field$A$, the representative$h^(0)$of the conformal structure$[h^(0)]$, and finally the boundary value$ϵ$of the bulk Killing spinor$ζ$(up to a Weyl transformation). In the remainder of this section, we show that the unintegrated superconformal anomaly vanishes on these backgrounds -- despite complex metrics and anti-periodic Killing spinors -- as expected for a Euclidean supersymmetric background that admits two supercharges of opposite R-charge Cassani:2013dba. Extending arguments in Papadimitriou:2005ii, this implies that the boundary conditions yield a well-posed variational problem, for both supergravity fillings$M_(5)$and$N_(5)$. We first consider the supersymmetric black holes of Gutowski:2004ezChong:2005hr. The horizon is generated by$V_H = ∂_t_E$. Their complex, non-extremal deformations are supersymmetric solutions of minimal gauged supergravity with topology$M_(5) = R^2 S^3$Cabo-Bizet:2018ehj. Choosing a regular gauge for$A$, such that$. V_H A |_ horizon = 0$where the$R^2$disc factor smoothly caps off, the asymptotics of the solution at leading order as$z → 0$is given by {{\rm d}}s^2_{(5)}\sim \frac{{{\rm d}}z^2}{z^2} + \frac{1}{z^2} [ \frac{\beta^2}{4\pi^2} {{\rm d}} t_E^2 + {{\rm d}} \vartheta^2\qquad \qquad \qquad + \sin^2 \vartheta \left({{\rm d}} \varphi_1 - \frac{\mathrm{i} \beta}{2\pi} \Omega_1 {{\rm d}} t_E \right)^2 + \cos^2 \vartheta \left({{\rm d}}\varphi_2 - \frac{\mathrm{i} \beta}{2\pi} \Omega_2 {{\rm d}} t_E \right)^2 ] \,,\mathcal{A}^{(0)}= \frac{\mathrm{i} \beta}{2\pi} \Psi \, {{\rm d}}t_E \,, with \frac{\beta}{2\pi\mathrm{i}} ( \Omega_1 + \Omega_2 - 2\Psi +1) = c_R \,, \qquad c_R = \pm 1 \,. Note we rescaled the coordinate$t_E$, such that$t_E ∼ t_E + 2π$. The sign$c_R$is associated to the two "branches" of the black hole solutions in the nomenclature of Aharony:2021zkr, where in BenettiGenolini:2025icr the discrete data was identified with the charge of the$D=5$Killing spinor$ζ$with respect to$V_H$: \mathcal{L}_{V_H} \zeta = c_R \frac{\mathrm{i}}{2} \zeta \,. Choosing$n = - c_R$in \ref{['KSE_4d_constraint_S3']}, the Killing spinor$ζ$of the black hole is precisely the bulk extension of the boundary (conformal) Killing spinor \ref{['S1S3KSchoice']}. Notice that the usual global AdS$_5$with \emph{periodic} Killing spinors around the Euclidean time circle does \emph{not} satisfy the boundary conditions specified by \ref{['asympS1S3']}-\ref{['asympS1S3cond']}, and hence cannot be$N_(5)$. Indeed, the regularised black hole on-shell action used in Cabo-Bizet:2018ehj was obtained by "AdS subtraction" Chen:2005zj, where the metric is obtained by setting the mass and charge to zero in the black hole metric. In the supersymmetric limit,$N_(5)$obtained this way has topology$S^1_β R^4$, and admits Killing spinors that are \emph{anti-periodic} around$S^1_β$. We remind the reader that by writing \ref{['asympS1S3']} we are only fixing the conformal structure, and not the representative Papadimitriou:2005ii. The boundary gauge field is flat, and from the boundary metric$h^(0)$we can easily check that E = C_{ijkl} C^{ijkl} = 0 \,, where$E$is the Euler density, and$C_ijklC^ijkl$is the square of the Weyl tensor, given by E= R_{ijkl} R^{ijkl} - 4 R_{ij} R^{ij} + R^2 \,,C_{ijkl}C^{ijkl}= R_{ijkl} R^{ijkl} - 2 R_{ij} R^{ij} + \frac{1}{3} R^2 \,. We next consider complex, supersymmetric, non-extremal black holes whose conformal boundary is given by the$S^1_β S^3_b_1,b_2$background constructed in section \ref{['subsection:S1M3']}. We shall refer to these as the$R^2 S^3_b_1,b_2$saddles. For$N_(5)$, we assume that there exist supersymmetric AlAdS$_5$solutions with the same conformal boundary, with$S^1_β R^4$topology, that admit Killing spinors that are anti-periodic around$S^1_β$. While both solutions are unknown, we will show that the boundary data is sufficient to compute the index via \ref{['ItotMN']}, which can be evaluated using equivariant localization. We demand that the solutions for both$M_(5)$and$N_(5)$takes the following form, at leading order as$z → 0$: {{\rm d}}s^2_{(5)}\sim \frac{{{\rm d}}z^2}{z^2} + \frac{1}{z^2} [ \frac{\beta^2}{4\pi^2} {{\rm d}} t_E^2 + f(\vartheta)^2 {{\rm d}} \vartheta^2\qquad \qquad \qquad + \frac{\sin^2 \vartheta}{\mathfrak{b}_1^2} \left({{\rm d}} \varphi_1 - \frac{\mathrm{i} \beta}{2\pi} \Omega_1 {{\rm d}} t_E \right)^2 + \frac{\cos^2 \vartheta}{\mathfrak{b}_2^2} \left({{\rm d}}\varphi_2 - \frac{\mathrm{i} \beta}{2\pi} \Omega_2 {{\rm d}} t_E \right)^2 ] \,,\mathcal{A}^{(0)}= \frac{\mathrm{i}\beta}{4\pi} \left[ \frac{1}{f(\vartheta)} \left( \frac{\Omega_1}{\mathfrak{b}_1} + \frac{\Omega_2}{\mathfrak{b}_2} +1 \right) - \left( 3 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi + \Omega_1 + \Omega_2 \right) \right] {{\rm d}} t_E+ \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_1}\right) {{\rm d}}\varphi_1 + \frac{1}{2} \left( 1- \frac{1}{f(\vartheta) \mathfrak{b}_2}\right) {{\rm d}}\varphi_2 \,, where \frac{\beta}{2\pi\mathrm{i}} \left( \Omega_1 + \Omega_2 - \mathfrak{b}_1 - \mathfrak{b}_2 - 2\Psi + 3 \right) = c_R \,,\qquad c_R = \pm 1 \,. Note we obtained$A^(0) = A^ nm - 32 V^ nm$from \ref{['AnmVnm_S1S3b']}. The five-dimensional metric \ref{['asympS1S3b']} is asymptotically locally AdS, and a straightforward calculation shows that these solve the equations of motion \ref{['sugraeom']} at leading order as$z → 0$. This, of course, is not sufficient to guarantee that such supersymmetric AlAdS$_5$solutions exist. While the sub-leading term$h^(2)$in the FG expansion is fixed by$h^(0)$deHaro:2000vlm, one needs to verify the existence of supersymmetric solutions$M_(5) = R^2 S^3_b_1,b_2$and$N_(5) = S^1_β R^4_b_1,b_2$, respectively at all orders. Equivariant localization is agnostic of such challenges; we will find that the gravity result precisely matches the elliptically squashed index \ref{['HondaSquashedIndexb']} in the large$N$limit, both of which are evaluated solely using boundary data. The squashing does not affect the Euler density, and one can check that we have$E=0$as expected from \ref{['anom_S1S3']}. From$h^(0)$and$A^(0)$, we can show that the superconformal anomaly vanishes, though in a non-trivial way: C_{ijkl} C^{ijkl} = \frac{8}{3} \mathcal{F}^{(0)}_{ij} \mathcal{F}^{(0)ij} = \frac{4 \left( \tan^2\vartheta + \csc^2\vartheta \right)f'(\vartheta)^2}{3 f(\vartheta)^6} \, . Finally, we consider the supersymmetric black holes of Cassani:2018mlhBombini:2019jhp, with biaxially squashed$S^1_β S^3_v$boundary. While complex, non-extremal deformations of these solutions have not been constructed, the physics of such Euclidean saddles have been explored restricting to equal angular momenta$Ω_1 = Ω_2$, based on an$SU(2) U(1)$invariant ansatz Ntokos:2021duk. Explicit bulk Killing spinors have been constructed in Ntokos:2021duk. Here, we assume that the supersymmetric non-extremal solutions exist, even for$Ω_1 ≠ Ω_2$. We shall refer to these as the$R^2 S^3_v$saddles. With regularity conditions imposed, the black hole solutions should have the following asymptotic behaviour in the leading order as$z → 0$, {{\rm d}}s^2_{(5)}\sim \frac{{{\rm d}}z^2}{z^2} + \frac{1}{z^2} [ \frac{\beta^2}{4\pi^2} {{\rm d}} t_E^2 + {{\rm d}} \vartheta^2\qquad \qquad \qquad + \sin^2 \vartheta \left({{\rm d}} \varphi_1 - \frac{\mathrm{i} \beta}{2\pi} \Omega_1 {{\rm d}} t_E \right)^2 + \cos^2 \vartheta \left({{\rm d}}\varphi_2 - \frac{\mathrm{i} \beta}{2\pi} \Omega_2 {{\rm d}} t_E \right)^2\qquad \qquad \qquad + (v^2-1) \left( \sin^2\vartheta ({{\rm d}}\varphi_1 - \frac{\mathrm{i} \beta}{2\pi} \Omega_1 {{\rm d}}t_E) + \cos^2\vartheta ({{\rm d}}\varphi_2 - \frac{\mathrm{i} \beta}{2\pi} \Omega_2 {{\rm d}}t_E) \right)^2 ] \,,\mathcal{A}^{(0)}= \frac{\mathrm{i} \beta}{4\pi} \left[ \Omega_1 + \Omega_2 + v - \left( 3 - \frac{2}{v} - 2\Psi + \Omega_1 + \Omega_2 \right) \right] {{\rm d}} t_E+ (v^2 - 1) [ \frac{\mathrm{i}\beta}{2\pi} (\Omega_1 \sin^2\vartheta + \Omega_2 \cos^2\vartheta) {{\rm d}} t_E - \sin^2\vartheta {{\rm d}}\varphi_1 - \cos^2\vartheta {{\rm d}}\varphi_2 ] \,, where \frac{\beta}{2\pi\mathrm{i}} \left( \Omega_1 + \Omega_2 - \frac{2}{v} - 2 \Psi + 3 \right) = c_R \,,\qquad c_R = \pm 1 \,. We obtained$A^(0) = A^ nm - 32 V^ nm$from \ref{['AnmVnm_S1S3v']}. As before, the metric \ref{['asympS1S3v']} is asymptotically locally AdS, and a straightforward calculation shows that these solve the equations of motion \ref{['sugraeom']} at leading order as$z → 0$. From$h^(0)$, one can check that$E=0$is unchanged from \ref{['anom_S1S3']}. The superconformal anomaly vanishes, again in a non-trivial way: C_{ijkl} C^{ijkl} = \frac{8}{3} \mathcal{F}^{(0)}_{ij} \mathcal{F}^{(0)ij} = \frac{64}{3} (v^2 - 1)^2 \, . There are no known supersymmetric AlAdS$_5$solutions with$S^1_β R^4$topology that asymptotically approach \ref{['asympS1S3v']}-\ref{['asympS1S3vcond']}. However, the results of Cassani:2013dbaCassani:2014zwa strongly suggest that they should exist. In the next subsection, we will show that the on-shell action of$R^2 S^3_v ∪ (- S^1 R^4_v)$, evaluated using equivariant localization (assuming these solutions exist), precisely match the biaxially squashed index \ref{['HondaSquashedIndex']} in the large$N$limit. The gluing of$M_(5)$and$N_(5)$has desirable properties that further simplify \ref{['ItotMN']}. Recall from \ref{['ItotFPbdy']} that$I^Total_(5)$\emph{a priori} includes boundary terms, in addition to fixed point contributions \ref{['I4FP']}. It is only the latter that can be computed without solving any supergravity equations, using equivariant localization. However, because we glued the two geometries along their common conformal boundary, the boundary terms that depend only on the intrinsic geometry automatically cancel. Another way of seeing this is that the gluing results in a closed manifold$M_(5) ∪ (- N_(5))$; this leaves us with fixed point contributions, and an integral of$Λ_4$over$M_(4) ∪ (-N_(4))$. As noted below \ref{['I4plusLambda4']},$Λ_4$is a closed four-form, that is globally defined (hence exact) whenever$A$is a globally defined one-form. We always choose the regular gauge for$A$on$M_(5)$, such that we have$. V_H A |_ horizon = 0$, where the$R^2$disc factor smoothly caps off. On$N_(5)$,$A$is globally defined by definition. Hence, we conclude that \ref{['ItotMN']} reduces to - \log \mathcal{I}_{S^1_\beta \times M_3} = I_{(4)}^{\rm FP} [M_{(4)}] - I_{(4)}^{\rm FP} [N_{(4)}] \,. The supersymmetric index is determined solely in terms of the topological data of$M_(5) ∪ (- N_(5))$, and the weights of the supersymmetric Killing vector$K$. As explained in the last subsection, we will assume that the boundary Killing spinor bilinears constructed on the backgrounds$S^1_β S^3$,$S^1_β S^3_b_1,b_2$, and$S^1_β S^3_v$extend to the respective supersymmetric Killing vectors$K$on the double-sided gravitational fillings$M_(5) ∪ (- N_(5))$. Noting that we rescaled the coordinate$t_E$such that$t_E ∼ t_E + 2π$, from \ref{['S1S3susyKV']}, \ref{['S1S3bsusyKV']}, and \ref{['S1S3vsusyKV']} we have \mathcal{K}= \partial_{t_E} - \frac{\beta (\Omega_1 -1)}{2\pi\mathrm{i}} \partial_{\varphi_1} - \frac{\beta (\Omega_2 -1)}{2\pi\mathrm{i}} \partial_{\varphi_2} \,,= \partial_{t_E} - \sigma \partial_{\varphi_1} - \tau \partial_{\varphi_2} \,, on$R^2 S^3 ∪ (- S^1 R^4)$, \mathcal{K}= \partial_{t_E} - \frac{\beta (\Omega_1 -\mathfrak{b}_1)}{2\pi\mathrm{i}} \partial_{\varphi_1} - \frac{\beta (\Omega_2 - \mathfrak{b}_2)}{2\pi\mathrm{i}} \partial_{\varphi_2} \,,= \partial_{t_E} - \sigma_{\mathfrak{b}} \partial_{\varphi_1} - \tau_{\mathfrak{b}} \partial_{\varphi_2} \,, on$R^2 S^3_b_1,b_2 ∪ (- S^1 R^4_b_1,b_2)$, and \mathcal{K}= \partial_{t_E} - \frac{\beta}{2\pi\mathrm{i}} \left( \Omega_1 - \frac{1}{v} \right) \partial_{\varphi_1} - \frac{\beta}{2\pi\mathrm{i}} \left(\Omega_2 - \frac{1}{v} \right) \partial_{\varphi_2} \,,= \partial_{t_E} - \sigma_v \partial_{\varphi_1} - \tau_v \partial_{\varphi_2} \,, on$R^2 S^3_v ∪ (- S^1 R^4_v)$, with the complex chemical potentials \ref{['susychempot_S3']}, \ref{['susychempot_S3b']}, and \ref{['susychempot_S3v']}. For convenience, we will collectively denote$K$as \mathcal{K} = \partial_{t_E} + \varepsilon_1 \partial_{\varphi_1} + \varepsilon_2 \partial_{\varphi_2} \,, express the fixed point contributions \ref{['I4FP']} in terms of$(ɛ_1, ɛ_2)$, and identify \begin{split} & \mathbb{R}^2 \times S^3 \, \cup \, (- S^1 \times \mathbb{R}^4) : \qquad (\varepsilon_1, \varepsilon_2) \leftrightarrow - (\sigma, \tau) \,, \\ & \mathbb{R}^2 \times S^3_{\mathfrak{b}_1,\mathfrak{b}_2} \, \cup \, (- S^1 \times \mathbb{R}^4_{\mathfrak{b}_1,\mathfrak{b}_2}) : \qquad (\varepsilon_1, \varepsilon_2) \leftrightarrow - (\sigma_{\mathfrak{b}}, \tau_{\mathfrak{b}}) \,, \\ & \mathbb{R}^2 \times S^3_v \, \cup \, (- S^1 \times \mathbb{R}^4_v) : \qquad (\varepsilon_1, \varepsilon_2) \leftrightarrow - (\sigma_v, \tau_v) \,, \end{split} at the end of all computations. To proceed, we consider a generic KK reduction along \ell = p \, \partial_{t_E} + \left( \partial_{\varphi_1} + \partial_{\varphi_2} \right) \,, where$Δ x^5 = 2π$, and$p ∈ Z$is an arbitrary integer. Given$M_(5) ≅ R^2 S^3$and$N_(5) ≅ S^1 R^4$, the base space of the circle fibrations are, respectively, M_{(4)} \cong \mathcal{O}(-p) \rightarrow S^2 \,, \qquad N_{(4)} \cong \mathbb{R}^4 / \mathbb{Z}_p \,. We emphasise that these are topological data, and hence are insensitive to the squashings. From the anti-periodic spin structure of the$R^2$disc factor, we have (cf. \ref{['LietE']}) \mathcal{L}_{\partial_{t_E}} \zeta = c_R \frac{\mathrm{i}}{2} \zeta \,, \qquad c_R = \pm 1\,, and from \ref{['app:LieKS']}, \ref{['app:LieKSb']}, and \ref{['app:LieKSv']}, we have \mathcal{L}_{\partial_{\varphi_1}} \zeta = \mathcal{L}_{\partial_{\varphi_2}} \zeta = c_J \frac{\mathrm{i}}{2} \zeta \,, \qquad c_J = \pm 1\,. The spinor charge$Q^(ℓ)$is then given by Q^{(\ell)} = \frac{p}{2} c_R + c_J \,. The remaining data that enters \ref{['indexfromFP']} via \ref{['I4FP']} are the signs$κ, χ ∈ { ± 1 }$and the weights$b_1, b_2, Φ_0^0$at the fixed points. These can be systematically computed as explained in BenettiGenolini:2024hyd. The weights explicitly depend on the squashings, however the functional dependence when expressed in terms of the variables$(ɛ_1, ɛ_2)$are identical, keeping in mind the identifications \ref{['weightsid']}. Hence we simply state the results, and refer the readers to BenettiGenolini:2025icr for a detailed derivation, noting that$ϑ^ here = ϑ^ there + π/2$, which effectively exchanges the "north" and "south" poles. On$M_(4)$, the Killing vector$ξ = π_*(K)$has fixed points at the north and south poles of the zero-section$S^2 ⊂ O(-p) → S^2$. In the conventions of BenettiGenolini:2024hydBenettiGenolini:2025icr, the standard toric data for$O(-p) → S^2$is given by \vec{v}_0 = (-1,0) \,,\qquad \vec{v}_1 = (0,-1) \,,\qquad \vec{v}_2 = (1,-p) \,, and the weights of$ξ$at the two poles are given by (cf. BenettiGenolini:2025icr) \begin{split} (b_1^N, b_2^N) & = \left( - \det (\vec{v}_1 , \xi) , \, \det (\vec{v}_0 , \xi) \right) \\ & = ( \varepsilon_1 - \varepsilon_2 , -1 + \varepsilon_2 p) \, , \\ (b_1^S , b_2^S) & = \left( -\det (\vec{v}_2,\xi) , \, \det (\vec{v}_1, \xi) \right) \\ & = ( -1 + \varepsilon_1 p , - \varepsilon_1 + \varepsilon_2 ) \,. \end{split} Next,$Φ_0^0$at the fixed points are determined by the weights of$K$. In our conventions, the circles generated by$φ_1$and$φ_2$degenerate at the north and south poles, respectively. Thus, from \ref{['susycK']} we have (cf. BenettiGenolini:2025icr) \left.\Phi_0^0\right\vert_N = \varepsilon_2 \,, \qquad \left.\Phi_0^0\right\vert_S = \varepsilon_1 \,, On$N_(4) ≅ R^4/Z_p$, there is a single fixed point at the centre. The toric data is specified by BenettiGenolini:2024hyd \vec{v}_0 = (-1,0) \,,\qquad \vec{v}_1 = (1,-p) \,, which gives the weights \begin{split} b_1 & = - \frac{1}{p} \det(\vec{v}_1,\xi) = -\frac{1}{p} + \varepsilon_1 \,, \\ b_2 & = \frac{1}{p} \det(\vec{v}_0,\xi) = -\frac{1}{p} + \varepsilon_2 \,. \end{split} To determine$Φ_0^0$at the fixed point, note the entire$S^3$collapses at the centre of$N_(5) ≅ S^1_β R^4$. Taking into account the orbifold singularity of$N_(4) ≅ R^4/Z_p$, from \ref{['susycK']} we have \left.\Phi_0^0\right\vert_{\rm centre} = \frac{1}{p} \,. Finally, our field theory conventions are such that$c_R = -1$and$c_J = +1$. This fixes the signs (cf. BenettiGenolini:2025icr) \chi_N = \chi_S = c_R c_J = -1 \,, \qquad \kappa_N = - c_J = -1 \,, \qquad \kappa_S = c_R = -1 \,, as well as$κ_ centre = - c_J = +1$,$ χ_ centre = -1$. Collecting everything, the fixed point contributions \ref{['I4FP']} evaluates to I_{(4)}^{\rm FP}[M_{(4)}]= I_{(4)N}^{\rm FP} + I_{(4)S}^{\rm FP}= \frac{\mathrm{i} \pi^2}{54 G_{(5)}} \left[ \frac{(1- \varepsilon_1 - \varepsilon_2)^3}{\varepsilon_2 (\varepsilon_1 - \varepsilon_2) (1-p\varepsilon_2)} - \frac{(1- \varepsilon_1 - \varepsilon_2)^3}{\varepsilon_1 (\varepsilon_1 - \varepsilon_2) (1-p\varepsilon_1)} \right] \,, on$M_(4)$, and I_{(4)}^{\rm FP}[N_{(4)}] = - \frac{\mathrm{i} \pi^2}{54 G_{(5)}} \frac{(1- \varepsilon_1 - \varepsilon_2)^3}{(1-p\varepsilon_1)(1-p\varepsilon_2)} p^2 \,, on$N_(4)$. From \ref{['indexfromFP']}, the supersymmetric index is then given by - \log \mathcal{I}_{S^1_\beta \times M_3} = \mathrm{i} \pi N^2 \frac{(1-\varepsilon_1 - \varepsilon_2)^3}{27 \varepsilon_1 \varepsilon_2} \,, where we used the AdS/CFT dictionary \frac{\pi}{2G_{(5)}} = N^2 \,. Substituting in the identifications \ref{['weightsid']}, we have - \log \mathcal{I}_{S^1_\beta \times S^3} = \mathrm{i} \pi N^2 \frac{(1+\sigma+\tau)^3}{27 \sigma \tau} \overset{{\ref{SYMconstraint_S3_1}}}{=} \mathrm{i} \pi N^2 \frac{8}{27} \frac{\Delta^3}{\sigma \tau} \,, on$R^2 S^3 ∪ (- S^1 R^4)$, - \log \mathcal{I}_{S^1_\beta \times S^3_{\mathfrak{b}_1,\mathfrak{b}_2}} = \mathrm{i} \pi N^2 \frac{(1+\sigma_{\mathfrak{b}}+\tau_{\mathfrak{b}})^3}{27 \sigma_{\mathfrak{b}} \tau_{\mathfrak{b}}} \overset{{\ref{SYMconstraint_S3b_1}}}{=} \mathrm{i} \pi N^2 \frac{8}{27} \frac{\Delta_{\mathfrak{b}}^3}{\sigma_{\mathfrak{b}} \tau_{\mathfrak{b}}} \,, on$R^2 S^3_b_1,b_2 ∪ (- S^1 R^4_b_1,b_2)$, and - \log \mathcal{I}_{S^1_\beta \times S^3_v} = \mathrm{i} \pi N^2 \frac{(1+\sigma_v+\tau_v)^3}{27 \sigma_v \tau_v} \overset{{\ref{SYMconstraint_S3v_1}}}{=} \mathrm{i} \pi N^2 \frac{8}{27} \frac{\Delta_v^3}{\sigma_v \tau_v} \,, on$R^2 S^3_v ∪ (- S^1 R^4_v)$. In the large$N$limit, these precisely match the field theory results \ref{['roundSCI']}, \ref{['HondaSquashedIndexb']}, and \ref{['HondaSquashedIndex']}, respectively. Notice the dependence on the KK vector$ℓ$, parametrised by the arbitrary integer$p$, dropped out of the final answer. While$I_(4)^ FP[M_(4)]$and$I_(4)^ FP[N_(4)]$individually depend on$ℓ$, their difference does not, and corresponds to a physical observable: the supersymmetric index on$S^1_β M_3$. More generally, it is possible to start with \emph{any} combination of the$U(1)^3$isometry, and show that the final answer is$ℓ$-independent Colombo:2025ihp. For example, setting$p=0$, the reduction \ref{['ptwist']} is simply along the Hopf fibre of the$S^3$, and in BenettiGenolini:2025icr it was shown that this reduction can be replaced by a countably infinite number of "spindle reductions", labelled by coprime integers$(n_N, n_S)$, such that$S^1 ⟶ S^3 π⟶ WCP^1_[n_N,n_S]$. The orbifold singularity at the two poles of the spindle dress up various fixed point formulae, however the dependence on$(n_N,n_S)$drops out of the final answer, as shown in BenettiGenolini:2025icr. The statement generalises for Lens space black holes with$M_(5) ≅ R^2 L(p,q)$and$N_(5) ≅ S^1 R^4/Z_p$. The circle action generated by$n_N ∂_φ_1 + n_S ∂_φ_2$commutes with the$Z_p$action \ref{['Lpqorbiid']}, and thus defines a Seifert fibration. According to Theorem 4.10 of geiges2017seifertfibrationslensspaces, the base space of the fibration is generally a "non-coprime" spindle, in the language of Arav:2025jee. The final answer \ref{['indexfromFP']} should then again be independent of the choice of fibration, with an overall factor of$1/p$produced from$Δ x^5$in \ref{['I4FP']}, in agreement with \ref{['HondaLensIndex']}. In this paper, we studied rigid supersymmetric backgrounds$S^1_β M_3$, comprising elliptically/biaxially squashed three-spheres and Lens spaces, that admit Killing spinors that are anti-periodic around the Euclidean time circle. Such backgrounds should arise from the conformal boundary of complex, supersymmetric, non-extremal Euclidean black holes, with topology$R^2 M_3$. We explicitly constructed non-trivial Killing spinors on these backgrounds, together with a choice of profile for the new minimal background fields. Using a 3d effective field theory approach similar to Cassani:2021fyvArabiArdehali:2021nsx, we computed the supersymmetric index of the$SU(N)$$N=4$SYM on these backgrounds in a Cardy-like limit Honda:2019cio. Moreover, we showed that the superconformal anomaly vanishes on these backgrounds, consistent with expectations for Euclidean supersymmetric backgrounds admitting two supercharges of opposite R-charge. Assuming supergravity fillings of these backgrounds exist, we demonstrated how the techniques of equivariant localization can be used to precisely recover the field theory results from a gravity computation, without solving any supergravity equations. Extending the "background subtraction" method utilised in Cabo-Bizet:2018ehjBenettiGenolini:2025icrColombo:2025ihp for asymptotically AdS$_5$black holes, we proposed the prescription \ref{['ItotMN']} for general AlAdS$_5$black holes, to compute directly the supersymmetric index, effectively removing the contribution from the supersymmetric Casimir energy from the partition function. We note that the sub-leading correction in the large$N$limit of the superconformal index Cassani:2021fyv has been recovered from four-derivative corrections to the$D=5$supergravity on-shell action Bobev:2022bjmCassani:2022lrkCassani:2024tvk. Restricting to$N=4$SYM, we computed the supersymmetric index in the Cardy-like limit on our complex$S^1_β S^3_v$and$S^1_β S^3_b_1,b_2$backgrounds, which is a finite$N$result. However, four-derivative corrections in supergravity trivially vanishes for$N = 4$SYM, since$ Tr R=0$Bobev:2022bjm. It would be thus interesting to revisit the computations in Cassani:2021fyvArdehali:2021irqOhmori:2021dzb, using the backgrounds constructed in this paper, namely \ref{['S1S3bmetric']} \& \ref{['AnmVnm_S1S3b']}, as well as \ref{['S1S3vmetric']} \& \ref{['AnmVnm_S1S3v']}, generalising our results to 4d$N=1$SCFTs for which$ Tr R$scales like$N$. We expect the results to take the same form as in Cassani:2021fyv, where the complex structure parameters are replaced by \ref{['susychempot_S3b']} and \ref{['susychempot_S3v']}, respectively. It will be interesting to see if such results can be recovered by incorporating higher derivatives in equivariant localization. It will also be interesting to extend our approach to AdS$_7$/CFT$_6$. Using holographic renormalisation, the authors of Bobev:2025xan computed the Euclidean on-shell action of the BDHM black holeBobev:2023bxl, and compared their result with the superconformal index of the 6d$N=(2,0)$theory on$S^1 S^5$. The on-shell action of the BDHM black hole regularised this way satisfies a quantum statistical relation Bobev:2025xan: I_{\rm reg} = - S + \beta \left[ E - \sum_{i=1}^3 \Omega_i J_i - \sum_{I=1}^2 \Phi_I Q_I \right] \,. If a particular choice is made for the finite counterterms, such that the energy$E = E^ BDHM$is zero for global AdS$_7$,$I_ reg = I_ reg^ BDHM$agrees with minus the logarithm of the index,$I_S^1 S^5$Bobev:2025xan. Subtleties in such counterterms are 6d analogues of the scheme dependence of the partition function of 4d$N=1$SCFTs (recall our footnote \ref{['footnote:scanomaly']}). In terms of \ref{['QSR_BDDH']}, there is a scheme dependence in the on-shell action$I_ reg$, which compensates that of$E$, while$(S, J_i, Q_I)$are scheme independent Papadimitriou:2017kzw. Indeed, there is another scheme, for which the energy of AdS$_7$is given by the supersymmetric Casimir energy$E_ susy$. Replacing$I_ reg^ BDHM$with$(- I_S^1 S^5)$, \ref{['QSR_BDDH']} can be written as - \log Z_{S^1 \times S^5} = - S + \beta \left[ \left( E^{\rm BDHM} + E_{\rm susy} \right) - \sum_{i=1}^3 \Omega_i J_i - \sum_{I=1}^2 \Phi_I Q_I \right] \,, where$ Z_S^1 S^5 = - β E_ susy + I_S^1 S^5$Bobev:2015kza. The holographic setup we proposed in section \ref{['gluing']} suggests that$I_ reg^ BDHM = - I_S^1 S^5$is the on-shell action of a seven-dimensional closed manifold, obtained by gluing the BDHM black hole with global AdS$_7$along their common$S^1 S^5$conformal boundary (analogously to figure \ref{['figure:MglueN']}). The statements should also apply for general classes of AlAdS$_7$black holes, including the$L^p,q,r$black holes studied in Bobev:2025xan. We expect the field theory analysis presented in this paper to generalise to supersymmetric$S^1_β M_5$backgrounds, realising Killing spinors that are anti-periodic around$S^1_β$. We hope to return to this problem in future work. Finally, in$D=5$gauged supergravity there exist uniqueness theorems Lucietti:2021bbhLucietti:2022fqjLucietti:2023mvj for (asymptotically locally) AdS$_5$black holes, which state that any supersymmetric toric solution that is timelike outside a \emph{smooth} horizon with compact cross sections is locally isometric to the known black hole of Gutowski:2004ezChong:2005hr (or its near horizon geometry). The construction of these theorems are based on the Lorentzian classification of Gauntlett:2003fk, and hence are not applicable for the complex Euclidean saddles studied in this paper. It would be interesting if the uniqueness results can be extended to cover such solutions. It is worth emphasising that the existence of Lorentzian BPS black hole solutions numerically constructed in Cassani:2018mlhBombini:2019jhp, with conformal boundary comprising a biaxially squashed three-sphere, are not counterexamples to these uniqueness theorems. It was shown in Lucietti:2021bbh that the squashed black holes have \emph{non-smooth} horizons: they are$C^1$, but not$C^2$. Whether supersymmetric black hole solutions with$Ω_1 ≠ Ω_2$and biaxially squashed$S^1 S^3_v$conformal boundary exist, or whether any supersymmetric black hole solution with elliptically squashed$S^1 S^3_b_1, b_2$conformal boundary exists, remain as an open problem. The uniqueness results imply that if the Lorentzian solutions were to exist, they will also have non-smooth horizons. Complex, non-extremal deformations of these solutions will \emph{not} have this feature Horowitz:2022mly, and should be taken seriously as dominant saddles of the Euclidean gravitational path integral BenettiGenolini:2025jwe, given the holographic match with supersymmetric indices as presented in this paper. The matching persists in the BPS limit, where the complex Euclidean saddles can be Wick-rotated to real Lorentzian black holes. While squashing the boundaries of supersymmetric AdS black holes result in singular horizons, holography seems to be suggesting that the black holes are just as physical. I am grateful to Masazumi Honda, Yusheng Jiao, Seok Kim, Rishi Mouland, Jesse van Muiden, Sameer Murthy, Pantelis Panopoulos, Ioannis Papadimitriou, and James Sparks for helpful discussions. Special thanks goes to Pietro Benetti Genolini and Jerome Gauntlett for inspiring discussions and valuable comments on a draft. I am also grateful to Davide Cassani, Vasil Dimitrov, and Luigi Tizzano for comments on an earlier version of this paper. I would like to thank the Royal Society under the International Collaboration Award Grant {\textbackslash}R2{\textbackslash}242058 and the RIKEN Interdisciplinary Theoretical and Mathematical Sciences Program. Discussions during the "Japan-UK Workshop on Quantum Gravity" were useful in completing this work. I am supported by a Dean's PhD studentship at Imperial College. We collect here some useful facts about Killing spinors on round, elliptically squashed, and biaxially squashed three-spheres. Consider$S^3$as a surface in$C^2$. Introducing complex coordinates$(z_1, z_2) ∈ C^2$, we parametrise the unit radius$S^3$by (z_1, z_2) = (\sin \vartheta e^{\mathrm{i} \varphi_1}, \cos \vartheta e^{\mathrm{i} \varphi_2}) \,, with$ϑ ∈ [0, π/2]$and$Δ φ_i = 2π$. The metric on$S^3$is then given by {{\rm d}}s^3(S^3)= {{\rm d}} z_1 {{\rm d}} \bar{z}_1 + {{\rm d}} z_2 {{\rm d}} \bar{z}_2= {{\rm d}}\vartheta^2 + \sin^2 \vartheta {{\rm d}}\varphi_1^2 + \cos^2 \vartheta {{\rm d}}\varphi_2^2 \,. Half of the 4 Killing spinors are constant in the left-invariant frame, and are solutions to the Killing spinor equation \left( \nabla_a^{(3)} - \frac{\mathrm{i}}{2} \gamma_a^{(3)} \right) \upsilon = 0\,, where$γ^(3)$are elements of Cliff(3), with$a = 1,2,3$. The two spinors satisfy, respectively, \mathcal{L}_{\partial_{\varphi_1}} \upsilon = \mathcal{L}_{\partial_{\varphi_2}} \upsilon = c_J \frac{\mathrm{i}}{2} \upsilon \,, \qquad c_J = \pm 1 \,. We consider elliptical squashing of the round$S^3$, specified by parameters$b_1, b_2$. Let (z_1, z_2) = \left( \frac{\sin \vartheta}{\mathfrak{b}_1} e^{\mathrm{i} \varphi_1}, \frac{\cos \vartheta}{\mathfrak{b}_2} e^{\mathrm{i} \varphi_2} \right) \,, such that |z_1|^2 + |z_2|^2 = \frac{\sin^2\vartheta}{\mathfrak{b}_1} + \frac{\cos^2\vartheta}{\mathfrak{b}_2} = 1 \,. We then obtain the metric on an ellipsoid, where {{\rm d}}s^3(S^3_{\mathfrak{b}_1, \mathfrak{b}_2})= {{\rm d}} z_1 {{\rm d}} \bar{z}_1 + {{\rm d}} z_2 {{\rm d}} \bar{z}_2= f(\vartheta)^2 {{\rm d}}\vartheta^2 + \frac{\sin^2 \vartheta}{\mathfrak{b}_1^2} {{\rm d}}\varphi_1^2 + \frac{\cos^2 \vartheta}{\mathfrak{b}_2^2} {{\rm d}}\varphi_2^2 \,, with f(\vartheta) = \sqrt{\frac{\cos^2\vartheta}{\mathfrak{b}_1^2} + \frac{\sin^2\vartheta}{\mathfrak{b}_2^2} } \,. As explained in Martelli:2011fu, the second line of \ref{['app:ellipsoidmetric']} is a non-singular metric on$S^3_b_1, b_2$for any smooth function$f(ϑ)$of definite sign, where for regularity one should demand that$|f(ϑ)| → 1/b_1$as$ϑ → 0$and$|f(ϑ)| → 1/b_2$as$ϑ → π2$. The following statements hold without having to specify$f(ϑ)$. In alignment with conventions chosen in the main text, we shall be interested in Killing spinors on$S^3_b_1, b_2$that are solutions to \left( \nabla_a^{(3)} - \mathrm{i} A_a^{(3)} - \frac{\mathrm{i}}{2 f(\vartheta)} \gamma_a^{(3)} \right) \upsilon = 0 \,, with the following background gauge field A^{(3)} = c_J \left[ \frac{1}{2} \left( 1 - \frac{1}{f(\vartheta) \mathfrak{b}_1} \right) {{\rm d}} \varphi_1 + \frac{1}{2} \left( 1 - \frac{1}{f(\vartheta) \mathfrak{b}_2} \right) {{\rm d}} \varphi_2 \right] \,. Taking the frame {\rm e}^1 = \frac{\sin\vartheta}{\mathfrak{b}_1} {{\rm d}} \varphi_1 \,,\qquad {\rm e}^2 = \frac{\cos\vartheta}{\mathfrak{b}_2} {{\rm d}} \varphi_2 \,,\qquad {\rm e}^3 = f(\vartheta) {{\rm d}}\vartheta \,, and Pauli matrices for$γ^(3)_a$, the explicit solutions are given by Hama:2011ea \upsilon = \frac{1}{\sqrt{2}} \begin{pmatrix} \mathrm{i} \exp \left[ \frac{\mathrm{i}}{2} (c_J (\varphi_1 + \varphi_2) + \vartheta) \right] \\ - c_J \exp \left[ \frac{\mathrm{i}}{2} (c_J (\varphi_1 + \varphi_2) - \vartheta) \right] \end{pmatrix} \,, \qquad c_J = \pm 1 \,. The two spinors satisfy, respectively, \mathcal{L}_{\partial_{\varphi_1}} \upsilon = \mathcal{L}_{\partial_{\varphi_2}} \upsilon = c_J \frac{\mathrm{i}}{2} \upsilon \,. We consider the$SU(2) U(1)$--invariant biaxial (Berger) squashing of the round$S^3$, with squashing parameter$v$. The metric can be written as {{\rm d}}s^2(S^3_v)= \frac{1}{4} \left( \sigma_1^2 + \sigma_2^2 + v^2 \sigma_3^2 \right)= {{\rm d}}\vartheta^2 + \sin^2\vartheta {{\rm d}}\varphi_1^2 + \cos^2\vartheta {{\rm d}}\varphi_2^2 + (v^2-1) \left( \sin^2\vartheta {{\rm d}}\varphi_1 + \cos^2\vartheta {{\rm d}}\varphi_2 \right)^2 \,, where$σ_a$are the standard left-invariant one-forms. The special case$v=1$corresponds to the round$S^3$. We shall be interested in Killing spinors that are solutions to \left( \nabla_a^{(3)} - \mathrm{i} A_a^{(3)} - \frac{\mathrm{i}}{2} v \, \gamma_a^{(3)} \right) \upsilon = 0 \,, where the background gauge field is given by A^{(3)} = - (v^2 - 1) \left( \sin^2\vartheta {{\rm d}} \varphi_1 + \cos^2\vartheta {{\rm d}} \varphi_2 \right) \,. In the left--invariant frame {\rm e}^1= \frac{1}{2} ( -2 \sin(\varphi_1 + \varphi_2) {{\rm d}}\vartheta + \cos(\varphi_1 + \varphi_2) \sin(2\vartheta) ({{\rm d}}\varphi_2 - {{\rm d}}\varphi_1) ) \,,{\rm e}^2= \frac{1}{2} ( 2 \cos(\varphi_1 + \varphi_2) {{\rm d}}\vartheta + \sin(\varphi_1 + \varphi_2) \sin(2\vartheta) ({{\rm d}}\varphi_2 - {{\rm d}}\varphi_1) ) \,,{\rm e}^3= \frac{v}{2} ( 2 \sin^2\vartheta {{\rm d}}\varphi_1 + 2 \cos^2\vartheta {{\rm d}} \varphi_2 ) \,, the two Killing spinors that solve \ref{['app:S3vKSE']} are constant spinors, and they satisfy, respectively, \mathcal{L}_{\partial_{\varphi_1}} \upsilon = \mathcal{L}_{\partial_{\varphi_2}} \upsilon = c_J \frac{\mathrm{i}}{2} \upsilon \,, \qquad c_J = \pm 1 \,. In section \ref{['gravity']}, we considered various black hole saddles that arise as bulk fillings of supersymmetric$S^1_β M_3$backgrounds at the conformal boundary, comprising of various squashings of the three-sphere. In this appendix, we conjecture charge relations for these black holes, based on expectations from the superalgebra \ref{['QQbarb']}, \ref{['QQbarv']}. We start by reviewing the argument of Cabo-Bizet:2018ehj. Naively, Lorentzian black holes of Chong:2005hr at the BPS (supersymmetric and extremal) point have a divergent Euclidean on-shell action, with$β → ∞$. To remedy this one instead considers a family of Euclidean deformations that are non-extremal and supersymmetric, by allowing the chemical potentials to be complex. At finite$β$, the quantum statistical relation Gibbons:1976ue can then be rearranged, such that I= \beta E - S - \beta \Omega_1 J_1 - \beta \Omega_2 J_2 - \beta \Phi Q= \beta ( E - \Omega_1^* J_1 - \Omega_2^* J_2 - \Phi^* Q )- S - \beta (\Omega_1 - \Omega_1^*) J_1 - \beta (\Omega_2 - \Omega_2^*) J_2 - \beta (\Phi - \Phi^*) Q \,. Here,$Ω_1, Ω_2$are the angular velocities of the black hole on the horizon, where the outer (Killing) horizon is generated by the Killing vector V_H = \partial_{t_L} + \Omega_1 \partial_{\phi_L} + \Omega_2 \partial_{\psi_L} \,, and$Φ$is the electrostatic potential on the horizon. The asterisks denote the values of these variables at the BPS point Chong:2005hr: \Omega_1^* = \Omega_2^* = 1 \,, \qquad \Phi^* = \frac{3}{2} \,. Note the coordinates$(t_L, ϕ_L, ψ_L)$are related to the Euclidean$(t_E, φ_1, φ_2)$coordinates used in the main text by the following coordinate transformation Hawking:1998kw: t_L = - \mathrm{i} t_E \,,\qquad \phi_L = \varphi_1 - \mathrm{i} \Omega_1 t_E \,,\qquad \psi_L = \varphi_2 - \mathrm{i} \Omega_2 t_E \,. In general, for supersymmetric non-extremal solutions the supersymmetric Killing vector$K$is not equal to$V_H$. However, at the BPS point, the two vectors are aligned and we have$K = V_H$. Indeed, notice that the vector$K$constructed as a bilinear in the boundary Killing spinor \ref{['S1S3susyKV']} agrees with \ref{['VHroundS3']} at the BPS point, up to a trivial rescaling. The conserved charges of the BPS black hole satisfy (cf. \ref{['QQbar']}) E - \Omega_1^* J_1 - \Omega_2^* J_2 - \Phi^* Q = 0 \,, and thus from \ref{['QSRroundS3']} we see that the following relation holds I = -S - 2\pi\mathrm{i} (\sigma J_1 + \tau J_2 + \Delta Q ) \,, where we substituted in the chemical potentials \ref{['susychempot_S3']}. This was referred to as the \emph{supersymmetric quantum statistical relation} in Cabo-Bizet:2018ehj, proposed as a limiting procedure approaching the BPS point along a supersymmetric trajectory to cure the naive$β → ∞$divergence; indeed, the entropy function \ref{['SCI']} and the relation \ref{['susyQSRroundS3']} are manifestly independent of$β$. Based on the numerical solutions of Cassani:2018mlhBombini:2019jhp (see also Blazquez-Salcedo:2017ghgBlazquez-Salcedo:2017kig), describing$SU(2) U(1)$--invariant AlAdS$_5$black holes with biaxially squashed$S^3_v$boundary, we can make a few observations on how the supersymmetric quantum statistical relation generalises for various squashed AlAdS$_5$black holes. The known supersymmetric numerical solutions are BPS (i.e. they are also extremal), and have equal angular momenta, with$Ω_1 = Ω_2$. Note when writing down the metric on the conformal boundary, we rescaled the time coordinate (before Wick rotation) in \ref{['S1S3vmetric']}, comparing with e.g. Cassani:2018mlh. It follows that, in our conventions, the supersymmetric Killing vector of their black hole solution is given by V_H= \partial_{t_L} + \frac{1}{v} \partial_{\phi_L} + \frac{1}{v} \partial_{\psi_L} \,. As emphasised above, since the black hole solution is BPS, we have$K = V_H$. It is then straightforward to read off the BPS angular velocities \Omega_{v,1}^* = \Omega_{v,2}^* = \frac{1}{v} \,. Notice that, via the coordinate transformation \ref{['twistidcoordtransform']} we may write \ref{['susyKVS3v']} as \mathcal{K} = \partial_{t_E} + \mathrm{i} \left( \Omega_1 - \frac{1}{v} \right) \partial_{\varphi_1} + \mathrm{i} \left( \Omega_2 - \frac{1}{v} \right) \partial_{\varphi_2} \,, up to a trivial rescaling. This is in precise agreement with the supersymmetric Killing vector constructed as a bilinear of the boundary Killing spinor \ref{['S1S3vsusyKV']}. It is then natural to conjecture that, even when the two angular momenta are unequal, with$Ω_1 ≠ Ω_2$, biaxially squashed supersymmetric black hole solutions should exist, where the supersymmetric Killing vector is given by \ref{['susyKVS3eucl']}. Of course, for$Ω_1 = Ω_2$this statement is obviously true. We thus conjecture that the conserved charges of the BPS biaxially squashed black holes should satisfy (see also Ntokos:2021duk) E - \Omega_{v,1}^*J_1 - \Omega_{v,1}^*J_2 - \Phi^* Q = 0 \,, as expected from \ref{['QQbarv']}. The supersymmetric quantum statistical relation is then modified as I= \beta ( E - \Omega_{v,1}^* J_1 - \Omega_{v,2}^* J_2 - \Phi^* Q )- S - \beta (\Omega_1 - \Omega_{v,1}^*) J_1 - \beta (\Omega_2 - \Omega_{v,2}^*) J_2 - \beta (\Phi - \Phi^* ) Q= -S - 2\pi\mathrm{i} (\sigma_v J_1 + \tau_v J_2 + \Delta_v Q ) \,, with biaxially squashed chemical potentials \ref{['susychempot_S3v']}. Now consider the elliptically squashed background \ref{['S1S3bmetric']} with squashing parameters$b_1, b_2$. Assuming supersymmetric AlAdS$_5$black hole solutions arising as regular bulk fillings of this background exist, it is natural to conjecture that the supersymmetric Killing vector of these solutions will be given by \mathcal{K} = \partial_{t_E} + \mathrm{i} (\Omega_1 - \mathfrak{b}_1)\partial_{\varphi_1} + \mathrm{i} (\Omega_2 - \mathfrak{b}_2) \partial_{\varphi_2} \,, as an extension of the boundary Killing vector \ref{['S1S3bsusyKV']} Anderson:2007jpe. Inverting the coordinate transformation \ref{['twistidcoordtransform']}, we can read off the BPS angular velocities, \Omega_{\mathfrak{b},1}^* = \mathfrak{b}_1 \,,\qquad \Omega_{\mathfrak{b},2}^* = \mathfrak{b}_2\,. We thus conjecture that the conserved charges of BPS elliptically squashed black holes should satisfy E - \mathfrak{b}_1 J_1 - \mathfrak{b}_2 J_2 - \Phi^* Q = 0\,, as expected from \ref{['QQbarb']}, and the quantum statistical relation will be modified as I= \beta ( E - \Omega_{\mathfrak{b},1}^* J_1 - \Omega_{\mathfrak{b},2}^* J_2 - \Phi^* Q )- S - \beta (\Omega_1 - \Omega_{\mathfrak{b},1}^*) J_1 - \beta (\Omega_2 - \Omega_{\mathfrak{b},2}^*) J_2 - \beta (\Phi - 1) Q= -S - 2\pi \mathrm{i} (\sigma_{\mathfrak{b}} J_1 + \tau_{\mathfrak{b}} J_2 + \Delta_{\mathfrak{b}} Q ) \,, with elliptically squashed supersymmetric chemical potentials \ref{['susychempot_S3b']}. @article{Aharony:2021zkr, author={Aharony, Ofer and Benini, Francesco and Mamroud, Ohad and Milan, Elisa}, title={A gravity interpretation for the Bethe Ansatz expansion of the $\mathcal{N}=4$ SYM index}, eprint={2104.13932}, archiveprefix={arXiv}, primaryclass={hep-th}, reportnumber={SISSA 01/2021/FISI}, doi={10.1103/PhysRevD.104.086026}, journal={Phys. 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