Vector + Chiral + (vector + hyper)${}_{\text{matter}}$.

The final conclusion from this subsection is that in a formalism where we have
off-shell supersymmetry with one complex supercharge and arrange the fields in
a manifestly $\mathcal{N}=2$ covariant manner, the spin-$\frac{3}{2}$
multiplet should be gauge-fixed using 2 $\mathcal{N}=2$ vectors + 1
$\mathcal{N}=2$ hyper multiplet in order to obtain an on-shell
spin-$\frac{3}{2}$ multiplet of Poincaré supergravity.

In the following subsection we perform a consistency check of this conclusion:
we calculate the index of all $\mathcal{N}=2$ supermultiplets in both the
superconformal theory and in the super-Poincaré theory. We will show that the
resulting one-loop determinants indeed agree with past one-loop calculations,
not around the localization locus, but rather around the on-shell geometries,
in a limit where we can do a consistent comparison.

####  4.4.2 The $Q_{\text{eq}}$-complex and the index of the $\mathcal{N}=2$
supermultiplets

In this subsection we decompose the $\mathcal{N}=4$ Weyl multiplet into
$Q_{\text{eq}}$-elementary and descendant fields (the
$Q_{\text{eq}}$-complex), using the procedure developed in [38]. Once we have
achieved this we use the separation of the $\mathcal{N}=4$ Weyl multiplet into
$\mathcal{N}=2$ multiplets as described in the previous subsection, in order
to read off the elementary and descendant fields of the $\mathcal{N}=2$
spin-$\frac{3}{2}$ multiplet. This then leads us to the calculation of the
index of the spin-$\frac{3}{2}$ multiplet.

The main ideas of the construction of the $Q_{\text{eq}}$-complex were
sketched in the previous subsection. The practical steps are as follows: we
begin by twisting all the fermionic fields using the Killing spinor
$\varepsilon^{i}$ in the AdS${}_{2}\times$ S2 background. We then write down
the supersymmetry variation of all the supergravity fields and keep track of
the charges under $SU(2)_{+}\times SU(2)_{-}\times SU(2)_{R}\times
SU(2)^{\prime}$ as described above. We start with the top component (which is
say bosonic) in a given representation and consider its
$Q_{\text{eq}}$-variation, which is typically a linear combination of products
of various twisted fields and Killing spinor bilinears. Among these terms we
isolate terms which are purely linear in the fields (in this case fermionic)
without derivatives (so that the change of variables to the twisted fields is
invertible). If we find such a fermionic field, it is a descendant and the
original bosonic field is elementary. Then we move down the supermultiplet,
considering the representations of fields which have not been classified and
repeat the process, interchanging bosons and fermions if needed. The
$c$-ghosts are classified according to the discussion sketched near Equation
(4.18).

In the off-shell superconformal formalism, the relevant part of the
supersymmetric transformation under $Q$ and $S$ of the fields in the
$\mathcal{N}=4$ Weyl multiplet are [90, 91]:

$\displaystyle\delta e_{\mu}\,^{a}$
$\displaystyle=\overline{\epsilon}^{i}\gamma^{a}\psi_{\mu i}+\text{h.c.}\,,$
$\displaystyle\delta\psi_{\mu}\,^{i}$ $\displaystyle\supset-
b_{\mu}\epsilon^{i}-2V_{\mu}\,^{i}\,{}_{j}\epsilon^{j}-\frac{1}{2}\gamma^{ab}T_{ab}{}^{ij}\gamma_{\mu}\epsilon_{j}-\gamma_{\mu}\eta^{i}\,,$
$\displaystyle\delta b_{\mu}$
$\displaystyle\supset-\frac{1}{2}\overline{\psi}_{\mu}{}^{i}\eta_{i}+\text{
h.c. }\,,$ $\displaystyle\delta V_{\mu}{}^{i}{}_{j}$
$\displaystyle\supset\overline{\epsilon}^{k}\gamma_{\mu}\chi^{i}{}_{kj}-\overline{\psi}_{\mu}{}^{i}\eta_{j}+\frac{1}{4}\delta^{i}{}_{j}\overline{\psi}_{\mu}{}^{k}\eta_{k}\,,$
$\displaystyle\delta\Lambda_{i}$ $\displaystyle\supset
E_{ij}\epsilon^{j}+\frac{1}{2}\varepsilon_{ijkl}T_{bc}{}^{kl}\gamma^{bc}\epsilon^{j}\,,$
$\displaystyle\delta E_{ij}$
$\displaystyle\supset-2\overline{\epsilon}^{k}\chi^{mn}_{(i}\varepsilon_{j)kmn}+2\overline{\eta}_{(i}\Lambda_{j)}\,,$
$\displaystyle\delta T_{ab}{}^{ij}$
$\displaystyle\supset-2\overline{\epsilon}^{[i}\left(\gamma_{[a}\phi_{b]}{}^{j]}+\frac{1}{2}\gamma\cdot
T^{j]k}\gamma_{[a}\psi_{b]k}\right)+\frac{1}{2}\overline{\epsilon}^{k}\gamma_{ab}\chi^{ij}{}_{k}-\frac{1}{4}\varepsilon^{ijkl}\overline{\eta}_{k}\gamma_{ab}\Lambda_{l}\,,$
$\displaystyle\delta\chi^{ij}{}_{k}$
$\displaystyle\supset-\frac{1}{2}{\ooalign{\hfil\hfil/\hfil\cr$D$}}T_{ab}^{ij}\epsilon_{k}+D^{ij}{}_{kl}\epsilon^{l}+\frac{1}{2}T_{ab}^{ij}\gamma^{ab}\eta_{k}+\frac{2}{3}\delta_{k}^{[i}T_{ab}{}^{j]l}\gamma^{ab}\eta_{l}-\frac{1}{2}\varepsilon^{ijlm}E_{kl}\eta_{m}\,,$
(4.34) $\displaystyle\delta D^{ij}{}_{kl}$
$\displaystyle\supset\varepsilon^{ijmn}\overline{\epsilon}^{p}T^{ab}{}_{kl}\left[2T_{abnp}\lambda_{m}+T_{abmn}\Lambda_{p}\right]\,,$
(4.35)

where we have only included the variations that are either linear in one field
or, when expanding around the localization background described in Section 3,
could be linear in the quantum fluctuations of a given field. One can
therefore then use (4.4.2) to write down the supersymmetric variations for the
quantum field fluctuations around the localization background described in
Section 3.1. Including ghosts one finds the BRST variations which then need to
be modified, as described in Section 4, to find the equivariant variation. To
exemplify how the cohomological partners are found in this case, we will
simply list the first few steps of the algorithm presented in Section 4.1. The
first few variations are as follows. The first variation tells us that

$\displaystyle e_{\mu}{}^{a}$
$\displaystyle\;=\;e_{(3,3,1)\oplus(1,3,1)\oplus(3,1,1)\oplus(1,1,1)}\in\Phi,$
$\displaystyle\psi_{\mu}^{a}$
$\displaystyle\;=\;\psi_{(2,2,1)\otimes(2,2,1)}=\psi_{(3,3,1)\oplus(1,3,1)\oplus(3,1,1)\oplus(1,1,1)}\notin\Psi\,.$
(4.36)

where we in the subscript we shall list the twisted $SU(2)_{\pm}\times
SU(2)_{R\mp}\times SU(2)^{\prime}$ representation associated to each field.
The next relations tell us that

$\displaystyle\psi_{(2,4,1)}\in\Psi\,,\qquad\psi_{(2,2,1)\otimes(1,1,1)}\in\Psi\,.$
(4.37)

and consequently,

$\displaystyle V_{(2,2,1)}\not\in\Phi\,,\qquad V_{(2,4,1)}\not\in\Phi\,.$
(4.38)

There is an additional component of the gravitino which is also in the
$(2,2,1)$ twisted representation. When modifying (4.4.2) to obtain the BRST
variation the term $\gamma_{\mu}\eta^{i}$ includes the fluctuation for the
ghost field $c_{S}$ which after twisting is also in the $(2,2,1)$
representation. Thus, we conclude that

$\displaystyle\psi_{(2,2,1)^{\prime}}\in\Psi\,,\qquad(c_{S})_{(2,2,1)}\not\in\Phi\,.$
(4.39)

Continuing on in this manner, we obtain the full classification of the
$\mathcal{N}=4$ Weyl supermultiplet. We present the results split into the
$\mathcal{N}=2$ multiplets following the decomposition in Table 5: the fields
that are part of the $\mathcal{N}=2$ Weyl multiplet are shown in Table 8,
those that are part of the spin-$\frac{3}{2}$ multiplet in Table 9, those in
the vector multiplet in Table 10, and finally those in the chiral multiplet in
Table 11.

$\Phi$ | $\subseteq Q_{eq}\Phi$
---|---
$e_{(3,3,1)}$ | $\psi_{(3,3,1)}$
$e_{(3,1,1)}$ | $\psi_{(3,1,1)}$
$e_{(1,3,1)}$ | $\psi_{(1,3,1)}$
$e_{(1,1,1)}$ | $\psi_{(1,1,1)}$
$V_{(2,2,1)}$ | $\chi_{(2,2,1)}$
$T_{(1,3,1)}$ | $\chi_{(1,3,1)}$
$(c_{Q})_{(1,3,1)}$ | $(c_{R})_{(1,3,1)}$
$(c_{Q})_{(1,1,1)}$ | $(c_{R})_{(1,1,1)}$
$(A_{\mu}^{D})_{(2,2,1)}$ | $(c_{K})_{(2,2,1)}$
$b_{*}$ | $B_{*}$

$\Psi$ | $\subseteq Q_{eq}\Psi$
---|---
$\psi_{(2,2,1)}$ | $(c_{S})_{(2,2,1)}$
$\psi_{(2,2,1)}$ | $V_{(2,2,1)}$
$\psi_{(2,4,1)}$ | $V_{(2,4,1)}$
$\chi_{(1,1,1)}$ | $D_{(1,1,1)}$
$(c_{M})_{(3,1,1)}$ | $T_{(3,1,1)}$
$(c_{M})_{(1,3,1)}$ | $(c_{S})_{(1,3,1)}$
$(c_{D})_{(1,1,1)}$ | $(c_{S})_{(1,1,1)}$
$(c_{\mu})_{(2,2,1)}$ | $(c_{Q})_{(2,2,1)}$
$b_{*}$ | $B_{*}$

Table 8: Cohomological primaries and descendants in the $\mathcal{N}=2$ Weyl
supermultiplet. The representation labels are the Cartans of
$SU(2)_{\mp}\times SU(2)_{R\pm}\times SU(2)^{\prime}$. All fermionic/bosonic
$b$-ghosts are cohomological primaries whose descendents are the
bosonic/fermionic $B$-ghosts with the same $SU(2)_{\mp}\times
SU(2)_{R\pm}\times SU(2)^{\prime}$ representations. All the fields in this
multiplet are singlets under $SU(2)^{\prime}$ and, if we suppress the notation
$1$, this table is the same as Table 6 of [38].

$\Psi$ | $\subseteq Q_{eq}\Psi$
---|---
$\psi_{(2,3,2)}$ | $V_{(2,3,2)}$
$\psi_{(3,2,2)}$ | $T_{(3,2,2)}$
$\psi_{(2,1,2)}$ | $V_{(2,1,2)}$
$\psi_{(1,2,2)}$ | $(c_{S})_{(1,2,2)}$
$\Lambda_{(1,2,2)}$ | $E_{(1,2,2)}$
$\chi_{(1,2,2)}$ | $D_{(1,2,2)}$
$\chi_{(1,2,2)}$ | $D_{(1,2,2)}$
$(c_{R})_{(1,2,2)}$ | $E_{(1,2,2)}$
$b_{*}$ | $B_{*}$

$\Phi$ | $\subseteq Q_{eq}\Phi$
---|---
$V_{(2,3,2)}$ | $\chi_{(2,3,2)}$
$V_{(2,1,2)}$ | $\chi_{(2,1,2)}$
$T_{(1,2,2)}$ | $\chi_{(1,2,2)}$
$T_{(1,4,2)}$ | $\chi_{(1,4,2)}$
$(c_{S})_{(2,1,2)}$ | $\chi_{(2,1,2)}$
$(c_{Q})_{(2,1,2)}$ | $\Lambda_{(2,1,2)}$
$(c_{Q})_{(1,2,2)}$ | $(c_{R})_{(1,2,2)}$
$b_{*}$ | $B_{*}$

Table 9: Cohomological primaries and descendants in the $\mathcal{N}=2$
spin-$3/2$ supermultiplet.

$\Phi$ | $\subseteq Q_{eq}\Phi$
---|---
$V_{(2,2,3)}$ | $\chi_{(2,2,3)}$
$E_{(1,1,3)}$ | $\chi_{(1,1,3)}$

$\Psi$ | $\subseteq Q_{eq}\Psi$
---|---
$\chi_{(1,3,3)}$ | $D_{(1,1,3)}$
$(c_{R})_{(1,1,3)}$ | $\overline{E}_{(1,1,3)}$
$(b_{R})_{(1,1,3)}$ | $(B_{R})_{(1,1,3)}$

Table 10: Cohomological primaries and descendants in the $\mathcal{N}=2$
vector supermultiplet.

$\Phi$ | $\subseteq Q_{eq}\Phi$
---|---
$\phi_{(1,1,1)}$ | $\Lambda_{(1,1,1)}$
$\phi_{(1,1,1)}$ | $\Lambda_{(1,1,1)}$
$E_{(3,1,1)}$ | $\chi_{(3,1,1)}$
$T_{(1,3,1)}$ | $\chi_{(1,3,1)}$

$\Psi$ | $\subseteq Q_{eq}\Psi$
---|---
$\Lambda_{(1,3,1)}$ | $\overline{E}_{(1,3,1)}$
$\Lambda_{(3,1,1)}$ | $T_{(3,1,1)}$
$\chi_{(1,1,1)}$ | $D_{(1,1,1)}$
$\chi_{(1,1,1)}$ | $D_{(1,1,1)}$

Table 11: Cohomological primaries and descendants in the $\mathcal{N}=2$
chiral supermultiplet.

Index calculation

Using the results in the tables above we can now easily calculate the index of
the various multiplets using the fixed-point formula (4.6), and the trace
formula (4.12). In the super-conformal formalism, the results for the various
$\mathcal{N}=2$ multiplets on the smooth AdS${}_{2}\times S^{2}$ are thus
given as follows:

$\displaystyle\text{Vector}:$
$\displaystyle\qquad\text{ind}_{\text{vect.}}^{\mathcal{N}=2\text{
s.c.}}(D_{10})(t)\big{|}_{\rm
NP/SP}\;=\;\frac{2(q+q^{-1})-4}{(1-q)^{2}\,(1-q^{-1})^{2}}\big{|}_{\rm
NP/SP}\,,$ (4.40) $\displaystyle\text{Hyper}:$
$\displaystyle\qquad\text{ind}_{\text{hyper}}^{\mathcal{N}=2\text{
s.c.}}(D_{10})(t)\big{|}_{\rm
NP/SP}\;=\;-\frac{2(q+q^{-1})-4}{(1-q)^{2}\,(1-q^{-1})^{2}}\big{|}_{\rm
NP/SP}\,,$ (4.41) $\displaystyle\text{Weyl}:$
$\displaystyle\qquad\text{ind}_{\text{Weyl}}^{\mathcal{N}=2\text{
s.c.}}(D_{10})(t)\big{|}_{\rm
NP/SP}\;=\;\frac{2(q^{2}+q^{-2})-6(q+q^{-1})+8}{(1-q)^{2}\,(1-q^{-1})^{2}}\big{|}_{\rm
NP/SP}\,,$ (4.42) $\displaystyle\text{Spin-$\frac{3}{2}$}:$
$\displaystyle\qquad\text{ind}_{\text{spin 3/2}}^{\mathcal{N}=2\text{
s.c.}}(D_{10})(t)\big{|}_{\rm
NP/SP}\;=\;\frac{-2(q^{2}+q^{-2})+4(q+q^{-1})-4}{(1-q)^{2}\,(1-q^{-1})^{2}}\big{|}_{\rm
NP/SP}\,,$ (4.43) $\displaystyle\text{Chiral}:$
$\displaystyle\qquad\text{ind}_{\text{chiral}}^{\mathcal{N}=2\text{
s.c.}}(D_{10})(t)\big{|}_{\rm NP/SP}\;=\;0\big{|}_{\rm NP}\,,$ (4.44)

where the “s.c.” upper-script indicates that we are still working in the off-
shell formulation of supergravity in the super-conformal formalism. As in the
example of the $\mathcal{N}=2$ vector supermultiplet given in Section 4.3, we
have separated the contribution of the north and south poles since one needs
to carefully treat them in a small $q$, respectively $q^{-1}$, expansion.

Following the discussion in Section 4.4.1, we can now write the indices of
$\mathcal{N}=2$ after fixing to super-Poincaré. We will denote such results
using the “s.P.” upperscript. Directly listing the the results for the
orbifold geometries using the sum over images discussed in Section 4.2, we
find

$\displaystyle\text{Vector}:$
$\displaystyle\qquad\text{ind}_{\text{vect.},\,c}^{\mathcal{N}=2\text{
s.P.}}(D_{10})(t)\big{|}_{\rm
NP}\;=\;\,\frac{2cq^{c}}{(1-q^{c})^{2}}\big{|}_{\rm NP},$ (4.45)
$\displaystyle\text{Hyper}:$
$\displaystyle\qquad\text{ind}_{\text{hyper},\,c}^{\mathcal{N}=2\text{
s.P.}}(D_{10})(t)\big{|}_{\rm
NP}\;=\;\frac{-2cq^{c}}{(1-q^{c})^{2}}\big{|}_{\rm NP}\,,$ (4.46)
$\displaystyle\text{graviton}:$
$\displaystyle\qquad\text{ind}_{\text{Graviton},\,c}^{\mathcal{N}=2\text{
s.P.}}(D_{10})(t)\big{|}_{\rm
NP}\;=\;\frac{2q^{2c}+2(c-2)q^{c}+2}{(1-q^{c})^{2}}\big{|}_{\rm NP}\,,$ (4.47)
$\displaystyle\text{gravitino}:$
$\displaystyle\qquad\text{ind}_{\text{Gravitino},\,c}^{\mathcal{N}=2\text{
s.P.}}(D_{10})(t)\big{|}_{\rm
NP}\;=\;-\frac{2q^{2c}+2(c-2)q^{c}+2}{(1-q^{c})^{2}}\big{|}_{\rm NP}\,,$
(4.48)

We can finally put together the $\mathcal{N}=2$ supermultiplets gauged fixed
to super-Poincaré, to form the $\mathcal{N}=4$ vector and Weyl or
$\mathcal{N}=8$ Weyl multiplets. These multiplets are found by putting these
together

$\displaystyle\mathcal{N}=4\;\text{vector}$
$\displaystyle\rightarrow\mathcal{N}=2\,\text{vector + hyper}:$
$\displaystyle\text{ind}_{\text{vector}}^{\mathcal{N}=4\text{
s.P.}}(D_{10})(t)\;=\;0\,,$ (4.49)
$\displaystyle\mathcal{N}=4\;\text{graviton}$
$\displaystyle\rightarrow\mathcal{N}=2\,\text{ graviton + 2~{}gravitini +
vector in s.P.}:$ $\displaystyle\text{ind}^{\mathcal{N}=4\text{
s.P.}}_{\text{graviton}}(D_{10})(t)\;=\;-2\,,$ (4.50)
$\displaystyle\mathcal{N}=8\;\text{graviton}$
$\displaystyle\rightarrow\mathcal{N}=2\,\text{ graviton + 6~{}gravitini + 15
vectors + 10 hypers in s.P.}:$ $\displaystyle\text{ind}^{\mathcal{N}=8\text{
s.P.}}_{\text{graviton}}(D_{10})(t)\;=\;-10\,.$ (4.51)

We emphasize that the results (4.4.2)-(4.4.2) are exact, and are not
expansions at leading order in $t$. Importantly, the indices of the
$\mathcal{N}=4$ and $\mathcal{N}=8$ supermultiplets are all independent of
$t$. Because of this on the orbifold geometry, the sum over images discussed
in Section 4.2 implies that the index of such super-multiplets is
$c$-independent. Since the result is $t$-independent we have also summed the
contribution of the north and south pole since we not longer have to
distinguish the two in the small $q$, respectively small $q^{-1}$, expansion.

The results (4.4.2)–(4.4.2) also agree with the leading quantum correction
found for the orbifold entropy [87], who noticed that the coefficient in front
of the logarithmic correction in the black hole area is independent of $c$.
Nevertheless, (4.4.2)–(4.4.2) are much stronger results: as we shall see, it
predicts that not only this coefficient in the quantum correction to the
entropy is $c$-independent but also the entire one-loop determinant of all
bulk modes is also $c$-independent.

To obtain the one-loop determinants from the results (4.45)–(4.48) for the
indices of the various supermultiplets that we have discussed we simply have
to follow the same procedure as in Section 4.3 and perform the integral over
$t$, appropriately keeping track of the expansion in $q$ and $q^{-1}$ at the
north and south poles, respectively. Below, we shall list the one-loop
determinants obtained through this procedure.

###  4.5 A summary of results

The one-loop determinants are given by a contribution captured by the index
$Z_{\rm index}$, and a correction from a proper treatment of zero-modes which
is explained in the next section. Since the results for the indices associated
to each supermultiplet had an intricate derivation, below we summarize the
contribution to the one-loop determinants from these indices for all
$\mathcal{N}=2$, $4$ or $8$ supergravity supermultiplets, when gauge fixed to
the super-Poincaré theory. For the AdS${}_{2}\times S^{2}$ geometries as well
as for the orbifold geometries, the one-loop determinants of all
$\mathcal{N}=2$ super-multiplets are:

$\displaystyle\log\left(Z_{\rm index}^{\mathcal{N}=2\,{\rm vector}}\right)$
$\displaystyle\;=\;-\frac{{\rm i}\pi}{12}\,c+\frac{c}{6}\log
c+2c\,\zeta^{\prime}(-1)-\frac{c}{12}\log\ell^{2}\,,$
$\displaystyle\log\left(Z_{\rm index}^{\mathcal{N}=2\,{\rm hyper}}\right)$
$\displaystyle\;=\;\frac{{\rm i}\pi}{12}\,c-\frac{c}{6}\log
c-2c\,\zeta^{\prime}(-1)+\frac{c}{12}\log\ell^{2}\,,$
$\displaystyle\log\left(Z_{\rm index}^{\mathcal{N}=2\,{\rm graviton}}\right)$
$\displaystyle\;=\;-\frac{{\rm i}\pi}{12}\,c+\frac{c}{6}\log
c+2c\,\zeta^{\prime}(-1)+\left(1-\frac{c}{12}\right)\log\ell^{2}$
$\displaystyle\log\left(Z_{\rm index}^{\mathcal{N}=2\,{\rm gravitino}}\right)$
$\displaystyle\;=\;\frac{{\rm i}\pi}{12}\,c-\frac{c}{6}\log
c-2c\,\zeta^{\prime}(-1)-\left(1-\frac{c}{12}\right)\log\ell^{2}\,.$ (4.52)

It is also useful to have the expression for the index ${\rm ind}(D_{10})(t)$
associated to each multiplet, which we gave above in equations (4.45)-(4.48).
These expression present complicated dependence with the orbifold parameter.
For $\mathcal{N}=4$ super-multiplets we have a much simpler answer:

$\displaystyle Z_{\rm index}^{\mathcal{N}=4\,{\rm vector}}$
$\displaystyle\;=\;1\,,$ $\displaystyle\log\left(Z_{\rm
index}^{\mathcal{N}=4\,{\rm graviton}}\right)$
$\displaystyle\;=\;-\log\ell^{2}\,.$ (4.53)

Finally, for $\mathcal{N}=8$ super-multiplets we obtain:

$\displaystyle\log\left(Z_{\rm index}^{\mathcal{N}=8\,{\rm graviton}}\right)$
$\displaystyle\;=\;-5\log\ell^{2}\,.$ (4.54)

Thus we see that the answer is independent of the orbifold parameter $c$ for
the $\mathcal{N}=4$ theory as well as the $\mathcal{N}=8$ theory.

##  5 The role of large gauge transformations

Large (super)diffeomorphisms that are nearly zero-modes have been found to
control the low-temperature limit of near-extremal (or near-BPS) black holes
(see our companion paper [58] and references therein). In this context, their
role within the localization framework presented in this paper should be re-
analyzed. At the beginning of Section 4, we noted that in addition to the
modes whose one-loop determinant we have considered in the previous section
there are additional large diffeomorphisms, some of which are zero-modes of
$Q_{\text{eq}}\mathcal{V}$ and some that are not. We denote the full one-loop
determinant by:

$\displaystyle
Z_{\text{1-loop}}^{Q_{\text{eq}}\mathcal{V}}\;\equiv\;Z_{\text{zero-
modes}}^{Q_{\text{eq}}\mathcal{V}}Z_{\text{non-
zero}}^{Q_{\text{eq}}\mathcal{V}}\,.$ (5.1)

$Z_{\text{zero-modes}}^{Q_{\text{eq}}\mathcal{V}}$ is given by the full
integral over the space of large super-diffeomorphisms that are zero-modes.
$Z_{\text{non-zero}}^{Q_{\text{eq}}\mathcal{V}}$ includes the contribution of
all other modes, part of it is reproduced by the index calculation of the
previous section but also has contributions from treating boundary non-zero-
modes carefully. In Section 5.1, we follow [38] and clarify how the presence
of such boundary non-zero modes affects the relation between the index
contribution $Z_{\text{index}}^{Q_{\text{eq}}\mathcal{V}}$ computed in the
previous sections and $Z_{\text{non-zero}}^{Q_{\text{eq}}\mathcal{V}}$.
Nevertheless, in [38] the path integral over the zero-modes and therefore the
entire moduli space of large super-diffeomorphisms was not considered and thus
the full $\phi^{0}$ and $c$-dependence of their contribution was left
undetermined beyond leading order. In Section 5.2 and 5.3, we compute this
path integral explicitly to finally obtain the full one-loop determinant
around the localization locus. A more general analysis about the role of these
large diffeomorphisms in the gravitational path integral for extremal black
holes is discussed in a companion paper [58].

###  5.1 Effect of large gauge transformations on the bulk one loop
determinant

Large gauge transformations or large diffeomorphisms are pure gauge modes
whose gauge parameters are not normalizable and thus, in our case, are non-
vanishing at the boundary of AdS2. Such modes are physical and should be
integrated over since they have a non-trivial effect on physically observables
quantities. For example, both large diffeomorphisms affect the extrisic
curvature of the boundary while large gauge transformations affects the value
of the electro-magnetic boundary term. This is true as long as such modes do
not correspond to isometries of the spacetime or of the gauge bundle. In fact,
as we shall see the exclusion of such isometry modes from the space of
physical large diffeomorphisms will play an important role below.

To analyze the role of all large diffeomorphisms modes in the relation between
$Z_{\text{index}}^{Q_{\text{eq}}\mathcal{V}}$ and $Z_{\text{non-
zero}}^{Q_{\text{eq}}\mathcal{V}}$, we will start by studying the illustrative
example of the simplest large-gauge transformation, that for the $U(1)$ gauge
fields $A_{\mu}^{I}$ that are part of the vector supermultiplets. They are
normalizable modes with $A_{\mu}^{\text{bdry}}=\partial_{\mu}\Lambda$ obtained
from non-normalizable scalar modes $\Lambda$ where for the purposes of this
section we will drop the $I$ label associated to each vector
supermultiplet252525Non-normalizable gauge parameters generate zero-modes only
for certain choice of the dependence on $S^{2}$. For example for vector zero-
modes they involve a constant gauge transformation along $S^{2}$. Some zero-
modes in the metric involve spin one harmonics, e.g. the ones realizing the
rotational $SU(2)$ isometries of $S^{2}$.. Such modes transform under the
equivariant variation as
$Q_{\text{eq}}A_{\mu}^{\text{bdry}}=\lambda_{\mu}^{\text{bdry}}$ where
$\lambda_{\mu}^{\text{bdry}}$ is a boundary mode associated to the gaugino.
Boundary modes that are zero-modes also satisfy
$Q_{\text{eq}}^{2}A_{\mu}^{\text{bdry}}=HA_{\mu}^{\text{bdry}}=n^{\nu}F_{\nu\mu}^{\text{bdry}}=0\,$.
More generally, all large bosonic gauge transformations and diffeomorphisms
(denoted by $\Phi^{\text{bdry}}$) that are zero-modes also satisfy

$\displaystyle
Q_{\text{eq}}^{2}\Phi^{\text{bdry}}\;=\;H\Phi^{\text{bdry}}\;=\;0\,.$ (5.2)

For all such modes $\mathcal{V}$ is vanishing—this can be easily seen for
$A_{\mu}^{\text{bdry}}$ since for such modes $\mathcal{V}$ solely contains
composites of the vanishing boundary field strength. This consequently implies
that $Q_{\text{eq}}\mathcal{V}|_{A_{\mu}^{\text{bdry}}}=0\,$ and, more
broadly, that

$\displaystyle Q_{\text{eq}}\mathcal{V}|_{\Phi^{\text{bdry}}}\;=\;0\,.$ (5.3)

Thus, all modes $\Phi^{\text{bdry}}$ are zero modes of the
$Q_{\text{eq}}$-exact deformation term in supergravity, confirming that they
are part of the localization locus. For extremal black holes it has also long
been known that such modes are zero modes of the original supergravity action
making their path integral somewhat more subtle as we explain in Section 5.3.
The superpartners $Q_{\text{eq}}\Phi^{\text{bdry}}$ are however not
necessarily zero-modes of the deformation term, even though they are boundary
modes. Explicitly, the deformation term for such superpartners can be
consistently chosen to be

$\displaystyle
Q_{\text{eq}}\mathcal{V}|_{Q_{\text{eq}}\Phi^{\text{bdry}}}=-(Q_{\text{eq}}\Phi^{\text{bdry}}){H}(Q_{\text{eq}}\Phi^{\text{bdry}})=-Q_{\text{eq}}\left[\Phi^{\text{bdry}}{H}(Q_{\text{eq}}\Phi^{\text{bdry}})\right]\,,$
(5.4)

which, from the last equality, is also clearly $Q_{\text{eq}}$-exact and thus
remains a valid deformation in the localization procedure. Thus, the
contribution of the modes $Q_{\text{eq}}\Phi^{\text{bdry}}$ for which the one-
loop determinant can be explicitly obtained from (5.4), should be
distinguished from that of $\Phi^{\text{bdry}}$ for which a full path integral
should be considered. To simplify notation we have rescaled $H$ by $\ell$ when
discussing the quadratic fluctuations of $Q_{\text{eq}}\mathcal{V}$.

Similarly, let us consider large fermionic gauge transformations, i.e. the
fermionic parts in the super-diffeomorphisms of supergravity. Such fermionic
modes are either cohomological primaries, which we denote by
$\Psi^{\text{bdry}}$, or they are cohomological descendants, in which case
they can be written as $Q_{\text{eq}}\Phi^{\text{pre-bdry}}$ for the
appropriate bosonic mode $\Phi^{\text{pre-bdry}}$.262626In supergravity, it
can be checked non-elementary bosons are not zero modes of either the
supergravity action or of the deformation term and, for this reason, they have
not been considered above. In both cases, for such fermionic modes
$Q_{\text{eq}}$ is nilpotent. Similarly, to the bosonic case, $\mathcal{V}$
vanishes for such modes, implying

$\displaystyle
Q_{\text{eq}}\mathcal{V}|_{\Psi^{\text{bdry}}}=Q_{\text{eq}}\mathcal{V}|_{Q_{\text{eq}}\Phi^{\text{pre-
bdry}}}=0\,,$ (5.5)

which means that both $\Psi^{\text{bdry}}$ and $Q_{\text{eq}}\Phi^{\text{pre-
bdry}}$ are part of the localization locus and should be fully integrated over
to obtain their contribution to $Z_{\text{zero-
modes}}^{Q_{\text{eq}}\mathcal{V}}$. The superpartners of such modes,
$Q_{\text{eq}}\Psi^{\text{bdry}}$ and $\Phi^{\text{pre-bdry}}$ are however not
zero-modes of the deformation term, given the definition (3.5) of
$\mathcal{V}$. Specifically, evaluating the deformation term for these
superpartners yields

$\displaystyle
Q_{\text{eq}}\mathcal{V}|_{Q_{\text{eq}}\Psi^{\text{bdry}}}=-(Q_{\text{eq}}\Psi^{\text{bdry}})1(Q_{\text{eq}}\Psi^{\text{bdry}})\,,\qquad
Q_{\text{eq}}\mathcal{V}|_{\Phi^{\text{pre-bdry}}}=\Phi^{\text{pre-
bdry}}H^{2}\Phi^{\text{pre-bdry}}\,,$ (5.6)

again $Q_{\text{eq}}$-exact due to the nilpotency of $\Psi^{\text{bdry}}$ and
$Q_{\text{eq}}\Phi^{\text{pre-bdry}}$.

Thus, we find that $Z_{\text{non-
zero}}^{Q_{\text{eq}}\mathcal{V}}=\sqrt{{\det^{*_{f}}K_{f}}/{\det^{*_{b}}K_{b}}}$,
where ${*_{b}}$ includes all bosonic modes with the exception of
$\Phi^{\text{bdry}}$, while ${*_{f}}$ includes all fermionic modes with the
exception of $\Psi^{\text{bdry}}$ and $Q_{\text{eq}}\Phi^{\text{pre-bdry}}$.
Using (5.4) and (5.6) we can explicitly separate out the contribution of the
superpartners of large super-diffeomophisms to the one-loop determinant:

$\displaystyle Z_{\text{non-
zero}}^{Q_{\text{eq}}\mathcal{V}}=\sqrt{\frac{\det^{*_{f}}K_{f}}{\det^{*_{b}}K_{b}}}=\sqrt{\frac{\text{det}_{Q_{\text{eq}}\Phi^{\text{bdry}}}(H)}{\text{det}_{\Phi^{\text{pre-
bdy}}}(H^{2})}\frac{\det^{\prime}K_{f}}{\det^{\prime}K_{b}}}=\sqrt{\frac{\text{det}_{\Phi^{\text{bdry}}}(H)}{\text{det}_{\Phi^{\text{pre-
bdy}}}(H^{2})}\frac{\det^{\prime}_{\Psi}H}{\det^{\prime}_{\Phi}H}}$ (5.7)

where the ′ indicates that we are only considering the determinant over the
bulk modes, no longer including the contribution of $\Phi^{\text{bdry}}$,
$Q_{\rm eq}\Phi^{\text{bdry}}$, $Q_{\rm eq}\Psi^{\text{bdy}}$, or
$\Phi^{\text{pre-bdy}}$. After the second equality sign, we have dropped the
one-loop determinant of the $Q_{\text{eq}}\Psi^{\rm bdry}$ modes since (5.6)
implies that this determinant equals $1$. Now, in the ′-space, all fields have
$Q_{\text{eq}}$superpartners. Consequently, the differential operators $K_{f}$
and $K_{b}$ are related and the third equality follows.

In total, writing the heat-kernel transformation associated to $Z_{\text{non-
zero}}^{Q_{\text{eq}}\mathcal{V}}$, we find

$\displaystyle\log Z_{\text{non-zero}}^{Q_{\text{eq}}\mathcal{V}}$
$\displaystyle\;=\;-\frac{1}{2}\int\frac{dt}{t}\left(\mathrm{Tr}_{\Phi^{\text{bdry}}}e^{\frac{tH}{\ell}}-2\mathrm{Tr}_{\Phi^{\text{pre-
bdry}}}e^{\frac{tH}{\ell}}+\mathrm{Tr}_{\Psi^{\prime}}{\rm
e}^{\frac{tH}{\ell}}-\mathrm{Tr}_{\Phi^{\prime}}{\rm
e}^{\frac{tH}{\ell}}\right)$
$\displaystyle\;=\;\int\frac{dt}{t}\left[-\mathrm{Tr}_{\Phi^{\text{bdry}}}e^{\frac{tH}{\ell}}+\frac{1}{2}\left(\mathrm{Tr}_{\Psi^{\text{bdry}}}e^{\frac{tH}{\ell}}+\mathrm{Tr}_{Q_{\text{eq}}\Phi^{\text{pre-
bdry}}}e^{\frac{tH}{\ell}}\right)\right]$
$\displaystyle\qquad-\frac{1}{2}\int\frac{dt}{t}\left(\mathrm{Tr}_{\Psi}{\rm
e}^{\frac{tH}{\ell}}-\mathrm{Tr}_{\Phi}{\rm e}^{\frac{tH}{\ell}}\right)\,,$
(5.8)

where we have reintroduced the scale $\ell$ in order to make the comparison
with the previous section easier. The spaces $\Psi$ and $\Phi$ is now the
entire space of normalizable modes associated to fields that are cohomological
primaries, including all boundary modes. The difference of traces for such
modes is precisely what the index of the operator $D_{10}$ computes,

$\displaystyle\text{ind}_{\text{Weyl}}^{\mathcal{N}=8\text{
s.P.}}(D_{10})(t)\;=\;\mathrm{Tr}_{\Phi}{\rm
e}^{\frac{tH}{\ell}}-\mathrm{Tr}_{\Psi}{\rm e}^{\frac{tH}{\ell}}\,.$ (5.9)

Additionally, since all modes $\Phi^{\text{bdry}}$, $\Psi^{\text{bdry}}$ and
$Q_{\text{eq}}\Phi^{\text{pre-bdry}}$ are nilpotent under the action of
$Q_{\text{eq}}$, the traces in the square paranthesis simply count the number
of zero modes in each set. Since the number of zero modes is technically
divergent, it has to be regularized by considering the ratio of bulk one-loop
determinants for different values of $\phi^{0}$. We will denote the number of
regularized zero modes by

$\displaystyle
n^{\text{bos}}_{\text{bdry}}\;\equiv\;\mathrm{Tr}_{\Phi^{\text{bdry}}}e^{{\tilde{t}H}}\,,\qquad
n^{\text{ferm}}_{\text{bdry}}\;\equiv\;\mathrm{Tr}_{\Psi^{\text{bdry}}}e^{{\tilde{t}H}}\,,\qquad\text{
with}\qquad\tilde{t}=\frac{t}{\ell}\,.$ (5.10)

The regularized number of zero modes can be concretely obtained by zeta
function regularization, excluding the contribution of the modes that
correspond to bosonic or fermionic isometries and thus should not be
integrated-over. On the smooth AdS${}_{2}\times S^{2}$ geometry the super-
isometry group is $H_{\rm Disk}=PSU(1,1|2)$ which has $6$ bosonic generators
and $8$ fermionic generators. Each generator has an associated large super-
diffeomorphism mode that has to be subtracted in the regularization of the
number of bosonic and fermionic zero-modes. One should similarly subtract the
contribution of modes with $\Lambda=$ constant for each vector supermultiplet.
Consequently, for the smooth AdS${}_{2}\times S^{2}$ geometry

$\displaystyle
n^{\text{bos}}_{\text{bdry}}\;\equiv\;n_{\text{bdry}}^{\text{1-form}}+n_{\text{bdry}}^{\text{grav}}\,,\qquad
n_{\text{bdry}}^{\text{grav}}\;=\;-6\,,\qquad
n_{\text{bdry}}^{\text{1-form}}\;=\;-n_{v}\,,\qquad
n^{\text{ferm}}_{\text{bdry}}\;=\;-8\,,\qquad\text{ for }c\;=\;1.$ (5.11)

For the orbifolded geometry the super-isometry group is $H_{\rm orb.}$ which
has $2$ bosonic generators and $4$ fermionic generators, which implies that

$\displaystyle n_{\text{bdry},\,c}^{\text{grav}}\;=\;-2\,,\qquad
n_{\text{bdry},\,c}^{\text{1-form}}\;=\;-n_{v}\,,\qquad
n^{\text{ferm}}_{\text{bdry},\,c}\;=\;-4\,,\qquad\text{ for }\,c>1\,.$ (5.12)

Thus, we find that $Z_{\text{bulk}}^{Q_{\text{eq}}\mathcal{V}}$ is related to
the index computed in the previous section through

$\displaystyle\log\,Z_{\text{non-
zero},\,c}^{Q_{\text{eq}}\mathcal{V}}=\left(-n^{\text{bos}}_{\text{bdry},\,c}+\frac{1}{2}n^{\text{ferm}}_{\text{bdry},\,c}\right)\log(\ell)+\log\,Z_{\text{index},\,c}^{Q_{\text{eq}}\mathcal{V}}\,,$
(5.13)

where

$\log\,Z_{\text{index},\,c}^{Q_{\text{eq}}\mathcal{V}}\;=\;\frac{1}{2}\int_{\epsilon}^{\infty}\frac{dt}{t}\text{ind}_{\text{Weyl},\,c}^{\mathcal{N}=8\text{
s.P.}}(D_{10})(t).$ (5.14)

The fact that
$-n^{\text{bos}}_{\text{bdry},c}+\frac{1}{2}n^{\text{ferm}}_{\text{bdry},\,c}=2+\frac{n_{v}}{2}$
for the $c=1$ smooth AdS${}_{2}\times S^{2}$ geometry is different than
$-n^{\text{bos}}_{\text{bdry},c}+\frac{1}{2}n^{\text{ferm}}_{\text{bdry},\,c}=\frac{n_{v}}{2}$
contribution for the orbifolded geometries with $c>1$ might seem problematic
at first sight since it seemingly suggests that the $\phi^{0}$-dependence (and
not only $c$-dependence) of $Z_{\text{index},\,c}^{Q_{\text{eq}}\mathcal{V}}$
on the two geometry types is different. However, as we shall see in the next
subsection, the measure of the integral over all physical modes is such that
$Z_{\text{1-loop}}^{Q_{\text{eq}}\mathcal{V}}$ has the same
$\phi^{0}$-dependence on all geometries that we sum over in the gravitational
path integral.

###  5.2 Integration measure in the space of large gauge transformations

Thus, what remains to be evaluated is $Z_{\text{zero-
modes},\,c}^{\text{bdy.}}$, the path integral over the localization locus of
the zero modes:

$\displaystyle Z_{\text{zero-modes},\,{c}}^{\text{bdy.}}$
$\displaystyle=\int_{\frac{\text{Large-diffeomorphisms}}{\text{Super-
isometries}}}[d(A_{\mu})^{\text{Large gauge
transf.}}][d{h}_{\mu\nu}^{\text{Large super-
diffs.}}][d{\psi}_{\mu}^{\text{Large super-diffs.}}]\,,$ (5.15)

where $(A_{\mu})^{\text{Large gauge transf.}}$ and $h_{\mu\nu}^{\text{Large
gauge transf.}}$ are part of $\Phi^{\text{bdry}}$ and, depending on their
twisted representation, twisted components of ${\psi}_{\mu}^{\text{Large
super-diffs.}}$ are either part of $\Psi^{\text{bdry}}$ or of
$Q_{\text{eq}}\Phi^{\text{pre-bdry}}$.

The integration space for large super-diffeomorphisms is given by
$\text{Diff}(S^{1|4})$. There are two types of bosonic modes in
$\text{Diff}(S^{1|4})$, which correspond to fluctuations of the metric that we
integrate in (5.15) when travelling along the AdS2 boundary: the first
captures fluctuations in the shape of the asymptotically AdS2 boundary and the
second captures rotational fluctuations around $S^{2}$. Similarly, the
fermionic modes in $\text{Diff}(S^{1|4})$ correspond to fluctuations of the
gravitino when travelling along the AdS2 boundary, that we also integrate in
(5.15). Finally, the integration space of the large gauge transformations in
(5.15) is $\text{Loop}(U(1))$ for each gauge field $A_{\mu}^{I}$.

From $\text{Diff}(S^{1|4})$ we have to quotient out the appropriate super-
isometry group, corresponding to the modes that are not physical. Thus, on the
smooth AdS${}_{2}\times S^{2}$ geometry the integration space is
$\text{Diff}(S^{1|4})/PSU(1,1|2)$ and on the orbifold geometry it is
$\text{Diff}(S^{1|4})/H_{\rm orb.}$. The integration space for the large gauge
transformations is similarly $\left[\text{Loop}(U(1))/U(1)\right]^{n_{v}}$ for
black holes in the grand canonical ensemble. If we instead want to consider
black holes in the canonical ensemble,272727Recall that we are still studying
an index in which angular momenta are not fixed. Here, by canonical ensemble
we mean that all $U(1)^{n_{v}+1}$ charges are fixed.  we need to perform an
additional integral over the holonomy around the AdS2 boundary in order to fix
the charges. Therefore, the integration space for large gauge transformations
in the canonical ensemble is $\left[\text{Loop}(U(1))\right]^{n_{v}}$.

We take the measure over all fields to be given by the ultralocal measure. For
example, for the gauge fields in the vector supermultiplets, we require that

$\displaystyle\int[d(A_{\mu})]{\rm e}^{-\int
d^{4}x\sqrt{g}g^{\mu\nu}A_{\mu}\,A_{\nu}}=1\,.$ (5.16)

In (5.16) and in the analogous equations for the metric and gravitino, we can
rescale the fields to fully extract the dependence in the integration measure
on the overall scale $\ell$ entering in the metric $g_{\mu\nu}$. (We remind
the reader that $\ell^{2}\sim 1/(\phi^{0})$ on the localization locus.) This
is a correct rescaling of the fields close to the localization locus and,
since we are only interested in computing the quadratic fluctuation
determinant around this locus, we do not have to consider fluctuations in the
metric within the definition of the integration measure (5.16). This rescaling
of the fields in (5.16) and in the analogous equations for the metric and
gravitino was discussed in [85, 92, 38] for the smooth AdS${}_{2}\times S^{2}$
geometry. A similar analysis can be performed on the orbifolded geometries
taking the different super-isometry group into account. The resulting path
integral (5.15) over zero-modes is

$\displaystyle Z_{\text{zero-modes},\,{c=1}}^{\text{bdy.}}$
$\displaystyle=(\ell)^{c_{\text{zero-
modes}}}\int_{\frac{\text{Diff}(S^{1|4})}{PSU(1,1|2)}}D(\text{Large-
diffs})\int_{\left[{\text{Loop}U(1)}\right]^{n_{v}}}D(\text{Large gauge})\,,$
(5.17) $\displaystyle Z_{\text{zero-modes},\,{c}}^{\text{bdy.}}$
$\displaystyle=(\ell)^{c_{\text{zero-
modes},\,c}}\int_{\frac{\text{Diff}(S^{1|4})}{H_{\rm orb.}}}D(\text{zero-
modes})\int_{\left[{\text{Loop}U(1)}\right]^{n_{v}}}D(\text{Large gauge})\,,$
(5.18)

where the coefficient $c_{\text{zero-modes},\,c}$ takes the form

$\displaystyle c_{\text{zero-
modes},\,c}=\beta^{\text{1-form}}n^{\text{1-form}}_{{\rm
bdry},\,c}+\beta^{\text{grav}}n^{\rm grav}_{{\rm
bdry},\,c}-\frac{1}{2}\beta^{\text{ferm}}n^{\rm ferm}_{{\rm bdry},\,c}\,.$
(5.19)

Here $\beta^{\text{1-form}}$, $\beta^{\text{grav}}$, and $\beta^{\text{ferm}}$
are coefficients that capture the number degrees of freedom of the 1-forms,
graviton and gravitino and were found to be [85, 92]

$\displaystyle\beta^{\text{1-form}}=1\,,\qquad\beta^{\text{grav}}=2\,,\qquad\beta^{\text{ferm}}=3\,,$
(5.20)

while the coefficients $n^{\text{1-form}}_{{\rm bdry},\,c}$, $n^{\rm
grav}_{{\rm bdry},\,c}$, and $n^{\rm ferm}_{{\rm bdry},\,c}$ are given in
(5.11) for the smooth AdS${}_{2}\times S^{2}$ geometry, and in (5.12) for the
orbifold geometry. The remaining integrals over zero-modes in (5.17)–(5.18) is
independent of $\ell$ but can still depend on the topology of the geometry,
and therefore on the parameter $c$ of the orbifold.

We now combine the powers of $\ell$ appearing in (5.17)–(5.18) with the powers
of $\ell$ appearing from counting the total number of zero-modes in (5.13). In
total we find a power $(\ell)^{c_{\text{non-index}}}$ coming from the zero-
modes, with $c_{\text{non-index}}$ given by

$\displaystyle c_{\text{non-
index}}\;=\;(\beta^{\text{1-form}}-1)\,n^{\text{1-form}}_{{\rm
bdry},\,c}+(\beta^{\text{grav}}-1)\,n^{\rm grav}_{{\rm
bdry},\,c}-\frac{1}{2}(\beta^{\text{ferm}}-1)\,n^{\rm ferm}_{{\rm
bdry},\,c}\;=\;2\,,~{}~{}\text{ for all }c\,.$ (5.21)

Notice that due to the particular values of $n^{\rm grav}_{{\rm bdry},\,c}$
and $n^{\rm ferm}_{{\rm bdry},\,c}$ in (5.11) for $c=1$ and (5.12) for $c>1$,
we find that (5.21) is the same for AdS${}_{2}\times S^{2}$ and its orbifolds.
Using the fact that the index of the $\mathcal{N}=8$ graviton supermultiplet
also has the same scaling with $\ell$ for all these geometries, this means
that the overall scaling with $\ell$ of the full one-loop determinant is also
precisely the same for all these geometries. As we shall see in Section 6 for
the $\frac{1}{8}$-BPS black hole index, the fact that $c_{\text{non-index}}$
is independent of the geometry around which the one-loop determinant is
computed is a crucial detail needed to obtain the correct form of the
Rademacher expansion.

###  5.3 The connection to the $\mathcal{N}=4$ super-Schwarzian

What remains to be done is to evaluate the path integrals in (5.17) and (5.18)
over the zero-modes, with their $\ell$-dependence stripped off. The integrals
that we would like to compute are schematically given by

$\displaystyle\mathrm{Vol}\left[\frac{\text{Diff}(S^{1|4})}{H_{\text{Disk}}}\right]$
$\displaystyle=\int_{\frac{\text{Diff}(S^{1|4})}{H_{\text{Disk}}}}D(\text{zero-
modes})\,,\qquad\mathrm{Vol}\left[\frac{\text{Diff}(S^{1|4})}{H_{\text{Orb}}}\right]=\int_{\frac{\text{Diff}(S^{1|4})}{H_{\text{Orb}}}}D(\text{zero-
modes})\,,$ $\displaystyle\hskip
14.22636pt\mathrm{Vol}\left[{\text{Loop}(U(1))}\right]=\int_{\text{Loop}(U(1))}D(\text{Large
gauge})$ (5.22)

where the boundary conditions for the zero modes will be clarified
momentarily. All integrals above are technically divergent and need to be
regularized. This brings up two issues that we need to address:

  * •

All volumes in (5.3) should have a non-vanishing regularized value; otherwise,
the gravitational path integral would incorrectly predict that BPS black holes
do not have a degeneracy. This should be contrasted to the case of non-
supersymmetric black holes discussed in the companion paper [58]. In that case
a similar integral, over zero-modes appears but there we find that
$\mathrm{Vol}\left[\text{Diff}(S^{1})/SL(2,\mathbb{R})\right]=0$ after a
consistent regularization, suggesting that non-supersymmetric extremal black
holes _do not_ have a degeneracy that scales as $\exp(\text{Area}/4G_{N})$.

  * •

$\mathrm{Vol}\left[{\text{Diff}(S^{1|4})}/{H_{\text{Orb}}}\right]$ can have a
$c$-dependence from that of the boundary conditions for the zero-modes that we
integrate over in (5.3). To complete the computation of the one-loop
determinant for all geometries we would thus like to determine this
$c$-dependence.

To regularize the quantities in (5.3) we would like to compare the partition
functions of black holes at zero temperature to the partition function of
black holes at some non-zero temperature. This is because in the latter case
the zero-modes (5.3) of the action and of the localizing deformation will now
gain a “mass” in the supergravity action and will thus no longer be zero-modes
[58]; instead, in the limit of low-temperatures we will refer to these modes
as “nearly zero-modes”. In this limit, these modes decouple from all other
bulk modes whose one-loop determinant we have computed in Section 4.4 and the
effective theory associated to these modes becomes strongly coupled.
Nevertheless, as we explain in detail in the companion paper [58] the
effective theory associated to these modes turns out to have a path integral
that is exactly solvable even at strong coupling, at small
temperatures.282828For $\mathcal{N}=0,1,2$ super-Schwarzian this has been
understood in [93] and [94, 95], while for the case of the $\mathcal{N}=4$
super-Schwarzian its path integral was studied in [96] using similar
techniques. Explicitly, the first two integrals in (5.3) are given by the
zero-temperature limit of the $\mathcal{N}=4$ super-Schwarzian path integral,
with appropriate boundary conditions for the disk and orbifold geometries
respectively. The final integral in (5.3) is given by the zero-temperature
limit of a particle moving on a circle. For black holes in the canonical
ensemble the path integral in (5.3) should be taken in a sector with fixed
$U(1)$ charge.

The volume of the space of gauge zero modes

We will first consider the path integral over the large gauge transformations,
$A_{\tau}=\partial_{\tau}\Lambda$, for the gauge fields in the vector
supermultiplets. At small temperatures the gauge transformations gain a non-
zero mass due to the electromagnetic boundary term $S_{\text{sugra,
bdy}}\supset\int_{\partial(\text{AdS}_{2}\times
S^{2})}d^{3}x\sqrt{h}A^{\mu}F_{\mu\nu}n^{\nu}$. This term simply evaluates to
[97, 98, 69]

$\displaystyle I_{\text{Particle on
}U(1)}=-\frac{T}{E_{U(1)}}\int_{0}^{2\pi}d\tau\,(\partial_{\tau}\Lambda(\tau))^{2}\,,$
(5.23)

where $T/E_{U(1)}$ is the coupling of the $U(1)$ nearly zero-mode where
$E_{U(1)}$ is determined by the background charge of the black hole. As we
shall see shortly, the exact value of $E_{U(1)}$ for the black holes that we
study in this paper is unimportant in the extremal limit.

While working in the grand canonical ensemble, with a chemical potential given
by $\mu_{U(1)}/T=\alpha_{U(1)}$, we need to impose the boundary condition
$\Lambda(\tau+2\pi)=\Lambda(\tau)+\alpha_{U(1)}$. The holonomy $\alpha_{U(1)}$
is conjugate not to the total charge of the higher dimensional $U(1)$ field,
but its fluctuation $\tilde{q}$ around the background value. However, since we
are interested in fixing the charges of the black hole for the gauge fields in
all the supermultiplets we should fix $\tilde{q}=0$. This can be achieved by
performing a Fourier transform in the holonomy associated to each gauge field,
setting $\tilde{q}=0$. Thus, the partition function for the theory of nearly
zero-modes associated to each gauge field is given by

$\displaystyle Z^{\text{Particle on }U(1)}(\beta,\tilde{q}=0)$
$\displaystyle=\int_{0}^{2\pi}d\alpha_{U(1)}\,Z(\beta,\alpha_{U(1)})$
$\displaystyle=\int_{0}^{2\pi}d\alpha_{U(1)}\int_{\frac{\text{Loop}(U(1))}{U(1)}}D\Lambda\,\,e^{-I_{\text{Particle
on }U(1)}[\Lambda,\alpha_{U(1)}]}\,,$ (5.24)

Since this is just the theory of a particle moving on a circle, (5.3) can be
computed easily. By taking the zero-temperature limit of the resulting
expression we obtain the following regularized volume of the space of gauge
zero modes

$\displaystyle\mathrm{Vol}\left[{\text{Loop}(U(1))}\right]=\lim_{T\to
0}\left[Z^{\text{Particle on }U(1)}(\beta,\tilde{q}=0)\right]=1\,.$ (5.25)

The volume of the space of disk zero modes

We now consider the path integral over the large super-diffeomorphisms on the
disk, which can be understood as fluctuations of the boundary curve inside
rigid AdS2 or as specific metric fluctuations with a fixed boundary. These
metric fluctuations are parametrized by a reparametrization mode $f(\tau)$,
where we label the boundary time by $\tau$. Similarly metric fluctuations
around $S^{2}$ by $g(\tau)\in SU(2)$ and fluctuations of the gravitino by
$\eta(\tau)$ and $\overline{\eta}(\tau)$. Together,
$(f,g,\eta,\overline{\eta})$ describe the elements of $\text{Diff}(S^{1|4})$
that we have to integrate in the first path integral in (5.3). At small but
non-zero temperature all these modes gain a non-zero mass just like in the
case of large gauge transformations discussed above. In [58], we show that the
action associated to these modes is given by that of the $\mathcal{N}=4$
super-Schwarzian theory,292929 The appearance of the $\mathcal{N}=4$ super-
Schwarzian should not come as a surprise. The same effective theory was shown
to arise when dimensionally reducing the 4D supergravity on $S^{2}$ in the
near-horizon region of near-BPS black holes to a 2D theory of $\mathcal{N}=4$
super-JT gravity [96].

$I_{\mathcal{N}=4\text{
Schw.}}=-\frac{T}{{E_{\text{gap}}}}\int_{0}^{2\pi}d\tau\left[\text{Sch}(f,\tau)+\mathrm{Tr}(g^{-1}\partial_{\tau}g)^{2}+({\rm
fermions})\right]\,.$ (5.26)

where ${E_{\text{gap}}}\sim\frac{1}{\ell_{\text{Pl}}\Delta^{3/2}}$. The ratio
$E_{\text{gap}}/T$ is identified as the coupling of the theory. At low
temperatures we indeed find that the effective theory is strongly coupled. The
nomenclature “$E_{\text{gap}}$” arises from the fact that this energy scale
was shown to give the gap scale between BPS states and the lightest non-BPS
states in the super-Schwarzian theory [96]. Nevertheless, just as in the
effective theory for large gauge transformations discussed above, the exact
value of $E_{\text{gap}}$ is unimportant if we are solely interested in the
degeneracy at extremality. Thus, to evaluate the first volume in (5.3) we will
take

$\displaystyle\mathrm{Vol}\left[\frac{\text{Diff}(S^{1|4})}{H_{\text{Disk}}}\right]=\lim_{T\to
0}\int_{\frac{\text{Diff}(S^{1|4})}{PSU(1,1|2)}}DfDgD\eta
D\overline{\eta}~{}{\rm e}^{-I_{\mathcal{N}=4\text{ super-Schw.}}}\,,$ (5.27)

To compute (5.27) we however need to specify the boundary conditions for the
fields $f(\tau)$, $g(\tau)$ and $\eta(\tau)$. These are given by

$\displaystyle f(\tau+2\pi)\;=\;f(\tau)\,,\qquad g(\tau+2\pi)\;=\;{\rm
e}^{{\rm i}2\pi\alpha\sigma_{3}}\,g(\tau)\,,\qquad\eta(\tau+2\pi)\;=\;-{\rm
e}^{{\rm i}{2\pi\alpha\sigma_{3}}}\eta(\tau)\,,$ (5.28)

with

$\displaystyle\alpha\;=\;\frac{{\beta\,\Omega_{E}}}{4\pi}\,,$ (5.29)

where $\Omega_{E}$ is the angular velocity on $S^{2}$.

The first boundary condition in (5.28) imposes that the interior of the AdS2
region is smooth. The second fixes that the black hole horizon rotates by an
angle $\beta\Omega_{E}$ when going around the boundary of AdS2,303030The
rotation is around an axis whose rotation generator we identify as
$\sigma_{3}$.. The third boundary condition, for the fermionic modes, comes
from identifying this angle of rotation as a chemical potential for the
rotational $SU(2)$ symmetry under which the fermionic fields are charged.
Since the gravitino and metric are related by $Q_{\text{eq}}$ the boundary
conditions should preserve supersymmetry which corresponds to computing an
index. This corresponds to setting $\alpha=1/2$, in which both the fermionic
boundary mode $\eta(u)$ in (5.28) is periodic. Nevertheless, in [43] it was
shown that even with these boundary conditions, bulk fermions (such as the
gravitino itself) are still anti-periodic around all contractible bulk cycles
which is necessary in order for the bulk to have a smooth spin structure.
Later, in Section 7, we will use the fact the path integral of these zero-
modes is computable for arbitrary $\alpha$ to show that, in fact, the
degeneracy and index of such black holes agree for all geometries that are
locally given by AdS${}_{2}\times S^{2}$.

Next, we want to find the saddle-points of (5.27), to start expanding the
fields $f$, $g$ $\eta$, and $\overline{\eta}$ around these saddles. Taking
(5.28) as our boundary conditions, we find an infinite number of solutions
which we label by $n\in\mathbb{Z}$,

$\displaystyle f(\tau)=\tan\left(\frac{\tau}{2}\right)\,,\qquad g(\tau)={\rm
e}^{i\tau\left[n+\alpha\right]\sigma_{3}}\,,\qquad\eta(\tau)=\overline{\eta}(\tau)=0\,.$
(5.30)

Because the space of field configurations
$\text{Diff}(S^{1|4})/H_{\text{Disk}}$ is symplectic, it turns out that
fluctuations in the path integral around each saddle in (5.30) are one-loop
exact. This yields a partition function [96]

$\displaystyle Z^{\mathcal{N}=4\text{
Schw.}}_{\text{Disk}}\left(\beta,\alpha\right)=\sum_{n\in\mathbb{Z}}\frac{{E_{\text{gap}}}}{T}\,\frac{2\cot(\pi\alpha)(\alpha+n)}{\pi^{3}(1-4(n+\alpha)^{2})^{2}}{\rm
e}^{2\pi^{2}\frac{T}{{E_{\text{gap}}}}\left(1-4(n+\alpha)^{2}\right)}.$ (5.31)

In the limit $\alpha\to 1/2$, all terms vanish with the exception of the $n=0$
and $n=-1$ saddles. In this case the sum of the two saddles gives

$\displaystyle Z^{\mathcal{N}=4\text{
Schw.}}_{\text{Disk}}\Big{(}\beta,\alpha=\frac{1}{2}\Big{)}=1\,,$ (5.32)

which is completely independent of $\beta$, as we would expect from the index.
Since we are computing an index in a supersymmetric theory, this temperature
independence should not come as a surprise. Thus, we find that the first
volume in (5.3) gets regularized to

$\displaystyle\mathrm{Vol}\left[\frac{\text{Diff}(S^{1|4})}{H_{\text{Disk}}}\right]=\lim_{T\to
0}\left[Z^{\mathcal{N}=4\text{
Schw.}}_{\text{Disk}}\left(\beta,\alpha=\frac{1}{2}\right)\right]=1\,.$ (5.33)

The volume of the space of orbifold zero modes

We now consider the path integral over the zero modes on the orbifolded
geometries. As explained above, on the orbifold we need to make the following
identification within the coordinate system (4.7),

$(\tau,\,\psi)\sim\Big{(}\tau+\frac{2\pi}{c},\,\psi+\frac{2\pi}{c}\Big{)}\sim\Big{(}\tau,\,\psi+2\pi\Big{)}\,.$
(5.34)

The identification (5.34) now dictates different boundary conditions for the
nearly zero-modes in (5.26) than those considered on the disk. We now have:

$\displaystyle
f(\tau+2\pi)=\frac{f(\tau)+\tan(\pi/c)}{1-f(\tau)\tan(\pi/c)}\,,\qquad
g(\tau+2\pi)={\rm e}^{2\pi i\alpha\sigma_{3}}\,g(\tau)\,{\rm e}^{2\pi
i\frac{2\pi}{c}\sigma_{3}}\,,\qquad\eta(\tau+2\pi)=-e^{2\pi
i\alpha\sigma_{3}}\eta(\tau)\,,$ (5.35)

where $\alpha$ once again corresponds to the chemical potential associated to
black hole rotation. The index is once again computed by taking the limit
$\alpha\to 1/2$. The effective theory arising by evaluating the gravitational
action at finite temperature remains the $\mathcal{N}=4$ super-Schwarzian
(5.26), unchanged from the case of the smooth geometry. Just like on the
smooth geometry, there are multiple saddles satisfying the equations of motion
of the $\mathcal{N}=4$ super-Schwarzian and satisfying the boundary conditions
(5.35). These saddles, that we label by $n$, come in pairs and are given by

$\displaystyle f(\tau)=\tan\left(\frac{\tau}{2c}\right)\,,\qquad g(\tau)={\rm
e}^{i\tau\left[n+\alpha\pm\frac{1}{2}\left(1-\frac{1}{c}\right)\right]\sigma_{3}}\,,\qquad\eta(\tau)=\overline{\eta}(\tau)=0\,,$
(5.36)

Using the fact that the path integral of the $\mathcal{N}=4$ super-Schwarzian
is one-loop exact, we can use (5.36) to find that the partition function with
a supersymmetric defect is given by

$Z_{\,(c)}^{\mathcal{N}=4\text{
Schw.}}(\beta,\alpha)=\sum_{n\in\mathbb{Z}}\frac{{E_{\text{gap}}}}{T}\frac{\cot(\pi\alpha)}{4\pi^{3}}\left(\frac{e^{2\pi^{2}\frac{T}{{E_{\text{gap}}}}\left[\frac{1}{c^{2}}-4(n+\frac{1}{2}-\alpha-\frac{1}{2c})^{2}\right]}}{(1-2\alpha+2n)^{2}}-\frac{e^{2\pi^{2}\frac{T}{{E_{\text{gap}}}}\left[\frac{1}{c^{2}}-4(n+\frac{1}{2}+\alpha-\frac{1}{2c})^{2}\right]}}{(1+2\alpha+2n)^{2}}\right)\,.$
(5.37)

When the partition function is computed for $\alpha\to 1/2$, the partition
function in (5.37) simplifies drastically since only the $n=0$ and $n=-1$
terms survive, giving

$\displaystyle Z_{\,(c)}^{\mathcal{N}=4\text{
Schw.}}\left(\beta,\alpha=\frac{1}{2}\right)=\frac{1}{c}$ (5.38)

at any temperature. Just like in the case of the disk, in Section 7 we will
confirm that the entire contribution to (5.38) comes from states with zero
angular momentum, indicating that for such geometries the degeneracy and index
agree. In summary, the integral over the zero-modes in a geometry with a
supersymmetric deficit at the origin is

$\mathrm{Vol}\left[\frac{\text{Diff}(S^{1|4})}{H_{\text{Orb}}}\right]=\lim_{T\to
0}\left[Z_{\,(c)}^{\mathcal{N}=4\text{
Schw.}}\left(\beta,\alpha=\frac{1}{2}\right)\right]=\frac{1}{c}\,.$ (5.39)

All of the above results confirm that all volumes are non-zero, which confirms
that, in contrast to non-supersymmetric black holes, $\frac{1}{8}$-BPS black
holes have a large degeneracy.

##  6 Putting it all together: $\frac{1}{8}$-BPS black hole index

Having analyzed the localization manifold, its action, and the one-loop
determinants around them including a careful treatment of boundary modes, we
can put everything together and reproduce the microscopic index of the
$\frac{1}{8}$-BPS black hole from supergravity.

There are four improvements over the previous results of [32] and its follow-
ups. Firstly, we work with the full $\mathcal{N}=8$ supergravity keeping all
vector multiplets, instead of using the truncation considered in [32], and we
work with arbitrary R-R charges $(q_{\Lambda},p^{\Lambda})$ in the type IIA
formulation. Note that the evaluation of the one-loop determinant even for
$c=1$ in [32] is incomplete and, as explained in [36], a more careful
treatment within this truncation leads to an answer inconsistent with the
large-charge limit. It is therefore crucial to consider the full theory.
Secondly, we work with arbitrary R-R charges $(q_{\Lambda},p^{\Lambda})$ in
the type IIA formulation and obtain a final black hole index, including all
charge-dependent prefactors. The resulting index is invariant under the
U-duality subgroup that shift only the R-R charges and thus provide a non-
trivial consistency check on the measure over the localization manifold
proposed in (3.21). Thirdly, we perform the exact path integral over the
moduli space of large super-diffeomorphisms. Finally, we perform the
calculation for the orbifolds $c\neq 1$ and verify the one-loop determinant
and the measure give the right $c$ dependence to reproduce the microscopic
Rademacher expansion of the index.

Now, we move on to the calculation. Firstly, we compute the action on the
localization manifold in terms of the prepotential given in (3.18) as
explained in Section 3.1. The result can be written in terms of the
intersection matrix and charges as

$\displaystyle I[\phi]$ $\displaystyle=$
$\displaystyle\frac{\pi\,q_{\Lambda}\,\phi^{\Lambda}}{c}-\frac{4\pi}{c}\,\mathop{\mathrm{Im}}{F\Big{(}\frac{\phi^{\Lambda}+{\rm
i}p^{\Lambda}}{2}\Big{)}}$ (6.1) $\displaystyle=$ $\displaystyle\frac{\pi
C_{IJK}p^{I}p^{J}p^{K}}{c\phi^{0}}-\frac{3\pi
C_{IJK}\phi^{I}\phi^{J}p^{K}}{c\phi^{0}}+\frac{\pi q_{0}\phi^{0}}{c}+\frac{\pi
q_{I}\phi^{I}}{c}.$ (6.2)

The fact that the action is quadratic in $\phi^{I}$ will be crucial later.

The second step is to compute the one-loop determinant, combining the
contribution from bulk and boundary modes found in Section 4.4 and regulating
the boundary zero-modes as explained in Section 5.3. First of all, the index
computed in Section 4.5 gives for $\mathcal{N}=8$ supergravity the
contribution
$Z_{\text{index},\,c}^{Q_{\text{eq}}\mathcal{V}}=e^{5\mathcal{K}}$. The full
non-zero mode determinant $Z_{\text{non-zero},\,c}^{Q_{\text{eq}}\mathcal{V}}$
(which combines the contribution of the index with that of the boundary non-
zero modes as in (5.13)) as well as the contribution of the zero modes
$Z_{\text{zero-mode},\,c}^{Q_{\text{eq}}\mathcal{V}}$ in (5.1) are given by

$\displaystyle Z_{\text{non-
zero},\,c}^{Q_{\text{eq}}\mathcal{V}}\;=\;\begin{cases}\;{\rm
e}^{5\mathcal{K}}\,{\rm e}^{-\frac{17}{2}\mathcal{K}}\,,&\quad c\;=\;1\,,\\\
\;{\rm e}^{5\mathcal{K}}\,{\rm e}^{-\frac{15}{2}\mathcal{K}}\,,&\quad
c\;>\;1\,,\end{cases},~{}~{}~{}Z_{\text{zero-
mode},\,c}^{Q_{\text{eq}}\mathcal{V}}\;=\;\begin{cases}\;\,{\rm
e}^{\frac{15}{2}\mathcal{K}}\,,&\quad c\;=\;1\,,\\\ \;\,\frac{1}{c}{\rm
e}^{\frac{13}{2}\mathcal{K}}\,,&\quad c\;>\;1\,.\end{cases}$ (6.3)

The full one-loop determinant is the product of these two contributions. The
final answer simplifies and can be written compactly for any $c$:

$\displaystyle Z^{Q_{\text{eq}}\mathcal{V}}_{\text{1-loop},\,c}(\phi)$
$\displaystyle=$ $\displaystyle\,Z_{\text{non-
zero},\,c}^{Q_{\text{eq}}\mathcal{V}}(\phi)\,Z_{\text{zero-
modes},\,c}^{Q_{\text{eq}}\mathcal{V}}(\phi)=\frac{e^{4\mathcal{K}}}{c},$
(6.4) $\displaystyle=$
$\displaystyle\frac{1}{c}\left(\frac{\phi^{0}}{-2C_{IJK}p^{I}p^{J}p^{K}}\right)^{4}.$
(6.5)

Upon putting together the action on the localization manifold, the one-loop
determinant, and the measure in (3.21), we obtain the full contribution of the
$\mathbb{Z}_{c}$ orbifolds to the index,313131Note that the measure and one-
loop determinants in $W^{(c)}(q,p)$ are independent of $\phi^{I}$ which will
allow us to easily perform the integral over these fields.

$\displaystyle W^{(c)}(q,p)$ $\displaystyle=$
$\displaystyle\sqrt{c}K_{c}(\Delta)\int{\rm
d}^{16}\phi~{}\sqrt{\frac{-2C_{IJK}p^{I}p^{J}p^{K}~{}{\rm
det}\left(3C_{IJ}\right)}{c^{16}(\phi^{0})^{18}}}\frac{1}{c}\left(\frac{\phi^{0}}{-2C_{IJK}p^{I}p^{J}p^{K}}\right)^{4}$
(6.6) $\displaystyle\times\exp{\left(-\frac{\pi
C_{IJK}p^{I}p^{J}p^{K}}{c\phi^{0}}+\frac{3\pi
C_{IJK}\phi^{I}\phi^{J}p^{K}}{c\phi^{0}}-\frac{\pi q_{0}\phi^{0}}{c}-\frac{\pi
q_{I}\phi^{I}}{c}\right)}\,.$

Here we have pulled out the Chern-Simons partition function
$\sqrt{c}K_{c}(\Delta)$ from the integral, as is independent of
$\phi^{\Lambda}$. We now integrate out the scalars $\phi^{I}$,
$I=1,\,\dots,\,15$ which have a Gaussian action in (6.6):

$\displaystyle\int d^{15}\phi~{}\exp\left(\frac{3\pi
C_{IJK}\phi^{I}\phi^{J}p^{K}}{c\phi^{0}}-\frac{\pi
q_{I}\phi^{I}}{c}\right)\;=\;\sqrt{-\frac{c^{15}(\phi^{0})^{15}}{{\rm
det}(3C_{IJ})}}\exp\left(-\frac{\pi}{12c}\phi^{0}q_{I}C^{IJ}q_{J}\right),$
(6.7)

The eigenvalues of $C_{IJ}$ are numbers fixed in terms of the charges which
can be positive of negative. We will see below the need to pick the contour
for the $\phi^{0}$-integral to be along the imaginary axis, both to make the
integral convergent and to match with the microscopic result. Therefore the
Gaussian integrals are oscillatory and not divergent.

Using this result we obtain the black hole index as a single integral over
$\phi^{0}$:

$\displaystyle W^{(c)}(q,p)\;=\;K_{c}(\Delta)\int
d\phi^{0}~{}\frac{1}{c}\frac{(\phi^{0})^{5/2}}{{\rm
i}(-2C_{IJK}p^{I}p^{J}p^{K})^{7/2}}\exp\left(-\frac{\pi
C_{IJK}p^{I}p^{J}p^{K}}{c\phi^{0}}-\frac{\pi\phi^{0}}{c}(q^{0}+\frac{1}{12}q_{I}C^{IJ}q_{J})\right),$
(6.8)

where we have identified $\Delta$ as in (3.19)
$\Delta=4C_{IJK}p^{I}p^{J}p^{K}\big{(}q^{0}+\frac{1}{12}q_{I}C^{IJ}q_{J}\big{)}$.
Performing the following change of variables $\sigma=\frac{-\pi
C_{IJK}p^{I}p^{J}p^{K}}{c\phi^{0}}$, we find that the integral (6.8) can be
finally rewritten in the same way it appears on the Rademacher expansion

$\displaystyle
W^{(c)}(q,p)\;=\;K_{c}(\Delta)~{}\frac{2\pi}{c^{9/2}}\left(\frac{\pi}{2}\right)^{7/2}~{}\frac{1}{2\pi{\rm
i}}\int\frac{d\sigma}{\sigma^{9/2}}\exp\left(\sigma+\frac{\pi^{2}\Delta}{4c^{2}\sigma}\right)\,.$
(6.9)

If we take a contour along the imaginary axis with ${\rm Re}(\sigma)>0$ then
we reproduce the integral representation of the Bessel function
$\widetilde{I}_{7/2}(z)$ appearing in the microscopic formula (2.7). As
explain in [32] the issues related to the precise integration contour of
$\phi^{\Lambda}$ can be traced to the indefiniteness of the Euclidean action
in gravity.

We see at this point a benefit of having kept all charges general. Since the
prefactors depend on charges it is a non-trivial result that the measure
proposed in (3.21) is such that all factors that would spoil the invariance
under the subgroup that acts on the 15 of the electric and magnetic RR-charges
cancel, resulting in a function only of $\Delta$.

In total we thus find from the localization calculation in supergravity that
the index of the $\frac{1}{8}$-BPS black hole is given by

$\displaystyle
W(q,p)=\sum_{c}W^{(c)}(q,p)=\sum_{c=1}^{\infty}\frac{2\pi}{c^{9/2}}\left(\frac{\pi}{2}\right)^{7/2}~{}K_{c}(\Delta)\tilde{I}_{7/2}\Big{(}\frac{\pi\sqrt{\Delta}}{c}\Big{)}=W_{\text{micro}}(\Delta)\,,$
(6.10)

and it precisely matches the Radamacher expansion of the microscopic index
obtained in (2.7).

##  7 The index vs. partition function:
why only orbifolds contribute to both

So far we have shown that the index obtained from the gravitational integral
using localization precisely matches the index obtained from the microstate
count of state in the D1/D5 construction. We would now like to argue that the
supergravity path integral implies that this index exactly agrees with the
actual degeneracy of $\frac{1}{8}$-BPS black holes, i.e. all extremal black
hole states are in fact bosonic. This statement applies to all configurations
that are asymptotically $AdS_{2}\times S^{2}$, and not necessarily to degrees
of freedom that live outside of this region, since those are out of our
computational control. In particular, a subtlety we will not address is how
the microscopic degeneracy of the D-brane system, whose degrees of freedom are
not clearly located in the $AdS_{2}\times S^{2}$ region, is related to the
black hole degrees of freedom.

These issues were addressed in [30]. The new ingredients in this section are
to carefully incorporate the large super-diffeomorphisms into this analysis
and show that no other geometry contributes to the degeneracy while having
vanishing index. This leads to a refined version of the previous arguments
made in the literature.

We first show the contribution of the gravitational path integrals in the
grand canonical ensemble when fixing the angular velocity to that
corresponding to the index (with $\Omega_{E}\beta=2\pi$) and in the canonical
ensemble with fixed angular momentum (with $J=0$), agree at zero temperature

$\displaystyle Z_{\frac{1}{8}\text{-BPS
BH},\,\Sigma}^{T=0}\left(\Delta,\Omega_{E}=\frac{2\pi}{\beta}\right)=Z_{\frac{1}{8}\text{-BPS
BH},\,\Sigma}^{T=0}\left(\Delta,J=0\right)\,,$ (7.1)

when expanding about a large class of geometries, with each geometry $\Sigma$
being locally given by AdS${}_{2}\times S^{2}$. (7.1) would not only indicate
that the index is equal to the partition function but also that all
$\frac{1}{8}$-BPS black hole states carry only zero spin. We will start by
first showing that (7.1) holds for manifolds $\Sigma$ that preserve
supersymmetry and that we have considered in our localization computation—the
smooth AdS2 geometry and its orbifolds. We will then show that for geometries
$\Sigma$ that do not admit the existence of a globally defined killing spinor,
such as manifolds of higher genus or with multiple defects, both terms in
(7.1) vanish due to path integral over large super-diffeomorphisms.

If studying the partition function instead of the index the only difference
are the boundary conditions that we impose at the edge of AdS${}_{2}\times
S^{2}$. In a co-rotating frame the circle at the boundary of AdS2 is
contractible, and therefore all fermionic fields are anti-periodic when going
around it for the computation of the index as well as the partition function.
The only difference in boundary conditions in such a case is for the large
super-diffeomorphism modes which fix the angular velocity of the black hole.
As we’ve seen above when computing the index we fixed $\Omega_{E}=2\pi/\beta$
while to compute the partition function we need $\Omega_{E}=0$. To extract
more information about what angular momenta contribute to the degeneracy
(whether states with $J\neq 0$ contribute), we should further keep
$\Omega_{E}$ arbitrary.

At zero-temperature, both the smooth AdS${}_{2}\times S^{2}$ geometry and the
orbifolds considered above have globally well-defined killing spinors which we
can use to define the supercharge $Q_{\text{eq}}$ with respect to which we
localize. Following the procedure discussed in Sections 2 and 4 and using the
fact that the boundary conditions for the bulk modes remain unchanged, the
path integral over such modes also yields the same result. Thus, what remains
to be done in order to confirm (7.1) is to perform the path integral over
large super-diffeomorphisms with arbitrary $\Omega_{E}$. In the case of smooth
AdS${}_{2}\times S^{2}$ geometry, taking the $T\to 0$ limit of (5.31) we find
(with $\alpha=\frac{\Omega_{E}\beta}{4\pi}$),

$\displaystyle\lim_{T\to 0}Z^{\mathcal{N}=4\text{
Schw.}}_{\text{Disk}}\left(\beta,\alpha\right)=1\,,\qquad\forall\,\,\,\,\alpha\,.$
(7.2)

Similarly, for the orbifold geometries, taking the $T\to 0$ limit of (5.37) we
find that

$\displaystyle\lim_{T\to 0}Z^{\mathcal{N}=4\text{
Schw.}}_{\text{Orb}}\left(\beta,\alpha\right)=\frac{1}{c}\,,\qquad\forall\,\,\,\,\alpha\,.$
(7.3)

From doing the Fourier transform in $\alpha$ for arbitrary $J$, this implies
that in the canonical ensemble for angular momenta, for the two types of
geometries that contribute to the index

$\displaystyle\lim_{T\to 0}Z^{\mathcal{N}=4\text{
Schw.}}_{\text{Disk}}\left(\beta,J\right)=\delta_{J,0}\,,\qquad\lim_{T\to
0}Z^{\mathcal{N}=4\text{
Schw.}}_{\text{Orb}}\left(\beta,J\right)=\frac{\delta_{J,0}}{c}\,.$ (7.4)

Thus, because we are able to localize the bulk modes and because the integral
over the boundary modes remains unchanged, we conclude that for all the
geometries that preserve supersymmetry

$\displaystyle Z_{\frac{1}{8}\text{-BPS BH},\text{
disk/orb}}^{T=0}\left(\Delta,\Omega_{E}=\frac{2\pi}{\beta}\right)=Z_{\frac{1}{8}\text{-BPS
BH},\text{disk/orb}}^{T=0}\left(\Delta,J=0\right)\,.$ (7.5)

We now consider the path integral over the zero modes on geometries that do
not preserve supersymmetry. We will start with the case of orbifolds that do
not preserve supersymmetry; for these, the angle of the defect is not
identified with the angle of rotation around $S^{2}$ when going around the
defect. Taking the coordinates of AdS${}_{2}\times S^{2}$ as in (4.7), we
consider the more general identification

$(\tau,\,\psi)\sim\Big{(}\tau+\frac{2\pi}{c},\,\psi-4\pi\varphi\Big{)}\sim\Big{(}\tau,\,\psi+2\pi\Big{)}\,,$
(7.6)

where $\varphi$ parametrizes the angle of rotation of $S^{2}$ as one goes
around the defect, and, unlike the supersymmetric case, we will take
$2\varphi\neq 1-1/c$. The identification (7.6) now dictate new boundary
conditions for the nearly zero-modes in (5.26), with $g(\tau+\beta)={\rm
e}^{i2\pi\alpha\sigma_{3}}\,g(\tau)\,{\rm e}^{-i{2\pi\varphi}\sigma_{3}}$.

For generic values of $\varphi$, the partition function in the grand canonical
ensemble is [43]

$Z^{\mathcal{N}=4\text{
Schw.}}_{(c,\varphi)}(\beta,\,\alpha)=\sum_{n\in\mathbb{Z}}\frac{T}{{E_{\text{gap}}}}\frac{\cot(\pi\alpha)}{2\pi}\left({\rm
e}^{2\pi^{2}\frac{T}{{E_{\text{gap}}}}\left(\frac{1}{c^{2}}-4\left(n-\alpha+\varphi\right)^{2}\right)}-e^{2\pi^{2}\frac{T}{{E_{\text{gap}}}}\left(\frac{1}{c^{2}}-4\left(n+\alpha+\varphi\right)^{2}\right)}\right)\,.$
(7.7)

Contributions of these defects to both the index and the degeneracy vanish. A
straightforward way to observe it is that now the overal factor of $T$ is in
the numerator and therefore $Z_{c,\varphi}(\beta\to\infty,\alpha)\to 0$ for
all $\alpha$. We thus conclude that orbifolds that do not preserve
supersymmetry

$\displaystyle\lim_{T\to 0}Z_{\frac{1}{8}\text{-BPS
BH},{(b,\varphi)}}^{T=0}\left(\Delta,J\right)=0\,,$ (7.8)

for all angular momenta $J$.

Next we consider higher genus geometries that could also possibly have defects
on them. If we want to be able to smooth out the geometry by considering its
fibration over the additional $S^{1}$ that we used to make the supersymmetric
orbifolds smooth in 5D, then the defect angle of all such defects needs to be
quantized with $2\pi\varphi_{i}=2\pi(1-1/c_{i})$ with $c_{i}\in\mathbb{Z}$,
for each defect $i\in\\{1,\dots,m\\}$. All such geometries have a closed
geodesic separating the asymptotic boundary from the higher genus multi-defect
component.323232This is the case as long as $g\geq 1$ and $m\geq 0$ or when
$g\geq 0$ and $m\geq 2$. We have already considered the case $g=0$ and $m=1$
in (7.8). This changes the boundary conditions for the large super-
diffeomorphism modes to

$\displaystyle f(\tau+2\pi)={\rm e}^{\frac{b}{2}}f(\tau)\,,\qquad
g(\tau+2\pi)={\rm e}^{2\pi i\alpha\sigma_{3}}\,g(\tau)\,{\rm e}^{2\pi
i\varphi\sigma_{3}}\,,\qquad\eta(\tau+2\pi)=-e^{i2\pi
i\alpha\sigma_{3}}\eta(\tau)\,.$ (7.9)

where $b$ parametrizes the length of the closed geodesic that is homotopic to
the asymptotic boundary. The space between the asymptotic boundary and this
geodesic is usually referred to as a trumpet. Above, $\varphi$ parametrizes
the angle of rotation of $S^{2}$ as one goes around the closed geodesic of the
trumpet. The path integral over these modes can now be computed to be [43]

$\displaystyle\lim_{T\to
0}Z_{\text{Trumpet}\,(b,\varphi)}^{\mathcal{N}=4\text{
Schw.}}\left(\beta,\alpha\right)=0\,,\qquad\forall\,\alpha\,,$ (7.10)

in the limit $T\to 0$, and upon taking the Laplace transform of the partition
function for general $T$ (which is the same as (7.7) upon replacing $1/c\to-
ib$) with respect to $\beta$ we find a continuous density of states that
starts at the energy ${E_{\text{gap}}}$ above the extremal energy
$E_{\frac{1}{8}\text{-BPS BH}}\sim\Delta^{1/4}$,

$\displaystyle\rho_{\text{trumpet},\,(b,\varphi)}\sim\Theta(E-E_{\frac{1}{8}\text{-BPS
BH}}-{E_{\text{gap}}})\,.$ (7.11)

We can now glue back the trumpet to a higher genus surface. To do this we in
principle need to include the contribution of matter fields (which we denote
by $Z^{\text{matter}}_{(b,\varphi)}$) as well as the integral over the moduli
space for a surface of genus $g$ with $m$ defects that is glued to the trumpet
(which we we denote by $\mathrm{Vol}_{g,m}$). While in principle the
contribution of matter fields is divergent for small values of the trumpet
geodesic length [1] and the volume of the moduli space for such surfaces is
unknown for the supersymmetry amount that we are interested in, (7.11) implies
that in the strict $T\to 0$ limit of partition function we have333333(7.10) is
insufficient to show that the partition function vanishes. For example, in the
non-supersymmetric theory, the partition function on the trumpet still
vanishes as $T\to 0$. Nevertheless, in the absence of matter, due to the fact
that $\mathrm{Vol}_{g,m}(b)$ are polynomial in $b$, the partition function for
surfaces whose genus is higher than $1$ is divergent as $T\to 0$. On the other
hand, (7.11) is a feature, albeit not obviously, of supersymmetry and is
sufficient to show that the partition function is indeed vanishing at zero
temperature.

$\displaystyle\lim_{T\to 0}\int
d\mu[b]\,d\mu[\varphi]\,Z_{\text{Trumpet}\,(b,\varphi)}^{\mathcal{N}=4\text{
Schw.}}\left(\beta,\,J\right)Z^{\text{matter}}_{(b,\varphi)}\,\mathrm{Vol}_{g,m}=0\,,\qquad$
(7.12)

for all angular momenta $J$. Now that we have seen that the partition function
of arbitrary non-supersymmetric orbifolds (7.8) and the partition function of
surfaces with arbitrary genus and an arbitrary number of defects (7.12) all
vanish, we conclude that no other geometries that are locally AdS${}_{2}\times
S^{2}$ contribute to the degeneracy. For this reason, we find that the
gravitational index computed in Section 6 (which we have denoted by $W(p,q)$)
is the same as the actual degeneracy of a single black hole (which we denote
by $d_{\frac{1}{8}\text{-BPS BH}}$):

$\displaystyle W(p,q)=Z_{\frac{1}{8}\text{-BPS
BH}}^{T=0}\left(\Delta,\,\,\Omega_{E}\beta=2\pi\right)=Z_{\frac{1}{8}\text{-BPS
BH}}^{T=0}\left(\Delta,\,\,\Omega_{E}=0\right)=d_{\frac{1}{8}\text{-BPS
BH}}\,.$ (7.13)

Since $d_{\frac{1}{8}\text{-BPS BH}}$ is not a protected quantity on the
string theory side, (7.13) is not necessarily valid along the entire moduli
space of the D1D5 CFT that is used to describe the black hole microstates. In
fact, at the orbifold point it is known that the degeneracy of BPS states is
much larger than the index, implying a large cancellation between bosonic and
fermionic degeneracies. This degeneracy between bosonic and fermionic states
is expected to be lifted when going away from the orbifold point in order for
(7.13) to be valid, and was explicitly verified in various examples in [30].
We discuss how this expectation is consistent with the path integral
description of near-BPS states in more detail in the following discussion.

##  8 Discussion and future directions

We have found that the the index of $\frac{1}{8}$-BPS states obtained in
string theory is the same as the index obtained from localizing the
gravitational path integral. On a technical level, this match is a culmination
of two directions of research. The first is the development of localization in
supergravity [30, 31, 32, 33, 34, 35, 36], and in particular the formalism for
BRST quantization of supergravity initiated in [37, 38] which we extensively
used to set up and calculate the functional integral. The second direction of
research which we have heavily relied upon has been that of the
(super)Schwarzian [99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110,
96, 69, 111, 112, 113]. Above, we have emphasized the role of this quantum
mechanical system in the proper quantization of the boundary modes. Without
the exact contribution of both the bulk modes and of the $\mathcal{N}=4$
super-Schwarzian around all the different geometries and field configurations
that we localized to, it would have been impossible to obtain the exact match
between the microscopic string theory computation and the macroscopic path
integral calculation.

Below we shall discuss how the above match, together with recent results in
the literature [96, 43], offer a comprehensive view for the quantum mechanical
properties of BPS black holes and, to a lesser (but still meaningful) extent,
near-BPS black holes. We also discuss a number of different future directions
of research.

###  8.1  BPS states vs. non-BPS states

In the introduction we motivated our work by asking whether the gravitational
path integral is sufficiently powerful to reproduce the exact number of black
hole microstates within an energy window. The results in Sections 3 to 6 show
that by localizing gravitational path integral the difference in bosonic and
fermionic degeneracies can be obtained exactly using localization. The fact
that we have managed to also compute the degeneracy (which is not obviously
protected by supersmmetry from the perspective of a putative boundary theory)
is a non-trivial fact. Below, we shall explain how this calculation can be
further extended to compute the total number of states within an energy window
that starts at extremality.

As discussed in Section 7, what is perhaps more surprising is that the
gravitational path integral allows us to probe observables (at strong string
coupling, away from the orbifold point of the D1D5 system) that are
unprotected, i.e. the exact degeneracy of states at the BPS energy level in a
sector of fixed charge and angular momentum. This is because from the
perspective of the gravitational path integral, the same supersymmetry
preserving geometries were shown to contribute to both the index and to the
degeneracy, while geometries preserving no supersymmetries contributed to
neither. The path integral over the nearly zero-modes of the super-Schwarzian
reveals even more if studying the density of states of such a system rather
than its partition function. For the geometries that preserve some amount of
supersymmetries the integral over the nearly zero modes gives, in addition to
the degenerate number of states at the BPS energy, a continous density of
states that starts at energies $E=E_{\frac{1}{8}\text{-BPS
BH}}+{E_{\text{gap}}}$ [96, 43]:

$\displaystyle\rho_{\text{Disk},\,(c,\varphi)}^{\text{cont.}}\,\sim\,\Theta(E-E_{\frac{1}{8}\text{-BPS
BH}}-{E_{\text{gap}}})\,,\qquad\rho_{\text{Orbifold},\,c}^{\text{cont.}}\,\sim\,\Theta(E-E_{\frac{1}{8}\text{-BPS
BH}}-{E_{\text{gap}}})\,,$ (8.1)

where for the black holes discussed above, we remind the reader that [96]

$\displaystyle{E_{\text{gap}}}\,\sim\,\frac{1}{\Delta^{3/2}}\,,\quad\Delta\to\infty$
(8.2)

up to stringy corrections which are subleading in $1/\Delta$. Similarly, on
geometries preserving no supersymmetries the integral over nearly zero modes
yields [43]

$\displaystyle\rho_{\text{orbifold},\,(c,\varphi)}\,\sim\,\Theta(E-E_{\frac{1}{8}\text{-BPS
BH}}-{E_{\text{gap}}})\,,\qquad\rho_{\text{trumpet},\,(b,\varphi)}\,\sim\,\Theta(E-E_{\frac{1}{8}\text{-BPS
BH}}-{E_{\text{gap}}})\,,$ (8.3)

In other words (8.1) and (8.3) indicate that the spectrum of near-BPS
microstates only starts at $E_{\frac{1}{8}\text{-BPS
BH}}+{E_{\text{gap}}}\Bigl{(}1+\text{O}\bigl{(}1/\Delta\bigr{)}\Bigr{)}$,
where one could imagine the correction to ${E_{\text{gap}}}$ arises, via a
mechanism similar to [9], from summing geometries with arbitrary topologies
obtained by gluing the answer in (8.3) as explained in Section 7. Going back
to the question that we posed in the introduction, we can expect that not only
is $d_{\frac{1}{8}\text{-BPS BH}}$ the degeneracy of BPS black holes, but
additionally343434Here, we assume that doubly non-perturbative effects in
$\Delta$, which contribute beyond the topological expansion which we have
considered when summing geometries that contain a “trumpet” geometry, might
yield a non-zero number of states between $E_{\frac{1}{8}\text{-BPS BH}}$ and
$E_{\frac{1}{8}\text{-BPS BH}}+\\#{E_{\text{gap}}}$. It would nevertheless be
valuable to understand whether this is indeed the case.

$\displaystyle\int_{E_{\frac{1}{8}\text{-BPS BH}}}^{E_{\frac{1}{8}\text{-BPS
BH}}+\\#E_{\text{gap}}}dE\rho(E)\;=\;d_{\frac{1}{8}\text{-BPS BH}}\,.$ (8.4)

for any number $\\#$ that is smaller than $1$ by an amount that does not scale
downwards with $1/\Delta$ as $\Delta$ is made large. As $\\#$ becomes larger
than 1 the path integral around any given saddle as well as the topological
expansion becomes out of control and we find can only estimate the total
number of states rather than recover an exact integer as in (8.4).
Nevertheless, for $E>E_{\frac{1}{8}\text{-BPS BH}}+E_{\text{gap}}$ the
continuum density of states found in [96] obtained from the expansion around
the leading AdS${}_{2}\times S^{2}$ offers a good course-grained approximation
to the densely spaced discrete spectrum of black hole mirostates that we
expect to exist for such energy values. Understanding how to resolve this
discretum requires full control over both stringy corrections and over the sum
of all geometries, a daunting task since the geometries in question no longer
preserve any supersymmetries.

The contrast in the cases with $\\#<1$ and $\\#>1$ is also captured by looking
at multi-boundary contributions to the gravitational path integral for the
total number of states between $E_{\frac{1}{8}\text{-BPS BH}}$ and
$E_{\frac{1}{8}\text{-BPS BH}}+\\#E_{\text{gap}}$. There are no connected
multi-boundary contributions to the number of states when $\\#<1$ and thus the
quantity (8.4) does not suffer from the factorization puzzle that affects
other observables—this is of course consistent with the fact that
$d_{\frac{1}{8}\text{-BPS BH}}\in\mathbb{Z}$ which points towards a
conventional Hilbert space interpretation. Counting states for $\\#>1$ (8.3)
shows that wormhole contributions exist, pointing towards the fact that
individual microstates cannot be distinguished but their chaotic energy
statistics might be computable through the contribution of such connected
geometries.

Finally let us comment on the role that stringy corrections can play in the
statements made above. In [113], it was shown that $E_{\text{gap}}$ receives
stringy corrections where higher derivative terms are include in the
supergravity action.353535In [113] the stringy corrections to $E_{\text{gap}}$
were determined in the context of near-BPS black holes in AdS${}_{5}\times
S^{5}$ with a broken $SU(1,1|1)$ near-horizon isometry. While it would be
informative to redo that analysis for the black holes discussed in this paper
which have a $PSU(1,1|2)$ at extremality, we expect that the role of stringy
correction on $E_{\text{gap}}$ in the two cases to be qualitatively similar.
Stringy corrections are not of concern however for the gravitational index—we
expect the value of the supergravity action along the localization locus to be
independent of the higher-derivative correction. Furthermore, we saw that the
integral over boundary zero-modes ended up being completely independent of
$E_{\text{gap}}$ for all geometries. Therefore we only need to be careful
about the upper bound of the integral in (8.4) which does receive stringy
corrections. At weak string coupling, we expect that $E_{\text{gap}}\to 0$ in
order to find agreement with the D1-D5 system at the orbifold point for which
there is a large difference between the degeneracy of states at the BPS energy
and the index of such states. A related issue is that at weak string coupling
we do not really know how to distill the would-be black hole degrees of
freedom from the rest of the D-brane quantum mechanics.

###  8.2  BPS black holes have a Type I von Neumann algebra

Understanding the algebra of observables in quantum gravity has been a problem
of recent interest [114, 115, 116, 117]. Recent results show that
semiclassical gravity can promote an algebra from type III to type II, and the
results in this paper indicate that observables for BPS black holes in fact
can be promoted to a type I algebra. While the fact that the gravitational
path integral reproduces an exact integer as $T\to 0$ meet seem sufficient to
claim a type I algebra, it is useful to verify that this integer indeed has
the interpretation of the dimension of the Hilbert space of BPS black holes.
To show this, we need to be able to be able to define density matrices in this
space and show that the rank of all matrices is finite (and, importantly,
integral). For this, we will check that the maximally mixed density matrix
indeed has a size equal to the degeneracy $d_{\frac{1}{8}\text{-BPS BH}}$
discussed in the previous section. We construct the thermofield double state
at zero temperature $|{\text{TFD}(T=0)}\rangle$ in a sector with fixed
charges, define its associated pure density matrix, and the reduced density
matrix on one of the sides (say, the left side),

$\displaystyle\rho_{\text{TFD}}\;=\;|{\text{TFD}(T=0)}\rangle\langle{\text{TFD}(T=0)}|\,,\quad\rho_{L}\;=\;\mathrm{Tr}_{R}(\rho_{\text{TFD}})\,.$
(8.5)

The Renyi entropy $S_{\alpha}$ associated to $\rho_{L}$ is obtained by gluing
$\alpha$ copies of a boundary segments, each with an infinite length to form a
closed boundary that also has $\beta\to\infty$. Filling in the bulk with the
geometries that contribute to the fixed charge partition function in the limit
of $\beta\to\infty$,

$\displaystyle S_{\alpha}$
$\displaystyle=\frac{1}{1-\alpha}\frac{\mathrm{Tr}\,(\rho_{L})^{\alpha}}{(\mathrm{Tr}\,\rho_{L})^{\alpha}}\;=\;\frac{1}{1-\alpha}\,\,\Biggl{[}{\overbrace{\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@transformcm{0.45}{0.0}{0.0}{0.45}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.35}]{Figures/loc3.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}^{\alpha\text{ copies
of segments of proper length }\beta}}\bigg{/}\,\,\,\,{\left(\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}\hbox{{\includegraphics[scale={0.35}]{Figures/loc4.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)^{\alpha}}\Biggr{]}$
$\displaystyle\;=\;\frac{\left(\leavevmode\hbox to0pt{\vbox
to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.45}{0.0}{0.0}{0.45}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.35}]{Figures/loc1.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\,\,\,+\,\,\,\leavevmode\hbox
to0pt{\vbox to0pt{\pgfpicture\makeatletter\hbox{\hskip 0.0pt\lower 0.0pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}}
{{{}}{{}}{{}}{{}}{{}}{{}}{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.45}{0.0}{0.0}{0.45}{0.0pt}{0.0pt}\pgfsys@invoke{
}\hbox{{\includegraphics[scale={0.35}]{Figures/loc2.pdf}}}\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}at (0,0); \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{
{}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right)^{1-\alpha}}{1-\alpha}\;=\;\frac{(d_{\frac{1}{8}\text{-BPS
BH}})^{1-\alpha}}{1-\alpha}\,.$ (8.6)

Here the third equality follows from the fact that the same geometries
contribute to both the numerator and denominator in the limit
$\beta\to\infty$. This confirms that $\rho_{L}$ is the maximally mixed density
matrix for the finite $d_{\frac{1}{8}\text{-BPS BH}}$-dimensional Hilbert
space of BPS black holes.363636 For a discussion involving matter correlators
see [118, 119].

An important point to emphasize is that in the path integral preparation of
the zero-temperature thermofield double state, the inclusion of orbifold
geometries in the bulk is crucial in order to recover a Renyi entropy that is
consistent with a maximally mixed state in a finite dimensional. One can
therefore wonder whether the presence of the orbifolds in the state
preparation plays an important role for other observables in the theory and,
in particular, whether one can construct observables for which the orbifold
geometries become dominant.

###  8.3  Future directions

Before jumping into possible future directions we would like to make a list of
technical questions regarding the calculations in this paper which would be
good to address in the future:

  * •

Perhaps the most important one is to find a principle that determines uniquely
the correct measure over the localization manifold. Our choice of ultra-local
measure turned out to be consistent with the U-duality invariance that is
known to exist in the UV string theory completion but is not in any way
explicit at the level of the AdS${}_{2}\times S^{2}$ supergravity path
integral. While we believe that this provides strong evidence for the validity
of our choice of ultra-local measure, it would be highly satisfying to give a
first-principles derivation of (3.20).

  * •

It is possible to give a 4d interpretation of the terms in the Kloosterman sum
as describing the topology of certain gauge fields. The evaluation of the
action and in particular the phases involved in the Kloosterman sum was done
by lifting from 4d to 5d and then reducing down to 3d Chern-Simons [34]. It
would be nice to compute the action in a fully 4d formalism to make the whole
calculation self-consistent.

  * •

We have done the calculation in a type IIA frame with only D0-D2-D4 charge. It
would be nice to extend the calculation to allow for D6 charge.

  * •

We have used an $\mathcal{N}=2$ formalism to treat $\mathcal{N}=8$
supergravity. This works as long as we study states with vanishing charge
under the NS-NS gauge fields in the gravitino multiplets, coming from the
isometries of $T^{6}$. It would be nice to extend this calculation to non-zero
charge and verify the full U-duality invariance.

The technical steps necessary for performing the localization computation rely
on several steps, none of which require working solely with the
$\mathcal{N}=8$ supergravity theory, and should be applicable in theories with
less supersymmetry. For instance, in theories of $\mathcal{N}=4$ supergravity
obtained by a dimensional reduction on $T^{2}\times K3$, one might hope to
reproduce the exact degeneracy of $\frac{1}{4}$-BPS and $\frac{1}{2}$-BPS
black holes. In the former case one has to remove the contribution of multi-
center black holes from the index. The resulting index for single-center black
holes is the Fourier coefficient of a mock Jacobi form [120], whose Rademacher
expansion is more complicated than the one for true Jacobi forms that we
considered in this paper [121, 122]. In the latter case, while the Rademacher
expansion of the associated modular form is, in fact, simpler than the one
studied in this paper, the difficulty is that $1/2$-BPS black holes have a
small horizon area and it is therefore difficult to define the gravitational
path integral to start with. For theories of $\mathcal{N}=2$ supergravity,
finding the exact index of black holes is even more difficult [56]—in such
cases the prepotential can receive corrections at arbitrary loop order and an
exact formula is unknown.

In this paper we have shown that the gravitational functional integral in AdS2
takes the form of the infinite series (1.1) which reproduces the convergent
microscopic expansion (2.7). The elements of the sum in the gravitational
picture are interpreted as contributions from a series of orbifolds. For
AdS3/CFT1 a very similar series of orbifolds are also known on both sides
[42]. In fact, such a series of orbifolds is also known to contribute in
higher dimensional AdS${}_{d}+1$/CFTd situations for $d=3,4$ [123, 124, 125,
126]. It would be interesting to make the link to the microscopic indices more
precise in all these cases.

The fact that we have managed to fully localize the gravitational path
integral for BPS black holes makes us hopeful that we might be able to perform
this procedure to match a variety of boundary quantities that are protected by
supersymmetry and do not rely on the asymptotics of the spacetime being
AdS${}_{2}\times S^{2}$. Two examples come immediately to mind. Firstly, as
mentioned above, it is known that multi-center center solutions can contribute
to the $1/4$-BPS index in $\mathcal{N}=4$ supergravity. It would be
interesting to understand whether the contribution of such configurations to
the gravitational path integral can also be computed from asymptotic flat
space and matched to the string theory answer. Secondly, one could also hope
to replicate the computation for spacetimes that are asymptotically
AdS${}_{d}\times\mathcal{M}$ with the hope of matching quantities that are
protected by supersymmetry in a boundary SCFTd-1. Progress in this direction
has been made in [127, 128], where the $S^{3}$ partition function of a
boundary theory has been matched by performing the localization procedure in
an asymptotically AdS4 bulk. Now, armed with a better understanding of how to
compute one-loop determinants around the localization saddles, we hope to
address a much greater variety of examples in near future.

### Acknowledgements

We thank Dan Butter, Alejandra Castro, Clay Cordova, Atish Dabholkar, Rajesh
Gupta, Zohar Komargodski, Juan Maldacena, Boris Pioline, Ashoke Sen, Stephen
Shenker, and Douglas Stanford for valuable discussions. GJT is supported by
the Institute for Advanced Study and the National Science Foundation under
Grant No. PHY-1911298, and by the Dipal and Rupal Patel funds. LVI was
supported by the Simons Collaboration on Ultra-Quantum Matter, a Simons
Foundation Grant with No. 651440. This work is supported by the ERC
Consolidator Grant N. 681908, “Quantum black holes: A macroscopic window into
the microstructure of gravity”, and by the STFC grant ST/P000258/1. This work
was performed in part at the Aspen Center for Physics, which is supported by
National Science Foundation grant PHY-1607611.

##  Appendix A A brief summary of Jacobi forms and the Rademacher expansion

We set $q={\rm e}^{2\pi{\rm i}\tau}$ and $\zeta={\rm e}^{2\pi{\rm i}z}$ with
$\tau\in\mathbb{H}$, $z\in\mathbb{C}$. The odd Jacobi theta function is given
by

$\displaystyle\vartheta_{1}(\tau,z)\;=\;\sum_{n\in\mathbb{Z}+\frac{1}{2}}(-1)^{n}\,q^{n^{2}/2}\,\zeta^{n}\,.$
(A.1)

The Dedekind eta function is

$\displaystyle\eta(\tau)\;=\;q^{\frac{1}{24}}\prod_{n=1}^{\infty}(1-q^{n})\,.$
(A.2)

We recall below a few relevant facts about Jacobi forms [129]. A Jacobi form
of weight $k$ and index $m$ is a holomorphic function $\varphi(\tau,z)$ from
$\mathbb{H}\times\mathbb{C}$ to $\mathbb{C}$ which transforms under the
modular group as

$\displaystyle\varphi\Bigl{(}\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\Bigr{)}\;=\;(c\tau+d)^{k}\,{\rm
e}^{\frac{2\pi
imcz^{2}}{c\tau+d}}\,\varphi(\tau,z)\qquad\forall\quad\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in SL(2;\mathbb{Z})$ (A.3)

and under the translations of $z$ by $\mathbb{Z}\tau+\mathbb{Z}$ as

$\displaystyle\varphi(\tau,z+\lambda\tau+\mu)\;=\;{\rm e}^{-2\pi
im(\lambda^{2}\tau+2\lambda
z)}\varphi(\tau,z)\qquad\forall\quad\lambda,\,\mu\in\mathbb{Z}\,.$ (A.4)

Here the integers $k,m$ are called the weight and index, respectively, of the
Jacobi form. The equations (A.3), (A.4) imply the periodicities
$\varphi(\tau+1,z)=\varphi(\tau,z)$ and $\varphi(\tau,z+1)=\varphi(\tau,z)$,
so that $\varphi$ has a double Fourier expansion

$\displaystyle\varphi(\tau,z)\;=\;\sum_{n,r\in\mathbb{Z}}c(n,r)\,q^{n}\,\zeta^{r}\,.$
(A.5)

The function is called a _weak_ Jacobi form if $c(n,r)=0$ unless $n\geq 0$.
Equation (A.4) is then equivalent to the periodicity property

$\displaystyle c(n,r)\;=\;C_{r}(4nm-r^{2})\
,\qquad\mbox{where}\;C_{r}(\Delta)\;\mbox{depends only
on}\;r\,\text{mod}\,2m\,.$ (A.6)

We have, for a Jacobi form $\varphi$ of weight $k$ and index $m$, the double
Fourier expansion

$\displaystyle\varphi(\tau,z)\;=\;\sum_{n,\ell\,\in\,\mathbb{Z}}c(n,\ell)\,q^{n}\,\zeta^{\ell}\,.$
(A.7)

The transformation property of a Jacobi form of index $m$ under elliptic
transformations implies that its Fourier coefficients obey

$\displaystyle
c(n,\ell)\;=\;C_{\nu}(\Delta)\,,\qquad\Delta\;=\;4mn-\ell^{2}\,,\qquad\text{$\nu\;=\;\ell$
mod~{}$2m$}\,,$ (A.8)

for some functions $C_{\nu}$. In effect, the elliptic symmetry reduces a
function of two variables to a $2m$-dimensional vector of functions of one
variable $\Delta=4mn-\ell^{2}$, which is invariant under elliptic
transformations. In the case of our interest, the Jacobi form (2.4) has index
$m=1$, and $\ell=4n-\ell^{2}$ mod 2, so that the two functions
$C_{\nu}(\Delta)$, $\nu=0,1$ mod 2 can be captured by one function, $C=C_{0}$
(or $C_{1}$) for $\Delta$ even (or odd), for all Fourier coefficients.

The Rademacher expansion is the following formula for the coefficients
$C_{\nu}(\Delta)$ of a Jacobi form of weight $k<0$ and index $m$ as in (A.7)
for $\Delta>0$,

$\displaystyle
C_{\nu}(\Delta)\;=\;\sum_{c=1}^{\infty}\,\Bigl{(}\frac{c}{2\pi}\Bigr{)}^{k-\frac{5}{2}}\,\sum_{\widetilde{\Delta}<0}\,C_{\mu}(\widetilde{\Delta})\,\text{Kl}(\Delta,\nu,\widetilde{\Delta},\widetilde{\nu};c)\,\biggl{\lvert}\frac{\widetilde{\Delta}}{4k}\biggr{\rvert}^{\frac{3}{2}-k}I_{\frac{3}{2}-k}\biggl{(}\frac{\pi}{c}\sqrt{|\widetilde{\Delta}|\Delta}\biggr{)}\,,$
(A.9)

where $\widetilde{I}_{\rho}$, the _modified Bessel function of index $\rho$_,
is given by the following integral formula,

$\widetilde{I}_{\rho}(z)\;=\;\frac{1}{2\pi{\rm i}}\int_{\epsilon-{\rm
i}\infty}^{\epsilon+{\rm
i}\infty}\,\frac{d\sigma}{\sigma^{\rho+1}}\exp\Bigl{(}\sigma+\frac{z^{2}}{4\sigma}\Bigr{)}\,.$
(A.10)

It is related to the standard $I$-Bessel function by

$\displaystyle\widetilde{I}_{\rho}(z)\;=\;\Bigl{(}\frac{z}{2}\Bigr{)}^{-\rho}I_{\rho}(z)\,,$
(A.11)

which admits the asymptotic expansion, with $\mu=4\rho^{2}$,

$\displaystyle I_{\rho}(z)\;\sim\;\frac{{\rm e}^{z}}{\sqrt{2\pi
z}}\biggl{(}1-\frac{(\mu-1)}{8z}+\frac{(\mu-1)(\mu-3^{2})}{2!(8z)^{3}}-\frac{(\mu-1)(\mu-3^{2})(\mu-5^{2})}{3!(8z)^{5}}+\ldots\biggr{)}\,.$
(A.12)

The coefficients $\text{Kl}(\Delta,\nu,\widetilde{\Delta},\widetilde{\nu};c)$
are generalized Kloosterman sums, equal to 1 for $c=1$ and defined for $c>1$
as

$\displaystyle\text{Kl}(\Delta,\nu;\widetilde{\Delta},\widetilde{\nu};c)\;\coloneqq\;{\rm
e}^{\frac{\pi{\rm i}}{2}(\frac{1}{2}-k)}\sum_{{-c\leq d<0}\atop{(d,c)=1}}{\rm
e}^{2\pi{\rm
i}\frac{d}{c}(\Delta/4m)}\;M(\gamma_{c,d})^{-1}_{\nu\widetilde{\nu}}\;{\rm
e}^{2\pi{\rm i}\frac{a}{c}(\widetilde{\Delta}/4m)}\,,$ (A.13)

where $\gamma=\bigl{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\in SL_{2}(\mathbb{Z})$ and $M$ is a
$2m$-dimensional representation of $SL_{2}(\mathbb{Z})$ which transforms the
standard theta functions of weight $\frac{1}{2}$ and index
$m\in\mathbb{Z}^{+}$,

$\vartheta_{m,\mu}(\tau,z)\;=\;\sum_{\ell=\mu\,\text{mod}\,2m}q^{\ell^{2}/{4m}}\,\zeta^{\ell}\,,$
(A.14)

as

$\vartheta_{m,\mu}\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right)\;=\;(c\tau+d)^{\frac{1}{2}}\,{\rm
e}^{2\pi
im\frac{cz^{2}}{c\tau+d}}\,M(\gamma)_{\nu\mu}^{-1}\,\vartheta_{\mu,\ell}(\tau,z)\,,$
(A.15)

where the square root is defined by choosing the principal branch of the
logarithm. The matrix $M(\gamma)$ is called the multiplier system.

An explicit formula for the value of $M^{-1}$ on an arbitrary element
$\gamma=\bigl{(}\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\bigr{)}\in
SL_{2}(\mathbb{Z})$ is given in [82] (with $r=m+2$). In the above conventions
we have [34]

$\displaystyle\begin{split}M^{-1}(\gamma)_{\nu\mu}&\;=\;\frac{{\rm
e}^{\frac{\pi{\rm
i}}{4}}\text{sign}(c)}{\sqrt{2r|c|}}\,\exp\Bigl{(}-\frac{{\rm
i}\pi}{6}\Phi(\gamma)\Bigr{)}\cr&\quad\sum_{\varepsilon\;=\;\pm}\sum_{n=0}^{c-1}\varepsilon\,\exp\Bigl{(}\,\frac{{\rm
i}\pi}{2rc}\bigl{(}d(\nu+1)^{2}-2(\nu+1)(2rn+\varepsilon(\mu+1))+a(2rn+\varepsilon(\mu+1))^{2}\bigr{)}\Bigr{)}\,.\end{split}$
(A.16)

Here the Rademacher $\Phi$-function is defined as

$\Phi(\gamma)\;=\;\frac{a+d}{c}-12\,\text{sign}(c)\,s(a,|c|)\,,$ (A.17)

where the Dedekind sum is, for $c>0$,

$s(a,c)\;=\;\frac{1}{4c}\sum_{j=1}^{c-1}\cot\bigl{(}\frac{\pi
j}{c}\bigr{)}\,\cot\bigl{(}\frac{\pi ja}{c}\bigr{)}\,.$ (A.18)

It can be shown from these expressions that each term in the the generalized
Kloosterman sum (A.13) depends only on the equivalence class
$\Gamma_{\infty}\backslash SL(2,\mathbb{Z})/\Gamma_{\infty}$.

##  Appendix B From BRST variation to equivariant variation

The main purpose of this appendix is to review the developments of the BRST
quantization in the off-shell formulation of supergravity, discussed in detail
in [37]. We will describe how to obtain the BRST variations of fields and
ghosts as well as how to deform these variations to obtain the equivariant
transformations with respect to the $Q_{\text{eq}}$ which we used in our
localization procedure. We start with the infinitesimal gauge transformation
of some bosonic or fermionic field $\phi^{i}$, whose gauge parameter is
$\xi^{\alpha}$

$\displaystyle\delta_{\xi}\phi^{i}=R(\phi)^{i}_{\,\alpha}\xi^{\alpha}\,,$
(B.1)

where $R(\phi)^{i}_{\,\alpha}$ can include derivatives acting on
$\xi^{\alpha}$, who itself might be bosonic or fermionic. Commuting or anti-
commuting two such transformations we need to obtain a third,

$\displaystyle\delta_{\xi_{1}}\delta_{\xi_{2}}-\delta_{\xi_{2}}\delta_{\xi_{1}}=\delta_{\xi_{3}}$
(B.2)

with
$\xi_{3}^{\alpha}=f(\phi)_{\beta\gamma}{}^{\alpha}\,\xi_{1}^{\beta}\,\xi_{2}^{\gamma}$.
In usual gauge theories, $f(\phi)_{\beta\gamma}{}^{\alpha}$ are the structure
constants of the gauge group and are thus independent of $\phi$. In the
superconformal formulation of supergravity $f(\phi)_{\beta\gamma}{}^{\alpha}$
indeed generically depend on some of the supergravity fields $\phi^{i}$ and
therefore we refer to them as “structure functions” and the gauge algebra is
called “soft”.

To quantize such theories, we need to add a set of ghost fields $c$, and write
down the BRST transformations acting on all fields

$\displaystyle\delta_{\rm brst}\phi^{i}=R(\phi)^{i}_{\,\alpha}\Lambda
c^{\alpha}\,,\qquad\delta_{\rm
brst}c^{\alpha}=\frac{1}{2}f(\phi)_{\beta\gamma}{}^{\alpha}c^{\beta}\Lambda
c^{\alpha}\,,$ (B.3)

where we can see that the BRST transformation $\delta_{\rm brst}$ is nilpotent
by acting on $\phi^{i}$ or $b$ repeatedly:

$\displaystyle\delta_{\rm brst}^{2}=0\,.$ (B.4)

Similar transformations apply to the anti-ghosts $b_{\alpha}$ and to the
Lagrange multipliers $B_{\alpha}$. We shall come back to the variations of
these fields after explaining how to deform the BRST transformations for
$\phi^{i}$ and $c^{\alpha}$.

To determine the one-loop determinant around each point on the localization
locus we need to define the action of the equivariant variation on all quantum
fluctuations of fields and ghosts in the theory. To do this, we separate the
fields and ghosts into their background values ${\overline{\phi_{i}}}$ and
${\overline{c^{\alpha}}}$, and their quantum fluctuations $\tilde{\phi}^{i}$
and $\tilde{c}^{\alpha}$. The BRST fluctuation acting on the variation is
therefore,

$\displaystyle\delta_{\rm brst}\tilde{\phi}^{i}=\delta_{\rm
brst}{{\phi^{i}}}-\delta_{\rm brst}{{\overline{\phi^{i}}}}\,,\qquad\delta_{\rm
brst}\tilde{c}^{\alpha}=\delta_{\rm brst}{c}^{\alpha}-\delta_{\rm
brst}{\overline{c}}^{\alpha}\,.$ (B.5)

What is this background for the fields $\phi^{i}$ and $c^{\alpha}$ along the
localization locus that we are interested in? Close to the boundary where the
spacetime is locally AdS${}_{2}\times S^{2}$, there exists a set of locally
defined Killing spinors associated to the $PSU(1,1|2)$ superisometry. We then
choose one of the Killing spinors associated and require that it can be
globally defined everywhere in the bulk. Field and metric configurations for
which a Killing spinor cannot be globally defined do not contribute to the
path integral when the boundary observable under study is protected. We then
fix the ghost associated to the supersymmetry transformation with respect to
which the Killing spinor to be equal to the Killing spinor itself. All other
ghosts will be set to zero. Due to the Killing spinor equation or due to the
vanishing of the other ghosts, we have that

$\displaystyle\delta_{\rm
brst}{{\overline{\phi^{i}}}}=\Lambda{\overline{c}}^{\alpha}\,R({\overline{\phi}})_{\alpha}{}^{i}=0\,,$
(B.6)

for all points on the localization locus. To properly define $Q_{\text{eq}}$
we however need to deform the BRSY transformation to an equivariant
transformation

$\displaystyle\delta_{\text{brst}}\,\,\Rightarrow\,\,\delta_{\text{eq}}$ (B.7)

in such a way that $\delta_{\text{eq}}$ is no longer nilpotent. This can be
done by taking the BRST transformation on the background (B.6) and on the
quantum fluctuations (which can be derived from (B.3)) to remain unchanged.
However, in contrast to the BRST transformation, we will fix the
transformations on the non-zero background ghost to also vanish. In other
words, we thus define

$\displaystyle\delta_{\rm eq}{\overline{\phi}}^{i}$
$\displaystyle=0\,,\qquad\delta_{\rm
eq}\tilde{\phi}^{i}=R(\phi)^{i}_{\,\alpha}\Lambda({\overline{c}}+\tilde{c})^{\alpha}\,,$
$\displaystyle\delta_{\rm eq}{\overline{c}}^{\alpha}$
$\displaystyle=0\,,\qquad\delta_{\rm
eq}\tilde{c}^{\alpha}=\frac{1}{2}f(\phi)_{\beta\gamma}{}^{\alpha}({\overline{c}}+\tilde{c})^{\beta}\Lambda({\overline{c}}+\tilde{c})^{\alpha}\,-\,\frac{1}{2}f({\overline{\phi}})_{\beta\gamma}{}^{\alpha}\,{\overline{c}}^{\beta}\Lambda{\overline{c}}^{\alpha}\,,$
(B.8)

Squaring the newly defined transformation, one finds

$\displaystyle\delta_{\rm eq}^{2}=\delta_{{\overline{\xi}}}\neq
0\,,\qquad[\delta_{\rm eq}\,,\,\,\,\delta_{{\overline{\xi}}}]=0\,,$ (B.9)

where the transformation $\delta_{{\overline{\xi}}}$ is given by

$\displaystyle\delta_{{\overline{\xi}}}{\overline{\phi}}=0\,,\qquad\delta_{{\overline{\xi}}}\tilde{\phi}^{i}={\overline{\xi}}^{\alpha}R(\phi^{i})_{\alpha}{}^{i}\,,\qquad\delta_{{\overline{\xi}}}{\overline{c}}^{\alpha}=0\,,\qquad\delta_{{\overline{\xi}}}\tilde{c}^{\alpha}=f(\phi)_{\beta\gamma}{}^{\alpha}({\overline{c}}+\tilde{c})^{\beta}{\overline{\xi}}^{\gamma}\,,$
(B.10)

with

$\displaystyle\tilde{\xi}=\Lambda_{[2}f({\overline{\phi}})_{\beta\gamma}{}^{\alpha}\,{\overline{c}}^{\beta}\Lambda_{1]}{\overline{c}}^{\gamma}\,,$
(B.11)

Since the only non-vanishing ghosts are fixed to be given by one of the
Killing spinors of the spacetime, the transformations (B.10) are exactly what
we expect under the bosonic isometry obtained by constructing the Killing
vector from the bilinear of the Killing spinor that we used as our ghost
background. In our case, this is the $U(1)$ generator $L_{0}-J_{0}$ in the
AdS${}_{2}\times S^{2}$ or orbifold geometries. To check that $\delta_{\rm
eq}$ is a proper equivariant transformation we finally need to define its
action on the anti-ghosts $b_{\alpha}$ and on the Lagrange multipliers
$B_{\alpha}$. To start, the background of these fields is set to zero,
${\overline{b}}_{\alpha}=0$ and ${\overline{B}}_{\alpha}=0$. The equivariant
variation on their quantum fluctuations is then defined to be

$\displaystyle\delta_{\rm eq}\tilde{b}_{\alpha}=\Lambda
B_{\alpha}\,,\qquad\delta_{\rm
eq}B^{\alpha}=\frac{1}{2}f({\overline{\phi}})_{\delta\epsilon}{}^{\beta}\,{\overline{c}}^{\delta}\,\Lambda\,{\overline{c}}^{\epsilon}\,f({\overline{\phi}})_{\alpha\beta}{}^{\gamma}\,.$
(B.12)

Acting once again on the anti-ghosts and on the Lagrange multipliers we find
$\delta_{\rm
eq}^{2}b_{\alpha}=\delta_{{\overline{\xi}}}b_{\alpha}={\overline{\xi}}f(\phi)_{\beta\alpha}{}^{\gamma}b_{\gamma}$,
$\delta_{\rm
eq}^{2}B_{\alpha}=\delta_{{\overline{\xi}}}B_{\alpha}={\overline{\xi}}f(\phi)_{\beta\alpha}{}^{\gamma}B_{\gamma}$
which is once again the expected transformation under the bosonic $U(1)$
transformation generated by $\delta_{{\overline{\xi}}}$. This indeed confirms
that $\delta_{\rm eq}^{2}=\delta_{{\overline{\xi}}}$ for all fields and ghosts
and therefore $\delta_{\rm eq}$ is a valid equivariant transformation.

##  Appendix C On possible truncations of $\mathcal{N}=8$ supergravity

The first calculation of the Bessel function appearing in the Rademacher
expansion of the $\mathcal{N}=8$ black hole index was outlined in [32]. The
calculation in that reference had some drawbacks which we addressed in this
paper. In this appendix we clarify the relation between the two approaches for
readers familiar with that work.

Firstly, instead of working with the full 4d $\mathcal{N}=8$ supergravity
theory obtained from a toroidal compactification of type IIA theory, the
calculation was done in a truncated theory. From the point of view of the low-
energy effective theory, the truncation is obtained as follows. All the fields
of the low-energy theory fall into a single representation of $\mathcal{N}=8$
supersymmetry containing the graviton, i.e. the $\mathcal{N}=8$ graviton
multiplet. We reduce this representation in terms of representations of a
particular $\mathcal{N}=4$ sub-algebra. This gives an $\mathcal{N}=4$ graviton
multiplet (= $\mathcal{N}=2$ graviton + 2 spin-$\frac{3}{2}$ \+ 1 vector),
four $\mathcal{N}=4$ spin-$\frac{3}{2}$ multiplets (each = $\mathcal{N}=2$
spin-$\frac{3}{2}$ \+ 2 vectors + 1 hyper), and 6 vector multiplets (each =
$\mathcal{N}=2$ vector + hyper). Next we throw away the spin-$\frac{3}{2}$
multiplets, and then reduce the into $\mathcal{N}=2$ representations. Now we
have an $\mathcal{N}=2$ graviton multiplet, 2 spin-$\frac{3}{2}$ multiplets, 7
vector multiplets, and 6 hyper multiplets. Finally, we throw away the the
hyper multiplets. The remaining $\mathcal{N}=2$ theory is precisely
$\mathcal{N}=2$ to $n_{\text{v}}=7$ vector multiplets for which we write a
prepotential below. The $T^{6}$ compactification has $SO(6,6;\mathbb{Z})$
T-duality which is a subgroup of the full $E_{7,7}$ U-duality. The 28 gauge
fields of the theory decompose as $\mathbf{28}=\mathbf{12}+\mathbf{16}$. The
vector $\mathbf{12}$ is in the NS-NS sector while the spinor $\mathbf{16}$ is
in the R-R sector. Now we can identify the $16$ gauge fields in the R-R sector
with the $4\times 4=16$ gauge fields in the four $\mathcal{N}=4$
spin-$\frac{3}{2}$ multiplets that we drop in the truncation. The NS-NS fields
behave as in usual CY3 compactifications, while the R-R fields decouple
classically.

The truncation consists on $\mathcal{N}=2$ supergravity coupled to $n_{v}=7$
vector multiplets and to perform the localization calculation we need to
specify its holomorphic prepotential. In the M-theory description, each vector
multiplet is associated with a 2-cycle (or its Hodge-dual 4-cycle) inside the
CY3. For concreteness we label them in the following way:

$\displaystyle\Sigma^{45}\to X^{1},~{}\Sigma^{67}\to X^{2},~{}\Sigma^{68}\to
X^{3},~{}\Sigma^{69}\to X^{4},~{}\Sigma^{78}\to X^{5},~{}\Sigma^{79}\to
X^{6},~{}\Sigma^{89}\to X^{7},$ $\displaystyle\Sigma^{46}\to
X^{8},~{}\Sigma^{47}\to X^{9},~{}\Sigma^{48}\to X^{10},~{}\Sigma^{49}\to
X^{11},~{}\Sigma^{56}\to X^{12},$ $\displaystyle\Sigma^{57}\to
X^{13},~{}\Sigma^{58}\to X^{14},~{}\Sigma^{59}\to X^{15}.$ (C.1)

With these conventions, in the truncation described above, the vector
multiplet $I=1$ is associated to the $T^{2}$ and the other 6 vector multiplets
$I=2,\dots,7$ are associated to the ${4\choose 2}=6$ 2-cycles of the $T^{4}$.
Now we can read off the holomorphic prepotential by considering the
intersection numbers of the 4-cycles on the $T^{6}$. We obtain the following
simple formula,

$\displaystyle F_{\rm
trunc.}(X)\;=\;-\frac{1}{2}\,\frac{X^{1}\,C_{ab}\,X^{a}X^{b}}{X^{0}}\,,\qquad
a=2,\dots,7,$ (C.2)

where $C_{ab}$ is the intersection matrix of the 2-cycles of the $T^{4}$. We
can use the conventions of Section 3.2.1 to compute since its a particular
case of $C_{IJK}$ when $I=1$ and $J,K=2,\ldots,7$.

In this appendix we want to use the fact that we did the calculation in the
full $\mathcal{N}=8$ supergravity theory to integrate out the extra fields and
reevaluate the validity of the truncation considered in [32]. Obtaining the
truncated theory is easy and involves turning off the hypermultiplets and
turning off the vector multiplets $X^{8}$ to $X^{15}$. This implies we should
also restrict to states with vanishing charges associated to these vectors
$p^{8}=\ldots=p^{15}=0$ and $q_{8}=\ldots=q_{15}=0$. It is easy to see that
the prepotential (3.18) reduces to (C.2) after identifying
$X^{a}=\\{X^{1},\ldots,X^{7}\\}$, and similarly for the charges.

Now we want to compare the result from applying localization to this
truncation to the result obtained from $\mathcal{N}=8$ supergravity after
setting to zero the extra charges. This is not trivial since we still need to
integrate out the extra localization manifold coordinates that the full theory
has compared to the truncation. In our parametrization we need to integrate
out $\phi^{8},\ldots,\phi^{15}$. Since the measure over the $\phi$’s and the
one-loop determinant both depend only on $\phi^{0}$ the integration is simply
Gaussian over the exponential of the action on the localization manifold. This
gives

$\displaystyle\int\prod_{I=8}^{15}{\rm
d}\phi^{I}~{}\exp\left(\sum_{I,J=8}^{15}\frac{3\pi
C_{IJK}\phi^{I}\phi^{J}p^{K}}{c\phi^{0}}\right)\Big{|}_{p^{8}=\ldots=p^{15}=0}=\frac{4c^{4}(\phi^{0})^{4}}{(P^{2})^{2}},~{}~{}~{}P^{2}=p^{a}C_{ab}p^{b}.$
(C.3)

This seems to indicate that integrating out the extra vector multiplets
involved in the truncation results in a non-trivial contribution to the
remaining integrals. Nevertheless this factor disappears if we correct the
measure over the $n_{v}=7$ vector multiplets to be over

$\displaystyle[{\rm d}^{8}\phi]_{c}\equiv{\rm d}^{8}\phi~{}\sqrt{2~{}{\rm
det}^{\prime}\Big{(}\frac{1}{2c}~{}{\rm Im}~{}F_{\Lambda\Sigma}\Big{)}},$
(C.4)

where the prime denotes that the determinant is taken only over the directions
$\Lambda=0,\ldots,7$ and only $\phi^{0},\ldots,\phi^{7}$ are turned on. Its
easy to check that the mismatch between this measure and the measure over all
the vector multiplets precisely cancels the integral over the remaining
vectors (C.3). The final answer is

$\displaystyle W^{(c)}(q,p)=\sqrt{c}K_{c}(\Delta)\int[{\rm
d}^{8}\phi]_{c}~{}c^{-1}{\rm e}^{4\mathcal{K}(\phi)}~{}e^{\frac{4\pi}{c}{\rm
Im}F_{\rm
trunc}\big{(}\frac{\phi^{\Lambda}+ip^{\Lambda}}{2}\big{)}-\frac{\pi}{c}q_{\Lambda}\phi^{\Lambda}},$
(C.5)

where we only turn on the charges and moduli in the truncation. Since this is
equivalent to the original integral over 15 moduli we already know this gives
the right answer to match the microscopic entropy.

The integral in (C.5) is precisely the starting point of [32] for $c=1$ (and
we extended it to the case $c\neq 1$ in this paper). This formula almost shows
that the truncation is consistent off-shell except for the one-loop
determinant. The action and measure is indeed the one corresponding to the
truncation (as well as the boundary-mode contribution to the c-dependence
$1/c$) but the bulk one-loop determinant $e^{4\mathcal{K}}$ corresponds only
to the full $\mathcal{N}=8$ supergravity theory.

One approach to justify the calculation in [32] can be to postulate we include
off-shell contributions from the $\mathcal{N}=8$ supergravity theory in the
one-loop determinant, but not include the extra coordinates on the
localization manifold. This is inconsistent since we would be treating two
sets of off-shell modes differently without a good reason. Another approach is
to see whether we can find an $\mathcal{N}=2$ theory (without gravitino
multiplet) that reproduces the bulk one-loop determinant $e^{4\mathcal{K}}$.
Imposing that the bulk one-loop determinant be c-independent restricts
$n_{H}-n_{V}=1$. Since in the truncation $n_{V}=7$ we can set $n_{H}=8$
without affecting the localization integral, other than the bulk one-loop
determinant. The problem is that this unique choice set by looking at the
c-dependence gives $e^{-2\mathcal{K}}$ for the bulk one-loop determinant.
Therefore we conclude there is no $\mathcal{N}=2$ supergravity theory able to
reproduce (C.5).

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