††institutetext: a Department of Mathematics, King’s College London,
The Strand, London WC2R 2LS, U.K††institutetext: b Asia Pacific Center for
Theoretical Physics, Postech, Pohang 37673, Korea ††institutetext: c
Department of Physics, Postech, Pohang 37673, Korea

# Localization on $\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ I: the 4d/5d
connection in off-shell Euclidean supergravity

Axel Ciceria, Imtak Jeonb,c, Sameer Murthya<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We begin to develop the formalism of localization for the functional integral
of supergravity on $\text{AdS}_{3}\times\text{S}^{2}$. We show how the
condition of supersymmetry in the Euclidean
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ geometry naturally leads
to a twist of the S2 around the time direction of AdS3. The twist gives us a
five-dimensional Euclidean supergravity background dual to the elliptic genus
of $(0,4)$ SCFT2 at the semiclassical level. On this background we set up the
off-shell BPS equations for one of the Killing spinors, such that the
functional integral of five-dimensional Euclidean supergravity on
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ localizes to its space of
solutions. We obtain a class of solutions to these equations by lifting known
off-shell BPS solutions of 4-dimensional supergravity on
AdS${}_{2}\\!\times\\!\text{S}^{2}$. In order to do this consistently, we
construct and use a Euclidean version of the off-shell 4d/5d lift of
arxiv:1112.5371, which could be of independent interest.

##  1 Introduction

In this paper we take the first steps to develop the formalism of localization
in supergravity on asymptotic $\text{AdS}_{3}\\!\times\\!\text{S}^{2}$. Our
main physical motivation is to study an exact sector of AdS3/CFT2 holography.
By exact, we mean the inclusion of all quantum effects in supergravity or,
equivalently, all finite-charge corrections to the results derived in the
limit of infinite central charge of CFT2. Our eventual goal is to calculate
observables protected by supersymmetry in the bulk gravitational theory using
localization applied to functional integrals in supergravity. The same
approach has yielded rich results in theories on AdS2 (times S2 or S3)
culminating in the calculation of the exact quantum entropy of black holes in
four dimensions Dabholkar:2010uh ; Dabholkar:2011ec ; Dabholkar:2014ema ;
Iliesiu:2022kny  and five-dimensions Gupta:2021roy . The idea is to extend
such calculations to higher dimensional AdS spaces.

In this paper we focus on the AdS3/CFT2 context and consider the bulk AdS3
calculation of the elliptic genus of the dual SCFT2. Many of the intermediate
calculations are motivated by the embedding of this problem in string theory,
wherein two-dimensional SCFTs are realized as the low-energy theory on the
world-volume of effective strings. Some canonical examples are the D1/D5
system in IIB string theory wrapped on $T^{4}$/$K3$ Strominger:1996sh ;
Maldacena:1999bp , $M5$-brane bound states in M-theory/CY3 Maldacena:1997de ,
and D3 branes wrapping curves on the base of elliptically fibered CY3 in
F-theory Vafa:1997gr . All these SCFTs have at least $(0,4)$ supersymmetry in
two dimensions with a corresponding $SU(2)$ R-symmetry. Focussing on the MSW
theory Maldacena:1997de , the gravitational dual in the generic theory has the
form $\text{AdS}_{3}\\!\times\\!\text{S}^{2}\times$CY3.111One has
AdS${}_{3}\times$S${}^{3}\times T^{4}/K3$ in the D1/D5 system and
AdS${}_{3}\times$S${}^{3}\times$CY3 in the F-theory case Haghighat:2015ega .
In either of these cases, we consider the $SU(2)\times\widetilde{SU(2)}$
action on S3, and we think of the S2 with $\widetilde{SU(2)}$ action as being
embedded in the S3. 222One could also consider black strings in AdS5 which
needs a different treatment using gauged supergravity which we will not
consider here.

The low energy theory is summarized by five-dimensional supergravity with
asymptotic $\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ boundary conditions, which
is the theory that we study in this paper in the off-shell conformal
formalism. The classical Lorentzian $\text{AdS}_{3}\\!\times\\!\text{S}^{2}$
theory in this formalism has been studied in a series of nice papers
Castro:2007sd ; Castro:2007hc ; Castro:2008ne . Our eventual goal is to
calculate functional integrals for quantum observables using localization in
the corresponding Euclidean supergravity theory defined on global
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ with a periodic Euclidean time
coordinate, i.e., the manifold
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$. The three-dimensional
part of the manifold has the topology of a solid torus, and we often refer to
it as such.

The localization problem in supergravity is substantially more difficult
compared to its counterpart in quantum field theories. To begin with, one
defines a rigid supercharge in the gravitational theory, using the background
field method in theories with soft gauge algebras as applied to supergravity
deWit:2018dix ; Jeon:2018kec . At a practical level the problem reduces to
finding all bosonic gravitational configurations that admit a Killing spinor
whose asymptotic limit is one of the supercharges of
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$. Secondly, one finds all matter
configurations invariant under this supercharge. Thirdly, one evaluates the
supergravity action at a generic point in this manifold. Finally one
calculates the one-loop determinant of the non-BPS fluctuations around the
localization manifold.

In this paper we address the first two questions whose solutions comprise the
so-called localization locus. We work in the context of 5d off-shell ${\cal
N}=1$ supergravity coupled to an arbitrary number of vector multiplets. Our
idea is to use the 4d/5d lift Banerjee:2011ts , which relates solutions of
off-shell 4d supergravity to those of off-shell 5d supergravity compactified
on a circle.333This is different from the 4d/5d lift of Gaiotto:2005gf  which
involves a lift on a Taub-NUT space. The localization manifold in 4d ${\cal
N}=2$ supergravity on asymptotically AdS${}_{2}\times$ S2 has been completely
determined, and we can lift those solutions to
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ . Although this is not guaranteed to
produce all supersymmetric solutions, it should give all solutions that are
independent of the circle of compactification. Similar ideas have been used
successfully to make progress in localization on AdS${}_{2}\times$ S3 theories
in Gomes:2013cca , Gupta:2019xac ; Gupta:2021roy .

As it turns out, implementing this idea is not quite straightforward. Firstly,
the 4d/5d map in Banerjee:2011ts  is given for Lorentzian backgrounds while we
need it for Euclidean backgrounds. To this end, we first work out a consistent
set of supersymmetry transformations in the five-dimensional Euclidean
supergravity theory. In four dimensions we use the Euclidean supergravity
discussed in Cortes:2003zd ; Cortes:2005uq ; Cortes:2009cs ; deWit:2017cle ,
Jeon:2018kec . We then map off-shell BPS solutions of the 4d theory to off-
shell BPS solutions of the 5d theory. Here there is an additional problem,
namely that the 4d Euclidean theory carries a redundancy of allowed reality
conditions which has no counterpart in the 5d Euclidean theory. We show that
this redundancy can be absorbed in a parameter whose role is to implement the
symmetry breaking $SO(1,1)_{R}\to\mathbb{I}$.444This parameter is the
Euclidean analog of the parameter that enforces $SO(2)_{R}\to\mathbb{I}$ in
Banerjee:2011ts . The second problem has to do with the global identifications
of the background that we are interested in, i.e.
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$, which is not a Kaluza-
Klein lift of Euclidean AdS${}_{2}\times$ S2. The Kaluza-Klein condition was
used in Banerjee:2011ts  for the off-shell 4d/5d lift and, indeed, general
off-shell configurations do not consistently lift from Euclidean
AdS${}_{2}\times$ S2 to $\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$.
Nevertheless, the class of off-shell solutions relevant for the 4d black hole
problem can be lifted to the supersymmetric $\mathbb{H}_{3}/\mathbb{Z}$, due
to their enhanced rotational symmetry.

In this manner we obtain an adaptation of the 4d/5d lift relevant for the
Euclidean AdS3/CFT2 problem, whose details we work out in Section 4. The final
part of the paper, in Section 5, is an application of this lift to find a
class of off-shell solutions in the AdS3 theory which contribute to the
elliptic genus problem in the boundary SCFT2. In a sequel to this paper we
study the subsequent steps in the localization problem, in particular the 5d
supergravity action and its associated boundary terms.

There is another subtlety that appears in the Euclidean functional integral
realization of the supersymmetric observables in the AdS3/CFT2 context even
before beginning the 4d/5d lift. Consider a 2d $(0,4)$ superconformal field
theory SCFT2 on the boundary torus with complex structure $\tau$. The elliptic
genus can be defined as a trace over the Hilbert space on S1 in the Ramond
sector with an insertion of $(-1)^{F}$, which can be calculated by reducing to
the free theory or by using index theorems Witten:1986bf ; Schellekens:1986xh
; Alvarez:1987wg . In the functional integral formalism, one correspondingly
chooses periodic boundary conditions for the fermions around both cycles of
the torus. This leads to a constant Killing spinor on the torus, which is used
to localize the SCFT2 functional integral Benini:2013nda . Now consider the
calculation of the same functional integral in the bulk theory. The bosonic
vacuum configuration is that of thermal AdS3. One of the circles (the space
circle in the thermal AdS3) is contractible and therefore the fermions should
have half-integer momentum around that circle at infinity. Demanding that a
spinor obeys the Killing spinor equation forces its momenta around the two
cycles to be equal, so that it has fixed non-zero momentum also along the non-
contractible (time) direction. On the other hand, the non-contractible
direction has periodicity $\text{Im}(\tau)$ which is arbitrary, and therefore
the above spinor with non-zero momentum is not well-defined on the torus. This
problem can be resolved by turning on a twist of the S2 around the non-
contractible circle in AdS3, which allows for spinors which are constant in
time and therefore well-defined. As we discuss in Section 3, this allows us to
set up a supersymmetric background of the form
$\mathbb{H}^{3}/\mathbb{Z}\times$ S2.555A related supersymmetric set-up has
been discussed in the literature in the context of supersymmetric black holes
in AdS space Cabo-Bizet:2018ehj  and, in particular, for BTZ black holes in
Larsen:2021wnu . The bulk calculation is the NS-sector calculation of the
elliptic genus, which is equivalent to the boundary Ramond sector trace by a
spectral flow in the $(0,4)$ algebra. We show how the twist reduces the final
asymptotic algebra to be a sub-algebra of the Brown-Henneaux-Coussaert
Brown:1986nw ; Coussaert:1993jp  $(0,4)$ algebra on AdS3.

We note that our study of localization of supergravity on
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ has similarities with localization of
3d supersymmetric gauge theories on AdS3, which has been studied in
Assel:2016pgi . Indeed the analysis of asymptotic boundary conditions on the
gauge fields is very similar in both cases. The main difference and new
challenge (apart from the two extra dimensions) is that the metric also
fluctuates in our analysis, as can be seen in our localization solutions. We
also note that the idea of exact holography has also been explored in the
literature in contexts different to the one that we study here. This includes
Koszul duality Costello:2020jbh , topological string theory Brennan:2017rbf ,
and classical (large-$N$) gravitational theories BenettiGenolini:2019jdz . It
would be interesting to explore possible connections with these approaches.

The plan of this paper is as follows. In Section 2 we discuss five-dimensional
supergravity coupled to matter multiplets. We then describe the classical
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ solution in Lorentzian signature, and
the Killing spinors and superalgebra on this background. In Section 3 we
present the Euclidean $\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$
solution, the Killing spinors and superalgebra on this background, and the
relation of the path integral to the trace definition of the elliptic genus.
In Section 4 we present the off-shell 4d/5d map modified to the Euclidean
signature and present the lift of AdS${}_{2}\times$S2 to
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$. In Section 5 we apply our
formalism to lift localization solutions from AdS${}_{2}\times$S2 localization
solutions on $\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$. In various
appendices, we present the spinor and Clifford algebra conventions, the
Killing spinors, Killing vectors, and Lie brackets of global
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$, and the four- and five-dimensional
Euclidean supersymmetry transformations in the respective conformal
supergravities.

##  2 Background and set-up of the problem

In this section we discuss five-dimensional supergravity coupled to matter
multiplets. We show that the Euclidean theory is obtained by redefinitions of
fields of the Lorentzian theory that follow simply from the Wick rotation. We
review the elements of the supergravity theory including the supermultiplets
and the supersymmetry transformations. We then describe the classical
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ solution in Lorentzian signature, and
present the Killing spinors on this background and the consequent
superalgebra.

###  2.1 Off-shell 5d supergravity

Off-shell supergravity in the superconformal formalism in Lorentzian signature
in various dimensions has been known for many decades (see the book
Freedman:2012zz ). Euclidean supergravity, on the other hand, is a less-
studied subject and few references exist (e.g. Cortes:2003zd ; Cortes:2005uq ;
Cortes:2009cs ; deWit:2017cle ). In these references the method of time-like
reduction from a higher-dimensional Lorentzian theory is used to
systematically construct the Euclidean-signature theory. In this section we
discuss the formalism of 5d conformal supergravity with ${\cal N}=1$ (minimal)
supersymmetry, i.e., 8 supercharges. The Lorentzian theory was constructed in
Bergshoeff:2001hc ; Bergshoeff:2002qk ; Bergshoeff:2004kh , and in Kugo:2000hn
; Kugo:2000af ; Fujita:2001kv , and it is reviewed in the more recent
deWit:2009de ; Banerjee:2011ts  whose conventions we follow. One potential
systematic approach to construct the Euclidean theory would be to perform a
timelike reduction on a 6d theory. However, we use a less formal approach
here: we start from a Wick rotation and make an appropriate set of
transformations on all the fields of the Lorentzian theory so that we obtain a
consistent 5d Euclidean theory.666Indeed, such an approach also works
successfully in four dimensions and the result agrees with the timelike
reduction Jeon:2018kec .

The starting point is the usual Wick rotation $t=-\mathrm{i}t_{E}$ relating
the Lorentzian and Euclidean time coordinates. This is followed by the
appropriate transformations of all tensors for this coordinate change.
Accordingly, the time directional gamma matrices are related by
$\gamma_{{t}}=\mathrm{i}\gamma_{{t}_{E}}$. Changing the signature of spacetime
by this Wick rotation, in general, demands changing the nature of irreducible
spinors. For instance, in four dimensions, while the Majorana representation
of irreducible spinors is allowed in the Lorentzian theory, it is not the case
in the Euclidean theory. Therefore, we need an appropriate field redefinition
of spinors. This can be achieved in a simple manner by going to the
symplectic-Majorana basis which exists in both the Lorentzian and Euclidean
theory. In this basis, the charge conjugation matrix is the same in both the
theories. Therefore the Euclidean action follows from the Lorentzian action in
this basis by simply implementing the Wick rotation (See Appendix B.4 of
Jeon:2018kec  for details of this map in four dimensions.) Under the Wick
rotation, the Lagrangian density is unchanged, i.e., ${\cal
L}_{\text{Lor.}}={\cal L}_{\text{Eulc.}}$, and so the action of the Lorentizan
theory and Euclidean theory are related by
$\mathrm{i}S_{\text{Lor.}}=S_{\text{Eucl.}}$. Our conventions are such that
the Euclidean action is to be negative definite when we impose well defined
path integral.

The reality property of the fermionic fields in the Lorentzian and Euclidean
theories are different. In the Lorentzian theory, an SU$(2)_{R}$ spinor
doublet $\psi^{i}$ with $i=1,2$ follows a symplectic-Majorana condition, as
reviewed in (167),

$(\psi^{i})^{\dagger}\gamma_{\hat{t}}\;=\;\varepsilon_{ij}(\psi^{j})^{T}C\,,$
(1)

where $C$ is the unique choice of the charge conjugation matrix in five
dimensions (this is more generally true in odd dimensions). Compatibility with
supersymmetry leads to the standard reality property on fluctuating bosons, in
which the gauge fields and the metric are real. In the Euclidean theory the
allowed reality condition for the spinors

$(\psi^{i})^{\dagger}\;=\;\varepsilon_{ij}(\psi^{j})^{T}C$ (2)

is not compatible with supersymmetry if we impose the usual reality conditions
for fluctuating bosons. Therefore, we treat $\psi^{1}$ and $\psi^{2}$ as two
independent Dirac spinors, formally doubling the fermionic degrees of freedom
and then choosing a middle-dimensional contour in the functional integral,
following the standard treatment of fermions in the Euclidean theory. This
allows us to impose the standard reality conditions on bosonic fluctuations,
so that the Euclidean action is negative-definite and the functional integral
is well-defined at the perturbative level.777Note that we still allow reality
conditions on the background values of the bosonic fields which are different
from the usual Lorentzian ones: these can be imposed on on-shell fields (e.g.
$A_{0}$ or $g_{0i}$ are typically imaginary as dictated by the Wick rotation),
or on off-shell BPS fluctuations (e.g. the localization manifold on AdS2
involves scalar fields with $\bar{X}\neq X^{*}$ Dabholkar:2010uh ;
Gupta:2012cy ; Gupta:2015gga .

Supermultiplets: For the $\mathcal{N}=1$ 5d conformal supergravity theory, we
follow the conventions of deWit:2009de . We consider the Weyl multiplet, which
couples to $N_{\text{v}}$ number of vector multiplets as well as a single
hyper multiplet. Independent fields in each multiplet is summarized in Table
3.

The Weyl multiplet consists of the gauge fields corresponding to all the
symmetry generators of ${\cal N}=1$ superconformal algebra
$\\{P^{A}\,,M^{AB}\,,D\,,K^{A}\,,Q^{i}\,,S^{i}\,,V_{j}{}^{i}\\}$. Among all
the gauge fields, the gauge fields associated with
$\\{M^{AB}\,,K^{A}\,,S^{i}\\}$ are composite, i.e. they are expressed in terms
of other gauge fields. The independent gauge fields in Weyl multiplet are the
vielbein $E_{M}{}^{A}$, dilatation gauge field $b_{M}$, gaugino
$\psi_{M}^{i}$, and the $SU(2)_{R}$ gauge field $V_{Mj}{}^{i}$. For the Weyl
multiplet to be off-shell supermultiplet, it includes auxiliary two-form
tensor $T_{AB}$, auxiliary fermion $\chi^{i}$, and auxiliary scalar $D$. Hence
the independent fields of the Weyl multiplet are summarized as

$\mbox{Weyl:~{}~{}~{}}\\{{E_{M}}^{A}\,,\,\Psi^{i}_{M}\,,\,b_{M}\,,\,{V_{M,\,i}}^{j}\,;\,T_{MN}\,,\,\chi^{i}\,,\,D\\}\,.$
(3)

Here, the indices $\\{A,B,\cdots\\}$, $\\{M,N,\cdots\\}$, $\\{i,j,\cdots\\}$
are five-dimensional flat tangent space, curved spacetime, and $SU(2)$
fundamental indices, respectively, which are summarized in appendix A. In the
Weyl multiplet fields, we use the special conformal symmetry (that acts only
on $b_{M}$) to gauge-fix $b_{M}=0$, so that from here on this field will not
appear. We consider $N_{\text{v}}$ vector multiplets labeled by $I$, each of
which consists of

$\mbox{Vector:~{}~{}~{}}\\{\sigma^{I}\,,\,W^{I}_{M}\,,\,\Omega^{I\,i}\,,\,Y^{I}_{ij}\\}\,,\qquad
I=1\,,2\,,\cdots\,,N_{\text{v}}\,.$ (4)

They corresponds to a scalar, a $U(1)$ gauge field, gaugini, and an auxiliary
symmetric $SU(2)$ triplet. The $i,j$ indices are raised and lowered using the
$SU(2)$ symplectic metric $\varepsilon$, where, explicitely,
$\varepsilon_{12}=\varepsilon^{12}=1$. In particular, we have
$Y_{ij}\;=\;\varepsilon_{ik}\,\varepsilon_{j\ell}\,Y^{k\ell}$. We finally
consider a single hypermultiplet, which consists of

$\mbox{Hyper:~{}~{}~{}}\\{A_{i}{}^{\alpha}\,,\,\zeta^{\alpha}\\},$ (5)

corresponding to the hyper scalar, and the hyper fermion, where $\alpha=1,2$.
There is no known off-shell Lorentz-covariant hypermultiplet with finite
number of fields. In the backgrounds that we consider below, the
hypermultiplet turns out to take its on-shell value, and we can therefore
integrate it out at the semi-classical level. For the full localization
calculation, one could construct an off-shell hypermultiplet for one complex
supercharge (see e.g. Hama:2012bg ; Murthy:2015yfa ). One of the
$N_{\text{v}}$ vector multiplets and the single hypermultiplet act as the two
compensators to be added to the Weyl multiplet to form a 5d ${\cal N}=1$
Poincaré supergravity multiplet.

Supersymmetry algebra: The supersymmetry transformations of the various spinor
fields under the $Q$ and $S$ supersymmetry transformations are given in (177).
Two $Q$-supersymmetry transformations, parametrized by spinors $\epsilon_{1}$
and $\epsilon_{2}$ respectively, close onto the bosonic symmetries of the
theory as

$[\delta_{Q}{(\epsilon_{1})}\,,\delta_{Q}{(\epsilon_{2})}]\;=\;\delta_{\text{gct}}(\xi^{\mu})+\delta_{M}(\lambda)+\delta_{S}(\eta)+\delta_{K}(\Lambda_{K})$
(6)

where $\delta_{\text{gct}}$ are the general coordinate transformations,
$\delta_{M}$ is a local Lorentz transformation, $\delta_{S}$ is a conformal
supersymmetry transformation, and $\delta_{K}$ is special conformal
transformation. The relevant parameters to this paper are

$\begin{split}\xi^{\mu}&\;=\;2\bar{\epsilon}_{2i}\gamma^{\mu}\epsilon^{i}_{1}\,,\\\
\lambda^{AB}&\;=\;-\xi^{\mu}\omega_{\mu}^{~{}AB}+\frac{\mathrm{i}}{2}T^{CD}\bar{\epsilon}_{2i}(6\gamma^{[A}\gamma_{CD}\gamma^{B]}-\gamma^{AB}\gamma_{CD}-\gamma_{CD}\gamma^{AB})\epsilon^{i}_{1}\,.\end{split}$
(7)

Action: The bosonic Lagrangian at two-derivative level is

$\begin{split}L_{\text{bulk}}\;=\;E\,\left(\mathcal{L}_{V}+\mathcal{L}_{VW}+\mathcal{L}_{H}+\mathcal{L}_{HW}+\mathcal{L}_{CS}\right)\,,\end{split}$
(8)

where $E\equiv\text{det}({E_{M}}^{A})$, $\mathcal{L}_{V}$ contains purely
vector multiplet terms, $\mathcal{L}_{VW}$ contains mixing between vector and
Weyl, $\mathcal{L}_{H}$ is the kinetic hyper scalar piece, $\mathcal{L}_{HW}$
contains coupling of hyper to Weyl, and $\mathcal{L}_{CS}$ is the five-
dimensional Chern-Simons action:

$\begin{split}\mathcal{L}_{V}&\;=\;\frac{1}{2}\,c_{IJK}\,\sigma^{I}\,\left(\frac{1}{2}\,D^{M}\,\sigma^{J}\,D_{M}\,\sigma^{K}+\frac{1}{4}\,F^{J}_{MN}\,F^{MN\,K}-3\sigma^{J}\,F^{K}_{MN}\,T^{MN}-Y_{ij}^{J}\,Y^{K\,ij}\right)\,,\\\
\mathcal{L}_{VW}&\;=\;-C(\sigma)\left(\frac{1}{8}\,R-4D-\frac{39}{2}\,T^{2}\right)\,,\\\
\mathcal{L}_{H}&\;=\;-\frac{1}{2}\Omega_{\alpha\beta}\varepsilon^{ij}D_{M}\,{A_{i}}^{\alpha}\,D^{M}\,{A_{j}}^{\beta}\,,\\\
\mathcal{L}_{HW}&\;=\;\chi\,\left(\frac{3}{16}\,R+2D+\frac{3}{4}\,T^{2}\right)\,,\\\
\mathcal{L}_{CS}&\;=\;-\frac{i}{48\,E}\varepsilon^{MNOPQ}c_{IJK}W^{I}_{M}F^{J}_{NO}F^{K}_{PQ}\,.\end{split}$
(9)

In the Chern-Simons Lagrangian $\mathcal{L}_{CS}$, the object
$\varepsilon^{MNOPQ}$ is a fully antisymmetric tensor density taking values in
$\\{-1,\,0,\,1\\}$. The scalar norms appearing in $\mathcal{L}_{VW}$ and
$\mathcal{L}_{HW}$ are:

$\displaystyle
C(\sigma)\mathrel{\mathop{:}}=\frac{1}{6\,}c_{IJK}\,\sigma^{I}\sigma^{J}\sigma^{K}\,,$
(10)
$\displaystyle\chi\mathrel{\mathop{:}}=\frac{1}{2}\,\Omega_{\alpha\beta}\,\varepsilon^{ij}{A_{i}}^{\alpha}{A_{j}}^{\beta}\,.$
(11)

The action of the theory is

$S_{\text{bulk}}\;=\;\frac{1}{8\pi^{2}}\int_{\mathcal{M}}d^{5}x\,L_{\text{bulk}}\,,$
(12)

for an appropriate coordinate chart on the 5d manifold $\mathcal{M}$.

###  2.2 Global Lorentzian AdS${}_{3}\times$ S2

We consider the fully supersymmetric AdS${}_{3}\times$ S2 solution of the
Lorentzian supergravity described above, corresponding to the near-horizon
geometry of the half-BPS magnetic black string Elvang:2004rt . The metric in
Lorentzian signature is

$ds^{2}\;=\;4\ell^{2}\,(-\cosh^{2}\rho\,dt^{2}+d\rho^{2}+\sinh^{2}\rho\,d\psi^{2})+\ell^{2}\,(d\theta^{2}+\sin^{2}\,\theta
d\phi^{2})\,,$ (13)

where the coordinates of the AdS3 have the ranges
$\rho\in[0,\,\infty),\,\psi\in[0,\,2\pi],\,t\in(-\infty,\,\infty)$ and the
angles on the S2 have ranges $\theta\in[0,\,\pi],\,\phi\in[0,\,2\pi]$. The
radii of the AdS3 and the S2 are $(2\ell)$ and $\ell$ respectively, where this
relative factor of $2$ is determined by supersymmetry. Note that in the off-
shell theory, $\ell$ is free and parametrizes the dilatations of the theory,
while in the on-shell theory (where dilatations are broken) it is determined
by the magnetic charges of the solution via the D-gauge condition. These
magnetic charges $p^{I}$ enter the solution through the vector multiplet. The
non-trivial fields of the vector multiplet are:

$\sigma^{I}\;=\;-\frac{p^{I}}{\ell}\,,\qquad
F^{I}_{\theta\phi}\;=\;p^{I}\,\sin\theta\,.$ (14)

Note that the solution does not have electric flux, which allows us to turn on
flat gauge connections on the AdS3. This aspect will become relevant in the
following. The five-dimensional Newton’s constant is
$G_{5}=6\pi\ell^{3}/p^{3}$ and the three-dimensional Newton’s constant is
obtained by setting $G_{5}=\text{Area}_{S^{2}}\times G_{3}=4\pi\ell^{2}G_{3}$.

In the off-shell formalism of Section 2.1, one requires additional auxiliary
fields. In the Weyl multiplet, the non-trivial fields are:

$T_{\theta\phi}\;=\;-\frac{\ell}{4}\,\sin\theta\,.$ (15)

In the compensating hypermultiplet, the BPS equation is solved by

$\displaystyle{A_{i}}^{\alpha}\;=\;{c_{i}}^{\alpha}\,,\quad$ (16)

where the constants ${c_{i}}^{\alpha}$ are determined in terms of the charge
$p^{I}$ by the field equation for the auxiliary field $D$ to be

$\Omega_{\alpha\beta}\,\varepsilon^{ij}c_{i}{}^{\alpha}c_{j}{}^{\beta}\;=\;\frac{2}{3\ell^{3}}\,c_{IJK}\,p^{I}p^{J}p^{K}\,.$
(17)

In this paper, we fix an explicit choice for the $c_{i}{}^{\alpha}$ as

$c_{1}{}^{2}\;=\;c_{2}{}^{1}\;=\;0\,,\qquad
c_{1}{}^{1}\;=\;c_{2}{}^{2}\;=\;\sqrt{\frac{p^{3}}{3\ell^{3}}}\,.$ (18)

###  2.3 Supersymmetry algebra in Lorentzian AdS${}_{3}\times$ S2

Killing spinors The $Q$\- and $S$\- supersymmetry parameters, $\epsilon^{i}$
and $\eta^{i}$ respectively, that are preserved by the bosonic fields of the
global AdS${}_{3}\times S^{2}$ background are determined by setting the
variation of gravitino and the variation the auxiliary fermion in (177) to
zero. These two equations are, respectively,

$\displaystyle
0\;=\;2\,\mathcal{D}_{M}\epsilon^{i}+\frac{\mathrm{i}}{2}T_{AB}(3\gamma^{AB}\gamma_{M}-\gamma_{M}\gamma^{AB})\epsilon^{i}-\mathrm{i}\gamma_{M}\eta^{i}\,.$
(19) $\displaystyle
0\;=\;\frac{1}{2}\epsilon^{i}D+\frac{1}{64}R_{MNj}{}^{i}(V)\gamma^{MN}\epsilon^{j}+\frac{3}{64}\mathrm{i}(3\gamma^{AB}\not{D}+\not{D}\gamma^{AB})T_{AB}\epsilon^{i}$
(20)
$\displaystyle~{}~{}~{}~{}~{}\quad-\frac{3}{16}T_{AB}T_{CD}\gamma^{ABCD}\epsilon^{i}+\frac{3}{16}T_{AB}\gamma^{AB}\eta^{i}\,.$

On our bosonic background, the second equation (20) immediately determines the
$S$-supersymmetry spinor as $\eta^{i}=0$. The first equation is referred to as
the Killing spinor equation. We analyze its solutions in Appendix C and
summarize the results below.

The complex basis of the Killing spinor on AdS${}_{3}\times$ S2 is given by
the the following four Killing spinors,

$\begin{array}[]{ll}\epsilon_{+}^{~{}+}=\sqrt{\frac{\ell}{2}}\,\epsilon^{+}_{\text{AdS}_{3}}\otimes\epsilon^{+}_{\text{S}^{2}}\,,&\epsilon_{+}^{~{}-}=\sqrt{\frac{\ell}{2}}\,\epsilon^{+}_{\text{AdS}_{3}}\otimes\epsilon^{-}_{\text{S}^{2}}\,,\\\
\epsilon_{-}^{~{}+}=\sqrt{\frac{\ell}{2}}\,\epsilon^{-}_{\text{AdS}_{3}}\otimes\epsilon^{+}_{\text{S}^{2}}\,,&\epsilon_{-}^{~{}-}=\sqrt{\frac{\ell}{2}}\,\epsilon^{-}_{\text{AdS}_{3}}\otimes\epsilon^{-}_{\text{S}^{2}}\,,\end{array}$
(21)

with

$\begin{split}\epsilon^{+}_{\text{AdS}_{3}}&\;=\;{\rm
e}^{\frac{\mathrm{i}}{2}(t+\psi)}\left(\begin{array}[]{c}\cosh\frac{\rho}{2}\\\
-\sinh\frac{\rho}{2}\end{array}\right)\,,\qquad\epsilon^{-}_{\text{AdS}_{3}}\;=\;{\rm
e}^{-\frac{\mathrm{i}}{2}(t+\psi)}\left(\begin{array}[]{c}-\sinh\frac{\rho}{2}\\\
\cosh\frac{\rho}{2}\end{array}\right)\,,\\\
\epsilon^{+}_{\text{S}^{2}}&\;=\;{\rm
e}^{\frac{\mathrm{i}}{2}\phi}\left(\begin{array}[]{c}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\end{array}\right)\,,\qquad\qquad\qquad\epsilon^{-}_{\text{S}^{2}}\;=\;{\rm
e}^{-\frac{\mathrm{i}}{2}\phi}\left(\begin{array}[]{c}-\sin\frac{\theta}{2}\\\
\cos\frac{\theta}{2}\end{array}\right)\,.\end{split}$ (22)

These 4 Killing spinors organize themselves into the 8 pairs of symplectic
Majorana spinors

$\begin{split}&\epsilon^{\,i}_{(1)}=(-\mathrm{i}\epsilon_{+}^{~{}+},\epsilon_{-}^{~{}-})\,,\quad\;\epsilon^{\,i}_{(2)}=(\epsilon_{+}^{~{}+},-\mathrm{i}\epsilon_{-}^{~{}-})\,,\quad\;\epsilon^{\,i}_{(3)}=\,-(\epsilon_{-}^{~{}-},\mathrm{i}\epsilon_{+}^{~{}+})\,,\quad\;\epsilon^{\,i}_{(4)}=\,-(\mathrm{i}\epsilon_{-}^{~{}-},\epsilon_{+}^{~{}+})\,,\\\
&\tilde{\epsilon}^{\,i}_{(1)}\;=\;(\epsilon_{+}^{~{}-},\mathrm{i}\epsilon_{-}^{~{}+})\,,\quad\;\tilde{\epsilon}^{\,i}_{(2)}\;=\;(\mathrm{i}\epsilon_{+}^{~{}-},\epsilon_{-}^{~{}+})\,,\quad\;\,\tilde{\epsilon}^{\,i}_{(3)}\;=\;(-\mathrm{i}\epsilon_{-}^{~{}+}\,,\epsilon_{+}^{~{}-})\,,\quad\;\tilde{\epsilon}^{\,i}_{(4)}\;=\;(\epsilon_{-}^{~{}+}\,,-\mathrm{i}\epsilon_{+}^{~{}-})\,,\end{split}$
(23)

to form the 8 real basis of the Killing spinor on AdS${}_{3}\times$S2. Each
pair satisfies the following symplectic Majorana condition (167) appropriate
to the 5d Lorentzian theory, i.e.
$(\epsilon^{i})^{\dagger}\gamma_{\hat{t}}\;=\;\varepsilon_{ij}(\epsilon^{j})^{T}{\cal
C}$ in the conventions of Appendix C.

Superconformal algebra: Let us denote

${\cal
Q}_{a}\;=\;\delta(\epsilon^{i}_{(a)})\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\widetilde{\cal
Q}_{a}\;=\;\delta(\tilde{\epsilon}^{i}_{({a})})\,,~{}~{}~{}~{}a=1\,,2\,,3\,,4\,,$
(24)

with the Grassmann even Killing spinors $\epsilon^{i}$. Then,

$\displaystyle\bigl{\\{}{\cal Q}_{a}\,,{\cal
Q}_{b}\bigr{\\}}\;=\;-2\mathrm{i}\delta_{ab}(L_{0}-J^{3})\,,\qquad\bigl{\\{}\widetilde{\cal
Q}_{a}\,,\widetilde{\cal
Q}_{b}\bigr{\\}}\;=\;-2\mathrm{i}\delta_{ab}(L_{0}+J^{3})\,,$ (25)
$\displaystyle\bigl{\\{}{\cal Q}_{a}\,,\widetilde{\cal
Q}_{b}\bigr{\\}}\;=\;\begin{pmatrix}-2\mathrm{i}J^{2}&2\mathrm{i}J^{1}&-(L_{+}-L_{-})&\mathrm{i}(L_{+}+\L_{-})\\\
-2\mathrm{i}J^{1}&-2\mathrm{i}J^{2}&-\mathrm{i}(L_{+}+L_{-})&-(L_{+}-L_{-})\\\
L_{+}-L_{-}&\mathrm{i}(L_{+}+L_{-})&-2\mathrm{i}J^{2}&-2\mathrm{i}J^{1}\\\
-\mathrm{i}(L_{+}+L_{-})&L_{+}-L_{-}&2\mathrm{i}J^{1}&-2\mathrm{i}J^{2}\end{pmatrix}\,,$
(26)

where $SL(2,R)$ generators $L_{0}\,,L_{\pm}$ and $SO(3)$ generators
$J^{\boldsymbol{a}}$ satisfy

$[L_{+},\,L_{-}]\;=\;-2L_{0}\,,\qquad[L_{0}\,,L_{\pm}]\;=\;\pm
L_{\pm}\,,\qquad[J^{\boldsymbol{a}}\,,J^{\boldsymbol{b}}]\;=\;\mathrm{i}\epsilon^{{\boldsymbol{a}}{\boldsymbol{b}}{\boldsymbol{c}}}J^{\boldsymbol{c}}\,.$
(27)

Their explicit representation on the $\text{AdS}_{3}\\!\times\\!\text{S}^{2}$
is given in Appendix D.

Let us define the supercharges $G^{i\alpha}_{\gamma}$

$\begin{split}&G_{+}^{++}\,\equiv\,\frac{\mathrm{i}{\cal Q}_{1}+{\cal
Q}_{2}}{2}\,,\quad G_{-}^{+-}\,\equiv\,\frac{-{\cal Q}_{3}+\mathrm{i}{\cal
Q}_{4}}{2}\,,\quad G_{+}^{+-}\,\equiv\,\frac{\widetilde{\cal
Q}_{1}-\mathrm{i}\widetilde{\cal
Q}_{2}}{2}\,,\quad\;\,G_{-}^{++}\,\equiv\,\frac{\mathrm{i}\widetilde{\cal
Q}_{3}+\widetilde{\cal Q}_{4}}{2}\,,\\\ &G_{-}^{--}\,\equiv\,\frac{{\cal
Q}_{1}+\mathrm{i}{\cal Q}_{2}}{2}\,,\quad
G_{+}^{-+}\,\equiv\,\frac{\mathrm{i}{\cal Q}_{3}-{\cal
Q}_{4}}{2}\,,\quad\;\;G_{-}^{-+}\,\equiv\,\frac{-\mathrm{i}\widetilde{\cal
Q}_{1}+\widetilde{\cal Q}_{2}}{2}\,,\quad
G_{+}^{--}\,\equiv\,\frac{\widetilde{\cal Q}_{3}+\mathrm{i}\widetilde{\cal
Q}_{4}}{2}\,,\end{split}$ (28)

where $\gamma$ is the sign of the $L_{0}$ eigenvalue, $i$ is the outer
automorphism from the SU(2) R-symmetry of the supergravity, and $\alpha$ is
the SU(2) R-symmetry index corresponding to isometries of the S2. Then, we
obtain the non-trivial commutation relations:

$\bigl{\\{}G^{+\alpha}_{\pm}\,,G^{-\beta}_{\mp}\bigr{\\}}\;=\;\epsilon^{\alpha\beta}L_{0}\pm(\epsilon{\boldsymbol{\tau}}_{\boldsymbol{a}})^{\beta\alpha}J^{\boldsymbol{a}}\,,\qquad\bigl{\\{}G^{+\alpha}_{\pm}\,,G^{-\beta}_{\pm}\bigr{\\}}\;=\;\mp\mathrm{i}\epsilon^{\alpha\beta}L_{\pm}\,,$
(29)

and

$\begin{array}[]{ll}\left[L_{0}\,,G^{i\alpha}_{\pm}\right]\;=\;\pm{\textstyle{\frac{1}{2}}}\,G^{i\alpha}_{\pm}\,,{}{}{}{}{}&\left[L_{\pm}\,,G^{i\alpha}_{\mp}\right]\;=\;-\mathrm{i}\,G^{i\alpha}_{\pm}\,,\\\
\left[J^{3}\,,G^{i\pm}_{\gamma}\right]\;=\;\pm{\textstyle{\frac{1}{2}}}\,G^{i\pm}_{\gamma}\,,{}{}{}&\left[J^{\pm}\,,G^{i\mp}_{\gamma}\right]\;=\;\,G^{i\pm}_{\gamma}\,,\end{array}$
(30)

where $J^{\pm}\equiv J^{1}\pm\mathrm{i}J^{2}$. The algebra (27), (29), (30) is
the global part $su(1,1|2)$ of the NS-sector chiral ${\cal N}=4$
superconformal algebra. Denoting the super Virasoro charges as ${\cal L}_{n}$,
$n\in\mathbb{Z}$ and ${\cal G}^{\alpha}_{\dot{A},r}$,
$r\in\mathbb{Z}+{\textstyle{\frac{1}{2}}}$, $\dot{A}=(+,-)$, the embedding
into the ${\cal N}=4$ superconformal algebra as presented e.g. in Guo:2022ifr
is given by ${L_{\pm}=\mp\mathrm{i}{{\cal L}}_{\mp 1}}$, $L_{0}={\cal L}_{0}$,
$G^{\pm\alpha}_{\pm}=\pm\,{{\cal G}}^{\alpha}_{\mp,\mp 1/2}$,
$G^{\mp\alpha}_{\pm}=\pm\,{{\cal G}}^{\alpha}_{\pm,\mp 1/2}$, and the $su(2)$
zero-modes are unchanged.

##  3 Supersymmetric $\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ and
twisting

In this section we move from the Lorentzian
$\text{AdS}_{3}\\!\times\\!\text{S}^{2}$ configuration to the Euclidean
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ geometry. We begin, in the
first subsection, by reviewing the thermal AdS3 (=$\mathbb{H}^{3}/\mathbb{Z}$)
geometry and some aspects of the AdS/CFT correspondence in this set up. In the
second subsection we move to the supersymmetric version of the thermal
geometry which requires a non-trivial twist. In the third subsection we
discuss the Hamiltonian trace interpretation of the functional integral on
this twisted configuration, and discuss how this is related to the elliptic
genus in the semi-classical limit.

###  3.1 Thermal compactification of AdS3

In this subsection we review the general set-up of the AdS3/CFT2 dictionary in
the context of our problem, following the treatment of Kraus:2006wn . We begin
with the three-dimensional pure Einstein-Hilbert action with a cosmological
constant:

$S_{\text{grav}}\;=\;\frac{1}{16\pi G_{3}}\int
d^{3}x\sqrt{g}(R-\frac{1}{2\ell^{2}})\,.$ (31)

AdS3 is a solution of the equations of motion of this theory with constant
negative curvature. The Wick rotation $t=-\mathrm{i}t_{E}$ leads to the
Euclidean ($\mathbb{H}^{3}$) metric

$ds^{2}\;=\;4\ell^{2}\left(\cosh^{2}\rho\,{dt_{E}}^{2}+d\rho^{2}+\sinh^{2}\rho\,d\psi^{2}\right)\,.$
(32)

When the range of $t_{E}$ is $(-\infty,\infty)$ we obtain the cylinder
geometry corresponding to the global solution. Thermal AdS3 corresponds to the
quotient $\mathbb{H}^{3}/\mathbb{Z}$, obtained by imposing the periodicities

$\bigl{(}t_{\scriptscriptstyle{E}}\,,\psi\bigr{)}\sim\bigl{(}t_{\scriptscriptstyle{E}}+2\pi\tau_{2}\,,\psi+2\pi\tau_{1}\bigr{)}\sim\bigl{(}t_{\scriptscriptstyle{E}}\,,\psi+2\pi\bigr{)}\,,$
(33)

so that the geometry is that of a solid torus. Physically, this corresponds to
setting the chemical potential conjugate to angular momentum (i.e. the angular
velocity) to $2\pi\tau_{1}$ and the chemical potential conjugate to energy
(i.e. the inverse temperature) to $2\pi\tau_{2}$.

It is convenient to introduce the coordinates $z\equiv\psi+\mathrm{i}\,t_{E}$,
$\bar{z}\equiv\psi\ -\mathrm{i}\,t_{E}$. Upon taking the large-$\rho$
expansion of the transverse metric tensor ${g}_{\alpha\beta}$, where
$x^{\alpha}=(z,\,\bar{z})$, one obtains the Fefferman-Graham form

${g}_{\alpha\beta}\;=\;{\rm
e}^{2\rho}{g}^{(0)}_{\alpha\beta}+{g}^{(2)}_{\alpha\beta}+\cdots\,.$ (34)

The conformal boundary metric ${g}^{(0)}_{\alpha\beta}$ is

${g}^{(0)}_{\alpha\beta}\;=\;\ell^{2}\,{dz\,d\bar{z}}\,,$ (35)

with the identifications

$(z,\bar{z})\sim(z+2\pi,\bar{z}+2\pi)\sim(z+2\pi\tau,\bar{z}+2\pi\bar{\tau})\,,$
(36)

consistent with the interpretation that the boundary CFT lives on the flat
torus with modular parameter $\tau$.

Boundary conditions on the gauge fields
More generally, we include constant chemical potentials $\mu^{I}$ for a number
of conserved $U(1)$ charges $q_{I}=\int J_{I}$ where $J_{I}$ are the
corresponding conserved currents in the boundary CFT. The partition function
of such a CFT is

$\text{Tr}_{\mathcal{H}_{\text{CFT}}}\,{\rm e}^{-\beta H+\mu^{I}q_{I}}\,.$
(37)

The dual gravitational theory (31) includes the same number of $U(1)$ gauge
fields888In this subsection, $W^{I}$ and all other fields are three-
dimensional. ${W}^{I}$. The most relevant term governing their dynamics at low
energies is given by the Chern-Simons action

$-\frac{\mathrm{i}}{8\pi}k_{IJ}\int W^{I}\wedge
dW^{J}\;=\;-\frac{\mathrm{i}}{8\pi}k_{IJ}\int
d^{3}x\,\varepsilon^{\mu\nu\lambda}\,W^{I}_{\mu}\,\partial_{\nu}W^{J}_{\lambda}\,.$
(38)

(Here, unlike in the rest of the paper, we employ the indices
$\mu,\,\nu\,\cdots$ to denote the 3d coordinates
${x^{\mu}=(\rho,\,x^{\alpha})}$.) In the gauge $W^{I}_{\rho}=0$, the gauge
fields admit a large-$\rho$ expansion analogous to (34) as:

${W}^{I}_{\alpha}\;=\;{W}^{I\,(0)}_{\alpha}+{\rm
e}^{-2\rho}\,{W}^{I\,(2)}_{\alpha}+\cdots\,.$ (39)

The asymptotic equations of motion imply that ${W}^{I\,(0)}_{\alpha}$ is flat.

As is well-known, the fact that the CS term has a first order kinetic term so
that the two legs ${W}^{I}_{z,\,\bar{z}}$ form canonical pairs in the
Hamiltonian theory Elitzur:1989nr . One should therefore impose Dirichlet
boundary conditions on only one of the legs:

$\delta{W}^{I\,(0)}_{z}\;=\;0\,,\qquad{W}^{I\,(0)}_{\bar{z}}\;\text{not
fixed.}$ (40)

Now, in accord with the bulk/boundary correspondence, the boundary source
$\mu^{I}$ must be identified with the asymptotic value of the gauge field
${W}^{I\,(0)}_{z}$. Since the $\psi$-cycle is contractible, any smooth
configuration must have $W^{I}_{\psi}=0$ at the origin. The saddle-point
configurations have flat gauge fields due to the equations of motion, and
therefore obey

${W}^{I}_{z}\;=\;-W^{I}_{\overline{z}}\;=\;-\mathrm{i}\,\mu^{I}\,.$ (41)

The AdS/CFT correspondence states that the trace (37) equals the following
bulk functional integral, up to a Casimir-energy-like term,

$Z_{\text{AdS}}(\tau,\,\mu)\;=\;\int D\phi_{\text{grav}}\;{\rm
e}^{-S_{\text{ren}}(\tau,\,\mu)}\,,$ (42)

where $\phi_{\text{grav}}$ denotes the gravitational fields of the theory, and

$S_{\text{ren}}\,\equiv\,S_{\text{bulk}}+S_{\text{bdry}}$ (43)

is the renormalized action of the gravitational theory. Here,
$S_{\text{bulk}}$ is the bulk Euclidean action of the Einstein-Hilbert-matter
theory and $S_{\text{bdry}}$ is the boundary action required to make the total
action finite and well-defined under our choice of boundary conditions. In
particular, it includes the Gibbons-Hawking boundary term, and a Chern-Simons
boundary term given by

$-\frac{\mathrm{i}}{8\pi}\,k_{IJ}\,\int
dz\,d\bar{z}\,\Big{[}{W}^{I}_{z}\,{W}^{J}_{\bar{z}}\Big{]}_{\text{bdry}}\,.$
(44)

This last term is required to ensure the consistency of the variational
principle of the gauge fields with the boundary conditions (40).

The semi-classical contribution: At leading order, the partition function (42)
is given by the value of $S_{\text{ren}}$ on the thermal AdS3 configuration
described above. The value of this action is Kraus:2006wn

$S_{\text{ren}}(\tau,\,\mu)\;=\;-\pi\tau_{2}k-\pi\tau_{2}\,k_{IJ}\,\mu^{I}\mu^{J}\,,$
(45)

where $6k=\frac{3(2\ell)}{2G_{3}}$ is the Brown-Henneaux central charge of the
gravitational theory for the AdS3 space (32), and $k_{IJ}$ is the level of the
Chern-Simons term (38). Here, note that the boundary $U(1)$ current obtained
from (38), (44) is right-moving. The choice of opposite relative sign between
(38) and (44) leads to the opposite chirality.

In the context of the five-dimensional theory of the previous section, recall
from the discussion around (14) that $G_{5}=6\pi\ell^{3}/p^{3}$, so that

$6k\;=\;\frac{3(2\ell)}{2G_{3}}\;=\;\frac{12\pi\ell^{3}}{G_{5}}\;=\;2p^{3}\,.$
(46)

To identify $k_{IJ}$, we reduce the five-dimensional Chern-Simons boundary
term in (72) onto the S2, and compare the resulting action with the three-
dimensional Chern-Simons boundary term (44) on thermal AdS3. This leads to:

$k_{IJ}\;=\;\frac{2}{3}c_{IJK}\,p^{K}\,.$ (47)

###  3.2 Twisted background and superalgebra

Now we consider the supersymmetric theory on the 5d geometry Under the Wick
rotation $t=-\mathrm{i}t_{E}$, the Lorentzian metric (13) rotates to that of
Euclidean $\mathbb{H}^{3}\times S^{2}$. The non-trivial fields in the Weyl
multiplet are:

$\displaystyle ds^{2}$ $\displaystyle\;=\;$ $\displaystyle
4\ell^{2}\,(\cosh^{2}\rho\,dt_{{E}}^{2}+d\rho^{2}+\sinh^{2}\rho\,d\psi^{2})+\ell^{2}\,(d\theta^{2}+\sin^{2}\theta\,d\phi^{2})\,,$
(48) $\displaystyle T_{\theta\phi}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\ell}{4}\,\sin\theta\,.$ (49)

If the Euclidean time coordinate $t_{E}$ runs from $(-\infty,\,\infty)$, the
topology is that of a solid cylinder times a sphere, which we call the
Euclidean cylinder frame. Although the Killing spinor equations (19) and (20)
are formally solved by the same set of eight spinors (23) in this background,
these spinors are no longer well-defined because they diverge at the ends of
the Euclidean cylinder. The solution to this problem involves compactifying
the Euclidean time on a circle and simultaneously rotating the S2 as we go
around the time circle. This twisted quotient makes for a well-defined
background, as we now describe.

We start from the configuration (48) describing an infinite solid cylinder
(times a sphere), and make the following identifications,

$(t_{E}\,,\psi\,,\phi)\;\sim\;(t_{E}\,,\psi+2\pi\,,\phi)\;\sim\;(t_{E}+2\pi\tau_{2}\,,\psi+2\pi\tau_{1}\,,\phi+\mathrm{i}2\pi\tau_{2}\Omega)\,.$
(50)

Equivalently, we can define a new set of “twisted” coordinates,

$t^{\prime}_{E}\;=\;t_{E}\,,\qquad\phi^{\prime}\;\equiv\;\phi-\mathrm{i}\Omega
t_{E}\,,$ (51)

which have the identification

$(t_{E}^{\prime}\,,\psi\,,\phi^{\prime})\;\sim\;(t_{E}^{\prime}\,,\psi+2\pi\,,\phi^{\prime})\;\sim\;(t_{E}^{\prime}+2\pi\tau_{2}\,,\psi+2\pi\tau_{1}\,,\phi^{\prime})\,.$
(52)

We denote the corresponding complex coordinates as
$z^{\prime}=\psi+\mathrm{i}t_{E}^{\prime}\,$,
$\bar{z}^{\prime}=\psi-\mathrm{i}t_{E}^{\prime}\,$, identified as
$(z^{\prime}\,,\bar{z}^{\prime})\;\sim\;(z^{\prime}+2\pi\tau\,,\bar{z}^{\prime}+2\pi\bar{\tau})$.

In the twisted frame, the on-shell background configuration is

$\begin{split}ds^{2}&\;=\;4\ell^{2}\left(\cosh^{2}\\!\rho\,dt^{\prime
2}_{E}+d\rho^{2}+\sinh^{2}\\!\rho\,d\psi^{2}\right)+\ell^{2}\Bigl{(}d\theta^{2}+\sin^{2}\theta\bigl{(}d\phi^{\prime}+\mathrm{i}\Omega
dt^{\prime}_{E}\bigr{)}^{2}\Bigr{)}\,,\\\
T_{\theta\phi^{\prime}}&\;=\;-\frac{\ell}{4}\,\sin\theta\,,\qquad T_{\theta
t_{E}^{\prime}}\;=\;-\mathrm{i}\frac{\ell}{4}\Omega\,\sin\theta\,,\\\
\sigma^{I}&\;=\;-\frac{p^{I}}{\ell}\,,\qquad
W^{I}_{{t_{E}}^{\prime}}\;=\;2\mu^{I}-\mathrm{i}\Omega\,p^{I}\cos{\theta}\,,\qquad
W^{I}_{\phi^{\prime}}\;=\;-p^{I}\cos\theta\,,\\\
A_{1}{}^{1}&\;=\;A_{2}{}^{2}\;=\;\sqrt{\frac{p^{3}}{3\ell^{3}}}\,.\end{split}$
(53)

The S2 in (53) is fibered over the time circle of AdS3, and we refer to this
configuration as the twisted torus background. We also note that in the
expression for $W^{I}_{{t_{E}}^{\prime}}$ we have introduced an arbitrary
constant $\mu^{I}$ which is allowed by the supersymmetry and equations of
motion, which we will interpret as the source of a $U(1)$ current in the
boundary CFT. The BPS equations may also allow $W^{I}_{\psi}$ to take a
constant value, but this constant is forced to be zero due to the
contractibility of the $\psi$-cycle.

To see that the twisted torus background (53) has well-defined supersymmetry,
we solve the Killing spinor equation from the variation of gravitino (177),
which is rewritten now as

$0\;=\;2{\cal
D}_{M}\varepsilon^{i}-\frac{\mathrm{i}}{4\ell}(3\gamma^{\hat{\theta}\hat{\phi}}\gamma_{M}-\gamma_{M}\gamma^{\hat{\theta}\hat{\phi}})\,\varepsilon^{i}\,.$
(54)

Here we use the following gamma matrices in the Euclidean theory, which follow
from the Wick rotation,

$\gamma_{\hat{t}_{E}}={\boldsymbol{\sigma}}_{3}\otimes{\boldsymbol{\tau}}_{3}\,,\quad\gamma_{\hat{\rho}}={\boldsymbol{\sigma}}_{1}\otimes{\boldsymbol{\tau}}_{3}\,,\quad\gamma_{\hat{\psi}}={\boldsymbol{\sigma}}_{2}\otimes{\boldsymbol{\tau}}_{3}\,,\quad\gamma_{\hat{\theta}}=\mathbb{I}\otimes{\boldsymbol{\tau}}_{1}\,,\quad\gamma_{\hat{\phi}}=\mathbb{I}\otimes{\boldsymbol{\tau}}_{2}\,,$
(55)

where we relate the $\sigma_{3}$ with the Lorentzian gamma matrix
${\boldsymbol{\sigma}}_{0}$ in (183) by
${\boldsymbol{\sigma}}_{3}\equiv-\mathrm{i}{\boldsymbol{\sigma}}_{0}$. We will
take the representation
$({\boldsymbol{\sigma}}_{3}\,,{\boldsymbol{\sigma}}_{1}\,,{\boldsymbol{\sigma}}_{2})=(-{\boldsymbol{\tau}}_{3}\,,{\boldsymbol{\tau}}_{1}\,,{\boldsymbol{\tau}}_{2})$
with the Pauli sigma matrix ${\boldsymbol{\tau}}_{a}$. Note that unlike the
case of global AdS${}_{3}\times$ S2 in the subsection 2.3, the Killing spinor
equation (54) does not split into the equations of AdS3 and S2. This is
because we have the following spin connections

$\omega_{t^{\prime}_{E}}^{12}=-\sinh\rho\,,~{}~{}~{}\omega_{t^{\prime}_{E}}^{45}=\mathrm{i}\Omega\cos\theta\,,~{}~{}~{}\omega_{\psi}^{23}=\cosh\rho\,,~{}~{}~{}\omega_{\phi^{\prime}}^{45}=\cos\theta\,,$
(56)

where there is mixing between AdS3 and S2 directions through the non-zero
twisting parameter $\Omega$.

The solution of Killing spinors can be easily found by following the twisting
construction. It is clear that the Euclidean continuation of the set of 8
Lorentzian Killing spinors (21), (23), followed by the coordinate
transformation (51) obeys the new Killing spinor equation. Upon setting the
parameter

$\Omega\;=\;1+\mathrm{i}\frac{\tau_{1}}{\tau_{2}}\,,$ (57)

the following999The choice $\Omega=-1-\mathrm{i}\frac{\tau_{1}}{\tau_{2}}$
also gives rise to a different 4 set of Killing spinors. 4 of the original 8
Killing spinors

$\varepsilon^{\,i}_{(1)}=(-\mathrm{i}\varepsilon_{+}^{~{}+}\,,\varepsilon_{-}^{~{}-})\,,\quad\,\varepsilon^{\,i}_{(2)}=(\varepsilon_{+}^{~{}+}\,,-\mathrm{i}\varepsilon_{-}^{~{}-})\,,\quad\,\varepsilon^{\,i}_{(3)}=\,-(\varepsilon_{-}^{~{}-}\,,\mathrm{i}\varepsilon_{+}^{~{}+})\,,\quad\,\varepsilon^{\,i}_{(4)}=\,-(\mathrm{i}\varepsilon_{-}^{~{}-}\,,\varepsilon_{+}^{~{}+})\,,$
(58)

where

$\begin{split}\varepsilon_{+}^{~{}+}&\;=\;\\!\sqrt{\frac{\ell}{2}}\;{{\rm
e}}^{\frac{1}{2}(1-\Omega)t^{\prime}_{E}+\frac{\mathrm{i}}{2}(\psi+\phi^{\prime})}\left(\\!\begin{array}[]{c}\cosh\frac{\rho}{2}\\\
-\sinh\frac{\rho}{2}\end{array}\right)\\!\otimes\\!\left(\begin{array}[]{c}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\end{array}\right)\,,\\\
\varepsilon_{-}^{~{}-}&\;=\;\\!\sqrt{\frac{\ell}{2}}\;{\rm
e}^{-\frac{1}{2}(1-\Omega)t^{\prime}_{E}-\frac{\mathrm{i}}{2}(\psi+\phi^{\prime})}\left(\\!\begin{array}[]{c}-\sinh\frac{\rho}{2}\\\
\cosh\frac{\rho}{2}\end{array}\right)\\!\otimes\\!\left(\\!\begin{array}[]{c}-\sin\frac{\theta}{2}\\\
\cos\frac{\theta}{2}\end{array}\right)\,,\end{split}$ (59)

respect the periodicity (52) (they are periodic around the non-conctractible
circle and anti-periodic around the contractible circle).

One could also directly solve the new Killing spinor equations (54). The only
non-trivial change compared to the untwisted case is the equation in the
$t^{\prime}_{E}$ direction,

$\displaystyle 0$ $\displaystyle\;=\;$
$\displaystyle\bigl{(}2\partial_{t^{\prime}_{E}}-\omega_{t^{\prime}_{E}}^{12}\gamma_{12}-\omega_{t^{\prime}_{E}}^{34}\gamma_{34}\bigr{)}\,\varepsilon_{\pm}{}^{\pm}-\frac{\mathrm{i}}{2\ell}E_{t^{\prime}_{E}}{}^{1}\gamma_{45}\gamma_{1}\,\varepsilon_{\pm}{}^{\pm}-\frac{\mathrm{i}}{\ell}E_{t^{\prime}_{E}}{}^{5}(\gamma_{45}\gamma_{5})\,\varepsilon_{\pm}{}^{\pm}\,.$
(60)

Comparing to the equation (54) that would be written in untwisted frame, The
difference with the equation in the untwisted frame is that
$2\partial_{t^{\prime}_{E}}$ acting on the Killing spinor (59) gives
$\pm(1-\Omega)$ instead of $\pm 1$. Also, the third and the last terms are
new. By the projection property along S2 direction of the Killing spinor
$(1\otimes{\rm
e}^{-\mathrm{i}\tau_{2}\theta}\tau_{3})\varepsilon_{\pm}{}^{\pm}=\pm\varepsilon_{\pm}{}^{\pm}$,
one can check that the effect of the third and the last term indeed cancels
the contribution of $\Omega$ in the time-derivative acting on the Killing
spinor.

Supersymmetry algebra:

The supercharges ${\cal Q}_{a}=\delta(\varepsilon^{i}_{(a)})$, with the
Killing spinors $\varepsilon^{i}_{(a)}$, $a=1\,,2\,,3\,,4$ defined in (58),
obey

$\\{{\cal Q}_{a}\,,{\cal
Q}_{b}\\}\;=\;-2\mathrm{i}\delta_{ab}(L_{0}-J^{3})\,,\qquad[L_{0}-J^{3}\,,{\cal
Q}_{a}]\;=\;0\,.$ (61)

Consider the following four supercharges $G^{i\alpha}_{\gamma}$,

$G_{+}^{++}\,\equiv\,\frac{\mathrm{i}{\cal Q}_{1}+{\cal
Q}_{2}}{2}\,,\quad\;G_{-}^{--}\,\equiv\,\frac{{\cal Q}_{1}+\mathrm{i}{\cal
Q}_{2}}{2}\,,\quad\;G_{-}^{+-}\,\equiv\,\frac{-{\cal Q}_{3}+\mathrm{i}{\cal
Q}_{4}}{2}\,,\quad\;G_{+}^{-+}\,\equiv\,\frac{\mathrm{i}{\cal Q}_{3}-{\cal
Q}_{4}}{2}\,,$ (62)

where $\gamma$ is the sign of the $L_{0}$ eigenvalue, $i$ is the doublet index
under the outer automorphism coming from the SU(2) R-symmetry of the
supergravity, and $\alpha$ is the doublet index under the SU(2) R-symmetry
arising from the isometry of the S2. They are charged under the bosonic
generators $L_{0}$ and $J^{3}$ as

$\left[L_{0}\,,G^{i\pm}_{\pm}\right]\;=\;\pm{\textstyle{\frac{1}{2}}}\,G^{i\pm}_{\pm}\,,\qquad\left[J^{3}\,,G^{i\pm}_{\pm}\right]\;=\;\pm{\textstyle{\frac{1}{2}}}\,G^{i\pm}_{\pm}\,,$
(63)

so that

$\bigl{[}L_{0}-J^{3}\,,G^{i\pm}_{\pm}\bigr{]}\;=\;0\,,$ (64)

and they obey the anticommutation relations

$\bigl{\\{}G^{+\pm}_{\pm}\,,G^{-\mp}_{\mp}\bigr{\\}}\;=\;\pm\left(L_{0}-J^{3}\right)\,,\qquad\bigl{\\{}G^{+\pm}_{\pm}\,,G^{-\pm}_{\pm}\bigr{\\}}\;=\;0\,.$
(65)

The above algebra (64), (65) forms a subalgebra of the global part of ${\cal
N}=4$ superconformal algebra in the NS sector given in Section 2.3. Note that
the subalgebra can also be thought of as the spectral flow101010The spectral
flow is taken on the charges with algebra in Guo:2022ifr  as ${\cal
L}_{n}\mapsto{\cal L}_{n}+2\eta J^{3}_{n}+\eta^{2}\frac{c}{6}\delta_{n,0}$,
${J^{3}_{m}\mapsto J^{3}_{m}+\eta\frac{c}{6}\delta_{n,0}}$,
$J^{\pm}_{m}\mapsto J^{\pm}_{m\pm 2\eta}$, ${\cal
G}^{\pm}_{\dot{A},r}\mapsto{\cal G}^{\pm}_{\dot{A},r\pm\eta}$., with parameter
$\eta=1/2$, to the following Ramond sector zero-modes as
$L_{0}-J^{3}+c/24\mapsto{{\cal L}}_{0}^{R}$, $G^{\pm\mp}_{\mp}\mapsto\mp{\cal
G}^{\mp}_{\mp,0}$, $G^{\pm\pm}_{\pm}\mapsto\pm{\cal G}^{\pm}_{\mp,0}$.

###  3.3 The trace interpretation and the semiclassical limit

We now interpret the functional integral on
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ as a trace on the Hilbert
space of the boundary CFT. First, a small change of conventions in this
section: we consider $(0,4)$ SCFT2 on the boundary, where supersymmetry
algebra acts on the right-moving sector, whose generators we denote with a bar
e.g. $\overline{L}_{0}$ and $\overline{J}^{\,3}$. This means we have to flip
the convention (regarding the bars) of the previous subsection so that we have
the following representation in the twisted coordinates,

${L}_{0}\;=\;\mathrm{i}\frac{1}{2}(\mathrm{i}\partial_{t^{\prime}_{E}}-\partial_{\psi}+\Omega\,\partial_{\phi^{\prime}})\,,\qquad\overline{L}_{0}\;=\;\mathrm{i}\frac{1}{2}(\mathrm{i}\partial_{t^{\prime}_{E}}+\partial_{\psi}+\Omega\,\partial_{\phi^{\prime}})\,,\qquad{\overline{J}}^{\,3}\;=\;\mathrm{i}\partial_{\phi^{\prime}}\,,$
(66)

with $\Omega=1+\mathrm{i}\tau_{1}/\tau_{2}$. Now consider the gravitational
functional integral $Z$ corresponding to the partition function on the twisted
torus (52), (53). Bosonic fields are periodic around both the cycles of the
torus, while fermionic fields are periodic around the $t^{\prime}_{E}$-circle
(which has periodicity $2\pi\tau_{2}$) and are anti-periodic around the
contractible $\psi$ circle (which has periodicity $2\pi$). In addition, we
have the chemical potential $2\pi\mathrm{i}\tau_{1}$ for the angular momentum
$\partial_{\psi}$ around the AdS3, and the chemical potentials $\mu^{I}$
coupling to $U(1)$ current(s) $q_{I}$. By the usual interpretation of the
functional integral, we have

$\displaystyle{\rm e}^{C(\tau,\mu)}\,Z(\tau,\mu)$ $\displaystyle\;=\;$
$\displaystyle{\rm
Tr}_{\text{NS}}\,(-1)^{F}\,\exp\bigl{(}2\pi\tau_{2}\,\partial_{t^{\prime}_{E}}+2\pi\tau_{1}\,\partial_{\psi}+\mu^{I}q_{I}\bigr{)}\,,$
(67) $\displaystyle\;=\;$ $\displaystyle{\rm
Tr}_{\text{NS}}\,(-1)^{F}\,\exp\bigl{(}-2\pi\tau_{2}\,(L_{0}+\overline{L}_{0}-\Omega\overline{J}^{\,3})+2\pi\mathrm{i}\tau_{1}\,({L}_{0}-\overline{L}_{0})+\mu^{I}q_{I}\bigr{)}\,,$
$\displaystyle\;=\;$ $\displaystyle{\rm
Tr}_{\text{NS}}\,(-1)^{F}\,{q}^{{L}_{0}}\,\overline{q}^{\overline{L}_{0}-\overline{J}^{\,3}}\,{\rm
e}^{\mu^{I}q_{I}}\,,$

with $q={\rm e}^{2\pi\mathrm{i}\tau}$, $\overline{q}={\rm
e}^{-2\pi\mathrm{i}\overline{\tau}}$, and $\tau=\tau_{1}+\mathrm{i}\tau_{2}$.
The Casimir-energy-type term $C(\tau,\mu)$ arises while relating the
functional integral to the Hamiltonian trace Kraus:2006wn .

We recognize the right-hand side of (67) as the elliptic genus. Indeed, from
the anticommutator (65) we see that the above trace can be written as

${\rm e}^{C(\tau,\mu)}\,Z(\tau,\mu)\;=\;{\rm
Tr}_{\text{NS}}\,(-1)^{F}\,{q}^{{L}_{0}}\,\overline{q}^{\mathrm{i}{\cal
Q}^{2}}\,{\rm e}^{\mu^{I}q_{I}}\,,$ (68)

where we have chosen one supercharge ${\cal Q}=\frac{1}{\sqrt{2}}{\cal
Q}_{1}=\frac{1}{\sqrt{2}}(G_{-}^{--}-\mathrm{i}G_{+}^{++})$ as in (61) and
(62). The pairing of all non-BPS modes with respect to the supercharge ${\cal
Q}$ enforces that the elliptic genus is a holomorphic function of $\tau$. Note
that there is no regularization scheme for the functional integral in which
the pre-factor $C(\tau,\mu)$ respects modular invariance and holomorphy. In
particular, if one chooses the pre-factor $C(\tau,\mu)$ to respect modular
invariance (and the gauge invariance associated to $\mu$), it suffers from a
holomorphic anomaly and cannot be purely holomorphic in $\tau$ Closset:2019ucb
.

On-shell action

Now that we have set up the twisted torus background, we can evaluate the
functional integral as explained in Section 3.1 for the 3d untwisted theory.
We have

$Z_{\text{AdS}}(\tau,\,\mu)\;=\;\int D\phi_{\text{grav}}\;{\rm
e}^{-S_{\text{ren}}(\tau,\,\mu)}\,,\qquad
S_{\text{ren}}\,\equiv\,S_{\text{bulk}}+S_{\text{bdry}}\,.$ (69)

as an integral over all field configurations $\phi_{\text{grav}}$ of the 5d
supergravity theory, with the renormalized action $S_{\text{ren}}$ of the
gravitational theory.

The bulk supergravity action (12) evaluated on the twisted torus (53) is

$\begin{split}S_{\text{bulk}}(\tau_{2},p,\mu)&\;=\;\frac{1}{8\pi^{2}}\int_{0}^{\rho_{0}}d\rho\int_{0}^{2\pi}d\psi\int_{0}^{\pi}d\theta\int_{0}^{2\pi}d\phi^{\prime}\int_{0}^{2\pi\tau_{2}}dt_{E}^{\prime}\,L_{\text{bulk}}\,,\\\
&\;=\;-\frac{\pi\tau_{2}}{3}p^{3}+\text{O}(e^{2\rho_{0}})\,.\end{split}$ (70)

The second term on the right-hand side denotes terms in the bulk action that
diverge when the radial cutoff $\rho_{0}\to\infty$, and is absorbed by
standard boundary terms.

The boundary terms in the action of the gauge fields behave essentially in the
same way as in the untwisted theory, but with slightly different details. In
the cylinder frame (48), the gauge fields $W_{z,\bar{z}}$ on the AdS3 factor
have the boundary conditions (40), while the components $W_{\theta,\phi}$ on
the S2 are fixed at the boundary. Twisting these boundary conditions using
(51) gives the boundary conditions for the gauge fields on the twisted torus:

$\delta{W}^{I\,(0)}_{z^{\prime}}\;=\;0\,,\qquad{W}^{I\,(0)}_{\bar{z}^{\prime}}\;\text{not
fixed,}\qquad\delta W^{I\,(0)}_{\theta,\phi^{\prime}}\;=\;0\,,$ (71)

where the $(0)$ indicates the boundary values in the large-$\rho$ expansion as
in (39). The Chern-Simons boundary action consistent with these boundary
conditions is:

$\displaystyle\begin{split}S_{\text{CS}}^{\text{bdry}}&=-c_{IJK}\frac{\mathrm{i}p^{I}}{48\pi^{2}}\int_{\partial{\cal
M}}dz^{\prime}d\bar{z}^{\prime}d\theta
d\phi^{\prime}\,\sin\theta\Big{[}(W_{z^{\prime}}^{J}-\frac{1}{2}\Omega
W^{J}_{\phi^{\prime}})W^{K}_{\bar{z}^{\prime}}\Big{]}_{\text{bdry}}\,.\end{split}$
(72)

which on the twisted torus (53) evaluates to111111Evaluating actions of this
type is more conviently done by transforming back from the
$(z^{\prime},\,\bar{z}^{\prime})$ to the $(\psi,\,t_{E}^{\prime})$ coordinates
where the ranges are as in (70).

$S_{CS}^{\text{bdry}}\;=\;-\frac{2\pi\tau_{2}}{3}c_{IJK}\,\mu^{I}\mu^{J}p^{K}\,.$
(73)

The boundary terms in the gravitational sector are more subtle. Recall that in
the standard treatment of AdS3 gravity, see e.g. Kraus:2006wn , the variation
of the boundary metric is fixed, but not that of its normal derivative. As is
well known, one requires the addition of a Gibbons-Hawking boundary term, in
order for the variational principle to be well-defined. In addition, one
requires additional local counterterms on the boundary to cancel the
divergences arising from the bulk action as in (70) as well as from the
Gibbons-Hawking term. These considerations are systematically summarized in
the procedure of holographic renormalization Bianchi:2001kw . As it turns out,
in the localization of the path integral for the elliptic genus, we need to
impose slightly different boundary conditions on the metric compared to the
standard ones, and correspondingly we have a different structure of boundary
terms. However, these differences are only relevant when the metric goes off-
shell, and do not change the on-shell background that we have discussed so
far. We postpone the details to a follow-up publication. Thus the value of the
full renormalized action on the twisted background (53) is

$S_{\text{ren}}\;=\;-\pi k\tau_{2}-\pi\tau_{2}\,k_{IJ}\,\mu^{I}\mu^{J}\,.$
(74)

Note that the twisting procedure only affects global properties and does not
change the Newton’s constant. Therefore the central charge continues to be
$c=6k=2p^{3}$ as in (46). Similarly, the level $k_{IJ}$ of the boundary
current algebra also does not change. To see this, note that the relation
between the twisted and cylinder-frame fields is:

$W_{z^{\prime}}\;=\;{W}_{z}+\frac{1}{2}\Omega\,W^{I}_{\phi}(\theta)\,,\qquad
W_{\bar{z}^{\prime}}\;=\;{W}_{\bar{z}}-\frac{1}{2}\Omega\,W^{I}_{\phi}(\theta)\,,\qquad
W_{\phi^{\prime}}=W_{\phi}(\theta)\,,$ (75)

where $W_{z,\bar{z}}$ only depends on the AdS3 coordinates
$(\rho,z,\bar{z})=(\rho,z^{\prime},\bar{z}^{\prime})$ while
$W^{I}_{\phi}=-p^{I}\cos\theta$. Substituting this into (72) gives:

$S_{\text{CS}}^{\text{bdry}}\;=\;-c_{IJK}\frac{\mathrm{i}p^{I}}{12\pi}\int_{\partial{\cal
M}}dz^{\prime}d\bar{z}^{\prime}\Big{[}W_{z}^{J}W^{K}_{\bar{z}}\Big{]}_{\text{bdry}}\,,$
(76)

which is the same as the 3d Chern-Simons boundary term (44), since the
integration ranges of $(z^{\prime},\,\bar{z}^{\prime})$ are the same as for
$(z,\bar{z})$. This shows that $k_{IJ}\;=\;\frac{2}{3}c_{IJK}\,p^{K}$ as in
the untwisted theory (47).

##  4 The Euclidean 4d/5d lift

In this section we present a formalism to obtain off-shell localization
solutions in 5d supergravity by lifting the localization manifold around
Euclidean AdS${}_{2}\,\times\,$S2. In particular, this allows us to obtain
localization solutions around the supersymmetric twisted torus
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ background presented in
(53).

We briefly recall the first step of the localization problem that the
formalism addresses. We define the localization supercharge ${\cal
Q}=\frac{1}{\sqrt{2}}{\cal
Q}_{1}=\frac{1}{\sqrt{2}}\delta(\varepsilon^{i}_{(1)})$ where the Killing
spinor $\varepsilon^{i}_{(1)}$ is given in (58). (Equivalently, ${\cal
Q}=\frac{1}{\sqrt{2}}(G_{-}^{--}-\mathrm{i}G_{+}^{++})$ in terms of the super-
Virasoro generators.) It obeys the algebra

${\cal
Q}^{2}\;=\;-\mathrm{i}(\overline{L}_{0}-\overline{J}^{\,3})\,,\qquad[\overline{L}_{0}-\overline{J}^{\,3}\,,{\cal
Q}]\;=\;0\,.$ (77)

We would like to study the space of solutions to the BPS equations given by
setting the supersymmetry variations generated by ${\cal Q}$ of all the
fermions (177) to zero.

The BPS equations form a system of matrix-valued partial differential
equations in terms of the bosonic fields of the theory. One systematic
approach to solve them, assuming no fermionic backgrounds, begins by forming
various Killing spinor bilinears Tod:1983pm ; Gauntlett:2002nw . The BPS
equations may then be expressed as a set of coupled first-order equations for
these tensor fields, which describe the bosonic background of the solution.
This approach was used in Gupta:2012cy ; Gupta:2019xac  to solve the off-shell
problem in the AdS${}_{2}\,\times\,$S2 (and S3) background. The general
solutions to the resulting equations are, however, typically difficult to
obtain, and we do not solve this problem of general classification in this
paper. Instead, we leverage what is already known about the localization
solutions in 4d supergravity around the Euclidean AdS${}_{2}\times$S2
background Dabholkar:2010uh ; Gupta:2012cy ; Gupta:2015gga , by lifting them
to five dimensions. This involves the KK lift of AdS${}_{2}\times$ S2 to
AdS${}_{3}\times$ S2, which we describe in Section 4.1. Note, however, that
while the 4d localization manifold has been determined completely, there may
be additional solutions in 5d that _do_ depend on the KK direction, and that
will therefore not emerge from the lift. We postpone the discussion of such
solutions to future work.

To lift the 4d localization solutions, we use the idea of the 4d/5d off-shell
connection of Banerjee:2011ts . However, as mentioned in the introduction,
implementing this idea is not straightforward for the following reasons.
Firstly, while the formalism in Banerjee:2011ts  was developed for Lorentzian
supergravities, our 4d/5d connection needs to be adapted to accommodate the
Euclidean supergravities in both four and five dimensions. A subtlety here, as
we will shortly see, is that the 4d Euclidean theory has a redundancy in the
choice of reality conditions and correspondingly a redundancy of
AdS${}_{2}\times$S2 backgrounds, which has no counterpart in the 5d theory.
Secondly, recall that the 4d/5d lift produces a five-dimensional background in
the Kaluza-Klein anstaz and so, in order to reach the five-dimensional theory
on the supersymmetric twisted torus
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ from the four-dimensional
theory on AdS${}_{2}\times$S2, we require a mapping of the twisted torus (53)
into the Kaluza-Klein frame of AdS${}_{3}\times$S2. In Section 4.1 we present
the mapping from the Kaluza-Klein frame to the cylinder frame. The twisted
frame can then easily be mapped to the cylinder frame (48) by the local
coordinate transformation (51). In Section 4.2 we review the 4d Euclidean
supergravity and the AdS${}_{2}\times$S2 background. In Section 4.3, we
present our construction of the Euclidean 4d/5d off-shell lift. Further, we
show that the redundancy of the 4d theory mentioned above can be absorbed into
the mapping parameter. We conclude the section by presenting the steps of
lifting the 4d off-shell solutions to the 5d twisted torus.

###  4.1 The Kaluza-Klein coordinate frame

In this subsection we map the cylinder frame to the Kaluza-Klein frame. This
mapping requires the local coordinate transformations as well as local Lorentz
transformations. After presenting the general mechanism, we find the specific
coordinate and Lorentz transformations, and the resulting background
configuration and supercharges for AdS${}_{3}\times$ S2 in the Kaluza-Klein
frame.

The general mechanism is as follows. Let $\\{\dot{M},\dot{N},\cdots\\}$ and
$\\{\dot{A},\dot{B},\cdots\\}$ be the spacetime and tangent indices,
respectively, in this Kaluza-Klein frame. The vielbein in the Euclidean
cylinder frame $E_{M}{}^{A}$ maps to the vielbein in the KK frame
$\dot{E}_{\dot{N}}{}^{\dot{A}}$ under a diffeomorphism together with some
local rotation $L_{{A}}{}^{\dot{A}}$ which acts on the frame as Scherk:1979zr

$E_{M}{}^{A}(x)\;=\;\frac{\partial\dot{x}^{\dot{N}}}{\partial
x^{{M}}}\dot{E}_{\dot{N}}{}^{\dot{A}}(\dot{x})L^{-1}_{~{}\dot{A}}{}^{{A}}\,.$
(78)

Correspondingly, the spin connection transforms as

$\omega_{{M}{A}}{}^{{B}}\;=\;\frac{\partial\dot{x}^{\dot{N}}}{\partial
x^{{M}}}\left(L_{{A}}{}^{\dot{A}}\dot{\omega}_{\dot{N}\dot{A}}{}^{\dot{B}}L^{-1}_{~{}\dot{B}}{}^{{B}}+(\partial_{N}L_{{A}}{}^{\dot{A}})\,L^{-1}_{~{}\dot{A}}{}^{{B}}\right)\,.$
(79)

Likewise, the remaining non-trivial background fields and the Killing spinors
are mapped into the KK frame using the same diffeomorphism and local rotation
$L_{{A}}{}^{\dot{A}}$, and a corresponding spinor rotation ${\cal L}$, as

$T_{{A}{B}}\;=\;L_{{A}}{}^{\dot{A}}L_{{B}}{}^{\dot{B}}\dot{T}_{\dot{A}\dot{B}}\,,\qquad
F_{{A}{B}}\;=\;L_{{A}}{}^{\dot{A}}L_{{B}}{}^{\dot{B}}\dot{F}_{\dot{A}\dot{B}}\,,\qquad\epsilon^{i}\;=\;{\cal
L}\,\dot{\varepsilon}^{j}\,,$ (80)

where $L_{{A}}{}^{\dot{A}}$ and ${\cal L}$ are related such that the gamma
matrix is preserved:

$L_{A}{}^{\dot{B}}\,{\cal L}\,\gamma_{\dot{B}}\,{\cal
L}^{-1}\;=\;\gamma_{A}\,.$ (81)

The diffeomorphism and local rotation in (78) should be chosen such that the
vielbein in KK frame $\dot{E}_{\dot{N}}{}^{\dot{A}}$ has the following
reduction ansatz. Decomposing the KK frame coordinate as
$x^{\dot{M}}=\\{x^{\mu},\,x^{5}\\}$ and $x^{\dot{A}}=\\{x^{a}\,,5\\}$, the
reduction ansatz of the vielbein is

${\dot{E}}_{\dot{M}}{}^{\dot{A}}\;=\;\Biggl{(}\,\begin{matrix}e_{\mu}{}^{a}&B_{\mu}\phi^{-1}\\\
0&\phi^{-1}\end{matrix}\,\Biggr{)}\,,\qquad\dot{E}_{\dot{A}}{}^{\dot{M}}\;=\;\Biggl{(}\,\begin{matrix}e_{a}{}^{\mu}&-e_{a}{}^{\mu}B_{\mu}\\\
0&\phi\end{matrix}\,\Biggr{)}\,,$ (82)

where all the fields in the KK frame are independent of the compactified
$x^{5}$ coordinate. Note that the KK ansatz (82) breaks the 5d diffeomorphisms
to 4d diffeomorphisms and a $U(1)_{\text{gauge}}$, and breaks the 5d local
rotation symmetry $O(5)$ to $O(4)\times\mathbb{Z}_{2}$. Using the
$\mathbb{Z}_{2}$ we can fix the $\phi$ to have a fixed sign, say, positive.
The vielbein (82) is equivalent to the following metric in the KK frame (with
$x^{5}\,\sim\,x^{5}+2\pi$),

$\dot{G}_{\dot{M}\dot{N}}\,dx^{\dot{M}}dx^{\dot{N}}\;=\;g_{\mu\nu}\,dx^{\mu}dx^{\nu}+\phi^{-2}(dx^{5}+B_{\mu}\,dx^{\mu})^{2}\,.$
(83)

We see from the (82) and (83) that the five-dimensional vielbein
$\dot{E}_{\dot{N}}{}^{\dot{A}}$ or metric $\dot{G}_{\dot{M}\dot{N}}$ are
related to the four-dimensional veilbein $e_{\mu}{}^{a}$ or metric
$g_{\mu\nu}$, a gauge field $B_{\mu}$ and a scalar $\phi$. The reduction
ansatz leads to the following reduction of the spin connection as

$\dot{\omega}_{\dot{A}}{}^{bc}\;=\;\Biggl{(}\,\begin{matrix}\omega_{a}{}^{bc}\\\
\frac{1}{2}\phi^{-1}F(B)^{bc}\end{matrix}\,\Biggr{)}\,\qquad\quad\dot{\omega}_{\dot{A}}{}^{b5}\;=\;\Biggl{(}\,\begin{matrix}-\frac{1}{2}\phi^{-1}F(B)_{a}{}^{b}\\\
-\phi^{-1}D^{b}\phi\end{matrix}\,\Biggr{)}\,.$ (84)

Here we see that the gauge field $B_{\mu}$ appears through its field strength
$F(B)_{ab}$ in four dimensions.

Now we find the coordinate transformations and the local rotation in (78) that
map the cylinder frame background in (48) to fit into the KK frame ansatz (82)
and (83). The cylinder frame coordinates $x^{M}$ and and the KK frame
coordinates $\dot{x}^{\dot{M}}$

$x^{M}\;=\;(\rho\,,\psi\,,\theta\,,\phi\,,t_{E})\,,~{}~{}~{}\dot{x}^{\dot{M}}\;=\;(\eta\,,\chi\,,\theta\,,\phi\,,x^{5})\,,$
(85)

are related as

$(\,\rho,\psi,t_{E})\;=\;\Bigl{(}\,\frac{\eta}{2},\,\chi+\mathrm{i}\frac{x^{5}}{2},\,\frac{x^{5}}{2}\,\Bigr{)}\quad\Leftrightarrow\quad(\eta\,,\chi\,,x^{5})\;=\;(2\rho\,,\psi-\mathrm{i}t_{E}\,,2t_{E})\,,$
(86)

with the coordinates $(\theta,\phi)$ remaining the same. Note that the global
conditions on the periodicities are not respected by this map (e.g. $x^{5}$ is
compact whereas $t_{E}$ is not). The corresponding local rotation matrix
$L_{A}{}^{\dot{A}}$ is given as a rotation in the $2-5$ plane (along
$\hat{\psi}$ and $\hat{t}_{E}$ direction) with angle
$\omega=-\mathrm{i}\eta/2$:

$L_{{1}}{}^{\dot{1}}=L_{{3}}{}^{\dot{3}}=L_{{4}}{}^{\dot{4}}\;=\;1\,,\qquad
L_{{2}}{}^{\dot{2}}=L_{{5}}{}^{\dot{5}}=\cosh\frac{\eta}{2}\,,\qquad
L_{{2}}{}^{\dot{5}}=-L_{{5}}{}^{\dot{2}}\;=\;\mathrm{i}\sinh\frac{\eta}{2}\,.$
(87)

In the exponential form, we have $L_{A}{}^{\dot{A}}=({\rm
e}^{\Omega})_{A}{}^{\dot{A}}$ , where the $2-5$ component of the matrix in the
exponent is $\Omega_{25}=-\Omega_{52}=-\omega=\mathrm{i}\eta/2$. 121212 The
rotation with angle $\omega$ is
$\exp\biggl{(}\omega\Bigl{(}\begin{smallmatrix}0&-1\\\
1&0\end{smallmatrix}\Bigr{)}\biggr{)}=\Bigl{(}\begin{smallmatrix}\cos\omega&-\sin\omega\\\
\sin\omega&\cos\omega\end{smallmatrix}\Bigr{)}$. Here, we take
$\omega=-\mathrm{i}\eta/2$ for the rotation in $2$–$5$ plane. Note that the
angle is imaginary, because the coordinate $x^{5}$ is Euclidean time. By the
relation (81), the corresponding spinor rotation is

$\begin{split}{\cal
L}&\;=\;\exp\left(\frac{1}{4}\Omega_{AB}\gamma^{AB}\right)=\exp\left(\frac{\mathrm{i}}{4}\eta\,\gamma_{25}\right)=\Biggl{(}\,\begin{matrix}\cosh\frac{\eta}{4}&-\sinh\frac{\eta}{4}\\\
-\sinh\frac{\eta}{4}&\cosh\frac{\eta}{4}\end{matrix}\,\Biggr{)}\otimes\mathbb{I}_{2}\,.\end{split}$
(88)

We note that although the spin connection in the cylinder frame has zero
component for $\omega_{\hat{\rho}}{}^{\hat{\psi}\hat{t}_{E}}$, as can be seen
in (181), the corresponding spin connection of KK frame mapped by (79),
$\dot{\omega}_{\hat{\eta}}{}^{\hat{\psi}\hat{t}_{E}}(=\dot{\omega}_{\dot{1}}{}^{\dot{2}\dot{5}})$,
is non-zero due to the contribution of the Lorentz transformation matrix in
the second term of (79). According to (84), this non-zero component gives the
non-zero value of the electric flux along AdS2 . This explains why there is
electric flux on AdS2 even though the AdS3 does not have any electric flux.

We now summarize the on-shell supersymmetric field configuration in the KK
coordinates. By using the transformations (86) (87) (88) on the background
Weyl multiplet as in (48) and matter multiplets as in (14) (16), we obtain the
following configuration:

$\displaystyle d{{s}_{5}}^{2}$
$\displaystyle\;=\;\ell^{2}\,(d\eta^{2}+\sinh^{2}{\eta}\,d\chi^{2}+d\theta^{2}+\sin^{2}{\theta}\,d\phi^{2})+\ell^{2}\,(dx^{5}+\mathrm{i}(\cosh{\eta}-1)\,d\chi)^{2}\,,$
$\displaystyle\dot{T}_{34}$ $\displaystyle\;=\;-\frac{1}{4\ell}\,,$ (89)
$\displaystyle\dot{\sigma}^{I}$
$\displaystyle\;=\;-\frac{p^{I}}{\ell}\,,~{}~{}~{}~{}~{}~{}~{}\dot{F}^{I}_{{\theta}{\phi}}\;=\;{p^{I}}\sin\theta\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{W}^{I}_{x^{5}}\;=\;\mu^{I}\,\,,$
$\displaystyle\dot{A}_{1}{}^{1}$
$\displaystyle\;=\;\dot{A}_{2}{}^{2}\;=\;\sqrt{\frac{p^{3}}{3\ell^{3}}}\,.$

We note that the background geometry has an S1 fibration over the four-
dimensional base, which is Euclidean AdS${}_{2}\times$S2. The angular
coordinate $\chi$ of Euclidean AdS2 has periodicity $2\pi$. By comparing the
metric with the KK ansatz (83), we identify the following values for the KK
one-form and scalar:

$B\;=\;\mathrm{i}(\cosh{\eta}-1)\,d\chi\quad,\quad\phi\;=\;\ell^{-1}\,.$ (90)

The background configuration given in (4.1) has well-defined supersymmetry. To
see that, we look for the Killing spinors. By the Eulidean continuation of the
Lorentzian Killing spinors (23) followed by the coordinate transformation (86)
and the Lorentz transformation (88) we obtain

$\begin{array}[]{ll}\dot{\varepsilon}^{\,i}_{(1)}\;=\;(-\mathrm{i}\dot{\varepsilon}_{+}^{~{}+}\,,\dot{\varepsilon}_{-}^{~{}-})\,,{}{}{}{}{}{}&\dot{\varepsilon}^{\,i}_{(2)}\;=\;(\dot{\varepsilon}_{+}^{~{}+}\,,-\mathrm{i}\dot{\varepsilon}_{-}^{~{}-})\,,\\\
\dot{\varepsilon}^{\,i}_{(3)}\;=\;(-\dot{\varepsilon}_{-}^{~{}-}\,,-\mathrm{i}\dot{\varepsilon}_{+}^{~{}+})\,,&\dot{\varepsilon}^{\,i}_{(4)}=(-\mathrm{i}\dot{\varepsilon}_{-}^{~{}-}\,,-\dot{\varepsilon}_{+}^{~{}+})\,,\\\
\dot{\tilde{\varepsilon}}^{\,i}_{(1)}\;=\;(\dot{\varepsilon}_{+}^{~{}-}\,,\mathrm{i}\dot{\varepsilon}_{-}^{~{}+})\,,&\dot{\tilde{\varepsilon}}^{\,i}_{(2)}\;=\;(\mathrm{i}\dot{\varepsilon}_{+}^{~{}-}\,,\dot{\varepsilon}_{-}^{~{}+})\,,\\\
\dot{\tilde{\varepsilon}}^{\,i}_{(3)}\;=\;(-\mathrm{i}\dot{\varepsilon}_{-}^{~{}+}\,,\dot{\varepsilon}_{+}^{~{}-})\,,&\dot{\tilde{\varepsilon}}^{\,i}_{(4)}\;=\;(\dot{\varepsilon}_{-}^{~{}+}\,,-\mathrm{i}\dot{\varepsilon}_{+}^{~{}-})\,.\end{array}$
(91)

where the spinors $\dot{\varepsilon}_{\pm}^{~{}\pm}$ and
$\dot{\varepsilon}_{\pm}^{~{}\mp}$ are

$\dot{\varepsilon}_{+}^{~{}+}\;=\;\\!\sqrt{\frac{\ell}{2}}\,{\rm
e}^{\frac{\mathrm{i}}{2}(\chi+\phi)}\Biggl{(}\\!\begin{array}[]{c}\cosh\frac{\eta}{2}\\\
-\sinh\frac{\eta}{2}\end{array}\Biggr{)}\\!\otimes\\!\Biggl{(}\begin{array}[]{c}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\end{array}\Biggr{)}\,,\quad\dot{\varepsilon}_{+}^{~{}-}\;=\;\\!\sqrt{\frac{\ell}{2}}\,{\rm
e}^{\frac{\mathrm{i}}{2}(\chi-\phi)}\Biggl{(}\\!\begin{array}[]{c}\cosh\frac{\eta}{2}\\\
-\sinh\frac{\eta}{2}\end{array}\Biggr{)}\\!\otimes\\!\Biggl{(}\\!\begin{array}[]{c}-\sin\frac{\theta}{2}\\\
\cos\frac{\theta}{2}\end{array}\Biggr{)}\,,$
$\dot{\varepsilon}_{-}^{~{}+}\;=\;\\!\sqrt{\frac{\ell}{2}}\,{\rm
e}^{-\frac{\mathrm{i}}{2}(\chi-\phi)}\Biggl{(}\\!\begin{array}[]{c}-\sinh\frac{\eta}{2}\\\
\cosh\frac{\eta}{2}\end{array}\Biggr{)}\\!\otimes\\!\Biggl{(}\begin{array}[]{c}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\end{array}\Biggr{)}\,,\quad\dot{\varepsilon}_{-}^{~{}-}\;=\;\\!\sqrt{\frac{\ell}{2}}\,{\rm
e}^{-\frac{\mathrm{i}}{2}(\chi+\phi)}\Biggl{(}\\!\begin{array}[]{c}-\sinh\frac{\eta}{2}\\\
\cosh\frac{\eta}{2}\end{array}\Biggr{)}\\!\otimes\\!\Biggl{(}\\!\begin{array}[]{c}-\sin\frac{\theta}{2}\\\
\cos\frac{\theta}{2}\end{array}\Biggr{)}\,.$ (92)

Note that they are well-defined with respect to the global structure of the
geometry (83) because they do not depend on the $x^{5}$ direction (the spinors
above are in fact precisely the four-dimensional Killing spinors on
AdS${}_{2}\times$ S2, as we spell out in Appendix 4). Note also that, as in
the twisted torus frame, we cannot impose any reality conditions on the
Euclidean spinors. This is because although they formally satisfy
$(\varepsilon^{i})^{\dagger}\mathrm{i}\gamma_{5}\;=\;\varepsilon_{ij}(\varepsilon^{j})^{T}C$,
which is the same symplectic-Majorana condition of the Lorentzian theory
(167), this condition is not compatible with the local rotation of the
Euclidean theory.

###  4.2 4d Euclidean supergravity and AdS${}_{2}\times$S2 background

The Kaluza-Klein formalism described in the previous subsection naturally
connects the 5d supergravity on the AdS${}_{3}\times$S2 background in KK
coordinates given in (4.1) to the 4d supergravity on an AdS${}_{2}\times$S2
background. In this subsection, we review the 4d Euclidean conformal
supergravity and the AdS${}_{2}\times$S2 background in more detail. In the 4d
Euclidean theory, there is a one-parameter redundancy for describing this
background that comes from the possible choice of reality condition for the
fermions. 4d ${\cal N}=2$ supergravity: For the 4d $\mathcal{N}=2$ Euclidean
conformal supergravity, we consider the Weyl multiplet, coupled to
$N_{\text{v}}+1$ vector multiplets and one hypermultiplet. One of the vector
multiplets and the single hypermultiplet act as the compensators to
consistently gauge-fix the dilatations of the off-shell theory. The fields of
the Weyl multiplet are

$\\{{e_{\mu}}^{a}\,,\,\psi^{i}_{a}\,,\,A_{\mu}^{D}\,,\,A_{\mu}^{R}\,,\,{{\cal
V}_{\mu}}^{i}{}_{j}\,,\,T_{ab}^{\pm}\,,\,{\cal D}\,,\,\chi_{4d}^{i}\\}\,,$
(93)

corresponding, respectively, to the vielbein, gravitino, dilatations gauge
field, the $SO(1,1)_{R}$ and $SU(2)_{R}$ gauge fields 131313The R-symmetry
group of the Euclidean theory is $SU(2)\times SO(1,1)$ compared to
$SU(2)\times U(1)$ in the Lorentzian case., auxiliary self-dual/anti-self-dual
two-form, auxiliary scalar, and the auxiliary fermion. As in the five-
dimensional case, we fix $A^{D}_{\mu}=0$ using the K-gauge. The fields of the
$N_{\text{v}}+1$ vector multiplets are

$\\{X^{\cal I}\,,\,\overline{X}^{\cal I}\,,\,A_{\mu}^{\cal
I}\,,\,\lambda^{{\cal I}\,i}\,,\,{\cal Y}^{{\cal I}\,ij}\\}\,,\qquad{\cal
I}\;=\;0\,,\,\cdots,\,N_{\text{v}}\,,$ (94)

corresponding to the complex scalar and its conjugate, the $U(1)$ gauge field,
the gaugino, and the auxiliary $SU(2)$ triplet. Finally, the hypermultiplet
consists of scalars and fermions,

$\\{{\cal A}_{i}{}^{\alpha}\,,\,\zeta_{4d}^{\alpha}\\}\,.$ (95)

The supersymmetry transformations on the spinor fields
$\psi^{i}_{a},\,\lambda^{{\cal I}\,i},\,\zeta_{4d}^{\alpha}$ are presented in
(206), following the conventions of Jeon:2018kec .

The 4d ${\cal N}=2$ supergravity is governed by the prepotential $F(X)$ which
is homogeneous of degree 2. Here, we choose the prepotential as
Banerjee:2011ts

$F(X)\;=\;-\frac{1}{12}\,c_{IJK}\frac{X^{I}X^{J}X^{K}}{X^{0}}\,,$ (96)

(the sum running over $I=1,\dots\,N_{\text{v}}$), such that the vector
multiplet sector of the 4d theory matches that of the 5d theory described in
the section 2.1, according to the 4d/5d map that we will present shortly in
Subsection 4.3.

Reality conditions: Note that in the Euclidean theory the fields $X^{\cal I}$
and $\overline{X}^{\cal I}$—and, more generally, fields related by complex
conjugation in the Lorentzian theory (e.g. $T^{+}_{ab}$ and $T^{-}_{ab}$)—are
independent in the Euclidean theory. In order to preserve the number of
degrees of freedom, we should impose reality conditions in the Euclidean
theory. This may be done by imposing an appropriate reality condition on the
spinors and using supersymmetry. The minimal spinors in the ${\cal N}=2$ four-
dimensional Euclidean theory can be chosen to obey the symplectic-Majorana
condition. We note that there are actually an infinite number of such
consistent conditions which, for any spinor $\psi^{i}$, are parametrized by a
real number $\alpha$ as

$(\psi^{i})^{\dagger}{\rm
e}^{\mathrm{i}\alpha\gamma_{5}}=\epsilon_{ij}(\psi^{i})^{T}C\,,\qquad\alpha\in\mathbb{R}\,.$
(97)

This infinite choice stems from the fact that the chiral and anti-chiral
spinors are independent in Euclidean 4d, and the symplectic-Majorana condition
for the chiral and anti-chiral spinors can be imposed with relatively
different phases. Two natural examples are:

$\alpha\;=\;\pi/2:\quad(\psi^{i})^{\dagger}\mathrm{i}\gamma_{5}=\epsilon_{ij}(\psi^{j})^{T}C\,,\qquad\quad\alpha=0:\quad(\psi^{i})^{\dagger}\;=\;\epsilon_{ij}(\psi^{j})^{T}C\,.$
(98)

A spinor satisfying the general reality condition (97) (which we denote by
$\psi^{i}(\alpha)$) is related to spinors satisfying (98)

$\psi^{i}(\alpha)\;=\;{\rm
e}^{\frac{\mathrm{i}}{2}\gamma_{5}(\alpha-\frac{\pi}{2})}\psi^{i}(\pi/2)\;=\;{\rm
e}^{\frac{\mathrm{i}}{2}\alpha\gamma_{5}}\psi^{i}(0)\,.$ (99)

Now, if we impose one such condition on all the spinors of the theory
(including the Killing spinors), then the consistency of the supersymmetry
transformations under this condition fixes specific reality conditions on the
bosonic fields. For the two examples above we have, respectively, the
following conditions for the relevant bosonic fields:

$\begin{split}\alpha\;=\;\pi/2:\qquad\bigl{(}T_{ab}^{\pm}\bigr{)}^{*}&\;=\;T_{ab}^{\pm}\;,\quad\quad\bigl{(}X^{\cal
I}\bigr{)}^{*}\;=\;X^{\cal I}\,,\qquad\bigl{(}\overline{X}^{\cal
I}\bigr{)}^{*}\;=\;\overline{X}^{\cal I}\,,\qquad\\\
\alpha\;=\;0:\qquad\bigl{(}T_{ab}^{\pm}\bigr{)}^{*}&\;=\;-T_{ab}^{\pm}\;,\quad\;\bigl{(}X^{\cal
I}\bigr{)}^{*}\;=\;-X^{\cal I}\,,\quad\,\bigl{(}\overline{X}^{\cal
I}\bigr{)}^{*}\;=\;-\overline{X}^{\cal I}\,.\qquad\end{split}$ (100)

However, note that imposing either reality condition in (100) causes the wrong
sign for the kinetic terms of the action, making path integral ill-defined. In
fact, this is the case for all bosonic reality conditions implied from
supersymmetry by (97). As was discussed in Section 2.1, the resolution is to
impose the standard reality condition on the bosonic fluctuations, e.g.
$(\delta X^{\cal I})^{\ast}=\delta\overline{X}^{\cal I}$, so that path
integral is well-defined, and to treat the fermion fluctuations $\psi^{1}$ and
$\psi^{2}$ as being independent. For the background, however, the effect of
the choice for $\alpha$ still remains: there is a one-parameter family of
Killing spinors that satisfy the reality condition (97), and the
supersymmetric bosonic background has a corresponding dependence on the choice
of $\alpha$ as we will shortly see below.

4d AdS${}_{2}\times$S2 background: Here we present the Euclidean
AdS${}_{2}\times$ S2 background, including the complete Weyl multiplet and
matter multiplets. This solution can be obtained by Wick rotation of the
Lorentzian AdS${}_{2}\times$ S2 solution, which carries both electric and
magnetic charges $\left(q_{{\cal I}},\,p^{{\cal I}}\right)$. The non-trivial
fields are:

$\begin{split}{ds_{4}}^{2}&\;=\;g_{\mu\nu}\,dx^{\mu}dx^{\nu}\;=\;\ell^{2}\left(d\eta^{2}+\sinh^{2}\eta\,d\chi^{2}+d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\,,\\\
T^{-}_{{1}{2}}&\;=\;-\mathrm{i}\,\omega\,,~{}~{}~{}~{}~{}~{}{T}^{+}_{{1}{2}}\;=\;-\mathrm{i}\,\overline{\omega}\,,~{}~{}~{}~{}~{}~{}T^{-}_{{3}{4}}\;=\;\mathrm{i}\,\omega\,,~{}~{}~{}~{}~{}~{}{T}^{+}_{{3}{4}}\;=\;-\mathrm{i}\,\overline{\omega}\\\
A^{{\cal I}}&\;=\;-\mathrm{i}\,e^{{\cal I}}(\cosh\eta-1)\,d\chi-p^{{\cal
I}}\cos\theta\,d\phi\,,\\\ {X^{{\cal I}}}&\;=\;\frac{\omega}{8}(e^{{\cal
I}}+\mathrm{i}p^{{\cal I}})\,,~{}~{}~{}~{}~{}\overline{X}^{{\cal
I}}\;=\;\frac{\overline{\omega}}{8}(e^{{\cal I}}-\mathrm{i}p^{{\cal I}})\,,\\\
{\mathcal{A}_{i}}^{\alpha}&\;=\;{a_{i}}^{\alpha}\;=\;\text{constant}\,,\end{split}$
(101)

By the field equation for the auxiliary scalar ${\cal D}$, the
${a_{i}}^{\alpha}$ are constrained to obey:

$\Omega_{\alpha\beta}\,\varepsilon^{ij}{a_{i}}^{\alpha}{a_{j}}^{\beta}\;=\;-4\mathrm{i}(F_{{\cal
I}}\overline{X}^{\cal I}-\overline{F}_{{\cal I}}X^{\cal I})\,.$ (102)

By the attractor equations Sahoo:2006rp , the electric field $e^{\cal I}$ is
related to the electric charge $q_{\cal I}$ as

$4\mathrm{i}\left(\overline{\omega}^{-1}\frac{\partial\overline{F}(\overline{X})}{\partial\overline{X}^{\cal
I}}-\omega^{-1}\frac{\partial{F}(X)}{\partial{X}^{\cal I}}\right)\;=\;q_{\cal
I}\,,$ (103)

and the two independent complex parameters $\omega$ and $\bar{\omega}$ (unlike
in the Lorentzian theory, they are not complex conjugate to each other) are
related to the length scale of the metric $\ell$ as

$\ell^{2}\;=\;\frac{16}{\omega\overline{\omega}}\,,$ (104)

which indeed scales consistently with Weyl weight $(-2)$ and $SO(1,1)_{R}$
weight $0$. Since the two complex parameters $\omega$ and $\bar{\omega}$ carry
opposite charges under the $SO(1,1)_{R}$ gauge symmetry, we can set their
magnitude to be same:

$|\omega|\;=\;|\overline{\omega}|\;=\;4/\ell\,.$ (105)

Note that to match our 5d set-up, we uphold the dilatational symmetry, which
is manifested here in the form of an arbitrary value for $\ell$ (one may break
the symmetry by fixing $\ell$ to $1$ for instance, as in Dabholkar:2010uh ).
The relation (104), (105) indicates that $\omega$ and $\bar{\omega}$ are now
formally conjugate to each other so that we can rewrite them using the
following parametrization:

$\omega(\alpha)\;=\;\frac{4}{\ell}e^{\mathrm{i}\alpha}\,,\qquad\overline{\omega}(\alpha)\;=\;\
\frac{4}{\ell}e^{-\mathrm{i}\alpha}\,,\qquad\alpha\in\mathbb{R}\,.$ (106)

Unlike in the 4d Lorentzian theory, where the phase $\alpha$ is fixed by the
$U(1)_{R}$ gauge symmetry, in the Euclidean theory it remains as a free
parameter. It is, in fact, precisely the parameter that determines the choice
of reality condition for the spinors as in (97), i.e. the background described
in (101) with generic $\alpha$ as in (106) preserves the supersymmetries
generated by Killing spinors obeying the reality condition (97). For the case
of $\alpha=\pi/2$ the 8 pairs of Killing spinors are presented in Appendix
(227) and the Killing spinors for a generic $\alpha$ can be read off from
(99). Note that the Killing spinors in (227) are exactly same Killing spinors
as those of the 5d KK frame given in (91).

Now, by comparing the 4d background (101) to the 5d KK-frame background (4.1),
it is clear that the AdS${}_{2}\times$ S2 metric in the former is the
reduction of the AdS${}_{3}\times$ S2 metric in the latter, as mentioned in
Section 4.1. However, it is not yet clear how the 4d/5d background values of
the other fields are related (beyond just the metric), and how off-shell
fluctuations are connected. In the next subsection, we will elucidate these
points by describing the full off-shell map between the Euclidean 4d and 5d
supergravity. Using this map, we will explicitly present how the 4d/5d
backgrounds are mapped.

###  4.3 The off-shell Euclidean 4d/5d lift

In this subsection, we describe the off-shell connection between the 4d
Euclidean and 5d Euclidean theory. We present how the AdS${}_{2}\times$ S2 on-
shell background in (101) maps to the AdS${}_{3}\times$ S2 on-shell background
in KK frame (4.1). This involves a choice of the relevant parameters of the 4d
background, specifically $(e^{0},p^{0})$ in (101), and depending on the choice
of parameter $\omega$ and $\bar{\omega}$ (101), a proper mapping parameter is
determined. We then show how to reach the 5d twisted torus background. We end
the section with the steps to lift off-shell localization solutions to the 5d
twisted torus.

To obtain the Euclidean 4d/5d connection, we use the Lorentzian 4d/5d
relations of Banerjee:2011ts  and map the two theories to their consistent
Euclidean counterparts. Getting the Euclidean 5d theory by the Wick rotation
is straightforward, as explained in Section 2.1. We follow the conventions of
the 4d Euclidean theory in Jeon:2018kec . Equivalently, one can start from the
relations between the 5d Lorentzian and 4d Euclidean theories of deWit:2017cle
, and Wick rotate the 5d theory. The map obtained in this approach differs
from ours only in the way that the conventions of the 4d Euclidean theory of
deWit:2017cle  differ from those of the 4d Euclidean theory of Jeon:2018kec
141414The mapping between these conventions is also presented in Jeon:2018kec
..

Under Kaluza-Klein reduction of the 5d conformal supergravity to 4d, the
vector multiplets $I=1,\dots,N_{\text{v}}$ reduce to the corresponding 4d
matter vector multiplets $\mathcal{I}=1,\dots,N_{\text{v}}$, and the Weyl
multiplet reduced to the 4d Weyl multiplet and the additional Kaluza-Klein
vector multiplet $\mathcal{I}=0$.

One can expect that the Kaluza-Klein scalar $\phi$ associated with the 5d
metric (82) falls into the scalar in the 4d Kaluza-Klein vector multiplet.
However, directly performing this reduction only gives one real scalar degree-
of-freedom, while there should be two real degree-of-freedom for the scalars
of the vector multiplet. Additionally, the 4d $SO(1,1)_{R}$ symmetry factor is
not realized in any of the multiplets. To recover the missing scalar d.o.f.,
an additional field $\varphi$ is introduced Banerjee:2011ts ; deWit:2017cle
to define the two 4d scalars in the KK vector multiplet as

$X^{0}=-\frac{\mathrm{i}}{2}{\rm
e}^{-\varphi}\phi\,,\qquad\overline{X}^{0}=\frac{\mathrm{i}}{2}{\rm
e}^{\varphi}\phi\,.$ (107)

The field $\varphi$ transforms locally under SO(1,1)R as

$\varphi\,\rightarrow\,\varphi+\Lambda^{0}\,,$ (108)

where $\Lambda^{0}$ is real. One can then consistently couple $\varphi$ to the
remaining 4d fields, so that the $SO(1,1)_{R}$ of the 4d theory is realized.

We now present the explicit 4d/5d mappings, up to quadratic order in the
fermions, keeping the general $\varphi$ dependence. The 4d Weyl multiplet is
related to the 5d Weyl multiplet as:

$\displaystyle e_{\mu}{}^{{a}}$ $\displaystyle\;=\;$
$\displaystyle\dot{E}_{\mu}{}^{{a}}\,,$ (109) $\displaystyle{\psi^{i}_{{a}}}$
$\displaystyle\;=\;$ $\displaystyle
e^{-\frac{1}{2}\varphi\gamma_{5}}\dot{\Psi}_{{a}}^{~{}i}\,,$ (110)
$\displaystyle A^{R}_{{a}}$ $\displaystyle\;=\;$
$\displaystyle-6\mathrm{i}\dot{T}_{{a}{5}}+{e_{a}}{}^{\mu}\partial_{\mu}\varphi$
(111) $\displaystyle{{\cal V}_{a}}^{i}{}_{j}$ $\displaystyle\;=\;$
$\displaystyle{\dot{V}_{a\,j}}{}^{i}\,,$ (112) $\displaystyle
T^{4d\pm}_{{a}{b}}$ $\displaystyle\;=\;$ $\displaystyle
e^{\pm\varphi}(24\,\dot{T}^{5d}_{{a}{b}}+\mathrm{i}\phi^{-1}\varepsilon_{{a}{b}{c}{d}}F(B)^{{c}{d}})^{\pm}\,,$
(113) $\displaystyle{\cal D}$ $\displaystyle\;=\;$ $\displaystyle
4\dot{D}+\frac{1}{4}\phi^{-1}e^{a\mu}{D}_{\mu}(e_{a}{}^{\nu}{D}_{\nu}\phi)+\frac{3}{32}\phi^{-2}F(B)^{ab}F(B)_{ab}$
$\displaystyle\quad\quad+\frac{3}{2}\dot{T}_{\dot{A}\dot{B}}\dot{T}^{\dot{A}\dot{B}}+\frac{1}{4}\phi^{2}\dot{V}_{x^{5}i}{}^{j}\dot{V}_{x^{5}j}{}^{i}\,,$
$\displaystyle\chi_{4d}^{i}$ $\displaystyle\;=\;$ $\displaystyle
8\dot{\chi}^{i}+\frac{1}{48}\gamma^{ab}F(B)_{ab}\dot{\Psi}^{i}_{x^{5}}-\frac{3\mathrm{i}}{4}\phi\dot{T}_{ab}\gamma^{5}\gamma^{ab}\dot{\Psi}_{x^{5}}^{i}$
$\displaystyle\quad\quad+\frac{1}{4}\phi^{-1}\gamma_{5}\not{{\cal
D}}(\phi^{2}\dot{\Psi}_{x^{5}}^{i})-\frac{1}{2}\phi^{2}{V_{x^{5}}}_{j}{}^{i}\dot{\Psi}_{x^{5}}^{j}-\frac{9}{4}\mathrm{i}\phi\dot{T}_{a5}\gamma^{a}\dot{\Psi}_{x^{5}}^{i}\,,$

where $\varepsilon_{{a}{b}{c}{d}}$ is the four-dimensional Levi-Civita symbol,
and the 4d conformally invariant D’Alembertian is
$e^{a\mu}D_{\mu}(e_{a}{}^{\nu}D_{\nu}\phi)=({\cal D}^{a}{\cal
D}_{a}+\frac{1}{6}R)\phi$, where $R$ is the 4d Ricci scalar and ${\cal D}_{a}$
is the 4d Lorentz and R-symmetry covariant derivative. The 4d supersymmetry
parameters are given in terms of the 5d supersymmetry parameters and 5d Weyl
multiplet fields as

$\displaystyle\varepsilon^{i}_{4d}$ $\displaystyle\;=\;$ $\displaystyle{\rm
e}^{-\frac{1}{2}\varphi\gamma_{5}}\dot{\varepsilon}^{i}\,,$ (116)
$\displaystyle\eta_{4d}^{i}$ $\displaystyle\;=\;$
$\displaystyle-\mathrm{i}\gamma_{5}{\rm
e}^{\frac{1}{2}\varphi\gamma_{5}}\left(\dot{\eta}^{i}-2\dot{T}_{{a}{5}}\gamma^{a}\gamma^{5}\dot{\varepsilon}^{i}+\frac{\mathrm{i}}{8}\phi^{-1}\gamma_{5}(F(B)_{{a}{b}}-4\mathrm{i}\phi\dot{T}_{{a}{b}}\gamma_{5})\gamma^{ab}\dot{\varepsilon}^{i}\right)\,.$
(117)

Moving on to the vector multiplets, the 4d KK vector multiplet fields in terms
of the 5d Weyl multiplet are:

$\displaystyle X^{0}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\mathrm{i}}{2}e^{-\varphi}\phi\,,~{}~{}~{}~{}\overline{X}^{0}\;=\;\frac{\mathrm{i}}{2}e^{\varphi}\phi\,,$
(118) $\displaystyle A^{0}_{{a}}$ $\displaystyle\;=\;$ $\displaystyle
e_{{a}}{}^{\mu}B_{\mu}\,,$ (119) $\displaystyle\lambda^{0\,i}$
$\displaystyle\;=\;$ $\displaystyle
e^{-\frac{1}{2}\varphi\gamma_{5}}\dot{\Psi}_{{5}}^{~{}i}\phi\,,$ (120)
$\displaystyle{\cal Y}^{0\,i}{}_{j}$ $\displaystyle=$
$\displaystyle\phi\,{\dot{V}_{5\,j}}{}^{i}\,,$ (121)

and the 4d matter vector multiplet fields in terms of the 5d vector multiplet
fields are:

$\displaystyle X^{I}$ $\displaystyle\;=\;$
$\displaystyle{\textstyle{\frac{1}{2}}}e^{-\varphi}(\sigma^{I}+\mathrm{i}\dot{W}^{I}_{{5}})\,,~{}~{}~{}\overline{X}^{I}\;=\;{\textstyle{\frac{1}{2}}}e^{\varphi}(\sigma^{I}-\mathrm{i}\dot{W}^{I}_{{5}})\,,$
(122) $\displaystyle A^{I}_{{a}}$ $\displaystyle\;=\;$
$\displaystyle\dot{W}^{I}_{{a}}\,,$ (123) $\displaystyle\lambda^{I\,i}$
$\displaystyle\;=\;$ $\displaystyle
e^{-\frac{1}{2}\varphi\gamma_{5}}\left(\dot{\Omega}^{I\,i}-\dot{W}^{I}_{{5}}\dot{\Psi}_{{5}}^{~{}i}\right)\,,$
(124) $\displaystyle{\cal Y}^{I\,i}{}_{j}$ $\displaystyle\;=\;$
$\displaystyle-2\left(Y^{I\,i}{}_{j}+{\textstyle{\frac{1}{2}}}\dot{W}^{I}_{{5}}{\dot{V}_{5\,j}}{}^{i}\right)\,.$
(125)

Finally, the 4d hypermultiplet in terms of the 5d hypermultiplet is

$\displaystyle{\cal
A}_{i}{}^{\alpha}\;=\;\phi^{-1/2}\dot{A}_{i}{}^{\alpha}\,.$ (126)

Using the above maps, the 4d supersymmetry transformation is obtained from the
5d supersymmetry transformation together with a 5d local rotation,

$\delta^{4d}\;=\;\delta^{5d}+\delta_{M}(\varepsilon)\,,\qquad\varepsilon_{5a}\;=\;-\varepsilon_{a5}\;=\;\overline{\dot{\varepsilon}}_{i}\gamma_{a}\Psi^{i}_{5}\,,$
(127)

where the rotation parameter $\varepsilon_{AB}$ is chosen to fix the gauge
$\dot{E}_{x^{5}}{}^{a}=\dot{E}_{5}{}^{\mu}=0$. We also need the supersymmetry
transformation rule of $\varphi$,

$\delta^{5d}\varphi\;=\;\overline{\dot{\varepsilon}}_{i}\dot{\Psi}^{i}_{5}\,.$
(128)

For the purpose of lifting the 4d configuration to 5d, we use the inverse map,
namely the 5d fields in terms of the 4d fields. The 5d Weyl multiplet fields
are given in terms of the 4d Weyl multiplet and 4d KK multiplet as:

$\displaystyle\dot{E}_{\mu}{}^{{a}}$ $\displaystyle\;=\;$
$\displaystyle{e}_{\mu}{}^{{a}}\,\,,\,\,\dot{E}_{\mu}{}^{{5}}\;=\;\phi^{-1}B_{\mu}\,\,,\,\,\,\dot{E}_{x^{5}}{}^{{5}}\;=\;\phi^{-1}\,,$
(129) $\displaystyle{\dot{\Psi}^{i}_{{a}}}$ $\displaystyle\;=\;$
$\displaystyle
e^{\frac{1}{2}\varphi\gamma_{5}}\psi_{{a}}^{i}\,\,,\,\,\,\dot{\Psi}_{{5}}^{~{}i}\;=\;\phi^{-1}e^{\frac{1}{2}\varphi\gamma_{5}}\lambda^{0\,i}\,,$
(130) $\displaystyle\dot{T}_{{a}{b}}$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{24}\left(e^{-\varphi}{T^{+}_{{a}{b}}}+e^{\varphi}T^{-}_{{a}{b}}-\mathrm{i}\phi^{-1}\varepsilon_{{a}{b}{c}{d}}\,F(B)^{{c}{d}}\right)\,,$
(131) $\displaystyle\dot{T}_{{a}{5}}$ $\displaystyle\;=\;$
$\displaystyle\frac{\mathrm{i}}{6}\left(A^{R}_{{a}}-e_{{a}}{}^{\mu}\partial_{\mu}\varphi\right)\,,$
(132) $\displaystyle{\dot{V}_{a\,j}}{}^{i}$ $\displaystyle\;=\;$
$\displaystyle{{\cal
V}_{a}}^{i}{}_{j}\,\,,\,\,\,{\dot{V}_{5\,j}}{}^{i}\;=\;\phi^{-1}{{\cal
Y}^{0\,i}{}_{j}}\,,$ (133) $\displaystyle\dot{D}$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{4}\Big{(}{\cal
D}-\frac{1}{4}\phi^{-1}e^{a\mu}{D}_{\mu}(e_{a}{}^{\nu}{D}_{\nu}\phi)-\frac{3}{32}\phi^{-2}F(B)^{ab}F(B)_{ab}$
(134)
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\frac{3}{2}\dot{T}_{\dot{A}\dot{B}}\dot{T}^{\dot{A}\dot{B}}-\frac{1}{4}\phi^{2}\dot{V}_{x^{5}i}{}^{j}\dot{V}_{x^{5}j}{}^{i}\Big{)}\,,$

where

$\phi\;=\;2\mathrm{i}e^{\varphi}X^{0}\;=\;-2\mathrm{i}e^{-\varphi}\overline{X}^{0}\,,\qquad\quad
B_{\mu}=A^{0}_{\mu}\,.$ (135)

The 5d supersymmetry parameters are:

$\displaystyle\dot{\varepsilon}^{i}$ $\displaystyle\;=\;$ $\displaystyle{\rm
e}^{\frac{1}{2}\varphi\gamma_{5}}\varepsilon_{4d}^{i}\,,$ (136)
$\displaystyle\dot{\eta}^{i}$ $\displaystyle\;=\;$
$\displaystyle\gamma_{5}\left(\mathrm{i}{\rm
e}^{-\frac{1}{2}\varphi\gamma_{5}}\eta_{4d}^{i}+2\dot{T}_{{a}{5}}\gamma^{a}\dot{\varepsilon}^{i}-\frac{\mathrm{i}}{8}\phi^{-1}(F(B)_{{a}{b}}-4\mathrm{i}\phi\dot{T}_{{a}{b}}\gamma_{5})\gamma^{ab}\dot{\varepsilon}^{i}\right)\,.$
(137)

The 5d vector multiplet is given in term of the 4d vector multiplet as:

$\displaystyle\dot{\sigma}^{I}$ $\displaystyle\;=\;$ $\displaystyle
e^{\varphi}X^{I}+e^{-\varphi}\overline{X}^{I}\,,$ (138)
$\displaystyle\dot{W}^{I}_{{a}}$ $\displaystyle\;=\;$ $\displaystyle
A^{I}_{{a}}\,,\quad\quad\dot{W}^{I}_{{5}}\;=\;-\mathrm{i}\left(e^{\varphi}X^{I}-e^{-\varphi}\overline{X}^{I}\right)\,,$
(139) $\displaystyle\dot{\Omega}^{I\,i}$ $\displaystyle\;=\;$ $\displaystyle
e^{\frac{1}{2}\varphi\gamma_{5}}\lambda^{i~{}I}+\dot{W}^{I}_{{5}}\dot{\Psi}_{{5}}^{~{}i},$
(140) $\displaystyle\dot{Y}^{I\,i}{}_{j}$ $\displaystyle\;=\;$
$\displaystyle-{\textstyle{\frac{1}{2}}}{\cal
Y}^{I\,i}{}_{j}-{\textstyle{\frac{1}{2}}}\dot{W}^{I}_{{5}}{\dot{V}_{5\,j}}{}^{i}\,.$
(141)

The 5d hyper scalar given in terms of the 4d hypermultiplet is

$\displaystyle\dot{A}_{i}{}^{\alpha}\;=\;\phi^{1/2}{\cal A}_{i}{}^{\alpha}\,.$
(142)

Mapping 4d/5d classical backgrounds: By the above 4d/5d map, the relation
between the 4d AdS${}_{2}\times$S2 backgrounds (101) and 5d AdS${}_{3}\times$
S2 background in (4.1) in KK coordinates becomes more manifest. One important
subtlety is about the choice of $\varphi$ in (135). In the case of the
Lorentzian 4d/5d connection, $\varphi$ is just a $U(1)_{R}$ gauge parameter
that fixes the gauge-redundant phase of $X^{0}$ and $\overline{X}^{0}$, making
the $\phi$ automatically real. However, in the Euclidean case, the 4d theory
has an $SO(1,1)_{R}$ gauge symmetry instead of $U(1)_{R}$, whereas the
background values for $X^{0}$ and $\overline{X}^{0}$ have a relative phase
coming from the choice of the parameter $\omega$ and $\bar{\omega}$ and value
of the charge $e^{0}$ and $p^{0}$. Therefore, unlike in the Lorentzian case,
the value of $\varphi$ is not a ‘gauge fixing’ to kill the phase of $X^{0}$
and $\overline{X}^{0}$, but rather a ‘choice’ to cancel the phase of $X^{0}$
and $\overline{X}^{0}$. (By the $SO(1,1)_{R}$ gauge redundancy and by the rule
(108), we shift the $\varphi$ to set the magnitude of $X^{0}$ and
$\overline{X}^{0}$ to be same.)

Recalling the background value of $X^{0}$ and $\overline{X}^{0}$ as given in
(101), where the $\omega$ and $\bar{\omega}$ are parametrized by $\alpha$ as
in (106), the value of the mapping parameter $\varphi$ is determined to be

$\varphi^{\pm}(\alpha\,,e^{0}\,,p^{0})=-\mathrm{i}\alpha\pm\mathrm{i}\frac{\pi}{2}-\mathrm{i}\arctan\Bigl{(}\frac{p^{0}}{e^{0}}\Bigr{)}\,,$
(143)

by the condition that $\phi$ be real. There remains an ambiguity of $\pm\pi/2$
that is related to an overall sign choice for $\phi$. We now consider specific
examples for two distinct choices of $(e^{0}\,,p^{0})$, keeping the choice of
$\alpha$ to be generic. These are:

$\begin{split}(1)\quad(e^{0},\,p^{0})&\;=\;(e^{0},\,0)\,,\qquad\varphi^{\pm}_{1}(\alpha)\;=\;-\mathrm{i}\alpha\pm\mathrm{i}\pi/2\,\Rightarrow\,\phi\;=\;\mp\frac{e^{0}}{\ell}\,,\\\
(2)\quad(e^{0},\,p^{0})&\;=\;(0,\,p^{0})\,,\qquad\varphi^{\pm}_{2}(\alpha)\;=\;-\mathrm{i}(\alpha+\pi/2)\pm\mathrm{i}\pi/2\Rightarrow\,\phi\;=\;\mp\frac{p^{0}}{\ell}\,.\end{split}$
(144)

Here we see that, by the mapping parameter $\varphi^{\pm}$, the background
value of the lifted 5d field $\phi$ is indeed real, but there is dependence on
the choice $\pm$. We note that for both cases in (144) and, more generally,
with any choice (143), all the lifted 5d fields are independent of the choice
of phase $\omega\equiv\exp(\mathrm{i}\alpha)$ in the 4d background (101).

The resulting 5d background fields are listed in Table 1. The 4d configuration
with $(e^{0},\,p^{0},\,\varphi)=(e^{0},\,0,\,\varphi_{1}^{\pm})$ as in (1)
lifts to an AdS${}_{3}\times$S2 background, while the one with
$(e^{0},\,p^{0},\,\varphi)=(0,\,p^{0},\,\varphi_{2}^{\pm})$ as in (2) lifts to
an AdS${}_{2}\times$S3 background. For the latter case, the localization
solutions were studied in Gupta:2019xac . In both cases, the choice of the
sign in $\varphi^{\pm}$ gives the opposite sign for the background values of
$\phi$, $\dot{T}_{\dot{A}\dot{B}},\dot{\sigma}$ and hyper norm $\dot{\chi}$.
At the level of the Killing spinor equation (that we review in Appendix C),
choosing either sign gives a set of Killing spinors corresponding,
respectively, to the right- or left-moving supercharges in terms of the 2d
chiral ${\cal N}=4$ super algebra.

| $\varphi=\varphi_{1}^{\pm}$
---|---
$(e^{0},\,p^{0})=(e^{0},\,0)$ |  $ds^{2}=$ AdS${}_{3}\times$S2 $\dot{T}_{34}=\mp 1/(4\ell)$ $\dot{\sigma}^{I}=\mp p^{I}/\ell\,,$ $\dot{F}^{I}_{34}=p^{I}/\ell^{2}$ $\dot{A}_{1,2}{}^{1,2}=\sqrt{\pm\frac{p^{3}}{3\ell^{3}}}$ $S_{\text{bulk}}=\frac{p^{3}}{12e^{0}}$

$\varphi=\varphi_{2}^{\pm}$ |
---|---
$ds^{2}=$ AdS${}_{2}\times$S3 $\dot{T}_{12}=\mp\mathrm{i}/(4\ell)$ $\dot{\sigma}^{I}=\pm e^{I}/\ell\,,$ $\dot{F}^{I}_{12}=-\mathrm{i}e^{I}/\ell^{2}$ $\dot{A}_{1,2}{}^{1,2}=\sqrt{\mp\frac{e^{3}}{3\ell^{3}}}$ $S_{\text{bulk}}=\frac{e^{3}}{12p^{0}}$ | $(e^{0},\,p^{0})=(0,\,p^{0})$

Table 1: The non-trivial 5d fields obtained by lifting the 4d backgrounds
(101) with $(\omega,\,\overline{\omega})$ as given in (106) and with different
choices for $(e^{0},\,p^{0})$ and $\varphi$. For the choice of
$(e^{0},\,p^{0})$ on the left and right panel, the 4d hyper scalar that is
lifted is determined by the ${\cal D}$-field equation constraint (102) as
${a_{1}{}^{1}=a_{2}{}^{2}={1}/{\ell}\sqrt{-{p^{3}}/{3e^{0}}}}$ and
${a_{1}{}^{1}=a_{2}{}^{2}={1}/{\ell}\sqrt{{e^{3}}/{3p^{0}}}}$ respectively. We
also include the value for the finite piece of the bulk action (12). The field
configurations on the right entry are solutions corresponding to the near-
horizon of the supersymmetric Euclidean 5d black hole. The field
configurations on the left entry are the Euclidean AdS${}_{3}\times$S2
solutions.

Now, for our problem, the full specification of parameters to lift the
Euclidean AdS${}_{2}\times$S2 backgrounds (101) to the 5d KK frame (4.1) is

$(e^{0},\,p^{0},\,\varphi)\;=\;(-1,\,0,\,\varphi^{+}_{1})\,,$ (145)

with identification $e^{I}=\mu^{I}$ and $\varphi^{+}_{1}$ given in (144). To
relate Euclidean AdS${}_{2}\times$S2 to the twisted torus (53), this lift is
then followed by the following steps: taking the lifted 5d KK frame background
(4.1) with $Q$\- Killing spinors (91), one applies the local coordinate
transformations (86), (51), the spinor Lorentz rotation in (80) with (88), and
finally one imposes the periodicity conditions (52) with $\Omega$ given in
(57). In this procedure, only four of the eight $Q$\- Killing spinors mapped
from (91) are well-defined on the twisted torus, as expected.

Mapping 4d localization solution to the 5d twisted torus frame: Having
identified the relevant 4d background, together with the correct mapping
parameter (145) that relates it to the 5d twisted torus background (53), we
now want to map the off-shell localization solution of 4d supergravity on that
background to the 5d localization solution around the twisted torus
background. The strategy for this mapping follows the same steps as the
mapping of the backgrounds presented above. Here, we assume that phase factors
in the quantum fluctuation of the scalars $X^{0}$ and $\overline{X}^{0}$ are
appropriately cancelled by a fluctuating value of $\varphi$ around its value
in (145), such that it makes the quantum fluctuation of the 5d field $\phi$
real.151515Since we choose the reality condition for the fluctuation of
$X^{0}$ and $\overline{X}^{0}$ to be complex conjugate to each other, as
explained after (100), and since this condition is the same as the condition
in the Lorentzian theory, it appears there may be some $U(1)_{R}$ gauge
symmetry hidden in the fluctuating field, and it may justify our assumption.
It will turn out that for our off-shell localization solution, we can use the
same value of $\varphi$ as was chosen in (145).

Here, we summarize the steps as follows:

  1. 1.

Start with the 4d localization manifold whose background is the Euclidean
AdS${}_{2}\times$S2 solution (101) with $(e^{0},\,p^{0})=(-1,0)$. Since the
result does not depend on the choice of $\alpha$ in (106), without loss of
generality we take $\alpha=\pi/2$ for convenience.

  2. 2.

Apply the 4d/5d lift with the mapping parameter
$\varphi=\varphi_{1}^{+}(\pi/2)=0$ to obtain 5d localization solutions in the
KK frame (4.1) of Euclidean AdS${}_{3}\times$S2.

  3. 3.

Transform these localization solutions to the twisted torus frame by applying
the local coordinate maps (86), (51), the spinor Lorentz rotation in (80) with
(88), and finally imposing the periodicity conditions (50) with $\Omega$ given
in (57).

Note that a consistent lift to the twisted torus requires that the lifted
solutions respect the periodicities (50). As an example of an inconsistent
lift, consider a scalar field fluctuation on AdS${}_{2}\times$S2 with non-zero
momentum on $\chi$, which therefore has $2\pi$-periodicity in $\chi$.
Recalling that $\chi=\psi-\mathrm{i}t_{E}$, we see that such a mode, lifted to
5d, does not respect the second periodicity condition in (50). As we discuss
in the next section, the fields in the four-dimensional localization manifold
depend only the radial coordinate $\eta=2\rho$ and therefore lift consistently
to the 5d twisted torus.

##  5 The lift of localization solutions on
AdS${}_{2}\\!\times\\!\text{S}^{2}$ to
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$

In this section, we apply the lifting procedure to obtain localization
solutions around the supersymmetric
$\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ background. We find a set
of solutions to the BPS equations parametrized by $N_{\text{v}}+1$ real
coordinates $C^{\cal I}$, ${\cal I}=0,\dots,N_{\text{v}}$. These coordinates
are inherited from the 4d localization manifold, where each $C^{{\cal I}}$
parametrizes the off-shell solution for the ${\cal I}^{th}$ vector multiplet.
In the 4d AdS${}_{2}\times$S2 problem, the boundary conditions fix all the
fields to their attractor values at infinity. The localization solution
consists of the scalar fields $X^{\cal I}$ going off-shell in the interior,
with a radially-decaying shape that is fixed by supersymmetry. The parameter
$C^{\cal I}$ labels the size of deviation at the origin. In 5d, the $C^{I}$,
$I=1,\,\cdots,\,N_{\text{v}}$ parametrize the size of the off-shell solution
in the vector multiplet, and $C^{0}$ parametrizes a certain excitation of the
Weyl multiplet. Here, we have an AdS${}_{3}\times$S2 background, where one leg
of the gauge field ($W_{{z^{\prime}}}$) is fixed at infinity to its on-shell
value while the other ($W_{\overline{z^{\prime}}}$) is free to fluctuate, as
we described in Section 3.3. The parameter $C^{I}$ labels the deviation of
both $W^{I}_{z^{\prime}}$ and $W^{I}_{\overline{z}^{\prime}}$ from their on-
shell value at the origin as well as the boundary fluctuation of
$W^{I}_{\overline{z}^{\prime}}$. The precise solutions are presented in
(151–154) for the Weyl multiplet, and in (158–161) for the vector multiplets.
The hypermultiplet also fluctuates, and the solution is given in (162).

4d localization solutions:

The most general solution in 4d around the AdS${}_{2}\times$S2 background is
parametrized by one real parameter in each vector multiplet and one real
parameter in the Weyl multiplet, before fixing the gauge for local scale
transformations Gupta:2012cy . The gauge can be chosen so that there is no
off-shell fluctuations in the Weyl multiplet Dabholkar:2010uh . The off-shell
solution in the vector multiplets takes the following form:

$\displaystyle X^{\mathcal{I}}$
$\displaystyle\;=\;\frac{\mathrm{i}}{2\ell}\,\left(e^{\mathcal{I}}+\mathrm{i}\,p^{\mathcal{I}}+\frac{C^{\mathcal{I}}}{\cosh\eta}\right)\quad,\quad\overline{X}^{\mathcal{I}}\;=\;-\frac{{\mathrm{i}}}{2\ell}\,\left(e^{\mathcal{I}}-\mathrm{i}\,p^{\mathcal{I}}+\frac{C^{\cal
I}}{\cosh\eta}\right)\,,$ (146) $\displaystyle A^{{\mathcal{I}}}$
$\displaystyle\;=\;-\mathrm{i}\,e^{\mathcal{I}}(\cosh\eta-1)\,d\chi-p^{\mathcal{I}}\cos\theta\,d\phi\,,$
(147) $\displaystyle{{\cal Y}^{{\cal I}\,1}}_{1}$ $\displaystyle\;=\;{\cal
Y}^{\cal I}_{12}\;=\;\frac{-C^{\mathcal{I}}}{\ell^{2}\cosh^{2}\eta}\,,$ (148)

where we use
$(\omega(\pi/2),\,\overline{\omega}(\pi/2))=(4\mathrm{i}/\ell,\,-4\mathrm{i}/\ell)$.
The $C^{\mathcal{I}}$ parametrize the off-shell fluctuations around the
background (101). Lift to the Weyl multiplet:

For the lift to the Weyl multiplet, the relevant fields of the 4d localization
solution (146) are those of the KK vector multiplet ${\mathcal{I}}=0$. Using
(135), we first obtain the off-shell values for the KK scalar and one-form:

$\phi\;=\;\frac{1}{\ell}\left(1-\frac{C^{0}}{\cosh\eta}\right)\,,\qquad
B_{\chi}\;=\;\mathrm{i}(\cosh\eta-1)\,.$ (149)

It is useful to define the function

$\phi(x)\,\mathrel{\mathop{:}}=\,1-\frac{C^{0}}{\cosh x}\,.$ (150)

Now, using the lifting equations (129 \- 134) with $(e^{0},\,p^{0})=(-1,\,0)$
and $\varphi=0$, we obtain the full Weyl multiplet configuration in the KK
frame. After applying the coordinate maps (86) and (51) to the twisted torus
frame, the non-trival fields are:

$\displaystyle{E_{M^{\prime}}}^{A}$ $\displaystyle\;=\;$
$\displaystyle\ell\begin{pmatrix}2&0&&0&&0&0\\\
0&\sinh\rho\left(1+\frac{1-C^{0}}{{\phi(2\rho)}}\right)&&0&&0&\mathrm{i}\cosh\rho\left(1-\frac{1-C^{0}}{{\phi(2\rho)}}\right)\\\
0&0&&1&&0&0\\\ 0&0&\,\,&0&\,\,&\sin\theta&0\\\
0&-\mathrm{i}\sinh\rho\left(1-\frac{1+C^{0}}{{\phi(2\rho)}}\right)&&0&&\mathrm{i}\Omega\sin\theta&\,\,\,\cosh\rho\left(1+\frac{1+C^{0}}{{\phi(2\rho)}}\right)\end{pmatrix}\,,$
(151) $\displaystyle T_{\theta\phi^{\prime}}$ $\displaystyle\;=\;$
$\displaystyle\ell\sin\theta\frac{1}{12}\left(\frac{1}{{\phi(2\rho)}}-4\right)\,,\qquad
T_{\theta{t_{E}}^{\prime}}\;=\;\mathrm{i}\ell\sin\theta\frac{\Omega}{12}\left(\frac{1}{{\phi(2\rho)}}-4\right)\,,$
(152) $\displaystyle{V}_{\psi}$ $\displaystyle\;=\;$
$\displaystyle\mathrm{i}\left(1-\frac{1}{\cosh
2\rho}\right)\frac{{\phi(2\rho)}-1}{\phi^{2}(2\rho)}{\boldsymbol{\tau}}_{3}\,,~{}~{}{V}_{t_{E}^{\prime}}\;=\;\left(1+\frac{1}{\cosh
2\rho}\right)\frac{{\phi(2\rho)}-1}{\phi^{2}(2\rho)}{\boldsymbol{\tau}}_{3}\,,$
(153) $\displaystyle D$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{8\ell^{2}{\phi(2\rho)}^{2}}\left((1-{\phi(2\rho)})\frac{\sinh^{2}2\rho}{\cosh^{2}2\rho}-\frac{2}{3}(1-{\phi(2\rho)})^{2}\right)\,.$
(154)

The line-element corresponding to the vielbein (151) is:

$\begin{split}ds^{2}\;=\;&4\ell^{2}d\rho^{2}+4\ell^{2}\left(\sinh^{2}\rho-\sinh^{4}\rho\left(\frac{1}{\phi^{2}(2\rho)}-1\right)\right)d\psi^{2}\\\
&+\ell^{2}\Bigl{(}d\theta^{2}+\sin^{2}\theta\bigl{(}d\phi^{\prime}+\mathrm{i}\Omega
dt^{\prime}_{E}\bigr{)}^{2}\Bigr{)}+2\mathrm{i}\ell^{2}\sinh^{2}2\rho\left(\frac{1}{\phi^{2}(2\rho)}-1\right)d\psi
d{t^{\prime}_{E}}\\\
&+4\ell^{2}\left(\cosh^{2}\rho+\cosh^{4}\rho\left(\frac{1}{\phi^{2}(2\rho)}-1\right)\right)d{t_{E}^{\prime}}^{2}\,.\end{split}$
(155)

Here, recall from Section 3.2 that $\Omega=1+\mathrm{i}\tau_{1}/\tau_{2}$ in
the twisted torus frame.

It remains to apply the lift to the $Q$\- and $S$-Killing spinors. In
principle, off-shell fluctuations in the bosonic fields of the Weyl multiplet
may induce off-shell fluctuations in the 5d Killing spinors such that the BPS
equations of the multiplet remain solved. Note however that the 4d Weyl
multiplet in the 4d localization solution does not fluctuate, and so the 4d
$Q$\- and $S$\- Killing spinors that we lift are just those of the 4d
background, namely the eight spinors $\varepsilon^{i}_{4d}(\pi/2)$, given
explicitly in (227), and $\eta^{i}_{4d}(\pi/2)=0$ (recall we have fixed
$\alpha=\pi/2$). Further note that the lifting equation (136) for the 5d $Q$\-
spinors only involves the 4d $Q$\- spinors (which are on-shell). We conclude
that the lift of the $Q$\- spinors is unchanged from the on-shell case, i.e.
we obtain, in the twisted torus frame, the four well-defined on-shell $Q$
-spinors $\varepsilon_{(a)}$, $a=1,2,3,4$, as given in (58). In contrast, the
lifting equation (137) of the $S$\- spinors $\eta_{4d}$ involves bosonic 5d
fields which do fluctuate. The 5d $S$\- spinors, which are zero on-shell,
therefore acquire a non-zero value off-shell. In the twisted frame, we obtain
four well-defined $S$\- spinors $\eta_{(a)}$, associated with the four $Q$\-
spinors $\varepsilon_{(a)}$. The one associated to the localization
supercharge $\varepsilon_{(1)}$ has value

$\displaystyle\eta^{1}_{(1)}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\frac{C^{0}}{\cosh\left(2\rho\right)}\,{\rm
e}^{\frac{\mathrm{i}}{2}(\psi+\phi^{\prime}+\mathrm{i}(\Omega-1)t_{E}^{\prime})}}{3\sqrt{2}\ell{\phi(2\rho)}}\begin{pmatrix}{\cos{\frac{\theta}{2}}\,\cosh{\frac{\rho}{2}}}\\\
-{\sin{\frac{\theta}{2}}\,\cosh{\frac{\rho}{2}}}\\\
-{\cos{\frac{\theta}{2}}\,\sinh{\frac{\rho}{2}}}\\\
{\sin{\frac{\theta}{2}}\,\sinh{\frac{\rho}{2}}}\end{pmatrix}\,,$ (156)
$\displaystyle\eta^{2}_{(1)}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\mathrm{i}\frac{C^{0}}{\cosh\left(2\rho\right)}\,{\rm
e}^{-\frac{\mathrm{i}}{2}(\psi+\phi^{\prime}+\mathrm{i}(\Omega-1)t_{E}^{\prime})}}{3\sqrt{2}\ell\,{\phi(2\rho)}}\begin{pmatrix}{\sin{\frac{\theta}{2}}\,\sinh{\frac{\rho}{2}}}\\\
{\cos{\frac{\theta}{2}}\,\sinh{\frac{\rho}{2}}}\\\
-{\sin{\frac{\theta}{2}}\,\cosh{\frac{\rho}{2}}}\\\
-{\cos{\frac{\theta}{2}}\,\cosh{\frac{\rho}{2}}}\end{pmatrix}\,.$ (157)

Lift of the vector multiplet:

The relevant 4d fields are those of (146) with ${\mathcal{I}}=I$. Using the
lifting equations (138 \- 141) followed by the coordinate transformations (86)
and (51), we obtain the following non-trivial fields of the vector multiplet
configuration in the twisted torus frame:

$\displaystyle\sigma^{I}$
$\displaystyle\;=\;-\frac{p^{I}}{\ell}\,,\qquad\qquad
W_{\phi^{\prime}}^{I}\;=\;-p^{I}\cos\theta\,,$ (158) $\displaystyle
W_{\psi}^{I}$
$\displaystyle\;=\;\frac{2\mathrm{i}\left({C^{I}/\mu^{I}}+C^{0}\right)\frac{\sinh^{2}(\rho)}{\cosh
2\rho}}{{\phi(2\rho)}}\mu^{I}\,,$ (159) $\displaystyle W^{I}_{t_{E}^{\prime}}$
$\displaystyle\;=\;-\mathrm{i}p^{I}\Omega\cos\theta+\frac{\frac{C^{I}/\mu^{I}-C^{0}}{\cosh
2\rho}+C^{I}/\mu^{I}+C^{0}+2}{{\phi(2\rho)}}\mu^{I}\,,$ (160) $\displaystyle
Y^{I}_{12}$
$\displaystyle\;=\;\frac{1}{2\ell^{2}{\phi(2\rho)}}\frac{C^{I}/\mu^{I}+C^{0}}{\cosh^{2}2\rho}\mu^{I}\,.$
(161)

Lift of the hypermultiplet:

Finally, the lift for the hypermultiplet (142) gives the following non-trivial
components for the off-shell hyper scalar:

${A}_{1}{}^{1}\;=\;{A}_{2}{}^{2}\;=\;\left(\frac{{\phi(2\rho)}}{\ell}\right)^{1/2}\sqrt{\frac{p^{3}}{3\ell^{3}}}\,.$
(162)

To summarize, the field configuration of the Weyl multiplet (151)–(154), the
vector multiplet (158)–(161), and the hypermultiplet (162) are the 5d
localization solutions. These configurations are off-shell fixed-points of the
variations generated by the supercharge ${\cal Q}$ given in (77), around the
supersymmetric $\mathbb{H}^{3}/\mathbb{Z}\\!\times\\!\text{S}^{2}$ given in
(53).

## Acknowledgements

We would like to thank Satoshi Nawata and Valentin Reys for useful discussions
and comments. This work is supported by the ERC Consolidator Grant N. 681908,
“Quantum black holes: A microscopic window into the microstructure of
gravity”, and by the STFC grant ST/P000258/1. A.C. is supported by the STFC
studentship ST/S505468/1. IJ is supported by an appointment to the JRG Program
at the APCTP through the Science and Technology Promotion Fund and Lottery
Fund of the Korean Government, by the Korean Local Governments -
Gyeongsangbuk-do Province and Pohang City, and by the National Research
Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No.
2021R1F1A1048531).

##  Appendix A Notations and conventions

We summarize the various index notations in Table 2.

Index | Range | Description
---|---|---
$M,\,N,\,\cdots$ | $(\rho,\,\psi,\,\theta,\,\phi,\,t_{E})$ | 5d cylinder coordinates
$M^{\prime},\,N^{\prime},\,\cdots$ | $(\rho,\,\psi,\,\theta,\,\phi^{\prime},\,t_{E}^{\prime})$ | 5d twisted torus coordinates
$A,\,B,\,\cdots$ | $(\hat{\rho},\,\hat{\psi},\,\hat{\theta},\,\hat{\phi},\,\hat{t_{E}})$ | tangent frame for cylinder and twisted torus coordinates
$\dot{M},\,\dot{N},\,\cdots$ | $(\eta,\,\chi,\,\theta,\,\phi,\,x^{5})$ | 5d Kaluza-Klein coordinates
$\dot{A},\,\dot{B},\,\cdots$ | $(1,\,2,\,3,\,4,\,5)$ | tangent frame for Kaluza-Klein coordinates
$\mu,\,\nu,\,\cdots$ | $(\eta,\,\chi,\,\theta,\,\phi)$ | Euclidean AdS${}_{2}\times$S2 coordinates
$a,\,b,\,\cdots$ | $(1,2,3,4)$ | tangent frame for Euclidean AdS${}_{2}\times$S2 coordinates
$\alpha,\,\beta,\,\cdots$ | $(z,\overline{z})$ | thermal AdS3 boundary coordinates
$i,\,j,\,\cdots$ |  $(1,\,2)$ or $(+,\,-)$ | Fundamental SU(2)
Table 2: Summary of index notation.

Spinors and gamma matrices:

We denote a basis for the the $d$-dimensional Clifford algebra as

$\left\\{\Gamma\;=\;\mathbb{I},\,\gamma^{A_{1}},\,\gamma^{A_{1}A_{2}},\cdots\gamma^{A_{1}A_{2}\cdots
A_{d}}\right\\}\,,$ (163)

where:

$\gamma^{A_{1}\cdots A_{k}}\;=\;\gamma^{[A_{1}}\cdots\gamma^{A_{k}]}\,.$ (164)

In five dimension with Lorentzian signature, a consistent choice of gamma
matrix satisfies the following relations:

$\begin{array}[]{llll}\gamma_{A}^{\dagger}\;=\;-A\gamma_{A}A^{-1}\,,{}&~{}A\;=\;\gamma_{0}\,,{}&~{}A^{\dagger}\;=\;A^{-1}\;=\;-\gamma_{0}\,,{}&\\\
\gamma_{A}^{T}\;=\;{\cal{C}}\gamma_{A}{\cal{C}}^{-1}\,,{}&~{}{\cal{C}}^{T}\;=\;-{\cal{C}}\,,{}&~{}{\cal{C}}^{\dagger}={\cal{C}}^{-1}\,,{}&\\\
\gamma_{A}^{*}\;=\;-B\gamma_{A}B^{-1}\,,{}&~{}B^{T}\;=\;{\cal{C}}A^{-1}\,,{}&~{}B^{\dagger}\;=\;B^{-1}\,,{}&~{}B^{*}B\;=\;-1\,.\end{array}$
(165)

This is followed by the property, regarding the charge conjugation matrix
${\cal C}$,

$({\cal{C}}\Gamma^{(r)})^{T}=-(-)^{r(r-1)/2}{\cal{C}}\Gamma^{(r)}\,,$ (166)

where $\Gamma^{(r)}$ is a matrix of the set (163) with rank $r$. Due to the
property of the charge conjugation matrix, we can use the spinor
representation satisfying the symplectic-Majorana condition

$\bar{\psi}_{i}\;=\;(\psi^{i})^{\dagger}\gamma_{0}\,,$ (167)

where $i$ is an SU(2)R index, and where $\bar{\psi}_{i}$ is the symplectic-
Majorana conjugate of $\psi^{i}$, defined as

$\bar{\psi}_{i}\mathrel{\mathop{:}}=\varepsilon_{ij}(\psi^{j})^{T}\mathcal{C}\,,$
(168)

with $\varepsilon_{ij}$ being the SU(2) symplectic metric
$\varepsilon_{12}=-\varepsilon_{21}=1$.

The five-dimensional Euclidean case is obtained by the Wick rotation of time
direction $x^{0}$, using the redefinition: $x^{0}=-\mathrm{i}x^{5}\,.$ This
consistently redefines the $0^{th}$ gamma matrix as the $5$-th directional one
as $\gamma_{0}=\mathrm{i}\gamma_{5}\,.$ The relations on the Lorentizan gamma
matrices (165) then become, for the Euclidean case:

$\begin{array}[]{lll}\gamma_{A}^{\dagger}\;=\;\gamma_{A}{}&{}\\\
\gamma_{A}^{*}\;=\;\gamma_{A}^{T}\;=\;{\cal{C}}\gamma_{A}{\cal{C}}^{-1}\,,{}&~{}{\cal{C}}^{\dagger}={\cal{C}}^{-1}\,,{}&~{}{\cal{C}}^{T}\;=\;-{\cal{C}}\,~{}\Leftrightarrow~{}{\cal{C}}^{*}{\cal{C}}\;=\;-1\,,\\\
\end{array}$ (169)

with charge-conjugation matrix property:

$({\cal{C}}\Gamma^{(r)})^{T}\;=\;-(-)^{r(r-1)/2}{\cal{C}}\Gamma^{(r)}\,.$
(170)

In the main text, we often consider Lorentz scalars of the type

$\bar{\lambda}_{i}\,\Gamma^{(r)}\epsilon^{j}\,.$ (171)

For two Grassman even spinors $\epsilon^{i}$, $\lambda^{j}$, the property
(170) leads to the following Majorana-flip relations:

$\bar{\lambda}_{i}\,\Gamma^{(r)}\epsilon^{j}\;=\;(-)^{r(r-1)/2}\left(\delta^{j}_{i}\,\bar{\epsilon}_{k}\,\Gamma^{(r)}\lambda^{k}-\bar{\epsilon}_{i}\,\Gamma^{(r)}\lambda^{j}\right)\,.$
(172)

Note some useful consequences of (172) for $\lambda=\epsilon$:

$\displaystyle\bar{\epsilon}_{i}\,\epsilon^{j}$
$\displaystyle\;=\;{\textstyle{\frac{1}{2}}}\left(\bar{\epsilon}_{k}\,\epsilon^{k}\right)\,\delta^{j}_{i}\,,$
(173) $\displaystyle\bar{\epsilon}_{i}\,\gamma^{A}\epsilon^{j}$
$\displaystyle\;=\;{\textstyle{\frac{1}{2}}}\left(\bar{\epsilon}_{k}\,\gamma^{A}\epsilon^{k}\right)\,\delta^{j}_{i}\,,$
(174) $\displaystyle\bar{\epsilon}_{k}\,\gamma^{AB}\epsilon^{k}$
$\displaystyle\;=\;0\,,\qquad\bar{\epsilon}_{k}\,\gamma^{ABC}\epsilon^{k}\;=\;0\,,$
(175)

where we used $r=0,1,2,3$ respectively. The spinors in the Euclidean theory
can also be chosen to be symplectic-Majorana, but differently from (167),
satisfying

$\bar{\psi}_{i}\;=\;(\psi^{i})^{\dagger}\,,$ (176)

with the same definition of the symmplectic Majorana conjugate
$\overline{\psi}_{i}$ as (168). However, we note that, as is commented in the
begining of section 2.1, we does not impose (176) for quantum theory.

##  Appendix B Supersymmetry transformations in Euclidean 5d supergravity

Weyl | ${E_{M}}^{A}\,,\,\Psi^{i}_{M}\,,\,b_{M}\,,\,{V_{M,\,i}}^{j}\,,\,T_{MN}\,,\,D\,,\,\chi^{i}$
---|---
Vector | $\sigma^{I}\,,\,W^{I}_{M}\,,\,\Omega^{I\,i}\,,\,Y^{I}_{ij}$
Hyper | $A_{i}{}^{\alpha}\,,\,\zeta^{\alpha}$
SUSY parameters | $\epsilon^{i}\,,\,\eta^{i}$
Table 3: Independent fields of the supersymmetric multiplets and $Q$,
$S$-supersymmetry parameters in five-dimensional $\mathcal{N}=1$ conformal
supergravity.

Up to higher order in fermions, the infinitesimal $Q$\- and $S$-supersymmetry
transformations on the spinor fields of the theory are as follows:

$\begin{split}\delta\Psi^{i}_{M}&\;=\;2\mathcal{D}_{M}\epsilon^{i}+\frac{i}{2}T_{AB}(3\gamma^{AB}\gamma_{M}-\gamma_{M}\gamma^{AB})\epsilon^{i}-\mathrm{i}\gamma_{M}\eta^{i}\,,\\\
\delta\chi^{i}&\;=\;{\textstyle{\frac{1}{2}}}\epsilon^{i}D+\frac{1}{64}R_{MNj}{}^{i}(V)\gamma^{MN}\epsilon^{j}+\frac{3\mathrm{i}}{64}(3\gamma^{AB}\gamma^{C}+\gamma^{C}\gamma^{AB})\epsilon^{i}{D}_{C}T_{AB}\\\
&\qquad-\frac{3}{16}T_{AB}T_{CD}\gamma^{ABCD}\epsilon^{i}+\frac{3}{16}T_{AB}\gamma^{AB}\eta^{i}\,,\\\
\delta\Omega^{i}&\;=\;-\frac{1}{2}(F_{AB}-4\sigma
T_{AB})\gamma^{AB}\epsilon^{i}-\mathrm{i}\gamma^{A}\epsilon^{i}{D}_{A}\sigma-2\varepsilon_{jk}Y^{ij}\epsilon^{k}+\sigma\eta^{i}\,,\\\
\delta\zeta^{\alpha}&\;=\;-\mathrm{i}\gamma^{A}\epsilon^{i}{D}_{A}{A_{i}}^{\alpha}+\frac{3}{2}{A_{i}}^{\alpha}\eta^{i}\,.\end{split}$
(177)

where the curvature $R_{MNi}{}^{j}(V)$ is given by:

$R_{MNi}{}^{j}(V)\;=\;2\,\partial_{[M}{{{{V_{N]}}}}_{i}}^{j}-2{{V_{[M}}_{i}}^{k}{{V_{N]}}_{k}}^{j}\,.$
(178)

The relevant supercovariant derivatives acting on each field are covariant
with respect to all bosonic gauge symmetries except conformal boosts:

$\begin{split}{D}_{M}\,\epsilon^{i}&\;=\;\left(\partial_{M}-\frac{1}{4}\,{\omega_{M}}^{AB}\,\gamma_{AB}+\frac{1}{2}b_{M}\right)\epsilon^{i}+\frac{1}{2}{{V_{M}}_{j}}^{i}\,\epsilon^{j}\,,\\\
D_{M}T_{AB}&\;=\;(\partial_{M}-b_{M})T_{AB}-\omega_{MA}{}^{C}T_{CB}-\omega_{MB}{}^{C}T_{AC}\,,\\\
D_{M}\,\sigma^{I}&\;=\;\left(\partial_{M}-b_{M}\right)\,\sigma^{I}\,,\\\
D_{M}\,{A_{i}}^{\alpha}&\;=\;\left(\partial_{M}-\frac{3}{2}b_{M}\right){A_{i}}^{\alpha}-\frac{1}{2}{{V}_{Mi}{}^{j}}{A_{j}}^{\alpha}\,.\end{split}$
(179)

##  Appendix C Killing spinors on AdS3 and S2

In this appendix, we present solution of the Killing spinor equation (19) on
the AdS${}_{3}\times$S2 background given in (13) and (15). Here, let us
decompose the spacetime and local indices into those for $3+2$ dimensions as
$M=\\{\mu\,,\text{m}\\}$ and $A=\\{a\,,{\text{a}}\\}$. Then the Killing spinor
equation (19) splits as

${\cal
D}_{\mu}\epsilon^{i}\;=\;s\frac{\mathrm{i}}{4\ell}\gamma^{\hat{\theta}\hat{\phi}}\gamma_{\mu}\epsilon^{i}\,,\quad\quad{\cal
D}_{\text{m}}\epsilon^{i}=s\frac{\mathrm{i}}{2\ell}\gamma^{\hat{\theta}\hat{\phi}}\gamma_{\text{m}}\epsilon^{i}\,,$
(180)

where we inserted the sign factor $s=\pm 1$ to keep track of the choice of the
background value of $T_{MN}$; $s=+1$ is for our background value of $T_{MN}$
in (15), and $s=-1$ is for another background value by changing
$T_{MN}\rightarrow-T_{MN}$ from the (15) (which involves changing
$\sigma\rightarrow-\sigma$ from (14) by the BPS equation of vector multiplet).
Note that, since the background metric (13) is direct product of 3 and 2
dimensions, the spin connection is also well separated as
$-\frac{1}{4}\omega_{\mu}^{AB}\gamma_{AB}=-\frac{1}{4}\omega_{\mu}^{ab}\gamma_{ab}$
and
$-\frac{1}{4}\omega_{\text{m}}^{AB}\gamma_{AB}=-\frac{1}{4}\omega_{\text{m}}^{{\text{a}}{\text{b}}}\gamma_{{\text{a}}{\text{b}}}$.
This can be seen explicitly by noting that the non-zero spin connection
components are

$\begin{array}[]{lll}\omega_{t}^{\hat{t}\hat{\rho}}=-\sinh\rho\,,&\omega_{\psi}^{\hat{\rho}\hat{\psi}}=\cosh\rho\,,&\omega_{\phi}^{\hat{\theta}\hat{\phi}}=\cos\theta\,.\end{array}$
(181)

We now decompose the spinor as

$\epsilon^{i}\;=\;\epsilon^{i}_{AdS_{3}}\otimes\epsilon^{i}_{S^{2}}\,,$ (182)

and take the following decomposition for the gamma matrices

$\begin{array}[]{llll}\gamma_{\hat{t}}\;=\;{\boldsymbol{\sigma}}_{0}\otimes{\boldsymbol{\tau}}_{3}\,,&\quad\gamma_{\hat{\rho}}\;=\;{\boldsymbol{\sigma}}_{1}\otimes{\boldsymbol{\tau}}_{3}\,,&\quad\gamma_{\hat{\psi}}\;=\;{\boldsymbol{\sigma}}_{2}\otimes{\boldsymbol{\tau}}_{3}\,,&\quad\gamma_{\hat{\theta}}\;=\;\mathbb{I}\otimes{\boldsymbol{\tau}}_{1}\,,\quad\gamma_{\hat{\phi}}=\mathbb{I}\otimes{\boldsymbol{\tau}}_{2}\,,\end{array}$
(183)

where ${\boldsymbol{\tau}}_{a}$, $a=1,2,3$, denotes the Pauli sigma matrix and
${\boldsymbol{\sigma}}_{a}$ with $a=0,1,2$ denotes the 3 dimensional gamma
matrix. Here we choose
${\boldsymbol{\sigma}}_{0}=-{\boldsymbol{\sigma}}_{1}{\boldsymbol{\sigma}}_{2}$
such that
$\gamma_{\hat{t}\hat{\rho}\hat{\psi}\hat{\theta}\hat{\phi}}\;=\;\mathrm{i}\,$
for our convention. The charge conjugation matrix can also be set to

${\cal
C}\;=\;-\mathrm{i}{\boldsymbol{\sigma}}_{2}\otimes{\boldsymbol{\tau}}_{1}\,,$
(184)

such that the gamma matrix relation (165) is satisfied. With this splitting of
spinors and gamma matrices, we arrive at the Killing spinor equations for AdS3
and S2 with radii $2\ell$ and $\ell$ respectively :

$0\;=\;\left({\cal
D}_{\mu}\epsilon^{i}_{AdS_{3}}+s\frac{1}{4\ell}{\boldsymbol{\sigma}}_{\mu}\epsilon^{i}_{AdS_{3}}\right)\otimes\epsilon^{i}_{S^{2}}\quad,\quad
0=\epsilon^{i}_{AdS_{3}}\otimes\left({\cal
D}_{\text{m}}\epsilon^{i}_{S^{2}}+s\frac{1}{2\ell}{\boldsymbol{\tau}}_{3}{\boldsymbol{\tau}}_{\text{m}}\epsilon^{i}_{S^{2}}\right)\,.$
(185)

The general solutions of these equations are well known Lu:1998nu , and the
solutions are given by

$\displaystyle\epsilon_{AdS_{3}}$ $\displaystyle\;=\;$ $\displaystyle{\rm
e}^{-s\frac{1}{2}{\boldsymbol{\sigma}}_{1}\rho}{\rm
e}^{-s\frac{1}{2}{\boldsymbol{\sigma}}_{0}t}{{\rm
e}}^{\frac{1}{2}{\boldsymbol{\sigma}}_{12}\psi}A\,,$ (186)
$\displaystyle\epsilon_{S^{2}}$ $\displaystyle\;=\;$ $\displaystyle{\rm
e}^{-s\mathrm{i}\frac{1}{2}{\boldsymbol{\tau}}_{2}\theta}{\rm
e}^{\mathrm{i}\frac{1}{2}{\boldsymbol{\tau}}_{3}\phi}B\,,$ (187)

where $A$ and $B$ are constant two-component complex spinors.

Let us write down the Killing spinor explicitly. We set the sign factor $s=1$,
denote the chiral and anti-chiral component of the constant spinors as
$A_{\pm}$ and $B_{\pm}$, and choose the 3 dimensional gamma matrix
representation as

${\boldsymbol{\sigma}}_{a}\;=\;(-\mathrm{i}\,{\boldsymbol{\tau}}_{3}\,,{\boldsymbol{\tau}}_{1}\,,{\boldsymbol{\tau}}_{2})\,.$
(188)

Then we can rewrite the solutions as

$\displaystyle\epsilon_{\text{AdS}_{3}}$ $\displaystyle\;=\;$ $\displaystyle
A_{+}\epsilon^{+}_{\text{AdS}}+A_{-}\epsilon^{-}_{\text{AdS}}\,,~{}~{}~{}~{}\quad\epsilon_{\text{S}^{2}}\;=\;B_{+}\epsilon^{+}_{\text{S}^{2}}+B_{-}\epsilon^{-}_{\text{S}^{2}}\,,$
(189)

where

$\displaystyle\epsilon^{+}_{\text{AdS}_{3}}\;=\;{\rm
e}^{\frac{\mathrm{i}}{2}(t+\psi)}\Biggl{(}\begin{array}[]{c}\cosh\frac{\rho}{2}\\\
-\sinh\frac{\rho}{2}\end{array}\Biggr{)}\,,~{}~{}~{}~{}~{}\epsilon^{-}_{\text{AdS}_{3}}\;=\;{\rm
e}^{-\frac{\mathrm{i}}{2}(t+\psi)}\Biggl{(}\begin{array}[]{c}-\sinh\frac{\rho}{2}\\\
\cosh\frac{\rho}{2}\end{array}\Biggr{)}\,,$ (194)
$\displaystyle\epsilon^{+}_{\text{S}^{2}}\;=\;{\rm
e}^{\frac{\mathrm{i}}{2}\phi}\Biggl{(}\begin{array}[]{c}\cos\frac{\theta}{2}\\\
\sin\frac{\theta}{2}\end{array}\Biggr{)}\,,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\epsilon^{-}_{\text{S}^{2}}\;=\;{\rm
e}^{-\frac{\mathrm{i}}{2}\phi}\Biggl{(}\begin{array}[]{c}-\sin\frac{\theta}{2}\\\