††institutetext:  1 Stanford Institute for Theoretical Physics, Stanford
University, Stanford, CA 94305, USA††institutetext: 2 Department of
Mathematics, King’s College London, The Strand, London WC2R 2LS,
UK††institutetext: 3 School of Natural Sciences, Institute for Advanced Study,
Princeton, NJ, USA††institutetext: 4 Physics Department, University of
Washington, Seattle, WA, USA

# Gravitational index of the heterotic string

Yiming Chen1, Sameer Murthy2,3, Gustavo J. Turiaci4

###### Abstract

The fundamental heterotic string has a tower of BPS states whose
supersymmetric index has an exponential growth in the charges. We construct
the saddle-point of the gravitational path integral corresponding to this
index. The saddle-point configuration is a supersymmetric rotating non-
extremal Euclidean black hole. This configuration is singular in the two-
derivative theory. We show that the addition of higher-derivative terms in
four-dimensional $\mathcal{N}=2$ supergravity resolves the singularity. In
doing so, we extend the recently-developed “new attractor mechanism” to
include the effect of higher-derivative terms. Remarkably, the one-loop, four-
derivative F-term contribution to the prepotential leads to a precise match of
the gravitational and microscopic index. We also comment, using the effective
theory near the horizon, on the possibility of a string-size near-extremal
black hole. Our results clarify the meaning of different descriptions of this
system in the literature. The thermal state transitions to a winding
condensate and a gas of strings without ever reaching a small black hole,
while the index is captured by the rotating Euclidean black hole solution and
is constant and thus smoothly connected to the microscopic ensemble.

##  1 Introduction

Consider a fundamental heterotic string of energy $M$, with $w$ units of
winding and $n$ units of momentum around a circle of finite size. In the
weakly-coupled theory the string has a tower of excited states whose
degeneracy grows exponentially in the excitation numbers. In the strongly-
coupled theory, the effective description of the string is as a source for the
metric and other low-energy fields of supergravity, and the solution of the
low-energy theory is generically a black hole in the $(d+1)$-dimensional non-
compact spacetime whose entropy corresponds to the exponentially growing
degeneracy of states Bowick:1985af ; Susskind:1993ws ; Horowitz:1996nw .

When all the right-movers are in their ground states, the heterotic string
configuration preserves half the supersymmetry of the theory and obeys
$M^{2}=Q_{R}^{2}/2$ where $Q_{R}$ is the right-moving momentum. The resulting
tower of $\frac{1}{2}$-BPS Dabholkar-Harvey states has a degeneracy which
grows as $\exp(4\pi\sqrt{nw})$ Dabholkar:1989jt . What is the strongly-coupled
description of this ensemble of supersymmetric states? One natural possibility
is that they form a supersymmetric black hole. Such a black hole would be
extremal with AdS${}_{2}\times S^{d-1}$ near-horizon geometry. In this paper
we mainly discuss four non-compact directions ($d=3$), although we make a few
comments on other dimensions as well.

As is well-known, this simple picture runs into various problems.111Notably,
this picture does hold in three- and four-charge systems, and leads to the
famous agreement of statistical and thermodynamic entropy of black holes in
string theory Strominger:1996sh . The zeroth-order problem, which was already
discussed in the early days, is that the two-derivative supergravity solutions
with a fundamental string source is singular, in that the event horizon lies
on top of the curvature singularity Dabholkar:1995nc . A more sophisticated
analysis suggested that including higher-derivative terms in the low-energy
supergravity stretches the horizon to string scale, thus resolving the
singularity Sen:1995in , Sen:2004dp ; Hubeny:2004ji ; Dabholkar:2004dq .

However, there are still problems with this picture. Firstly, if we think of
obtaining the extremal black hole solution by lowering the mass to its
extremal value, there is good evidence that there is a transition at low
temperatures to the so-called Horowitz-Polchinski solution Horowitz:1997jc
instead of obtaining an extremal black hole Chen:2021dsw .222See also
Cano:2018hut  that studies the validity of the small black hole analysis.
Secondly, there is no known AdS${}_{2}\times S^{d-1}$ solution in supergravity
which admits 16 Killing spinors. This could be an issue of simply not having
found such a solution but, in fact, we can sharpen this argument by analyzing
possible superconformal groups in the near-horizon region. As we explain in
Section 6, by studying the quantum super Schwarzian theory near the horizon,
one can rule out the existence of a decoupled near-horizon AdS2 region with 16
or more supercharges describing the heterotic string with all its symmetries.

In order to clarify the situation, it is useful to focus on particular
observables that are amenable to precise calculation. In particular, the
supersymmetric index Witten:1981nf  is independent of coupling, and can be
precisely compared in the two descriptions. For the $\frac{1}{2}$-BPS states
that we are considering, the relevant index is a helicity supertrace in flat
spacetime (see e.g. Kiritsis:1997gu ). This quantity is protected against
changes of coupling, for the usual reason of bosonic and fermionic
supermultiplets appearing in pairs. In the free theory, the generating
function of this index is the chiral partition function of 24 free bosonic
fields i.e. $1/\eta(\tau)^{24}$ Dabholkar:1989jt , whose coefficient at level
$nw$ grows as ${\rm exp}\,(4\pi\sqrt{nw})$ as mentioned above. The question we
would like to address is: what is the saddle-point of the dual gravitational
path integral of this index?

The aim of this paper is to present and analyze this gravitational saddle-
point. Here, by saddle-point we mean a Euclidean solution of the low-energy
supergravity, treating $G_{N}$ in perturbation theory. The main subtlety—and
difference from previous treatments—is the inclusion of $(-1)^{\sf F}$ in the
gravitational functional integral. We use the construction of the saddle-point
for the gravitational index discussed in Cabo-Bizet:2018ehj ; Iliesiu:2021are
. The basic idea is that $(-1)^{\sf F}$ can be included in the gravitational
functional integral by adding an imaginary part to a background bosonic field
that couples to an $R$-symmetry. This implies that the solution is necessarily
complex. The Euclidean solution is non-extremal and has the topology of a
cigar times $S^{d-1}$, with the horizon at the tip of the cigar.

In asymptotic flat space, the relevant background field that implements
$(-1)^{\sf F}$ is the angular velocity $\Omega$ which obeys $\beta\Omega=2\pi
i$. The solution has non-zero rotation along one axis, which defines a north
and south pole at the horizon. Applying this idea to four-dimensional
$\mathcal{N}=2$ supergravity coupled to an arbitrary number of vector
multiplets leads to the _new attractor mechanism_ Boruch:2023gfn . The metric
of these solutions can be written in the Israel-Wilson-Perjés (IWP) form
Perjes:1971gv ; Israel:1972vx , which is sourced by harmonic functions with
two simple poles at the north and south poles, respectively. Near the poles,
the values of the scalars of supergravity get fixed in terms of the charges of
the black hole. The complex nature of the solution is reflected in that the
holomorphic scalars are fixed at the north pole and vanish at the south pole.
(The anti-holomorphic scalar has the opposite behavior.)

In this paper we construct the rotating Euclidean black hole as a solution to
the effective four-dimensional supergravity describing the heterotic string
theory on $T^{6}$. We also write the solution in the variables of the ten-
dimensional heterotic theory. As it turns out, the solution is singular at the
two-derivative level. In the Einstein frame in four dimensions, the metric is
smooth but the string coupling goes to zero and so the string-frame metric is
singular. This opens the possibility that higher-derivative effects can
resolve this singularity.

In order to resolve this singularity, we study the effect of higher-derivative
corrections in the supergravity action on the Euclidean rotating black hole
solution. We discuss a class of higher-derivative corrections that are
summarized in the holomorphic prepotential of four-dimensional $\mathcal{N}=2$
supergravity, and can be calculated using topological string methods in the
dual Type II theory. In particular, the one-loop term in the prepotential
leads to the supersymmetric completion of a particular four-derivative term
proportional to the square of the Weyl tensor. This leads to an extension of
the new attractor mechanism to include higher-derivative terms.

In the two-charge system, we find that the singularity is resolved in the
higher-derivative theory and, in particular, the string coupling at the tip is
now non-zero. Note that the resolution involves a play-off between
2-derivative and 4-derivative effects. This would lead to an uncontrollable
approximation for generic observables. However, the calculation for
supersymmetric quantities such as the index has a better chance of leading to
a precise result, along the lines of the non-renormalization theorem discussed
in Butter:2014iwa ; Murthy:2013xpa . In particular, if $D$-terms vanish on our
supersymmetric configurations, the fact that the topological string expansion
for Type II on $K3\times T^{2}$ is one-loop exact implies that our result
would be rigorous. Quite remarkably we find that, with these assumptions, the
gravitational free-energy of the index agrees precisely with that of the
microscopic index to give the entropy $4\pi\sqrt{nw}$ including the numerical
coefficient333To evaluate the action we assume it is temperature independent
and compute it in the large $\beta$ limit. At two derivative level we have
proved this earlier Boruch:2023gfn . The calculation of the on-shell action
(the Gibbons-Hawking free energy of the system) at finite $\beta$ including
the effect of 4-derivative terms is complicated and we postpone it to future
work..

Now we briefly summarize the overall picture that we propose. First consider
the thermal partition function and states in the Lorentzian theory. Away from
extremality the fundamental heterotic string is described by the two-charge
black hole solution with a macroscopic horizon. As we lower the mass towards
extremality there is a transition to a gas of free strings, without ever
reaching the extremal black hole solution Chen:2023mbc . Now consider the
gravitational path integral for the supersymmetric index as in the present
paper. The saddle-point is a supersymmetric non-extremal Euclidean geometry
valid at all temperatures. The saddle-point is really like an instanton rather
than describing Lorentzian states. At the two-derivative level the solution
has a large horizon area, but contains curvature singularities near the poles
of the horizon. These singularities are resolved by the inclusion of string-
scale higher-derivative corrections in the effective action coming from string
theory. The index calculated in this manner is independent of coupling, and
agrees with the index of a gas of strings at weak-coupling.444There may even
exist a Horowitz-Polchinski-type supersymmetric solution at intermediate
coupling. The details of this would be interesting and may involve a perhaps
suitably complexified sigma-model. We comment further on this possibility in
Section 3.1. In this respect, the index behaves in a manner similar to the
supersymmetric indices corresponding to big black holes555Big black holes
refer to black holes with with horizon curvatures parameterically smaller than
the string and Planck scale. Cabo-Bizet:2018ehj ; Iliesiu:2021are ;
Boruch:2023gfn . They do not depend on coupling, and have continuously
connected dual pictures in the gravitational and microscopic regimes, rather
than a sharp transition.

The plan of paper is as follows. In Section 2 we review the family of two-
charge black hole solutions in heterotic string theory, their thermodynamics,
and the transition to the gas of free strings via the Horowitz-Polchinski
solution. In Section 3 we discuss the gravitational saddle-point for the
index, i.e. the Euclidean rotating black hole, in the heterotic string
variables. In Section 4 we switch to the four-dimensional effective
supergravity description at the two-derivative level. We review the new
attractor mechanism and the embedding of the Euclidean rotating black hole in
this description. In Section 5 we include a four-derivative term in the 4d
supergravity coming from the string compactification. We show how it
desingularizes the Euclidean rotating black hole, and analyze its effect on
the entropy. In Section 6 we analyze the superconformal groups that can
possible arise in the near-horizon geometry and the quantum effects of the
respective super-Schwarzians. We show that AdS2 cannot arise at the quantum
level in the two-charge system. In Section 7 we discuss further possible
directions. In the appendix we discuss how the solution generating technique
Narain:1985jj ; Narain:1986am  relates the rotating charged and uncharged
solutions.

##### Note added

While this paper was in preparation we received Chowdhury:2024ngg , which
contains overlap on the two-derivative solutions that calculate the index.

##  2 The two-charge black hole and its transition to free strings

In this section we review the properties of the two-charge black hole in the
heterotic string theory on $T^{6}$. In particular, we discuss some peculiar
features of the naive extremal limit of this solution. Then we review the
construction of the charged Horowitz-Polchinski solution and how it modifies
the approach to extremality.

###  2.1 Review of the two-charge black hole

We begin by reviewing the two-charge black hole, constructed by Sen in
Sen:1994eb , as a solution to the four dimensional supergravity theory arising
from a toroidal compactification of heterotic string theory.

####  2.1.1 Heterotic string on $T^{6}$

The four dimensional theory has $\mathcal{N}=4$ supersymmetry and contains the
supergravity multiplet, and 22 $\mathcal{N}=4$ vector multiplets. The
supergravity multiplet has six graviphotons, resulting in a total gauge group
$\mathrm{U}(1)^{28}$. We collectively denote the potentials by $A^{a}$ with
$a=1,\ldots,28$. The moduli space consists of the dilaton $\Phi$ and 132
scalar fields organized in a symmetric $28\times 28$ dimensional matrix $M$
that satisfies $MLM^{T}=L$. $L$ is a diagonal matrix $L_{aa}=-1$ for
$a=1,\ldots,22$ and $L_{aa}=+1$ for $a=23,\ldots,28$. The scalars $M$ encode
deformations of the torus, of the background $B$-field, and of the Wilson
lines. Besides these fields, we also have a four dimensional $B$-field
$B_{\mu\nu}$, which is dual to a scalar.

The action, in Lorentzian signature, is given by

$\displaystyle I$ $\displaystyle\;=\;$
$\displaystyle\frac{1}{16\pi}\int\mathrm{d}^{4}x\,\sqrt{-g}\,e^{-\Phi}\,\Big{[}R+(\partial\Phi)^{2}+\frac{1}{8}{\rm
Tr}\left[\partial_{\mu}M\,L\,\partial^{\mu}M\,L\right]$ (1)
$\displaystyle~{}~{}~{}-\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}-F^{a}_{\mu\nu}\,(LML)_{ab}\,F^{b}{}^{\mu\nu}\Big{]}+({\rm
fermions})+({\rm bdy~{}terms})\,,$

where we denote the $\mathrm{U}(1)^{28}$ field strength by
$F^{a}_{\mu\nu}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}$ and the
$B$-field field strength by
$H_{\mu\nu\rho}=\partial_{\mu}B_{\nu\rho}+2A^{a}_{\mu}L_{ab}F^{b}_{\nu\rho}+({\rm
cyclic})$. The action is invariant under an $O(6,22,\mathbb{R})$
transformation which acts on $a,b$ indices only, so it does not act on the
dilaton, metric or $B$ field. Consistency with quantization of electromagnetic
charge restricts it to $O(6,22;\mathbb{Z})$.

The solutions we discuss in the following have boundary conditions at infinity
such that locally

$M_{ab}|_{\infty}\;=\;\delta_{ab}\,,\qquad\Phi|_{\infty}\;=\;0\,,\qquad
g_{\mu\nu}|_{\infty}\;=\;\eta_{\mu\nu}\,.$ (2)

For the gauge fields, both the electromagnetic ones and the $B$-field, we can
choose to fix their holonomies or their charges at infinity.

####  2.1.2 Sen’s black hole solution

The procedure to find the electrically charged black holes described in
Sen:1994eb  is the following. First, one realizes that for metrics with a
$\mathrm{U}(1)$ isometry (corresponding to Euclidean time), the symmetry of
the action gets enhanced to $O(7,23)$. One can think about this in terms of a
$T^{7}=T^{6}\times S^{1}$ compactification of the heterotic string, but
Sen:1994eb  verifies it by explicit construction of the $O(7,23)$ action. We
discuss such transformations more generally in Appendix A, not necessarily
restricting our attention to black hole solutions.

Let us present the final results. The metric in the Einstein frame
$g_{\mu\nu}^{E}=e^{-\Phi}g_{\mu\nu}$ and Lorentzian signature is given by

$\displaystyle e^{-\Phi}\,\mathrm{d}s^{2}$ $\displaystyle\;=\;$
$\displaystyle\Delta^{1/2}\Big{[}-\frac{r^{2}+a^{2}\cos^{2}\theta-2mr}{\Delta}\mathrm{d}t^{2}+\frac{\mathrm{d}r^{2}}{r^{2}+a^{2}-2mr}+\mathrm{d}\theta^{2}$
(3)
$\displaystyle\qquad+\frac{\sin^{2}\theta}{\Delta}[\Delta+a^{2}\sin^{2}\theta(r^{2}+a^{2}\cos^{2}\theta+2mr\cosh\upalpha\cosh\upbeta)]\,\mathrm{d}\phi^{2}$
$\displaystyle\qquad-\frac{2mra\sin^{2}\theta(\cosh\upalpha+\cosh\upbeta)}{\Delta}\,\mathrm{d}t\mathrm{d}\phi\Big{]}\,,$

where we define the function

$\Delta\;=\;(r^{2}+a^{2}\cos^{2}\theta)^{2}+2mr(r^{2}+a^{2}\cos^{2}\theta)(\cosh\upalpha\cosh\upbeta-1)+m^{2}r^{2}(\cosh\upalpha-\cosh\upbeta)^{2}\,.$
(4)

The dilaton is given by

$\Phi\;=\;\frac{1}{2}\log\frac{(r^{2}+a^{2}\cos^{2}\theta)^{2}}{\Delta}.$ (5)

We can decompose the vector fields into a 22-dimensional vector of one-forms
$\vec{A}_{L}$, made up of the first 22 components of $A$, and a 6-dimensional
vector of one-forms $\vec{A}_{R}$ made of the last 6 components of $A$. These
gauge potentials and the two-form are given by

$\displaystyle\vec{A}_{L}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\vec{n}}{\sqrt{2}}\frac{mr\sinh\upalpha((r^{2}+a^{2}\cos^{2}\theta)\cosh\upbeta+mr(\cosh\upalpha-\cosh\upbeta))}{\Delta}\mathrm{d}t$
(6)
$\displaystyle+\frac{\vec{n}}{\sqrt{2}}\frac{mra\sinh\upalpha\sin^{2}\theta(r^{2}+a^{2}\cos^{2}\theta+mr\cosh\upbeta(\cosh\upalpha-\cosh\upbeta))}{\Delta}\mathrm{d}\phi\,,$
$\displaystyle\vec{A}_{R}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\vec{p}}{\sqrt{2}}\frac{mr\sinh\upbeta((r^{2}+a^{2}\cos^{2}\theta)\cosh\upalpha+mr(\cosh\upbeta-\cosh\upalpha))}{\Delta}\mathrm{d}t$
(7)
$\displaystyle+\frac{\vec{p}}{\sqrt{2}}\frac{mra\sinh\upbeta\sin^{2}\theta(r^{2}+a^{2}\cos^{2}\theta+mr\cosh\upalpha(\cosh\upbeta-\cosh\upalpha))}{\Delta}\mathrm{d}\phi\,,$
$\displaystyle B$ $\displaystyle\;=\;$
$\displaystyle\frac{mra\sin^{2}\theta(\cosh\upalpha-\cosh\upbeta)}{\Delta}(r^{2}+a^{2}\cos^{2}\theta+mr(\cosh\upalpha\cosh\upbeta-1))\,\mathrm{d}t\mathrm{d}\phi\,.$
(8)

The scalar moduli are given by

$M\;=\;I_{28}+\begin{pmatrix}{\sf P}nn^{T}&{\sf Q}np^{T}\\\ {\sf Q}pn^{T}&{\sf
P}pp^{T}\end{pmatrix}\,,$ (9)

where

$\displaystyle{\sf P}$ $\displaystyle\;=\;$
$\displaystyle\frac{2m^{2}r^{2}\sinh^{2}\upalpha\sinh^{2}\upbeta}{\Delta},$
(10) $\displaystyle{\sf Q}$ $\displaystyle=$
$\displaystyle-\frac{2mr\sinh\upalpha\sinh\upbeta}{\Delta}(r^{2}+a^{2}\cos^{2}\theta+mr(\cosh\upalpha\cosh\upbeta-1))\,.$
(11)

Here $\vec{n}$ and $\vec{p}$ are arbitrary 22 and 6 dimensional unit vectors.
It will be important to know the charges and chemical potentials as functions
of the parameters appearing in the solution, which are
$(m,a,\upalpha,\upbeta,\vec{n},\vec{p})$. Define $\vec{Q}_{L}$ to be a
22-dimensional vector made out of the first 22 components of $Q^{a}$.
Similarly define $\vec{Q}_{R}$ to be the last 6 components. To compute the
charges we can evaluate the large $r$ limit of the gauge potentials and read
them off from $A\sim Q/r\,\mathrm{d}t$. Using that $\Delta\sim r^{4}$ as
$r\to\infty$ we obtain

$\vec{Q}_{L}\;=\;\frac{m}{\sqrt{2}}\sinh\upalpha\cosh\upbeta\,\vec{n}\,,\qquad\vec{Q}_{R}=\frac{m}{\sqrt{2}}\sinh\upbeta\cosh\upalpha\,\vec{p}\,.$
(12)

We will also use $Q_{L},Q_{R}$ to denote the modulus of the respective charge
vectors. Since the black holes are electrically charged and rotating there
will also be a non-trivial magnetic dipole moment, but it will not be
important for the discussion here.

Let us now move on to the other charges encoded in the metric. The mass and
angular momentum of the black hole can be read off from the large $r$ behavior
of $g_{tt}$ and $g_{t\phi}$ and are given by

$M\;=\;\frac{m(1+\cosh\upalpha\cosh\upbeta)}{2}\,,\qquad
J\;=\;\frac{ma(\cosh\upalpha+\cosh\upbeta)}{2}.$ (13)

The metric has an outer horizon $r_{+}$ at the locations where $g_{rr}^{-1}$
vanishes

$r^{2}-2mr+a^{2}=0\quad\Rightarrow\quad r_{\pm}\;=\;m\pm\sqrt{m^{2}-a^{2}}\,.$
(14)

The temperature and angular velocities are

$\displaystyle\beta$ $\displaystyle\;=\;$ $\displaystyle\frac{2\pi
m(\cosh\upalpha+\cosh\upbeta)(m+\sqrt{m^{2}-a^{2}})}{\sqrt{m^{2}-a^{2}}}\,,$
(15) $\displaystyle\Omega$ $\displaystyle\;=\;$
$\displaystyle\frac{a}{m(\cosh\upalpha+\cosh\upbeta)(m+\sqrt{m^{2}-a^{2}})}\,.$
(16)

We see that for the Lorentzian solution to make sense we need $m\geq a$ and
the limit $m\to a$ is an extremal limit (in the sense that inner and outer
horizons have the same area) although the temperature does not vanish. We will
expand on this later. When $a\to 0$ and $\upalpha,\upbeta\to 0$ then the
solution becomes the Schwarszchild black hole. We can check this by looking at
the potentials, for example $\beta\to 8\pi m$.

Having found the location of the horizon and the angular velocity, we can
evaluate the electric chemical potentials. These quantities are given by
$\mu={\sf i}_{V}A|_{r\to\infty}-{\sf i}_{V}A|_{\rm hor.}$ where
$V=\partial_{t}+\Omega\partial_{\phi}$. This is a gauge invariant statement,
but to be more rigorous we should apply a large gauge transformation such that
${\sf i}_{V}A|_{\rm hor.}=0$. The electric potentials are

$\vec{\mu}_{L}\;=\;\frac{\vec{n}}{\sqrt{2}}\frac{\sinh\upalpha}{\cosh\upalpha+\cosh\upbeta}\,,\qquad\vec{\mu}_{R}\;=\;\frac{\vec{p}}{\sqrt{2}}\frac{\sinh\upbeta}{\cosh\upalpha+\cosh\upbeta}\,.$
(17)

Finally we can compute the area of the event horizon, with respect to the
Einstein frame metric,

$A\;=\;4\pi m(\cosh\upalpha+\cosh\upbeta)(m+\sqrt{m^{2}-a^{2}}).$ (18)

The on-shell action of the black hole is, in the grand canonical ensemble
where we set Dirichlet boundary conditions for all fields, given by

$\log Z\;=\;-I_{\rm on-shell}\;=\;\frac{A}{4}-\beta
M+\beta\vec{\mu}_{L}\cdot\vec{Q}_{L}+\beta\vec{\mu}_{R}\cdot\vec{Q}_{R}+\beta\Omega
J\,.$ (19)

We can compare the supersymmetric limit with the extremal limit. We already
mentioned the extremal limit is $m=a$. The solution is supersymmetric when

${\rm SUSY}:\qquad M^{2}\;=\;\frac{\vec{Q}_{R}^{2}}{2}.$ (20)

Combining with (13) and (12) we see that it implies

$m^{2}(1+\cosh\upalpha\cosh\upbeta)^{2}\;=\;m^{2}\cosh^{2}\upalpha\sinh^{2}\upbeta\,.$
(21)

If all parameters are finite and real we cannot satisfy (21) as we need either
(i) $m=0$, implying $M=0$ and $\vec{Q}_{R}=\vec{0}$ and, moreover,
$A=0$—implying a singularity at the horizon; or (ii) we have $m\neq 0$ but the
solution is complex, for example we can see that if we solve for
$\cosh\upalpha$ as a function of $\cosh\upbeta$ then $\cosh\upalpha<1$ for all
$\cosh\upbeta\in(+1,\infty)$. As explained by Sen the supersymmetric solutions
have $m\to 0$ and either $\upbeta$ going to infinity or $\upalpha,\upbeta$
going to infinity together, but it is not always extremal in the sense of
being zero temperature.

####  2.1.3 Thermodynamics without $(-1)^{\sf F}$ insertion

We want to describe the termodynamics of non-rotating electric black holes in
this setup, without any insertion of $(-1)^{\sf F}$ (which we will discuss in
later sections). We then set $a=0$ and the expressions in the previous section
simplify. The mass and entropy of the black hole are given by

$\displaystyle M=\frac{m(1+\cosh\upalpha\cosh\upbeta)}{2},\quad S=2\pi
m^{2}(\cosh\upalpha+\cosh\upbeta),$ (22)

and the temperature

$\beta=4\pi m(\cosh\upalpha+\cosh\upbeta).$ (23)

The expressions for the charges and electric potentials are unmodified.

There are several distinct limits one can consider. In this section we assume
the validity of (22) and (23), and discuss the thermodynamics implied by them.
However, we should keep in mind that these relations have $\alpha^{\prime}$
corrections which can become important when the black hole becomes small and,
in fact, as we will discuss in Section 2.2, the correct description in certain
regime is in fact given by horizonless geometries.

##### Low temperature limit

The simplest limit one can consider is the low temperature limit, i.e.
$\beta\to\infty$, while holding the charges fixed. In this limit we have

$\displaystyle\sinh\upalpha\;\sim\;\sqrt{2}\frac{8\pi
Q_{L}}{\beta}\;\sim\;0\,,\qquad\cosh\upalpha\;\sim\;1\,,$ (24)
$\displaystyle\sinh\upbeta\;\sim\;\sqrt{2}\frac{8\pi
Q_{R}}{\beta}\;\sim\;0\,,\qquad\cosh\upbeta\;\sim\;1\,.$ (25)

For this reason, if we compute the mass and entropy as functions of
temperature we get

$M\;\sim\;\frac{\beta}{8\pi}\,,\qquad S\;\sim\;\frac{\beta^{2}}{16\pi}.$ (26)

We simply recover the behavior of a regular Schwarszchild black hole, the
bigger the black hole the bigger the mass and the colder it gets. Since the
black hole becomes large and weakly curved, the black hole description is
trustworthy even though the ensemble is thermodynamically unstable.

##### The naive extremal limit

Since we are considering here the special case $a=0$, what we mean by the
extremal limit is $m\rightarrow 0$. However, to keep the charges (12) fixed,
the limit is not so straightforward. The only solution to this problem is to
take either $|\upalpha|\to\infty$ or $|\upbeta|\to\infty$. For simplicity,
here we will focus on the cases where $\upalpha,\upbeta>0$. The generalization
of the discussion to the cases where one or both of them are negative is
straightforward.

We will consider first the case of $\upbeta\to\infty$, which as it turns out
corresponds to the case of $Q_{R}>Q_{L}$. To keep the charges fixed, we take
$\upbeta\to\infty$ and $m\to 0$ while keeping

$m\cosh\upbeta\to m_{0}$ (27)

finite. The first surprise, under the assumption that we can trust the black
hole solution, is that this extremal limit does not imply vanishing
temperature. Instead

$\beta_{\rm ext}\;=\;4\pi m_{0}\,,\qquad
Q_{L}\;=\;\frac{m_{0}}{\sqrt{2}}\sinh\upalpha\,,\qquad
Q_{R}\;=\;\frac{m_{0}}{\sqrt{2}}\cosh\upalpha.$ (28)

From the last two equations of (28) we see that $\tanh\upalpha=Q_{L}/Q_{R}$
and $m_{0}=\sqrt{2}\sqrt{Q_{R}^{2}-Q_{L}^{2}}$. Therefore the extremal limit
has a finite, charge dependent, temperature

$T_{\textrm{max}}\;=\;\frac{1}{4\sqrt{2}\pi\sqrt{Q_{R}^{2}-Q_{L}^{2}}}.$ (29)

This solution is precisely Sen’s small black hole since the area of the event
horizon now vanishes

$A\;\sim\;m^{2}\cosh\upbeta\sim m_{0}m\to 0.$ (30)

We saw earlier that at low temperature the black hole is very large since it
behaves as the Schwarszchild solution. As the temperature increases the mass
decreases until it reaches extremality at $\beta_{\rm ext}$. This is the
situation if $Q_{R}>Q_{L}$. At this point the entropy vanishes since the area
of the horizon vanishes. If we further extrapolate the expressions to
temperature higher than $T_{\textrm{max}}$, the mass parameter becomes
negative $m<0$, leaving a naked singularity. Moreover, the parameter $\upbeta$
goes to the complex plane. At the temperature $T_{\textrm{max}}$ the value of
the mass is given by $M_{\textrm{ext}}^{2}=Q_{R}^{2}/2$ which is precisely the
BPS mass which is the lowest possible from the dual quantum mechanical point
of view. In Figure 1, we show how the mass and entropy behave as functions of
temperature. We should however stress that we should not trust the curves when
the temperature becomes too close to $T_{\rm max}$ and the black hole becomes
string scale. We will discuss the correct behavior in Section 2.2.

$(a)$

$(b)$

Figure 1: (a) An illustration of the black hole mass (divided by its extremal
value $M_{\textrm{ext}}$) as a function of temperature (divided by
$T_{\textrm{max}}=1/(4\pi\sqrt{2}\sqrt{|Q_{R}^{2}-Q_{L}^{2}|})$). The mass
decreases until the temperature reaches maximum. The dashed part of the blue
line means that the black hole is of string scale and the curve is just an
extrapolation. (b) An illustration of the black hole entropy (divided by the
extremal entropy $S_{\textrm{ext}}$ from microstate counting) as a function of
temperature. The naive extrapolation of the entropy simply vanishes when we
reach the maximum temperature.

Above we considered the case of $\upbeta\rightarrow\infty$ and therefore
$Q_{R}>Q_{L}$. The discussion of the case $Q_{L}>Q_{R}$ essentially mirrors
the previous case, but the interpretation is slightly different. In this case
as the black hole temperature rises, the solution eventually has
$\upalpha\to\infty$ instead. The expressions above are still true, but now the
lowest mass black hole has $M=Q_{L}/\sqrt{2}$ which is not supersymmetric.
Since $Q_{L}>Q_{R}$ there is now a range of masses, namely $Q_{L}^{2}/2\geq
M^{2}\geq Q_{R}^{2}/2$ which is allowed by unitarity in the microscopic
theory, but where black holes do not exist. Therefore depending on the sign of
$Q_{R}-Q_{L}$ the ground state might be BPS or not.

Finally, if $Q_{L}=Q_{R}$ then $\upalpha=\upbeta$. We will not go into the
details of this special case, but just note that we need to take both of them
to infinity, and the extrapolation of (23) suggests that the extremal limit
corresponds to infinite temperature $\beta_{\rm ext}=0$.

We note that the black hole solution in the two derivative theory (1) is
symmetric with respect to turning on the left or right moving charges.
However, from the microscopic theory point of view, we wouldn’t expect such
symmetry. For one, the solution should be supersymmetric only when
$M=Q_{R}/\sqrt{2}$ but not when $M=Q_{L}/\sqrt{2}$. As it turns out, this
symmetry is broken once one considers the leading $\alpha^{\prime}$
corrections to the black hole Massai:2023cis .

###  2.2 Transition to free strings via the charged Horowitz-Polchinski
solution

As we saw in the previous section, the extrapolation of the gravity solution
(3) to the extremal limit has some peculiar features. The black hole solution
becomes singular as the horizon shrinks to zero size. Of course, once the size
of the horizon becomes comparable to string scale, the higher curvature
corrections would become important and there is no reason to trust the
solution (3). In this section, we will review the result in Chen:2021dsw ,
which states that the approach to extremality of the two-charge system is in
fact described by a charged analogue of the Horowitz-Polchinski solution
Horowitz:1997jc .

Here we will only sketch the idea behind the construction of the solution as
details can be found in Chen:2021dsw . The original uncharged Horowitz-
Polchinski solution can be constructed explicitly by considering the effective
action of the thermal winding modes coupled to gravity and dilaton. The
uncharged solution is trustworthy when the temperature is below but very close
to the Hagedorn temperature, $(\tilde{\beta}-\beta_{H})/\beta_{H}\ll 1$,666We
will use tilded quantities to denote the quantities in the uncharged solution,
which are distinct from those of the charged solution but related via the
solution generating procedure, see Appendix. A for a more detailed discussion.
where we can focus on only the winding modes with winding number $\pm 1$ and
the fluctuation of the $g_{tt}$ component of the metric. The solution carries
a classical entropy, which agrees with the Hagedorn density of states for a
gas of strings close to the Hagedorn temperature. This suggests that the
physical interpretation of the solution is describing a gas of highly excited
strings, weakly interacting via gravity (see also Damour:1999aw ). It is
further suggested in Chen:2021dsw  that in Heterotic string theory, this
classical solution could potentially be smoothly connected to the
Schwarzschild black hole solution as we lower the temperature.777While in the
Type II theories, a worldsheet index argument suggests that the two solutions
cannot be smoothly interpolated.

An important property of the solution which differs from the Schwarzschild
black hole is that as we increase the temperature, the physical size
$\ell_{\textrm{size}}$ of the solution expands as
$1/\sqrt{\tilde{\beta}-\beta_{H}}$. Physically, this means the classical
solution is describing a more and more dilute string gas. When the temperature
is very close to the Hagedorn temperature, the classical solution breaks down
and the correct description is simply the quantum description of free string
gas. In four dimensions, the solution is only valid when

$\tilde{g}_{s}^{\frac{4}{3}}\lesssim(\tilde{\beta}-\beta_{H})/\beta_{H}\ll 1.$
(31)

Given the uncharged solution, Chen:2021dsw  performed a solution generating
transformation to find the charged solution. The explicit form of the solution
can be found in Chen:2021dsw , while here we will only mention a few main
properties. It turns out that the $\tilde{\beta}\rightarrow\beta_{H}$ limit of
the “seed” solution is mapped to the extremal limits $M\sim Q_{R}/\sqrt{2}$ or
$M\sim Q_{L}/\sqrt{2}$ of the generated solutions. Differing from the naive
extrapolation of the black hole results in Section 2.1.3, where the
temperature rises and reaches a maximum, the temperature of the generated
Horowitz-Polchinski solution goes to zero, i.e. $\beta\rightarrow\infty$ in
the extremal limit. Furthermore, the solution correctly reproduces the entropy
as expected from counting microstates of long strings carrying momentum and
winding in the extra directions

$\displaystyle
S\;\sim\;\sqrt{2}\pi\ell_{s}\sqrt{Q_{R}^{2}-Q_{L}^{2}}\,,\qquad$
$\displaystyle M\;\sim\;\frac{Q_{R}}{\sqrt{2}}\,,$ (32) $\displaystyle
S\;\sim\;\pi\ell_{s}\sqrt{Q_{L}^{2}-Q_{R}^{2}}\,,\qquad$ $\displaystyle
M\;\sim\;\frac{Q_{L}}{\sqrt{2}}\,.$

The thermodynamics of the charged solution can be determined purely from the
thermodynamics of the seed solution plus the form of the solution generating
transformation at spatial infinity. We will discuss the derivation of such
relations in Appendix A, extending the analysis in Chen:2021dsw  to the cases
where the seed solution has rotation. Specifying (156)–(160) to the case
without rotation, and expanding the temperature of the seed solution around
$\beta_{H}=2\pi R_{H}=2\pi(1+1/\sqrt{2})\ell_{s}$, we get

$\displaystyle M$ $\displaystyle\;=\;\frac{\tilde{S}^{\prime}}{2\pi
R_{H}}\cosh\upalpha\cosh\upbeta\,,$ (33) $\displaystyle S$
$\displaystyle\;=\;\frac{\tilde{S}^{\prime}}{2}\left[\left(1-\frac{\alpha^{\prime}}{2R_{H}^{2}}\right)\cosh\upalpha+\left(1+\frac{\alpha^{\prime}}{2R_{H}^{2}}\right)\cosh\upbeta\right]\,,$
$\displaystyle\beta$
$\displaystyle\;=\;\frac{\beta_{H}}{2}\left[\left(1+\frac{\alpha^{\prime}}{2R_{H}^{2}}\right)\cosh\upalpha+\left(1-\frac{\alpha^{\prime}}{2R_{H}^{2}}\right)\cosh\upbeta\right]\,,$
$\displaystyle Q_{L}$ $\displaystyle\;=\;\frac{\tilde{S}^{\prime}}{\sqrt{2}\pi
R_{H}}\cosh\upbeta\sinh\upalpha\,,$ $\displaystyle Q_{R}$
$\displaystyle\;=\;\frac{\tilde{S}^{\prime}}{\sqrt{2}\pi
R_{H}}\cosh\upalpha\sinh\upbeta\,,$

where $\tilde{S}^{\prime}$ is the entropy of the seed solution after stripping
off the dependence on the string coupling, and is given by

$\tilde{S}^{\prime}\;\approx\;c_{0}\sqrt{\ell_{s}^{3}(\tilde{\beta}-\beta_{H})}.$
(34)

Here $c_{0}\approx 2.41$ is an order one number coming from the properties of
the seed solution.

In summary, (33) and (34) describe the thermodynamic quantities of the charged
Horowitz-Polchinski solution, parameterized by three parameters
$\\{\tilde{\beta},\upalpha,\upbeta\\}$. They are only valid when we satisfy
(31). Note that the lower end of the inequliaty in (31) is expressed in terms
of the string coupling constant in the seed solution, rather than the
generated solution.

With (33), we can reexamine how the naive approach to extremality in section
2.1.3 gets modified. As we mentioned, now the extremal limit corresponds to
$\tilde{\beta}\rightarrow\beta_{H}$, and to fix the charges, we need to
combine it with the limits $\upalpha\rightarrow\infty$ or
$\upbeta\rightarrow\infty$. For the case of $\upalpha\rightarrow\infty$, we
have $M\rightarrow Q_{L}/\sqrt{2}$, and for $\upbeta\rightarrow\infty$, we
have $M\rightarrow Q_{R}/\sqrt{2}$. From the expression of $\beta$ in (33), we
clearly see that both would correspond to the limit where
$\beta\rightarrow\infty$, i.e. the temperature goes to zero in the extremal
limit.

We can consider some special cases for which we can write down simpler
formulas for how the energy and entropy depends on the temperature. Consider
the case $Q_{L}=0$ ($\upalpha=0,\,\upbeta\rightarrow\infty$), at low
temperature we have

$\frac{M-M_{\textrm{ext}}}{M_{\textrm{ext}}}\;\sim\;\frac{S-S_{\textrm{ext}}}{S_{\textrm{ext}}}\;\sim\;2e^{-2\upbeta}\;\sim\;(\pi\ell_{s}T)^{2},$
(35)

where $M_{\textrm{ext}}=Q_{R}/\sqrt{2}$ and
$S_{\textrm{ext}}=\sqrt{2}\pi\ell_{s}Q_{R}$. Translating (31) into the
parameters of the generated solution, we find that the solution is trustworthy
when

$\left(\frac{\ell_{s}}{Q_{R}}\right)^{\frac{1}{2}}\ll Q_{R}T\ll 1\,.$ (36)

On the other hand, for the case of $Q_{R}=0$
($\upbeta=0,\,\upalpha\rightarrow\infty$), at low temperature we have

$\frac{M-M_{\textrm{ext}}}{M_{\textrm{ext}}}\;\sim\;\frac{S-S_{\rm
ext}}{S_{\rm ext}}\;\sim\;2e^{-2\upalpha}\sim 2(\pi\ell_{s}T)^{2}\,,$ (37)

where $M_{\textrm{ext}}=Q_{L}/\sqrt{2}$ and
$S_{\textrm{ext}}=\pi\ell_{s}Q_{L}$. The region of validity of the solution is
similar to (36), but with $Q_{R}$ replaced by $Q_{L}$. Note that we have
$\delta S\propto\delta M$, which is expected from the microscopic entropy
formula

$S\;=\;2\pi\ell_{s}\left[\sqrt{M^{2}-\frac{Q_{L}^{2}}{2}}+\frac{1}{\sqrt{2}}\sqrt{M^{2}-\frac{Q_{R}^{2}}{2}}\right]$
(38)

when we expand around extremality.

$(a)$

$(b)$

Figure 2: (a) The blue curves are from Figure 1 for comparison. The orange
curve is the mass of the charged Horowitz-Polchinski solution as a function of
the temperature. It is trustworthy when $\sqrt{\ell_{s}/|Q|}\ll
T/T_{\textrm{max}}\ll 1$, where $|Q|$ can be taken to be the greater of
$|Q_{L}|,|Q_{R}|$. The dashed lines are extrapolations. We see that the
temperature goes to zero in the extremal limit. (b) The orange curve is the
entropy of the Horowitz-Polchinski as a function of the temperature. We see
that $S/S_{\textrm{ext}}$ goes to one in the extremal limit. The precise shape
of the curves depends on the values of $Q_{R}$ and $\ell_{s}$ in Planck units.
In the microcanonical ensemble, where we vary the mass, we expect the two
curves to be joined together near where their extrapolations cross.

We note that the low temperature thermodynamics is qualitatively different
from what one would expect if there were an AdS2 region in the geometry. In
Section 6 we will argue that the fact that the low energy physics is not given
by AdS2 could be argued from more general principle. In Figure 2 we illustrate
that after considering the charged Horowitz-Polchinski solution, the naive
extremal limit in Figure 1 is modified significantly.

We close this section by mentioning that other than the thermodynamic
quantities, one can also look at the physical size of the Horowitz-Polchinski
solution, and it matches with the size of the BPS string expected from a
random walk picture Chen:2021dsw ,

$\ell_{\textrm{phys}}\sim\left(nw\right)^{\frac{1}{4}}\ell_{s}\,.$ (39)

##  3 Gravitational configurations that contribute to the index

We now initiate our study of the gravitational path integral that evaluates
the index of two-charge states. We first construct supersymmetric solutions by
taking a non-trivial limit of the two-charge black hole analyzed in the
previous section. The situation is more subtle than its ‘big black hole’
counterpart and we emphasize some puzzling features. Those features will go
away later when we include higher derivative corrections.

###  3.1 Imposing supersymmetry

We want to construct supersymmetric finite temperature black hole solutions
that contribute to the gravitational path integral for the index. We actually
need to consider the helicity supertrace, but one can simply absorb the
fermion zero modes and we will not worry about this here. Imposing
supersymmetry at the boundary (i.e. the asymptotic region of flat space)
requires fermions to be periodic. As explained in Iliesiu:2021are  this
constrains the angular velocity $\Omega$. Supersymmetry in the bulk requires
the existence of a globally defined Killing spinor. For the black hole
geometries presented in Section 2 this constrains the mass parameter $M$.
Explicitly these two conditions are given by

$\displaystyle{\rm
Boundary~{}Supersymmetry:}~{}~{}~{}~{}\,\beta\Omega\;=\;2\pi i\,~{}{\rm
mod}~{}4\pi\mathbb{Z}\,,$ $\displaystyle{\rm
Bulk~{}Supersymmetry:}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}M\;=\;Q_{R}/\sqrt{2}.$

The first condition fixes the fermion periodicity conditions because this
choice of angular velocity amounts to an insertion of $e^{\beta\Omega J}\to
e^{2\pi iJ}=(-1)^{\sf F}$. Since all excitations have integral or half-
integral angular momentum this condition defines the angular velocity only up
to a shift multiple of $4\pi/\beta$. Bulk and boundary supersymmetry are in
principle independent. For example, in the case of Reissner-Nordström, if
$\beta\Omega=2\pi i$ then this implies that $M=Q$ and a globally defined
Killing spinor exists, but if $\beta\Omega=6\pi i$ or any other odd multiple
of $2\pi i$, then $M\neq Q$ and supersymmetry is not preserved in the bulk.
One unexpected feature of the two-charge system is that even for
$\beta\Omega=2\pi i$ bulk supersymmetry is not guaranteed.

The discussion here will assume that in the presence of $(-1)^{\sf F}$ the
gravitational path integral is dominated by a black hole-like saddle. Can
there be another saddle similar to the Horowitz-Polchinski one that
contributes to the index? One option, since the thermal circle is non-
contractible, is to take the Horowitz-Polchinski solution of the previous
section and impose fermions to be periodic. The problem with this proposal is
that the solution is not supersymmetric and therefore does not contribute to
the index due to fermion zero-modes.888Even if it did, it produces a
contribution that depends on temperature in the free energy, inconsistent with
the behavior of an index. Note also that the Horowitz-Polchinski solution also
breaks down before we reach exact extremality due to quantum corrections. Even
if this had worked, the problem is that the equations of motion that Horowitz-
Polchinski solves are not correct in the presence of $(-1)^{\sf F}$. At least
in flat space, periodic fermions lead to the standard GSO projection and
therefore the light winding mode that condenses in the Horowitz-Polchinski
solution is not part of the low energy spectrum Atick:1988si . It is therefore
not clear which equations we are supposed to solve to find such a solution. It
is possible that by turning on an imaginary angular potential, similar to the
black hole case, one can evade the flat space GSO projection argument and
arrive at a supersymmetric solution, but we are not sure how to construct such
a solution explicitly (see related discussion in Ceplak:2023afb ). We should
emphasize that even if such a solution was found and produces the correct
index, it would not invalidate our discussion below. On the contrary, it would
be in harmony with the conjecture that there is no phase transition between
the Horowitz-Polchinski solution and the black hole in the heterotic theory.
Since the supergravity sector is part of the low energy spectrum when fermions
are periodic, and since there are black hole solutions in this sector, this
will be our focus in the rest of the paper.

After these introductory remarks let us impose $\beta\Omega=2\pi i$ and see
what the implications are for the two-charge black hole. Since the angular
velocity is imaginary it is convenient to introduce $a=i{\sf a}$, so ${\sf a}$
will then be real. The condition becomes then

$\frac{\beta\Omega}{2\pi i}\;=\;\frac{{\sf a}}{\sqrt{m^{2}+{\sf
a}^{2}}}\;=\;1\,.$ (40)

It is convenient sometimes to replace the variables $(m,{\sf a})$ by
$(m,x={\sf a}/m)$. Then the boundary supersymmetry condition becomes
$x=\sqrt{1+x^{2}}$, and the only solution is to take $x\to\infty$. We are
working in the thermal ensemble and the black hole temperature, in the limit
$x={\sf a}/m\to\infty$, becomes

$\beta\;=\;\frac{2\pi m(\cosh\upalpha+\cosh\upbeta)(m+\sqrt{m^{2}+{\sf
a}^{2}})}{\sqrt{m^{2}+{\sf a}^{2}}}\Big{|}_{{\sf a}/m\to\infty}\;=\;2\pi
m(\cosh\upalpha+\cosh\upbeta)\,.$ (41)

Since the temperature is a free parameter set by our choice of boundary
conditions at asymptotic flat space, the right hand side of this expression
should remain finite. We analyze below two ways to implement this condition.

##### Setting ${\sf a}\to\infty$ and $m$ finite

As a first possibility we can keep $m$, $\upalpha$ and $\upbeta$ finite, with
one combination of them set by the temperature, and send ${\sf a}\to\infty$ to
implement boundary supersymmetry. We will see now this option is ruled out
after imposing bulk supersymmetry. For $m$ finite, combining (13) with (12)
one can show that setting $M^{2}=Q_{R}^{2}/2$ implies the following relation
between the parameters $\upalpha$ and $\upbeta$

$(1+\cosh\upalpha\cosh\upbeta)^{2}\;=\;\cosh^{2}\upalpha\sinh^{2}\upbeta\,.$
(42)

There is no real solution of this equation for finite $\upalpha$ and
$\upbeta$, since the constraint implies $e^{\upalpha+\upbeta}<0$. It is not
clear that the black hole solves the correct equations of motion since for
complex values of $\upalpha$ and $\upbeta$ the transformation used to generate
the solution is not in $\mathrm{O}(7,23)$ any longer. There is one more
problem with this scenario that is worth mentioning. We evaluate the area of
the event horizon, given in (18), in the large ${\sf a}$ limit

$A\to 4\pi m(\cosh\upalpha+\cosh\upbeta)mx\quad\Rightarrow\quad A\;=\;2\beta
mx\,.$ (43)

We see that at fixed temperature and keeping $m$ finite, the area is
$A\propto{\sf a}\to\infty$. Now instead of vanishing, the area actually
diverges! It is not obvious that this feature is necessarily problematic, as
explained in Chen:2023mbc , but we would like to avoid it if possible since
the interpretation of such a saddle is unclear. For these reasons we will not
consider this direction any further.

##### Setting $m\to 0$ and ${\sf a}$ finite

Another way of implementing the bulk supersymmetry condition is to keep ${\sf
a}$ finite while taking the mass parameter $m$ to zero. This is the regime
considered in Sen:1994eb , in the absence of the $(-1)^{\sf F}$ insertion.
Looking at the expression for the temperature given in (41) we see that this
limit requires taking $(\cosh\upalpha+\cosh\upbeta)\to\infty$. How this limit
is taken is fixed by bulk supersymmetry. For this purpose let us take the
$m\to 0$ limit of the charges

$\vec{Q}_{L}\;=\;\frac{\beta}{2\pi\sqrt{2}}\frac{\sinh\upalpha\cosh\upbeta}{\cosh\upalpha+\cosh\upbeta}\,\vec{n}\,,\qquad\vec{Q}_{R}\;=\;\frac{\beta}{2\pi\sqrt{2}}\frac{\sinh\upbeta\cosh\upalpha}{\cosh\upalpha+\cosh\upbeta}\,\vec{p}\,.$
(44)

and similarly the mass

$M\;=\;\frac{\beta}{4\pi}\frac{1+\cosh\upalpha\cosh\upbeta}{\cosh\upalpha+\cosh\upbeta}\,.$
(45)

One can check that $\upbeta\to\infty$ while keeping $\upalpha$ finite does
provide a solution of $M^{2}=Q_{R}^{2}/2$, as pointed out in Sen:1994eb . To
guarantee both boundary and bulk supersymmetry, as well as a finite
temperature, we then take

$m\to 0,\quad\upbeta\to\infty\,,\qquad{\rm such~{}that}\qquad m\cosh\upbeta\to
m_{0}\,,$ (46)

where we introduced the finite parameter $m_{0}$. In terms only of parameters
that are finite in the limit, the temperature and charges are

$\beta\;=\;2\pi m_{0}\,,\qquad
Q_{L}\;=\;\frac{m_{0}}{\sqrt{2}}\sinh\upalpha\,,\qquad
Q_{R}\;=\;\frac{m_{0}}{\sqrt{2}}\cosh\upalpha\,.$ (47)

We can use the second and third equations to solve for $m_{0}$ and $\upalpha$
in terms of the charges. Explicitly, we obtain

$m_{0}\;=\;\sqrt{2(Q_{R}^{2}-Q_{L}^{2})}\,,\qquad\tanh\upalpha\;=\;\frac{Q_{L}}{Q_{R}}\,.$
(48)

Using these values, we see that the temperature is fixed in terms of the
charges! Moreover, the parameter ${\sf a}$ is undetermined. We will comment
more on this later. This solution only exists if $Q_{R}>Q_{L}$. No
supersymmetric solution of this kind exists if $Q_{L}>Q_{R}$ instead.999What
happens with the supergravity solution when $Q_{L}=Q_{R}$? In this case
$\upalpha=\upbeta$ and bulk supersymmetry demands that
$\upalpha=\upbeta\to\infty$. But it is evident from (44) that in this case
keeping the temperature finite leads to $Q_{L},Q_{R}\to\infty$, therefore no
solution exists with finite $Q_{L}=Q_{R}$. Another option is to take $\beta\to
0$ such that $\beta e^{\upbeta}$ is finite. This implies now that $m\sim
e^{-2\upbeta}$, vanishing faster than before. To keep the black hole area
finite we need to send ${\sf a}\to\infty$. Since its not clear whether this is
a physical solution or not we will not consider it further and restrict
ourselves to $Q_{R}>Q_{L}$. This is consistent with the spectrum of heterotic
strings compactified on a torus, and consistent with the spectrum of
Lorentzian black holes in the absence of the $(-1)^{\sf F}$ insertion
discussed in Section 2. Finally, the chemical potentials of the left and right
charges have magnitudes $\mu_{L}=0$ and $\mu_{R}=1/\sqrt{2}$, independent of
the parameters $m_{0}$ and $\upalpha$.

###  3.2 Explicit form of the solution

Now that we have determined the scaling of all parameters in the
supersymmetric limit we can write down explicitly how the solution looks like.
The same solution was written down in Horowitz:1995tm ; Dabholkar:1995nc
although it was interpreted differently. In the supersymmetric limit the
metric simplifies slightly,

$\displaystyle e^{-\Phi}\,\mathrm{d}s^{2}$ $\displaystyle\;=\;$
$\displaystyle\Delta^{\frac{1}{2}}\Big{\\{}\frac{r^{2}-{\sf
a}^{2}\cos^{2}\theta}{\Delta}\mathrm{d}t_{E}^{2}+\frac{\mathrm{d}r^{2}}{r^{2}-{\sf
a}^{2}}+\mathrm{d}\theta^{2}$
$\displaystyle+\frac{\sin^{2}\theta}{\Delta}[\Delta-{\sf
a}^{2}\sin^{2}\theta(r^{2}-{\sf
a}^{2}\cos^{2}\theta+2m_{0}r\cosh\upalpha)]\,\mathrm{d}\phi^{2}-\frac{2m_{0}r{\sf
a}\sin^{2}\theta}{\Delta}\,\mathrm{d}t_{E}\mathrm{d}\phi\Big{\\}}\,,$

where the supersymmetric limit of the function $\Delta$ becomes

$\Delta\;=\;(r^{2}-{\sf a}^{2}\cos^{2}\theta)^{2}+2m_{0}r(r^{2}-{\sf
a}^{2}\cos^{2}\theta)\cosh\upalpha+m_{0}^{2}r^{2}\,.$ (50)

Here, we remind the reader that $\mathrm{d}s^{2}$ is defined in the string
frame. The dilaton is still given by $\Phi=\log(r^{2}-{\sf
a}^{2}\cos^{2}\theta)/\Delta^{1/2}$. The solution is real in Euclidean space.
From the second term in the first line we see that the horizon is located at

$r_{+}\;=\;{\sf a}\,,$ (51)

giving a geometric interpretation to ${\sf a}$. One can verify directly with
the supersymmetric solution that smoothness of the metric at the horizon
imposes values of the temperature and angular velocity consistent with the
previous analysis, namely $\beta=2\pi m_{0}$ and $\Omega=i/m_{0}$, such that
$\beta\Omega=2\pi i$. In fact, the solution is smooth everywhere on the sphere
except at the poles, which we comment on later. The area of the event horizon
in Einstein frame is

$A\;=\;\int\mathrm{d}\theta\mathrm{d}\phi\sqrt{g^{E}_{\theta\theta}g^{E}_{\phi\phi}}\;=\;\int\mathrm{d}\theta\mathrm{d}\phi\,m_{0}{\sf
a}\sin\theta\;=\;4\pi m_{0}{\sf a}\,,$ (52)

which is finite and non-vanishing! The rotation necessary to implement the
$(-1)^{\sf F}$ has turned the small black hole (at extremality) into a
macroscopic one at finite temperature for the index. This seems to suggest we
are now able to use the gravitational path integral to evaluate the index of
the two-charge state avoiding small black holes, but this is too naive. Even
though the solution has a macroscopic horizon, the black hole geometry is
actually singular at the poles of the sphere $\theta=0,\pi$ so the rotation
alone has not fixed all issues. This is clear when looking in the Einstein
frame metric which is given by the string frame one multiplied by $e^{\Phi}$.
Near the horizon

$e^{\Phi}\big{|}_{r={\sf a}}\;=\;\Delta^{-\frac{1}{2}}{\sf
a}^{2}\sin^{2}\theta\,,$ (53)

and $\Delta|_{r={\sf a}}$ is a non-vanishing finite function of $\theta$.
Therefore we see that $e^{\Phi}$ vanishes at the two poles of the sphere. A
more detailed analysis shows that the string frame metric is singular, while
the vanishing of $e^{\Phi}$ at the poles makes the Einstein frame metric
smooth, although higher derivative corrections in the Einstein frame can be
large due to dilaton diverging.

The solution is specified by the gauge potentials besides the metric. These
also have a finite limit in the supersymmetric limit given by

$\displaystyle\vec{A}_{L}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\vec{n}}{\sqrt{2}}\frac{m_{0}r\sinh\upalpha(r^{2}-{\sf
a}^{2}\cos^{2}\theta)}{\Delta}\mathrm{d}t-i\frac{\vec{n}}{\sqrt{2}}\frac{m_{0}^{2}r^{2}{\sf
a}\sinh 2\upalpha\sin^{2}\theta}{2\Delta}\mathrm{d}\phi\,,$
$\displaystyle\vec{A}_{R}$ $\displaystyle\;=\;$
$\displaystyle-\frac{\vec{p}}{\sqrt{2}}\frac{m_{0}r((r^{2}-{\sf
a}^{2}\cos^{2}\theta)\cosh\upalpha+m_{0}r)}{\Delta}\mathrm{d}t$
$\displaystyle~{}~{}~{}~{}~{}+i\frac{\vec{p}}{\sqrt{2}}\frac{m_{0}r{\sf
a}\sin^{2}\theta(r^{2}-{\sf
a}^{2}\cos^{2}\theta+m_{0}r\cosh\upalpha)}{\Delta}\mathrm{d}\phi\,,$
$\displaystyle B$ $\displaystyle\;=\;$ $\displaystyle-i\frac{m_{0}r{\sf
a}\sin^{2}\theta}{\Delta}(r^{2}-{\sf
a}^{2}\cos^{2}\theta+m_{0}r\cosh\upalpha)\,\mathrm{d}t\mathrm{d}\phi\,.$ (54)

These gauge potentials are smooth and one can extract from them the chemical
potentials, verifying that $\mu_{L}=0$ and $\mu_{R}=1/\sqrt{2}$. Finally, the
functions appearing in the moduli scalar profiles $M$ also simplify

${\sf P}\;=\;\frac{2m_{0}^{2}r^{2}\sinh^{2}\upalpha}{\Delta}\,,\qquad{\sf
Q}\;=\;-\frac{2m_{0}r\sinh\upalpha}{\Delta}(r^{2}-{\sf
a}^{2}\cos^{2}\theta+m_{0}r\cosh\upalpha)\,.$ (55)

These functions are smooth and finite at the horizon.

In general, the Euclidean action of the two-charge heterotic black hole is
given by $I_{\rm on-shell}=\beta M-A/4-\beta\Omega J$ and comes solely from
the boundary terms in the action. This is the result in an ensemble of fixed
electric charges. If we have a supersymmetric solution then the first term is
$\beta Q_{R}^{2}/2$. In the rest of the paper we will always include an
appropriate boundary term that has the effect of shifting the zero-point
energy such that $I\to I-\beta Q_{R}^{2}/2$. After this modification, $-I_{\rm
on-shell}$ simplifies to $A/4+\beta\Omega J$. We can show now that whenever a
supersymmetric saddle exists, then these two terms cancel each other and the
classical two-derivative contribution to the on-shell action, and therefore to
$\log Z_{\rm index}$ vanishes. The area and the angular momentum can be
written as

$\frac{A}{4}\;=\;\frac{\beta{\sf a}}{2}\,,\qquad J\;=\;i\frac{\beta{\sf
a}}{4\pi}\,.$ (56)

We already discussed the area. The angular momentum can be read off directly
from the supersymmetric solution or taking the appropriate limit of (13).
Using that $\beta\Omega=2\pi i$ we get the desired cancellation

$\frac{A}{4}+\beta\Omega J\;=\;0\,,\qquad\Rightarrow\qquad\log Z_{\rm
index}\;=\;-I_{\rm on-shell}\;=\;0\,.$ (57)

We found a black hole with a macroscopic area that nevertheless gives a
vanishing contribution to the index to leading order. This is the desired
conclusion, since a non-zero action at the two-derivative level would, by
dimensional analysis, grow too fast with the charges and would not reproduce
the microscopic index.

###  3.3 Summary of properties

To recap, we have found a supersymmetric finite temperature black hole with
two-charges that evaluates the index avoiding the small black hole regime.
Even after including the rotation necessary to implement the $(-1)^{\sf F}$
some puzzles remain to be resolved. We finish this section by summarizing and
expanding on some of them. In the next sections we will resolve them by
including higher derivative corrections.

  1. 1.

The solution we found has a curvature singularity at the poles of the $S^{2}$
horizon $r=r_{+}={\sf a}$ at $\theta=0$ and $\pi$. This forces us to include
higher derivative corrections even though the event horizon is macroscopic.

  2. 2.

For a given set of charges, the solution only exists for a unique value of the
temperature. This is surpring since charges and temperatures can be
independently determined by boundary conditions at infinity. After taking the
supersymmetric limit we found

$\beta\;=\;2\pi\sqrt{2(Q_{R}^{2}-Q_{L}^{2})}\,.$

For comparison, in the Reissner-Nordström case analyzed in Iliesiu:2021are
the inverse temperature is $\beta=\pi Q(Q+2{\sf a})/{\sf a}$, while
$r_{+}=Q+{\sf a}$. A solution therefore exists for any temperature and the
horizon radius depends on its value. For the case considered here, the
singular feature is that the dependence on ${\sf a}$ disappeared completely
from $\beta$, making it a function only of the charge. What solution
contributes to the gravitational path integral when $\beta\neq
2\pi\sqrt{2(Q_{R}^{2}-Q_{L}^{2})}$?

  3. 3.

In the Reissner-Nordström case quoted in the previous item, not only is the
temperature a tunable parameter but it also determines a unique value of ${\sf
a}$. In the two-charge case another consequence of the fact that ${\sf a}$
drops out of the temperature101010The fact that ${\sf a}$ drops out, and the
temperature is fixed in terms of the charges, is special to compactifications
to four dimensions. Similar black hole solutions in toroidal compactifications
to $D>4$ dimensions do not have this problem Chowdhury:2024ngg . The reason we
insist on working in $D=4$ is the fact that supersymmetric higher derivative
solutions are known and we use them in Section 5.  is that there is a moduli
space of solutions labeled by ${\sf a}$ from $0$ to $\infty$. All values of
${\sf a}$ contribute equally and have to be integrated over. This is a
disturbing state of affairs and we will see that higher derivative corrections
will determine a unique value of ${\sf a}$.

Before addressing the effect of higher derivative corrections, we will show in
the next section that these features are common to a class of two-charge black
holes in $\mathcal{N}\geq 2$ supergravity.

##  4 The index and new attractor mechanism in 4d $\mathcal{N}=2$ supergravity

In this section we move to the $\mathcal{N}=4$ four-dimensional effective
theory that describes the compactification of the heterotic string on $T^{6}$.
From the string-string duality, this is equivalent to Type II theory on
$K3\times T^{2}$. More generally, we can consider four-dimensional
$\mathcal{N}=2$ theories that arise from the compactification of Type II on a
Calabi-Yau 3-fold. These theories are described by $\mathcal{N}=2$
supergravity coupled to vector and hyper multiplets. The fields that
participate in the black hole solutions live in the graviton multiplet and the
vector multiplet, and we focus on this sector. In this section we discuss the
theory with a two-derivative action, and in the next section we consider
higher-derivative corrections.

The conformal supergravity formalism is an elegant and powerful formalism to
describe the theory of $\mathcal{N}=2$ supergravity coupled to a number of
vector multiplets deWit:1979dzm ; deWit:1980lyi . In this formalism, local
conformal invariance is a gauge symmetry of the theory. Accordingly, each
field is assigned a conformal or Weyl weight. Eventually, the conformal
symmetry is gauge-fixed by assigning a dimensionful number like
$m_{\text{Pl}}$ to a field with non-zero Weyl weight. This gauge-fixing leads
to Poincaré supergravity.

The bosonic physical field content of each vector multiplet consists of a
vector field $A^{I}$ and a complex scalar $X^{I}$, $I=0,1,\dots,n_{\text{v}}$.
The scalars have Weyl weight 1. Solutions of the theory carry electric and
magnetic charges $(Q_{I},P^{I})$ under these gauge fields. The two-derivative
action is completely specified by the prepotential $F(X)$, which is a
homogeneous function of degree two and has Weyl weight 2. An important role is
played by the first derivatives $F_{I}=\partial F/\partial X^{I}$ which have
Weyl weight 1. At the two-derivative level the equations of motion of the
theory have $Sp(2n_{\text{v}}+2;\mathbb{R})$ symmetry, under which the _charge
vector_ $\Gamma=(P^{I},Q_{I})$ and the _period vector_
$\Omega_{\text{hol}}=(X^{I},F_{I})$ transform as
$2(n_{\text{v}}+1)$-dimensional vectors.

Two important invariants of the symplectic symmetry are111111Here, and below,
the symplectic product of two vectors $A=(\widetilde{A}^{I},A_{I})$ and
$B=(\widetilde{B}^{I},B_{I})$ is given by $\langle
A,B\rangle=\widetilde{A}^{I}B_{I}-\widetilde{B}^{I}A_{I}$. the generalized
Kähler potential (of Weyl weight 2)

$e^{-\mathcal{K}(X,\overline{X})}\;=\;i\bigl{(}X^{I}\overline{F}_{I}-\overline{X}^{I}F_{I}\bigr{)}\;=\;i\langle\,\Omega_{\text{hol}}\,,\,\overline{\Omega}_{\text{hol}}\,\rangle\,,$
(58)

and the central charge function (of Weyl weight 0) ,

$Z(\Gamma;\Omega)\;=\;\langle\,\Gamma\,,\,\Omega\,\rangle\;=\;e^{\frac{1}{2}\mathcal{K}(X,\overline{X})}\,(P^{I}F_{I}-Q_{I}X^{I})\,.$
(59)

The conjugate central function
$\overline{Z}(\Gamma;\Omega)\;=\;Z(\Gamma;\overline{\Omega})$ is also used
below.

###  4.1 $\frac{1}{2}$-BPS solutions of $\mathcal{N}=2$ supergravity

The general $\frac{1}{2}$-BPS stationary solution of this supergravity theory
is described as follows Behrndt:1997ny ; LopesCardoso:2000qm . The input to
the solution the vector
$H(\textbf{x})=\bigl{(}\widetilde{H}^{I}(\textbf{x}),H_{I}(\textbf{x})\bigr{)}$
of harmonic functions on base space. These functions are sources of electric
and magnetic fields.

The metric takes the form

$\mathrm{d}s^{2}\;=\;e^{2U}(\mathrm{d}t_{E}+i\omega)^{2}+e^{-2U}\mathrm{d}x^{m}\mathrm{d}x^{m}\,,$
(60)

where $U(\textbf{x})$ is a function of the base space coordinates $x^{m}$,
$m=1,2,3$, and $\omega=\omega_{m}\mathrm{d}x^{m}$ is a one-form on the base
space that governs the angular rotation of the solution. In order to have
asymptotically flat space, one has $e^{-2U(\infty)}=1$. Note that we have
directly described the Euclidean solution with compact time coordinate
$t_{E}$. The warping of the base space and the connection $\omega$ in (60) are
given by

$e^{-2U}\;=\;i\bigl{(}\mathcal{Y}^{I}\,\overline{\mathcal{G}}_{I}-\overline{\mathcal{Y}}^{I}\,\mathcal{G}_{I}\bigr{)}\,,\qquad\star\,\mathrm{d}\,\omega\;=\;i\langle\mathrm{d}H,H\rangle\,,$
(61)

where the weight-0 normalized fields $(\mathcal{Y}^{I},\mathcal{G}_{I})$ are
given by

$\bigl{(}\mathcal{Y}^{I},\mathcal{G}_{I}\bigr{)}\;=\;e^{\frac{1}{2}\mathcal{K}(X,\overline{X})}\,\overline{Z}(H;\Omega)\,\bigl{(}X^{I},F_{I}\bigr{)}\,$
(62)

and similarly $\overline{\mathcal{Y}}^{I}$, $\overline{\mathcal{G}}_{I}$ by
the complex conjugate equations. The scalar fields obey the _generalized
stabilization equations_ 121212Note that the signs in these equations are
opposite to the one in Boruch:2023gfn .

$\mathcal{Y}^{I}-\overline{\mathcal{Y}}^{I}\;=\;i\widetilde{H}^{I}\,,\qquad\mathcal{G}_{I}-\overline{\mathcal{G}}_{I}\;=\;iH_{I}\,,$
(63)

The above equations of supersymmetry for the
$(\mathcal{Y}^{I},\mathcal{G}_{I})$ fields can be presented in terms of a
prepotential function $\mathcal{G}(\mathcal{Y})$ with
$\mathcal{G}_{I}=\partial\mathcal{G}/\partial\mathcal{Y}^{I}$, where
$\mathcal{G}$ is the same mathematical function as $F$.131313One should note
that the rescaling between $X^{I}$ and $\mathcal{Y}^{I}$ is not holomorphic,
but one can still apply the concept of holomorphic and anti-holomorphic to the
$\mathcal{Y}$ variables directly. Indeed, we encounter holomorphic (and anti-
holomorphic) functions of $\mathcal{Y}^{I}$ in the discussion of the entropy.

Besides the metric and the scalars, the solution is specified by the
$n_{\text{v}}+1$ vector multiplet ${\rm U}(1)$ gauge fields. These are also
determined by the harmonic functions in a specific way, presented for example
in equation (3.21) of Boruch:2023gfn .

###  4.2 Euclidean black holes and near-horizon behavior

In this subsection we review the “new attractor mechanism” that was found in
Boruch:2023gfn . Here we present all the equations in coordinate language. We
refer the reader to Boruch:2023gfn  for details of the coordinate-invariant
equations.

Extremal black holes Supersymmetric extremal solutions are obtained by
choosing the harmonic functions in the above discussion to have electric and
magnetic sources at one point, say the origin, i.e.,

$H(\textbf{x})\;=\;h+\frac{\Gamma}{r}\,,\qquad r=|\textbf{x}|\,.$ (64)

The equations (61) show that the solution is spherically symmetric and static,
i.e. $\omega=0$ in (60). The vector $h\equiv H(\infty)$ is the asymptotic
value of sources. The generalized stabilization equations (63) imply that the
asymptotic behavior of the scalars as $r\to 0$ is

$\mathcal{Y}^{I}\,\sim\,\frac{Y^{I}_{*}}{r}\,,\qquad\mathcal{G}_{I}\,\sim\,\frac{G_{I*}}{r}\,,$
(65)

where the constant values $Y^{I}_{*},G_{I*}$ (and therefore $Z_{*}$) are
determined by the _extremal attractor equations_

$Y^{I}_{*}-\overline{Y}^{I}_{*}\;=\;iP^{I}\,,\qquad
G_{I*}-\overline{G}_{I*}\;=\;iQ_{I}\,.$ (66)

The behavior of the metric as $r\to 0$ is

$e^{-2U}\,\sim\,\frac{i\bigl{(}Y^{I}_{*}\,\overline{G}_{I*}-\overline{Y}^{I}_{*}\,G_{I*}\bigr{)}}{r^{2}}\,.$
(67)

From the form of the metric (60) we see that the Bekenstein-Hawking entropy of
the black hole is given by $S_{\text{BH}}=\pi
i\bigl{(}Y^{I}_{*}\,\overline{G}_{I*}-\overline{Y}^{I}_{*}\,G_{I*}\bigr{)}$.

Non-extremal supersymmetric solutions In order to obtain non-extremal
solutions relevant for the index, we replace the double pole behavior (67) of
the extremal metric by two single poles. Accordingly, we split the sources
into a pair of sources at two different points called the north and south
pole. The total charge is split as

$\gamma_{N}\;=\;\frac{1}{2}\bigl{(}P^{I}+in^{I},Q_{I}+im_{I}\bigr{)}\,,\qquad\gamma_{S}\;=\;\frac{1}{2}\bigl{(}P^{I}-in^{I},Q_{I}-im_{I}\bigr{)}\,,\qquad\Gamma\;=\;\gamma_{N}\,+\,\gamma_{S}\,,$
(68)

and the corresponding harmonic functions are given by

$H(\textbf{x})\;=\;h+\frac{\gamma_{N}}{|\textbf{x}-\textbf{x}_{N}|}+\frac{{\gamma}_{S}}{|\textbf{x}-{\textbf{x}}_{S}|}\,.$
(69)

Compared to the extremal solution we have introduced electric and magnetic
dipole charges $(m_{I},n^{I})$ and the distance
$|\textbf{x}_{N}-\textbf{x}_{S}|$ between the north and south poles. The
equation for $\omega$ in (61) shows that the metric necessarily has angular
momentum. All these parameters of the solution are determined by the _new
attractor mechanism_ Boruch:2023gfn  that we now recall.

The main point is that the spinning non-extremal solution near one of the
poles has a simple scaling behavior, and the values of the scalars at the
poles are therefore determined completely by the charges, similar to the
extremal solution. There is, however, an important difference in that now we
only have simple poles. As we briefly review below, one of the consequences is
that the solution is necessarily complex.

The full solution has a symmetry which exchanges the north with the south pole
and the scalars with their complex conjugates. Therefore we focus on the north
pole and let the parameter $\rho=|\textbf{x}-\textbf{x}_{N}|$ be very small.
From (69) we have, as $\rho\to 0$,

$H(\textbf{x})\;\sim\;\frac{\gamma_{N}}{\rho}+H^{(0)}+\text{O}(\rho)\,,\qquad
H^{(0)}\;=\;h+\frac{\gamma_{S}}{|\textbf{x}_{N}-\textbf{x}_{S}|}\,.$ (70)

The generalized stabilization equations (63) then imply that

$\mathcal{Y}^{I}\;\sim\;\frac{Y^{I}_{N}}{\rho}+Y^{I(0)}+\text{O}(\rho)\,,\qquad\mathcal{G}_{I}\;\sim\;\frac{G_{NI}}{\rho}+G^{(0)}_{I}+\text{O}(\rho)\,,\qquad$
(71)

and similarly for the conjugate scalars, with

$\bigl{(}Y^{I}_{N}-\overline{Y}^{I}_{N}\,,\,G_{IN}-\overline{G}_{IN}\bigr{)}\;=\;i\gamma_{N}\,,\qquad\bigl{(}Y^{I(0)}-\overline{Y}^{I(0)}\,,\,G^{(0)}_{I}-\overline{G}^{(0)}_{I}\bigr{)}\;=\;iH^{(0)}\,.$
(72)

Note the Laurent expansion (71) combined with the definition of the scalar
fields (62) and the homogeneity property of the prepotential leads to the
following identity that we use below,

$G^{(0)}_{I}\;=\;\frac{\partial\mathcal{G}_{I}}{\partial\mathcal{Y}^{J}}(\textbf{x}_{N})\,Y^{J(0)}\quad\Rightarrow\quad
Y^{I}_{N}\,G^{(0)}_{I}\;=\;G_{JN}\,Y^{J(0)}\,.$ (73)

Now, the equation (61) for the metric implies that

$e^{-2U}\;\sim\;i\Bigl{(}\frac{Y^{I}_{N}}{\rho}+Y^{I(0)}\Bigr{)}\Bigl{(}\frac{\overline{G}_{IN}}{\rho}+\overline{G}^{(0)}_{I}\Bigr{)}-i\Bigl{(}\frac{\overline{Y}^{I}_{N}}{\rho}+\overline{Y}^{I(0)}\Bigr{)}\Bigl{(}\frac{G_{IN}}{\rho}+G^{(0)}_{I}\Bigr{)}+\text{O}(\rho)\,.$
(74)

Demanding a simple pole at $\textbf{x}_{N}$ implies that

$\bigl{(}\overline{Y}^{I}_{N}\,,\overline{G}_{IN}\bigr{)}\;=\;0\quad\Rightarrow\quad\bigl{(}Y^{I}_{N}\,,\,G_{IN}\bigr{)}\;=\;i\gamma_{N}\,,$
(75)

where we use the first equation of (72) to obtain the second equality. The
real part of these equations,

$Y^{I}_{N}-(Y^{I}_{N})^{*}\;=\;iP^{I}\,,\qquad
G_{IN}-(G_{IN})^{*}\;=\;iQ_{I}\,,$ (76)

are recognized as the extremal attractor equations (66). Assuming a unique
solution, we obtain $(Y^{I}_{N},G_{IN})=(Y^{I}_{*},G_{I*})$. The scalars at
the north pole are thus fixed by the solution to the extremal attractor
equations. Further, the dipole charges can then be read off to be

$-n^{I}\;=\;Y^{I}_{N}+(Y^{I}_{N})^{*}\,,\qquad-
m_{I}\;=\;G_{IN}+(G_{IN})^{*}\,.$ (77)

The metric now only has a simple pole,

$\begin{split}e^{-2U}&\;\sim\;\frac{i}{\rho}\Bigl{(}Y^{I}_{N}\,\overline{G}^{(0)}_{I}-G_{IN}\,\overline{Y}^{I(0)}\Bigr{)}+\text{O}(1)\,,\\\
&\;=\;-\frac{1}{\rho}\Bigl{(}Y^{I}_{N}\,H^{(0)}_{I}-G_{IN}\,\widetilde{H}^{I(0)}\Bigr{)}+\frac{i}{\rho}\Bigl{(}Y^{I}_{N}\,G^{(0)}_{I}-G_{IN}\,Y^{I(0)}\Bigr{)}+\text{O}(1)\,,\\\
&\;=\;-\frac{1}{\rho}\Bigl{(}Y^{I}_{N}\,H^{(0)}_{I}-G_{IN}\,\widetilde{H}^{I(0)}\Bigr{)}+\text{O}(1)\,.\end{split}$
(78)

Here, we use the second equation in (72) to reach the second line, and the
identity (73) in order to obtain the final equality.

Finally, the behavior of the connection $\omega$ near the poles is given by
expanding the equation (61). Using (71) we obtain

$\star\,\mathrm{d}\,\omega\;=\;\frac{\widehat{n}}{\rho^{2}}\langle\gamma_{N},H^{(0)}\rangle+\text{O}(1/\rho)\,,$
(79)

where $\widehat{n}=(\textbf{x}-\textbf{x}_{N})/|\textbf{x}-\textbf{x}_{N}|$,
the three-dimensional unit vector with the north pole as origin. This equation
implies that there is a three-dimensional Dirac string associated with
$\omega$ emanating from the north pole. There is a similar condition at the
south pole. The condition of smoothness implies that this Dirac string should
end smoothly at the poles. This regularity condition leads to (see the
discussion in Hartle:1972ya ; Whitt:1984wk  for more details)

$\frac{i\langle\gamma_{N},{\gamma}_{S}\rangle}{|\textbf{x}_{N}-{\textbf{x}}_{S}|}\;+\;i\langle\gamma_{N},h\rangle\;=\;\frac{\beta}{4\pi}\,.$
(80)

This equation fixes the distance between the north and south poles in terms of
the temperature, charges at the poles, and the asymptotic values of the
moduli. As $\beta\rightarrow\infty$, we obtain the attractor solution with
$\Omega\rightarrow 0$, $|\textbf{x}_{N}-{\textbf{x}}_{S}|\to 0$, and an
infinite throat.

To summarize, we obtain a Euclidean metric with a cigar-like topology. The
consistency of the periods of angular and time coordinates, leads to the
condition $\beta\Omega=2\pi i$. The values of the scalars at the tip of the
cigar at the north and south poles are fixed by the charges in a manner
similar to the extremal attractor mechanism, but now all the equations are
split into a holomorphic piece at the north pole and an anti-holomorphic piece
at the south pole. The holomorphic scalar fields take their attractor values
at the north pole, and the anti-holomorphic scalars vanish. At the south pole
we have the conjugate equations, and the anti-holomorphic scalars take their
attractor values and the holomorphic scalars vanish.

###  4.3 Two-charge solution

With this set-up, we study the behavior of these $\frac{1}{2}$-BPS rotating
Euclidean solutions with two-charges. We begin by making some general comments
that apply to any theory of $\mathcal{N}=2$ supergravity. We will make contact
with the black hole studied in sections 2 and 3 at the end. The classical on-
shell action of the non-extremal supersymmetric solution reviewed above is
given, after removing a factor of $\beta M_{\rm BPS}$ by a boundary term, by

$\displaystyle-I_{\rm on-shell}$ $\displaystyle\;=\;$
$\displaystyle\pi(q_{I}\mathcal{Y}^{I}+2i\mathcal{G})\big{|}_{N}+\pi(q_{I}\bar{\mathcal{Y}}^{I}-2i\bar{\mathcal{G}})\big{|}_{S}\,,$
(81) $\displaystyle\;=\;$ $\displaystyle\pi
i\langle\gamma_{N},\gamma_{S}\rangle\,,$ (82)

where $\mathcal{G}$ is the same function of $\mathcal{Y}^{I}$ as $F(X^{I})$.
The result in the first line comes from showing that the on-shell action is a
total derivative up to delta functions localized at the poles Boruch:2023gfn .
Using the new attractor equations it can be put in the form of the second
line141414The rewriting in the second line is only valid at the two-derivative
level.. This quantity $\pi i\langle\gamma_{N},\gamma_{S}\rangle$ is _not_ the
area of the event horizon of the non-extremal solution, which is instead equal
to

$A\;=\;\beta|\textbf{x}_{N}-\textbf{x}_{S}|\,.$ (83)

This can be easily derived (see Boruch:2023gfn ) from the metric (60) together
with the equation for $\omega$ in (61) and we will not repeat it here. The on-
shell action and the area only coincide at zero temperatures, such that $\log
Z_{\rm index}$ is proportional to the extremal area. This is due to the fact
that these black holes satisfy the quantum statistical relation, i.e. the on-
shell action is equal to $A/4+2\pi iJ$ (since $\beta\Omega=2\pi i$) and at low
temperatures the angular momentum vanishes.

Given the discussion in the previous paragraph, in a general $\mathcal{N}=2$
theory of supergravity, solutions that have vanishing area at extremality are
characterized by

$\langle\gamma_{N},\gamma_{S}\rangle\;=\;0\,.$ (84)

This condition guarantees that the extremal area of the black hole is zero at
the two-derivative level, and the index of the black hole is small. Since
$\gamma_{N/S}$ are determined in terms of the solution to the extremal
attractor equations, this can be rephrased as a condition on the choice of
charges that the black hole carries. Finally, as emphasized above, the
condition (84) still allows for the finite temperature supersymmetric solution
of Boruch:2023gfn  to have a finite size event horizon with area (83).

For the choice of charges that satisfy
$\langle\gamma_{N},\gamma_{S}\rangle=0$, this might suggest that we have
succeeded in finding a smooth solution that contributes to the gravitaitonal
path integral for the index. However, this is too quick. Let us recall the
regularity condition at the horizon (80). When charges are chosen to satisfy
$\langle\gamma_{N},\gamma_{S}\rangle=0$, the condition degenerates to:

$\beta\;=\;4\pi i\langle\gamma_{N},h\rangle\,.$ (85)

This is problematic for the following reasons

  1. 1.

First, the temperature (85) is fixed in terms of the black hole charges and
asymptotic moduli, while we should be free to set them independently.

  2. 2.

The second problem is that $|\textbf{x}_{N}-\textbf{x}_{S}|$ has disappeared
completely from the equation. There is nothing fixing the separation between
the two poles and therefore there is a moduli space of solutions,
characterized for example by the area of the event horizon. In particular
$|\textbf{x}_{N}-\textbf{x}_{S}|=0$ is part of the moduli space which gives
rise to the original singular small black hole we were trying to avoid.

  3. 3.

The extremal area is also proportional to
$Z_{*}(\Gamma)\overline{Z}_{*}(\Gamma)$, which implies $Z_{*}(\Gamma)=0$ for
the small black hole. In the non-extremal solution $Z=0$ and $\bar{Z}=0$ at
the north and south poles. The solution is then singular whenever
$\langle\gamma_{N},\gamma_{S}\rangle=0$.

These are precisely the shortcomings enumerated in Section 3.3 for the two-
charge black hole in the heterotic string. We see here all those features are
universal to $\mathcal{N}=2$ supergravity with vanishing extremal area at the
two-derivative level.

To provide some more intuition we will consider some special cases, and
connect to the black hole in Section 3. To leading order, compactifications of
type II or heterotic string theory to four dimensions can be described by the
following prepotential

$F(X)\;=\;-\frac{1}{6}\frac{C_{IJK}X^{I}X^{J}X^{K}}{X^{0}}\,.$ (86)

The value of $C_{IJK}$ depends on the model. For example for type IIA
compactified on $K3\times T^{2}$ the coefficients $C_{IJK}$ are the
intersection numbers of a basis of 4-cycles. This case is related by duality
to the toroidally compactified heterotic string theory in Sections 2 and 3.
Instead of studying this theory we will analyze a simpler version where only
$X^{1},X^{2},X^{3}$ are non-zero and moreover $X^{2}=X^{3}$. Therefore
consider a theory with $n_{\text{v}}=2$ and a prepotential of the form

$F(X)\;=\;-\frac{X^{1}(X^{2})^{2}}{X^{0}}\,.$ (87)

This theory is equivalent to a sector of (1), as explained in Sen:2004dp . We
make more comments on this below. A concrete solution with vanishing extremal
area involves turning on only the following two charges

$Q_{I}\;=\;(Q_{0},0,0)\,,\qquad P^{I}\;=\;(0,P^{1},0)\,.$ (88)

We take $Q_{0}<0$ and $P^{1}>0$. In the context of the toroidally compactified
heterotic string these charges have the interpretation of momentum $n$ and
winding $w$ along a circle $S^{1}\subset T^{6}$, namely

$Q_{0}\;=\;-n\,,\qquad P^{1}\;=\;w\,.$ (89)

Moreover the interpretation of the moduli is $X^{1}/X^{0}=ie^{-\Phi}$ while
$X^{2}/X^{0}$ parametrizes the volume of a cycle inside $T^{6}$. The solution
to the extremal attractor equations (66) is given by

$Y_{*}^{I}\;=\;\Big{(}0,\frac{iP^{1}}{2},0\Big{)}\,,\qquad
G_{I*}\;=\;\Big{(}\frac{iQ_{0}}{2},0,-\sqrt{P^{1}|Q_{0}|}\Big{)}\,.$ (90)

This result should be surprising. Since
$G_{2*}=-2Y^{1}_{*}Y^{2}_{*}/Y^{0}_{*}$, how come $X^{0}=X^{2}=0$ and $G_{2*}$
is finite? What is the physical value of the moduli $X^{2}/X^{0}$ that
describes the geometry of $T^{6}$? To see how this comes about let us turn a
small magnetic charge $P^{2}$. (Physically the way this is regulated is by
turning on $\widetilde{H}^{2}=h^{2}\neq 0$ and going slightly away from the
horizon, keeping $P^{2}=0$. In practice it is easier to turn on a small
$P^{2}$ at the attractor point, both lead to the same answer.) Then the
attractor equation becomes

$Y_{*}^{I}\;=\;\Big{(}\frac{P^{1}P^{2}}{2\sqrt{P^{1}|Q_{0}|}},\frac{iP^{1}}{2},\frac{iP^{2}}{2}\Big{)}\,.$
(91)

We see that as $P^{2}\to 0$ the ratio between $Y^{2}_{*}$ and $Y^{0}_{*}$ is
finite

$\frac{Y^{2}_{*}}{Y^{0}_{*}}\;\to\;i\sqrt{\frac{|Q_{0}|}{P^{1}}}\;=\;i\sqrt{\frac{n}{w}}\,.$
(92)

In the application to the heterotic string this guarantees that near the
horizon (or near $\textbf{x}_{N}$ and $\textbf{x}_{S}$ for the non-extremal
metric) the volume of $T^{6}$ is finite, and also explains why $G_{2*}$ is
non-vanishing. The other moduli controlling the string coupling does diverge

$\frac{Y^{1}_{*}}{Y^{0}_{*}}\;=\;i\sqrt{P^{1}|Q_{0}|}\frac{1}{P^{2}}\;\to\;i\sqrt{nw}\,\times\infty\,.$
(93)

This divergence will be regulated by higher derivative corrections. In string
theory this implies that the string coupling vanishes at the horizon
(extremal) or at the poles $\textbf{x}_{N/S}$ (non-extremal).

The new attractor equations imply that $(Y^{I}_{*},G_{I*})$ and
$(\bar{Y}^{I}_{*},\bar{G}_{I*})$ are precisely the local charge at the north
and south poles respectively

$\gamma_{N}\;=\;\Big{(}0,\frac{P^{1}}{2},0;\frac{Q_{0}}{2},0,i\sqrt{P^{1}|Q_{0}|}\Big{)},~{}~{}~{}\gamma_{S}=(\gamma_{N})^{*}\,.$
(94)

We can replace these local charges into equation (69). Once these harmonic
functions are known, the full non-extremal solution can be constructed. Let us
assume for simplicity that the moduli at infinity are chosen such that
$H_{1}=\widetilde{H}^{0}=0$ and $\widetilde{H}^{2}=h^{2}$. We also choose
$h_{2}=0$, although this does not imply $H_{2}=0$ since $\gamma_{N/S}$ include
a dipole in this direction. For concreteness we also pick
$X^{1}/X^{0}|_{\infty}=i2$ and $X^{2}/X^{0}|_{\infty}=i$, implying
$h^{1}=-h_{0}=1$ and $h^{2}=1/2$.151515This specific choice of asymptotic
moduli is done mainly to connect later to the solution in Section 3. The
solution can be easily generalied to arbitrary $X^{1,2}/X^{0}|_{\infty}$. The
emblackening factor is

$\displaystyle e^{-2U}$ $\displaystyle\;=\;$ $\displaystyle
2\tilde{H}^{2}\sqrt{-\widetilde{H}^{1}H_{0}}$ (95) $\displaystyle\;=\;$
$\displaystyle\Big{\\{}\Big{(}1+\frac{\frac{1}{2}P^{1}}{|\textbf{x}-\textbf{x}_{N}|}+\frac{\frac{1}{2}P^{1}}{|\textbf{x}-{\textbf{x}}_{S}|}\Big{)}\Big{(}1+\frac{\frac{1}{2}|Q_{0}|}{|\textbf{x}-\textbf{x}_{N}|}+\frac{\frac{1}{2}|Q_{0}|}{|\textbf{x}-{\textbf{x}}_{S}|}\Big{)}\Big{\\}}^{\frac{1}{2}}\,.$
(96)

Note that $e^{-2U}=-i\mathcal{Y}^{1}/\mathcal{Y}^{0}=e^{-\Phi}$. Curiously
this is independent of $H_{2}$, which carries the information of the dipole.
(Of course the dipole does appear in the scalars and gauge fields.) One can
readily verify that the charge vector at the poles satisfies
$\langle\gamma_{N},\gamma_{S}\rangle=0$. All the previous considerations then
apply to this black hole. A solution only exist with a unique temperature, it
has a moduli space of solutions labeled by the horizon area, and the scalars
are singular at the poles of the horizon.

Let us now show that this solution is precisely the same as the one analyzed
in Section 3 earlier161616The reader might wonder why we used two different
formalism to describe the solution. The presentation in Section 3 is useful
since it can be extended outside of the supersymmetric regime as in Section 2.
The advantage of the formalism in this section is that, within supersymmetric
solutions, it will allow us to incorporate higher derivative corrections, as
in Section 5. . This has to be true since it was proven in Sen:2004dp  that
the action (1) can be obtained from $\mathcal{N}=2$ supergravity with
prepotential (87), so we will be brief. Consider the metric given in
(LABEL:metricsentcbh) and change variables $(t_{E},r,\theta,\phi)$ to
$(t_{E},x^{1},x^{2},x^{3})$ according to

$\displaystyle x^{1}$ $\displaystyle\;=\;$ $\displaystyle\sqrt{r^{2}-{\sf
a}^{2}}\sin\theta\cos\phi\,,$ (97) $\displaystyle x^{2}$ $\displaystyle\;=\;$
$\displaystyle\sqrt{r^{2}-{\sf a}^{2}}\sin\theta\sin\phi\,,$ (98)
$\displaystyle x^{3}$ $\displaystyle\;=\;$ $\displaystyle r\cos\theta\,.$ (99)

In this coordinates the metric (LABEL:metricsentcbh) has the form
Chowdhury:2024ngg

$\mathrm{d}s^{2}\;=\;e^{2\Phi}(\mathrm{d}t_{E}+i\omega)^{2}+\mathrm{d}x^{m}\mathrm{d}x^{m}\,.$
(100)

where $i\omega={\sf a}m_{0}(r^{2}-{\sf
a}^{2}\cos^{2}\theta)^{-1}r\sin^{2}\theta\mathrm{d}\phi$ and the dilaton
becomes

$e^{-2\Phi}\;=\;\Big{(}1+\frac{m_{0}e^{\upalpha}}{2|\textbf{x}-\textbf{x}_{N}|}+\frac{m_{0}e^{\upalpha}}{2|\textbf{x}-\textbf{x}_{S}|}\Big{)}\Big{(}1+\frac{m_{0}e^{-\upalpha}}{2|\textbf{x}-\textbf{x}_{N}|}+\frac{m_{0}e^{-\upalpha}}{2|\textbf{x}-\textbf{x}_{S}|}\Big{)}\,,$
(101)

with $\textbf{x}_{N}=(0,0,{\sf a})$ and $\textbf{x}_{S}=(0,0,-{\sf a})$. This
metric looks different than the one in (60). The solution is to recall
(LABEL:metricsentcbh) is given in string frame while the formalism used here
for $\mathcal{N}=2$ supergravity generates an action with canonical Einstein-
Hilbert term. Therefore we should translate (LABEL:metricsentcbh) into
Einstein frame, which gives

$e^{-\Phi}\mathrm{d}s^{2}\;=\;e^{\Phi}(\mathrm{d}t_{E}+i\omega)^{2}+e^{-\Phi}\mathrm{d}x^{m}\mathrm{d}x^{m}\,.$
(102)

It is possible now to put this metric in the form (60) with $2U=\Phi$ for a
suitable choice of the constants $h$. This also implies that
$P^{1}=m_{0}e^{\upalpha}$ while $Q_{0}=-m_{0}e^{-\upalpha}$. This is
consistent with the interpretation of $P^{1}$ as winding and $Q_{0}$ as
momentum. More importantly, the separation between the poles is identified
with

$|\textbf{x}_{N}-\textbf{x}_{S}|\;=\;2{\sf a}\,.$ (103)

For example, (83) implies that the horizon metric in IWP form is $A=2\beta{\sf
a}$ which matches the result in Section 3. Moreover, for our choice of $h$ and
$\gamma_{N}$ we obtain
$i\langle\gamma_{N},h\rangle=\sqrt{P^{1}|Q_{0}|}/2=m_{0}/2$ which implies from
(85) a value of the temperature $\beta=2\pi m_{0}$, matching Section 3 as
well.

##  5 Higher derivative corrections to the gravitational index

In this section we present the rotating black hole solution including higher-
derivative corrections in the $\mathcal{N}=2$ supergravity coming from string
theory. The higher-derivative corrections in $\mathcal{N}=2$ supergravity are
of two types: chiral superspace integrals of the holomorphic prepotential
function, and full superspace integral, sometimes called F-terms and D-terms,
respectively, in the $\mathcal{N}=2$ context. We only consider the first type
of terms, which are summarized by corrections to the tree-level holomorphic
prepotential function $\mathcal{G}(\mathcal{Y}^{I})$ discussed in the previous
section. We then make comments on D-terms and the controllability of the
solution.

The treatment begins by introducing another chiral field $\widehat{A}$ of
weight $2$, whose top component is proportional to the Weyl-squared tensor in
four-dimensions. The prepotential is extended to a function
$F(X^{I},\widehat{A})$ and, in addition to the first derivatives
$F_{I}=\partial F/\partial X^{I}$, we have $F_{\widehat{A}}=\partial
F/\partial\widehat{A}$. The particular form of the prepotential that arises in
Calabi-Yau three-fold compactifications of Type II string theory is

$F(X^{I},\widehat{A})\;=\;-\frac{1}{6}\frac{C_{IJK}X^{I}X^{J}X^{K}}{X^{0}}-c\frac{\widehat{A}}{64}\frac{X^{1}}{X^{0}}+\dots\,.$
(104)

Here $\widehat{A}$ effectively counts the loops in the topological string
expansion. As $\widehat{A}$ contains four derivatives, we can also think of
the series as an expansion in the number of derivatives.

The cubic or tree-level term was discussed in the previous section, and we
have shown the one-loop term. Here $c=c_{2}/24$ where $c_{2}$ is the second
Chern class of the Calabi-Yau. In addition there are typically an infinite
series of terms given by the higher loops of the topological string expansion.
In Type II theory on $K3\times T^{2}$, the term $X^{1}/X^{0}$ above is
completed by worldsheet instanton effects to a modular function of
$X^{1}/X^{0}$. Importantly, the expansion terminates at this order due to the
fermion zero modes on $T^{2}$.

In our case of interest here, i.e. the two-charge black hole, the two-
derivative result is singular, so that in the full solutions the correction
terms are as important as the two-derivative term. This means that, a priori,
one does not have a controllable expansion. Nevertheless it is useful to see
explicitly that including higher-derivative terms can lead to a smooth
solution, as we show below.

A proof that D-terms vanish on configurations that preserve supersymmetry
would justify the approach where we only keep F-terms.171717As opposed to
previous work Sen:2004dp ; Hubeny:2004ji ; Dabholkar:2004dq , we do not expect
the generic expansion to be controllable for the thermal partition function
which does not include an insertion of $(-1)^{\sf F}$. In some situations,
like the attractor black hole entropy, one can indeed show that D-terms do not
affect the quantity of interest Butter:2014iwa ; Murthy:2013xpa , and
therefore one obtains an exact formula Dabholkar:2010uh ; Dabholkar:2011ec ;
Dabholkar:2014ema ; Iliesiu:2022kny . In our present context, as we see below,
the entropy arising from the prepotential (104) agrees with the microscopic
string theory index including the numerical coefficient, pointing to a non-
renormalization result.

The analysis of supersymmetry equations and solutions of the higher-derivative
theory was performed in the impressive set of papers LopesCardoso:1999za ;
LopesCardoso:2000qm ; LopesCardoso:2000fp . Just as we define the rescaled
fields $\bigl{(}\mathcal{Y}^{I},\mathcal{G}_{I}\bigr{)}$ in (62), we also
define a rescaled field
$\Upsilon=e^{\mathcal{K}(X,\overline{X})}\,\overline{Z}^{2}\widehat{A}$. The
higher-derivative equations are, generally speaking, more complicated than the
two-derivative theory. The exception is the equations governing the scalar
fields, which are still given by the generalized stabilization equations (63),
although they now depend on $\Upsilon$.

### New attractor with higher-derivative corrections

The solution is specified by a vector of harmonic functions
$H=(\widetilde{H}^{I},H_{I})$. The generalized stabilization equations are now

$\mathcal{Y}^{I}-\overline{\mathcal{Y}}^{I}\;=\;i\widetilde{H}^{I}\,,\qquad\mathcal{G}_{I}(\mathcal{Y},\Upsilon)-\overline{\mathcal{G}}_{I}(\mathcal{Y},\Upsilon)\;=\;iH_{I}\,,$
(105)

and they determine $\mathcal{Y}^{I}$ in terms of $H$ and $\Upsilon$. The
metric is of the IWP form as in the two-derivative theory

$\mathrm{d}s^{2}\;=\;e^{2U}(\mathrm{d}t_{\rm
E}+i\omega_{m}\mathrm{d}x^{m})^{2}+e^{-2U}\mathrm{d}x^{m}\mathrm{d}x^{m}\,.$
(106)

The functions $U$ and $\omega$ are now governed by

$\begin{split}e^{-2U}&\;=\;i\bigl{(}\mathcal{Y}^{I}\overline{\mathcal{G}}_{I}-\mathcal{G}_{I}\overline{\mathcal{Y}}^{I}\bigr{)}+128i\,e^{U}\nabla^{p}\bigl{(}(\nabla_{p}e^{-U})(\mathcal{G}_{\Upsilon}-\overline{\mathcal{G}}_{\Upsilon})\bigr{)}\\\
&\quad-32i\,e^{4U}(R(\omega)_{p})^{2}(\mathcal{G}_{\Upsilon}-\overline{\mathcal{G}}_{\Upsilon})-64e^{2U}R(\omega)_{p}\nabla^{p}(\mathcal{G}_{\Upsilon}+\overline{\mathcal{G}}_{\Upsilon})\,.\end{split}$
(107)
$\begin{split}-iR(\omega)_{p}&\;=\;H_{I}\overset{\leftrightarrow}{\nabla}_{p}H^{I}-128i\,\nabla^{q}\left[\nabla_{[p}(e^{2U}R(\omega)_{q]}(G_{\Upsilon}-\overline{G}_{\Upsilon}))\right]\\\
&\quad-128\,\nabla^{q}\bigl{(}2\nabla_{[p}U\,\nabla_{q]}(G_{\Upsilon}+\overline{G}_{\Upsilon})\bigr{)}\,.\end{split}$
(108)

In the two-derivative theory $\mathcal{G}_{\Upsilon}=0$ and the above
equations reduce to the simpler equations (61). Finally, the value of
$\Upsilon$ is

$\Upsilon\;=\;-64\Bigl{(}\nabla_{p}U-\frac{i}{2}e^{2U}R(\omega)_{p}\Bigr{)}^{2}\,.$
(109)

Here we follow the notation of LopesCardoso:1999za  with $\nabla_{a}$ the
derivative in the three-dimensional flat base-space, and
$R(\omega)^{p}=\varepsilon^{pqs}\nabla_{q}\omega_{s}$. For the non-extremal
rotating saddle we set, as in Section 4

$H(\textbf{x})\;=\;h+\frac{\gamma_{N}}{|\textbf{x}-\textbf{x}_{N}|}+\frac{{\gamma}_{S}}{|\textbf{x}-{\textbf{x}}_{S}|}\,,\qquad\gamma_{N}+\gamma_{S}\;=\;\Gamma\,.$
(110)

Therefore part of the problem is to determine $\gamma_{N}$, $\gamma_{S}$ and
$|\textbf{x}_{N}-\textbf{x}_{S}|$ using smoothness, just like we did at the
two derivative level Boruch:2023gfn .

As an exercise we can study the behavior of all the fields in an extremal
attractor background as in the discussion near (64), as $r\to 0$,

$\text{Extremal:}\qquad
R(\omega)\;=\;0\,,\qquad\mathcal{Y}^{I}\;\sim\;\frac{Y_{*}}{r}+\dots\,,\qquad
e^{-2U}\;\sim\;\frac{a}{r^{2}}+\dots\,,\qquad\Upsilon\;\sim\;\frac{\Upsilon_{*}}{r^{2}}+\dots\,.$
(111)

The fact that $U\sim\log r$ together with (109) implies that
$\Upsilon_{*}=-64$. The scalars $Y_{*}^{I}$ are determined by
$Y_{*}^{I}-\bar{Y}_{*}^{I}=iP^{I}$ and $G_{I*}-\bar{G}_{I*}=iQ_{I}$. The
scalar fields therefore behave as in the two-derivative theory. The scaling
behavior of the other fields can be read off from their conformal weights—this
can be explained either by using a conformal compensator, or by thinking of
the other fields as functions of the scalar fields. The only thing to be
careful of is that coefficients can be zero, as happens for $R(\omega)$
because of the spherical symmetry. The value $\Upsilon_{*}=-64$ is precisely
the one required for the Wald entropy of the extremal black hole to equal the
logarithm of the growth of microscopic states of the fundamental heterotic
string Dabholkar:2004yr .

Now we repeat the analysis for the rotating non-extremal solution, and
determine the value of $\gamma_{N/S}$. We focus on the behavior of the
solution near the north pole, since the behavior at the south pole is related
by complex conjugation. Recall that the idea in the rotating non-extremal
solution is to demand that the metric factor $e^{-2U}$ has only a single pole,
leading to a finite temperature horizon. Accordingly, we assume the following
behavior for the non-extremal solution, as
$\rho=|\textbf{x}-\textbf{x}_{N}|\to 0$,

$\mathcal{Y}^{I}\;\sim\;\frac{Y_{N}}{\rho}+\dots\,,\qquad
e^{-2U}\;\sim\;\frac{a_{1}}{\rho}+\dots\,,\qquad
R(\omega)_{p}\;=\;\frac{a_{2}}{\rho^{2}}\,\widehat{n}+\dots\,,\qquad\Upsilon\;\sim\;\frac{\Upsilon_{*}}{\rho^{2}}+\dots\,,$
(112)

where $\widehat{n}=(\textbf{x}-\textbf{x}_{N})/|\textbf{x}-\textbf{x}_{N}|$.
Now we analyze the equations (107)–(109). The upshot is that the solution to
the extremal attractor $(Y_{*}^{I},G_{I*})$ together with $\Upsilon_{*}=-64$
determines $\gamma_{N}$. Together with the smoothness condition, this
completely determines the sources $H$. Now the full solution can be determined
in terms of the boundary data ($\Gamma$ and $\beta$)181818We assume that,
given $H$, a solution to the partial differential equations (107) and (108)
exists and is unique. This has been verified in the low temperature limit
numerically..

We begin with (108). $\mathcal{G}_{\Upsilon}$ has Weyl weight 0 and so
$\mathcal{G}_{\Upsilon}\sim\text{O}(1)$ as $\rho\to 0$. The leading behavior
of the second term on the right hand side is therefore proportional to
$\nabla_{[p}(e^{2U}R(\omega)_{q]})$. Now,
$e^{2U}R(\omega)_{q}\sim\frac{a_{2}}{\rho}\widehat{n}_{q}+\dots$, and
therefore the anti-symmetry of the action of the derivatives annihilates this
term so as to give no contribution at $\text{O}(1/\rho^{2})$. A similar
analysis applies to the last term in (108). This shows that the value of
$R(\omega)$ at the pole in the higher-derivative theory is the same as its
value given in (79) in the two-derivative theory. We conclude that
$a_{2}=i\langle\gamma_{N},H^{(0)}\rangle$. In particular, this means that the
condition of there being no Dirac-string singularity implies the same relation
(80) as in the two-derivative theory, which we repeat here for convenience

$\frac{i\langle\gamma_{N},\gamma_{S}\rangle}{|\textbf{x}_{N}-\textbf{x}_{S}|}+i\langle\gamma_{N},h\rangle\;=\;\frac{\beta}{4\pi}\,.$
(113)

Now we move to (107). Let us start by considering the first term on the right
hand side. This is the value of $e^{-2U}$ in the two-derivative theory. We
consider a solution that cancels the double pole coming from this term by
imposing $\bar{Y}_{N}^{I}=\bar{G}_{IN}=0$ (and $Y^{I}_{S}=G_{IS}=0$ for the
south pole). The stabilization equation, together with the old attractor,
would then imply

$i\gamma_{N}\;=\;(Y_{*}^{I},G_{I*}).$ (114)

This solution is not yet complete, as we have not yet determined
$\Upsilon_{*}$ which appears implicitly in $G_{I}$. We will come back to this
point shortly. First, we explain how the other terms in (107) lead to a
vanishing double pole. As $\rho\to 0$, calling the constant
$\mathcal{G}_{\Upsilon}=C_{1}$, the leading behavior of the second and third
terms is governed by

$C_{1}\bigl{(}128i\,e^{U}\nabla^{2}e^{-U}-32i\,e^{4U}(R(\omega)_{p})^{2}\bigr{)}\;\sim\;-32i\,C_{1}\Bigl{(}1+\frac{a_{2}^{2}}{a_{1}^{2}}\Bigr{)}\frac{1}{\rho^{2}}+\dots\,.$
(115)

We see that in order to have a vanishing double pole we must have $a_{2}=\pm
ia_{1}$. In the two-derivative theory we have, from (78) and (75), the value
$a_{1}=-\langle\gamma_{N},H^{(0)}\rangle$ so that $a_{2}=ia_{1}$. We choose
this solution also in the higher-derivative theory. This implies a condition
of vanishing of other $\text{O}(1/\rho)$ terms of the right-hand side of (107)
involving higher derivative terms. From the equation (109), we deduce that

$\Upsilon\;\sim\;-\frac{64}{\rho^{2}}\Bigl{(}\frac{1}{2}-\frac{i}{2}\frac{a_{2}}{a_{1}}\Bigr{)}\;\;=\;\;-\frac{64}{\rho^{2}}\,.$
(116)

This completes the derivation since now we determined $\Upsilon_{*}=-64$. As
in the two-derivative solution, we see that the holomorphic (anti-holomorphic)
scalar fields take the attractor values at the north (south) pole. Furthermore
they determine the dipole charges $\gamma_{N}$ ($\gamma_{S}$) through the
relation (114).

So far we described the metric and scalar profiles. The gauge fields are also
determined by the harmonic sources through the equations
$F^{I}_{0p}=-\nabla_{p}[e^{2U}(\mathcal{Y}^{I}+\bar{\mathcal{Y}}^{I})]$ and
$G_{0p}^{I}=-\nabla_{p}[e^{2U}(\mathcal{G}_{I}+\bar{\mathcal{G}}_{I})]$ where
$F^{I}_{\mu\nu}$ and $G^{I}_{\mu\nu}$ determine the field strength and its
dual. These equations are the same as in the two-derivative theory.

To conclude the characterization of the solution, we propose a formula for the
on-shell action of such configurations, including boundary terms that remove
the zero-point energy $\beta M_{\rm BPS}$, to be

$-I_{\rm on-
shell}\;=\;\pi(q_{I}\mathcal{Y}^{I}+2i\mathcal{G})|_{N}+\pi(q_{I}\bar{\mathcal{Y}}^{I}-2i\bar{\mathcal{G}})|_{S}.$
(117)

We have two reasons that justify this expression. First, it reduces to the
two-derivative action (81) when $\mathcal{G}$ is independent of $\Upsilon$,
for any temperature. (Note that when $\mathcal{G}$ depends on $\Upsilon$ (82)
is different from (81).) Second, one can prove this formula in the presence of
arbitrary higher-derivative corrections, under the assumption that the on-
shell action is temperature independent. The action in the zero-temperature
limit was explicitly evaluated in Dabholkar:2010uh  and matches (117) (as well
as the Wald entropy derived in LopesCardoso:1999za ).

The reason we have not attempted an explicit evaluation of the action is that
it requires solving the differential equations (107) and (108). Nevertheless,
there might be a more elegant way to obtain (117). The split form of the
action in (117) is similar to observations of the factorization of the
partition function in BenettiGenolini:2019jdz ; Hosseini:2019iad . An
efficient way to prove this could be to apply the techniques of
BenettiGenolini:2023kxp , we leave this for future work.

### The corrected two-charge solution

We now come back to the specific theory relevant to the two-charge black hole
in heterotic string theory, with only $X^{1},X^{2}=X^{3}$, with prepotential

$F(X^{I},\widehat{A})\;=\;-\frac{X^{1}(X^{2})^{2}}{X^{0}}-c\frac{\widehat{A}}{64}\frac{X^{1}}{X^{0}}\,.$
(118)

For the heterotic string $c=1$ but we find it useful to keep it general to
keep track of the effect of higher derivative corrections. The non-zero
monopole charges in the solution are taken to be $Q_{0},P^{1}$ and recall that
$Q_{0}<0$ and $P^{1}>0$. According to the discussion in Sections 4.2, 4.3, we
choose sources
$(\widetilde{H}^{0},\widetilde{H}^{1},\widetilde{H}^{2},H_{0},H_{1},H_{2})$.
Let us begin by solving the new attractor equations which determine
$\gamma_{N/S}$. The starting point is the extremal attractor solution

$Y_{*}^{I}\;=\;\left(\sqrt{\frac{cP^{1}}{|Q_{0}|}},\frac{iP^{1}}{2},0\right)\,,\qquad
G_{I*}\;=\;\left(\frac{iQ_{0}}{2},-\sqrt{\frac{cP^{1}}{|Q_{0}|}},0\right)\,,\qquad\Upsilon_{*}\;=\;-64\,,$
(119)

which is equal to $Y_{N}$ at the north pole while $\bar{Y}_{N}=0$. The
divergence in the dilaton (near the north and south pole) at the horizon is
now regulated since $Y_{*}^{0}$ is finite. The fact that now $G_{2*}=0$ and
$G_{1*}\neq 0$ might seem like a major modification compared to the two-
derivative level discussion. To see how this comes about one can turn a small
charge $P^{2}$ (or, equivalently, go slightly away from the pole, as explained
in Section 4.3). For example, this produces a term
$G_{2*}\sim\sqrt{P^{1}|Q_{0}|}\frac{P^{2}}{4c+(P^{2})^{2}}$. We see explicitly
that the limits $c\to 0$ and $P^{2}\to 0$ do not commute and this is the
source of discrepancy. This is clearly due to the singular nature of the two-
derivative solution and would not happen for a big black hole.

According to the new attractor solution (114) the charge vectors associated to
the north and south pole contributions are

$\gamma_{N}\;=\;\left(i\sqrt{\frac{cP^{1}}{|Q_{0}|}},\frac{P^{1}}{2},0,\frac{Q_{0}}{2},-i\sqrt{\frac{cP^{1}}{|Q_{0}|}},0\right)$
(120)

and $\gamma_{S}=(\gamma_{N})^{*}$. Notice again that this is not a small
correction to the two-derivative solution for $\gamma_{N}$! One can check that
now we have

$i\langle\gamma_{N},\gamma_{S}\rangle\;=\;2\sqrt{cP^{1}|Q_{0}|}\,.$ (121)

The final ingredient to determine the sources (110) and therefore the full
solution is to choose $h$. At infinity we have $\Upsilon\to 0$ and therefore
we can repeat the same analysis as in the two-derivative theory to establish
the values $h^{I}=\widetilde{H}^{I}|_{\infty}$ and $h_{I}=H_{I}|_{\infty}$
that allow us to compare with the solution in Sections 2 and 3. The answer
found in Section 4.3 is $h_{I}=(-1,0,0)$ and $h^{I}=(0,1,\frac{1}{2})$. The
solution to the generalized stabilization equations (63) with prepotential
(104) becomes then an algebraic equation that relates
$(\mathcal{Y}^{I},\mathcal{G}_{I})$ to $(\widetilde{H}^{I},H_{I})$ and
$\Upsilon$, which can be solved easily191919It is important to emphasize that
the stabilization equations are not enough to determine $\mathcal{Y}$ in terms
of $H$ in the higher-derivative theory. The reason is that they involve
$\Upsilon$, which depends on $U$ and $\omega$. Therefore the stabilization
equations are not decoupled from (107) and (108). .

The relation (80) fixes the parameter $2{\sf
a}=|\textbf{x}_{N}-\textbf{x}_{S}|$ in terms of $\beta$, as anticipated
earlier in Section 3. Since, in the language of that section,
$i\langle\gamma_{N},\gamma_{S}\rangle=2\sqrt{cP^{1}|Q_{0}|}=2\sqrt{c}m_{0}$
and
$i\langle\gamma_{N},h\rangle=\frac{\sqrt{c}(P^{1}-Q_{0})}{\sqrt{P^{1}|Q_{0}|}}=2\sqrt{c}\cosh\upalpha$,
the relation (80) gives

$\frac{2\sqrt{c}m_{0}}{2{\sf
a}}+2\sqrt{c}\cosh\upalpha\;=\;\frac{\beta}{4\pi}\,,\qquad\Rightarrow~{}~{}{\sf
a}\;=\;\frac{4\pi m_{0}}{\frac{\beta}{\sqrt{c}}-8\pi\cosh\upalpha}\,.$ (122)

This relation resolves the problem appearing in the two-derivative expression
(85), namely that the temperature was apparently fixed in terms of the charges
and asymptotic moduli. Now we see that the temperature and the moduli are
independent quantities. The relation (122) shows the solution is only valid as
long as the denominator is positive. Curiously, the denominator changes sign
at a finite value of $\beta$. (The same phenomenon is found even for the big
black hole where $|\textbf{x}_{N}-\textbf{x}_{S}|$ becomes negative above a
charge-dependent temperature. Explaining this remains an open question.)

In summary, we find a solution that is smooth with the dilaton finite
everywhere, all the parameters of the solution are determined in terms of
boundary data which are independent values of the temperature and moduli.

To conclude this section we evaluate the on-shell action of the solution we
constructed, using (117). Since $Z_{\rm index}\sim e^{-I_{\rm on-shell}}$ we
find that it makes the following contribution to the gravitational index

$\displaystyle\log Z_{\rm index}$ $\displaystyle\;=\;$
$\displaystyle\pi(q_{I}\mathcal{Y}^{I}+2i\mathcal{G})\big{|}_{N}+\pi(q_{I}\bar{\mathcal{Y}}^{I}-2i\bar{\mathcal{G}})\big{|}_{S}\,,$
(123) $\displaystyle\;=\;$ $\displaystyle 4\pi\sqrt{cP^{1}|Q_{0}|}\,.$ (124)

This is now finite. To compare with the heterotic string action we recall that
$c=1$. Therefore we find a gravitational index given, in terms of the charges
$n=-Q_{0}$ and $w=P^{1}$, by

$\log Z_{\rm index}\;=\;-I_{\text{on-shell}}\;=\;4\pi\sqrt{nw}\,.$ (125)

(In terms of the parameter $m_{0}$ of Section 3 this is $4\pi m_{0}$.) This
result precisely matches the microscopic evaluation of the index counting BPS
fundamental heterotic strings, as reviewed in the introduction.

Note that the same phenomenon is known to occur if one assumes an attractor
AdS2 geometry Dabholkar:2004yr . What is remarkable here is that one obtains
the same result even though the rotating geometry for the index preserves much
less symmetry, and is compatible with the discussion in Section 2 on the
partition function.

##  6 Can the low-temperature physics be nearly conformal?

In Section 2 we studied two-charge black holes in string theory. At low
temperatures the black hole becomes small, i.e. it has a horizon area that
vanishes at the two-derivative level. Following Chen:2021dsw , we explained
how a transition takes place to a winding condensate without a horizon, before
we reach the small black hole, and ultimately the ground state is described by
a gas of strings and not a black hole.

Is the behavior described above universal? Could there be a string theory
solution that does not transition to a gas of strings or other stringy objects
without a horizon, such that the ground state is really described by a small
black hole? In this section we use effective theory arguments to comment on
the possibility of a string-size near-extremal black hole. Our arguments also
apply for big black holes with 16 or more supercharges.

By a small black hole we mean a gravitational solution with a finite (string
scale) area that exists at very low temperatures, which develops an ${\rm
AdS}_{2}$ throat. This assumption can be justified in some cases where there
is a smooth limit to the two-derivative theory Kunduri:2007vf ;
Figueras:2008qh . We also assume that the index is a function that grows
exponentially with the charges. The latter requirement is satisfied by most
examples in string theory (an exception being e.g. the D0 brane black hole).
Finally, we assume that the extremal limit of the small black hole preserves
at least 16 supercharges.

The near-extremal AdS2 phase of such black holes implies the presence of a
nearly-conformal symmetry of the quantum system describing the black hole
microstates. Its low energy description is given by the Schwarzian theory.
Since we assume the system preserves a certain number of supercharges, we have
a supersymmetric generalization of the Schwarzian theory.

It is possible to label the different supersymmetric extensions of the
Schwarzian theory by the superconformal group of symmetries broken by the
near-AdS2 solution. For example the bosonic Schwarzian theory would be labeled
by ${\rm PSL}(2,\mathbb{R})$, the one describing the near-BPS limit of the
1/16-BPS black holes in ${\rm AdS}_{5}$ would be labeled by
$\mathrm{SU}(1,1|1)$ and the one describing the near-BPS limit of large black
holes in flat space by ${\rm PSU}(1,1|2)$. Therefore we can address this
problem by analyzing the possible superconformal groups. To perform our
analysis we use that the supersymmetric Schwarzian action acts as a ${\rm
U}(1)$ generator in the space of super-reparametrization, such that we can
apply the localization principle of Stanford:2017thb . We assume that the
R-symmetry of the superconformal group is realized in a geometric manner in
spacetime.202020These assumptions do not rule out the possibility of a sigma
model description. See Giveon:2006pr ; Dabholkar:2007gp ; Johnson:2007du ;
Lapan:2007jx ; Kraus:2007vu  for a related discussion on the AdS3 near-horizon
configurations of a fundamental string. We now show that no Schwarzian theory
exists with these properties.

Given the choice of a superconformal group we can derive the partition
function of such a theory including quantum corrections. These corrections are
one-loop exact and have the following form. Denote by $\vec{\mu}$ the vector
of chemical potentials that couple to the $r={\rm rank}(G)$ Cartan generators
of the $R$-symmetry group $G$, with the periodicity $\mu\sim\mu+1$. The
partition function then takes the form

$Z\;=\;\sum_{\vec{n}\in\mathbb{Z}^{r}}\beta^{\frac{N_{F}-N_{B}}{2}}\,Z_{\rm
one-
loop}(\vec{\mu}+\vec{n})\,e^{S_{0}+\frac{1}{\beta}I(\vec{\mu}+\vec{n})}\,,$
(126)

where $Z_{\rm one-loop}(\vec{\mu}+\vec{n})$, and $I(\vec{\mu}+\vec{n})$, are
functions of chemical potentials only arising from the one loop determinant,
and on-shell action, respectively. The sum over integers $\vec{n}$ is a sum
over saddles related by large gauge transformations. $S_{0}$ is the extremal
entropy while $N_{F}$ and $N_{B}$ correspond to the number of fermionic
generator and bosonic generators in the superconformal group. The overall
prefactor of $\beta^{\frac{N_{F}-N_{B}}{2}}$ in the one-loop determinant comes
from subtracting a contribution from zero-modes.

When $N_{F}\geq N_{B}$ the theory can predict the presence of a large number
of ground states of order $e^{S_{0}}$. For example for $\mathrm{SU}(1,1|1)$,
$N_{B}-N_{F}=0$ so that the one-loop determinant is temperature independent
and in the zero-temperature limit the action becomes temperature independent,
resulting in $Z(\beta\to\infty)\sim e^{S_{0}}+O(e^{-\beta})$. For ${\rm
PSU}(1,1|2)$, instead, $N_{F}=8>N_{B}=6$ and the one-loop determinant diverges
as $\beta\to\infty$. In this case the sum over saddles is important in order
to regulate this divergence and once again we find $Z(\beta\to\infty)\sim
e^{S_{0}}$.

When $N_{B}>N_{F}$ the one-loop determinant vanishes in the $\beta\to\infty$
limit and its unavoidable to obtain $Z(\beta\to\infty)=0$. This implies that
the quantum corrections due to the Schwarzian mode would destroy the classical
degeneracy of ground states. In other words $\log Z\sim
S_{0}-\frac{N_{F}-N_{B}}{2}\log\beta\to-\infty$ in the large $\beta$ limit.
Since the index gives a lower bound on the number of ground states, we can use
this criterion to rule out the existence of a small black hole horizon with an
${\rm AdS}_{2}$ factor. In other words, there is no mass-gap and therefore no
decoupled AdS2 geometry in such theories.

By the arguments in the previous paragraph we reduced the problem to an
analysis of the number of bosonic vs fermionic generators of superconformal
groups Nahm:1977tg . The list of such groups can be found e.g. in Table 1 of
the review Britto-Pacumio:1999dnb , see Frappat:1996pb  for an extensive list.

Let us begin with superconformal groups with more than $16$ supercharges.
There are three possible families. The first family are $\mathrm{SU}(1,1|n)$
with $n>4$, which has a $\mathrm{SU}(n)\times{\rm U}(1)$ $R$-symmetry. This
group has $N_{B}=n^{2}+3$ and $N_{F}=4n$. For all $n>1$ one has $N_{B}>N_{F}$
and therefore the ground states do not survive quantum corrections. The same
is true for ${\rm OSp}(n|2)$ for $n>8$ since $N_{B}=\frac{1}{2}(n^{2}-n)+3$
and $N_{F}=2n$ and for ${\rm OSp}(4^{*}|2n)$ with $n>2$ which has
$N_{B}=2n^{2}+n+6$ and $N_{F}=8n$. Therefore, if an index predicts a large
ground state degeneracy, those states cannot describe small black holes
preserving more than 16 supercharges.

Next, we consider the case of systems with 16 supercharges. In this case there
are four superconformal groups, which are

Superconformal Group |  $N_{B}$/$N_{F}$ |  $R$-symmetry
---|---|---
${\rm OSp}(4^{*}|4)$ | $16/16$ | $\mathrm{SU}(2)\times{\rm Spin}(5)$
$\mathrm{SU}(1,1|4)$ | $19/16$ | $\mathrm{SU}(4)\times{\rm U}(1)$
$F(4)$ | $24/16$ | $\mathrm{SO}(7)$
${\rm OSp}(8|2)$ | $31/16$ | $\mathrm{SO}(8)$
(127)

The last three of these groups have $N_{B}>N_{F}$ and therefore do not have a
ground state degeneracy. The case ${\rm OSp}(4^{*}|4)$ is special since
$N_{B}=N_{F}$. Here, the ${\rm Spin}(5)$ R-symmetry does not act as a
geometric rotation of any subspace of $T^{6}$.212121It is possible that the
supercharges transform as a spinor of the tangent space ${\rm Spin}(5)$. This
possibility was discussed in related AdS3 near-horizon configurations in
Dabholkar:2007gp .

Thus, there is no superconformal group, and therefore no Schwarzian theory,
which could describe the low energy phase of such small black holes. This
argument can be extended to theories with 14 supercharges. Below 14 there are
multiple possibilities of nearly-conformal theories with a large number of
ground states such as $\mathrm{SU}(1,1|1)$ or ${\rm PSU}(1,1|2)$, and other
superconformal groups which haven’t found an application yet. Another theory
that deserves further study along these lines is ${\rm D}(2,1;\alpha)$.

##  7 Discussion

We would like to conclude with some comments and future directions.

We have evaluated the gravitational path integral that corresponds to the
index of the two-charge states in heterotic string theory. This calculation
involved constructing a solution that combines two-derivative with four-
derivative interactions. An obvious future direction is to make this
construction more precise. The advantage of the present paper compared with
previous attempts is that we turned this question into a concrete program,
starting with proving that D-terms vanish due to supersymmetry. Besides this,
it would be necesasry to prove that the action coming from F-terms is
temperature independent and reproduces (117) for any prepotential.

Another natural direction is to compute quantum corrections around the black
hole solution evaluating the index. We hope that one can use the recent
techniques in H:2023qko  and Anupam:2023yns  to reproduce the logarithmic
correction to the index in the large $N=\sqrt{nw}$ limit. An even more
ambitious goal can be to use supergravity localization to reproduce the exact
generating function $1/\eta^{24}(\tau)$.

Finally, it is interesting to consider whether this approach can be extended
to black holes with one charge. An example is a 10 dimensional type IIA string
theory black hole with D0-charge $N$. This leads to a small black hole in the
extremal limit and its near-extremal excitations are dual to the BFSS matrix
model. In this case even a microscopic calculation of the index is hard and
has only been completed for the $N=2$ theory Sethi:1997pa , although there is
a simple M-theory prediction for it for all $N$. To attack this problem using
the gravitational path integral, one can easily find a rotating D0-brane black
hole starting with the uncharged Myers-Perry solution in 10 dimensions
$(t,x^{1},\ldots,x^{9})$, uplifting to 11 dimensions by adding an extra circle
$x^{10}$, performing a boost in the $t-x^{10}$ coordinates, and finally
dimensionally reducing back to 10 dimensions. Imposing the boundary conditions
relevant for the index one can find, at the two-derivative level, a
supersymmetric rotating non-extremal black hole with a macroscopic area and
zero on-shell action, similar to its two-charge counterpart of Section 3
gjtwip . The solution is singular at the poles of the horizon and the string
coupling becomes large at those locations. An evaluation of the index of the
BFSS model using the gravitational path integral for type IIA seems to require
a new idea and cannot be computed by a simple extension of the techniques used
here.

## Acknowledgements

It is a pleasure to thank Pietro Benetti-Genolini, Jan Boruch, Matt Heydeman,
Luca Iliesiu, Juan Maldacena, Ashoke Sen, and Edward Witten for interesting
and useful discussions. Y.C. acknowledges the KITP program “What is String
Theory? Weaving Perspectives Together”, during which this work is completed.
S.M. acknowledges the support of the J. Robert Oppenheimer Visiting
Professorship at the Institute for Advanced Study, Princeton. The work of GJT
is supported by the University of Washington and the DOE award DE-SC0024363.

##  Appendix A Solution generation for rotating two-charge systems

In this appendix, we discuss how one can use the solution generating procedure
to work out the properties of the rotating two-charge system from the
uncharged (but rotating) solution. We will derive the thermodynamic quantities
of the charged solution with the information of the thermodynamic quantities
of the uncharged solution as well as the transformation at infinity. The
derivation applies to general classical solutions in string theory, including
black hole solutions as well as potential rotating Horowitz-Polchinski
solutions that have not yet been constructed. One application of our
derivation, which we discuss in appendix. A.1, is that one can work out the
$\alpha^{\prime}$ corrections to the rotating two-charge black hole in
Sen:1994eb , without needing to find the corrected geometry explicitly. The
discussion here mostly follows the derivation in Chen:2021dsw , with the extra
ingredient that the seed solution can carry rotation.

The starting point is a seed solution in $D$ dimensions which has an
asymptotic time circle with length $\tilde{\beta}=2\pi\tilde{R}$. For the
purpose of applying to the discussion in the main text, we have $D=4$, though
the discussion is more general. We will assume the existence of a
$\mathrm{U}(1)$ isometry, time translation, everywhere in the solution. For
simplicity, we consider the transformations that generate charges along one
extra circle, and the relevant symmetry group is $O(2,2)$ which acts on fields
in $d=D-1$ dimensional space once we dimensional reduce on both the time
circle and the extra circle. This is a subgroup of the $O(7,23)$
transformation of heterotic string theory compactified on $T^{6}$ we
considered in the main text, but it also applies to the Bosonic/Type II
theories compactified on a torus. We parametrize the transformation by the
parameters $(\upalpha,\upbeta)$.

The main equation, which we refer to Chen:2021dsw  for justification, is the
invariance of the action under the solution generating transformation.

$\log
Z(R,\Omega,r,\mu_{n},\mu_{w},\Phi_{D})=\log\tilde{Z}(\tilde{R},\tilde{\Omega},\tilde{\Phi}_{D}).$
(128)

Let’s unpack this equality by explaining the notations. All the tilded
quantities refer to the seed solution, which has an angular potential
$\tilde{\Omega}$ other than the inverse temperature (over $2\pi$) $\tilde{R}$
and asymptotic value of the dilaton $\tilde{\Phi}_{D}$. The quantities without
tildes are those for the generated solution, which also has $r$ being the size
of the extra circle, and $\mu_{n},\mu_{w}$ being the chemical potentials for
the momentum and winding charges $Q_{n}$ and $Q_{w}$. We will fix $r=1$ in the
following discussion.222222This is only valid since we are using $r$ to denote
the asymptotic size of the circle. Inside the spacetime it will be varying. We
can factor out the dependence of the dilaton and define the primed
quantities232323For the dilaton, here we follow the convention used in the
main text, see (1).

$e^{-\Phi_{D}}\log Z^{\prime}(R,\Omega,\mu_{n},\mu_{w})\equiv\log
Z(R,\Omega,\mu_{n},\mu_{w},\phi_{D}),\quad
e^{-\tilde{\Phi}_{D}}\log\tilde{Z}^{\prime}(\tilde{R},\tilde{\Omega})\equiv\log\tilde{Z}(\tilde{R},\tilde{\Omega},\tilde{\Phi}_{D})$
(129)

Using the invariance of the $d=D-1$ dimensional dilaton, the asymptotic value
of the dilaton transforms as

$e^{-\tilde{\Phi}_{D}}\tilde{R}=e^{-\Phi_{D}}R$ (130)

and therefore from (128) we have

$\log Z^{\prime}=\frac{R}{\tilde{R}}\log\tilde{Z}^{\prime}.$ (131)

For the generated solution, we have the general statistical expression for the
action

$\log Z=e^{-\Phi_{D}}\log Z^{\prime}=S-2\pi RM+2\pi R\Omega J+2\pi
R\mu_{n}Q_{n}+2\pi R\mu_{w}Q_{w}$ (132)

where various charges and entropy can be computed as

$\displaystyle 2\pi RQ_{n}=e^{-\Phi_{D}}\partial_{\mu_{n}}\log
Z^{\prime},\quad 2\pi RQ_{w}=e^{-\Phi_{D}}\partial_{\mu_{w}}\log
Z^{\prime},\quad 2\pi RJ=e^{-\Phi_{D}}\partial_{\Omega}\log Z^{\prime},$ (133)
$\displaystyle 2\pi
M=e^{-\Phi_{D}}\left(-\partial_{R}+\frac{\mu_{n}}{R}\partial_{\mu_{n}}+\frac{\mu_{w}}{R}\partial_{\mu_{w}}+\frac{\Omega}{R}\partial_{\Omega}\right)\log
Z^{\prime},\quad S=e^{-\Phi_{D}}(1-R\partial_{R})\log Z^{\prime}.$

On the other hand, for the seed solution, we have

$\displaystyle\log\tilde{Z}^{\prime}=\tilde{S}^{\prime}-2\pi\tilde{R}\tilde{M}^{\prime}+2\pi\tilde{R}\tilde{\Omega}\tilde{J}^{\prime},$
(134) $\displaystyle
2\pi\tilde{R}\tilde{J}^{\prime}=\partial_{\tilde{\Omega}}\log\tilde{Z}^{\prime},\quad
2\pi\tilde{M}^{\prime}=\left(-\partial_{\tilde{R}}+\frac{\tilde{\Omega}}{\tilde{R}}\partial_{\tilde{\Omega}}\right)\log\tilde{Z}^{\prime},\quad\tilde{S}^{\prime}=\left(1-\tilde{R}\partial_{\tilde{R}}\right)\log\tilde{Z}^{\prime}.$

Note that here similar to $\tilde{Z}^{\prime}$ we have also defined primed
thermodynamic quantities which are simply themselves but without the dilaton
dependence (for example, $\tilde{S}=e^{-\tilde{\Phi}_{D}}\tilde{S}^{\prime}$).
Now, combining the expression for $Q_{n}$ in (133) with (134), we get

$\displaystyle 2\pi RQ_{n}$
$\displaystyle=e^{-\Phi_{D}}\partial_{\mu_{n}}\left(\frac{R}{\tilde{R}}\log\tilde{Z}^{\prime}\right)=e^{-\Phi_{D}}R\left[\frac{\partial\tilde{R}}{\partial\mu_{n}}\partial_{\tilde{R}}\left(\frac{1}{\tilde{R}}\log\tilde{Z}^{\prime}\right)+\frac{1}{\tilde{R}}\frac{\partial\tilde{\Omega}}{\partial\mu_{n}}\partial_{\tilde{\Omega}}\left(\log\tilde{Z}^{\prime}\right)\right]$
(135)
$\displaystyle=e^{-\Phi_{D}}R\left[-\frac{1}{\tilde{R}^{2}}\frac{\partial\tilde{R}}{\partial\mu_{n}}\tilde{S}^{\prime}+2\pi\frac{\partial\tilde{\Omega}}{\partial\mu_{n}}\tilde{J}^{\prime}\right].$

Similar derivations lead to

$2\pi
Q_{w}=e^{-\Phi_{D}}\left[-\frac{1}{\tilde{R}^{2}}\frac{\partial\tilde{R}}{\partial\mu_{w}}\tilde{S}^{\prime}+2\pi\frac{\partial\tilde{\Omega}}{\partial\mu_{w}}\tilde{J}^{\prime}\right],$
(136) $\displaystyle 2\pi J$
$\displaystyle=e^{-\Phi_{D}}\left[-\frac{1}{\tilde{R}^{2}}\frac{\partial\tilde{R}}{\partial\Omega}\tilde{S}^{\prime}+2\pi\frac{\partial\tilde{\Omega}}{\partial\Omega}\tilde{J}^{\prime}\right],$
(137) $\displaystyle S$
$\displaystyle=e^{-\Phi_{D}}\left[\frac{R^{2}}{\tilde{R}^{2}}\frac{\partial\tilde{R}}{\partial
R}\tilde{S}^{\prime}-2\pi R^{2}\frac{\partial\tilde{\Omega}}{\partial
R}\tilde{J}^{\prime}\right],$ (138)

and finally, for the energy, we have

$\displaystyle 2\pi M$ $\displaystyle=2\pi
e^{-\Phi_{D}}\tilde{M}^{\prime}+e^{-\Phi_{D}}\left[-1+\frac{R}{\tilde{R}}\frac{\partial\tilde{R}}{\partial
R}-\frac{\mu_{n}}{\tilde{R}}\frac{\partial\tilde{R}}{\partial\mu_{n}}-\frac{\mu_{w}}{\tilde{R}}\frac{\partial\tilde{R}}{\partial\mu_{w}}-\frac{\Omega}{\tilde{R}}\frac{\partial\tilde{R}}{\partial\Omega}\right]\frac{\tilde{S}^{\prime}}{\tilde{R}}$
(139) $\displaystyle\quad+2\pi
e^{-\Phi_{D}}\left[-1-\frac{R}{\tilde{\Omega}}\frac{\partial\tilde{\Omega}}{\partial
R}+\frac{\Omega}{\tilde{\Omega}}\frac{\partial\tilde{\Omega}}{\partial\Omega}+\frac{\mu_{n}}{\tilde{\Omega}}\frac{\partial\tilde{\Omega}}{\partial\mu_{n}}+\frac{\mu_{w}}{\tilde{\Omega}}\frac{\partial\tilde{\Omega}}{\partial\mu_{w}}\right]\tilde{\Omega}\tilde{J}^{\prime}.$

(135) - (138) constitute as generalizations of the formulas in Chen:2021dsw .
These relations are general and we didn’t use the specific form of the
solution generating transformation yet. All the information of the specific
solution generation is contained in the map between

$\\{\tilde{R},\tilde{\Omega}\\}\leftrightarrow\\{R,\Omega,\mu_{n},\mu_{w}\\}.$
(140)

To find these relations, only the asymptotic form of the solution generating
transformation is needed. To zeroth order in $\alpha^{\prime}$, the
transformation is the same in the bosonic/Type II and heterotic case, while
they are different at order $\alpha^{\prime}$, which we defer to A.1.
Following the derivation of Chen:2021dsw  (which uses the formalism of
Maharana:1992my ), we have242424Here we follow the convention of Chen:2021dsw
for the normalization of the gauge fields and charges, which leads to a
$\sqrt{2}$ factor difference compared to the main text, i.e. $Q_{\rm
appendix}=Q_{\textrm{main text}}/\sqrt{2}$.

$\tilde{R}=R\sqrt{1-\mu_{n}^{2}}\sqrt{1-\mu_{w}^{2}},\quad\quad\tilde{\Omega}=\frac{\Omega}{\sqrt{1-\mu_{n}^{2}}\sqrt{1-\mu_{w}^{2}}}.$
(141)

An easy way to understand the relation between $\Omega$ and $\tilde{\Omega}$
is the following. Let’s say that the rotation is in $\varphi$ direction, then
in the seed solution we have periodic identification
$(t_{E},\varphi)\sim(t_{E}+1,\varphi+2\pi i\tilde{R}\tilde{\Omega})$, where
we’ve chosen the Euclidean time coordinate such that its periodicity is one.
This identification should remain invariant under the transformation, meaning
that $\tilde{R}\tilde{\Omega}$ should be invariant, which explains (141). To
connect with the discussion in the main text, we can further parameterize
$\mu_{n}$ and $\mu_{w}$ in terms of $\upalpha,\upbeta$

$\mu_{n}=\tanh\frac{\upalpha+\upbeta}{2},\quad\quad\mu_{w}=\tanh\frac{\upalpha-\upbeta}{2}.$
(142)

Using (141) and (142), as well as $Q_{L,R}=Q_{n}\pm Q_{w}$, we can simplify
(135) - (138) and get

$2\pi
Q_{L}=e^{-\Phi_{D}}\cosh\upbeta\sinh\upalpha\left[\frac{\tilde{S}^{\prime}}{\tilde{R}}+2\pi\tilde{\Omega}\tilde{J}^{\prime}\right],$
(143) $2\pi
Q_{R}=e^{-\Phi_{D}}\cosh\upalpha\sinh\upbeta\left[\frac{\tilde{S}^{\prime}}{\tilde{R}}+2\pi\tilde{\Omega}\tilde{J}^{\prime}\right],$
(144) $2\pi J=2\pi
e^{-\Phi_{D}}\frac{\cosh\upalpha+\cosh\upbeta}{2}\tilde{J}^{\prime},$ (145)
$S=e^{-\Phi_{D}}\tilde{S}^{\prime}\frac{\cosh\upalpha+\cosh\upbeta}{2},$ (146)
$2\pi M=2\pi
e^{-\Phi_{D}}\tilde{M}^{\prime}+e^{-\Phi_{D}}(\cosh\upalpha\cosh\upbeta-1)\left(\frac{\tilde{S}^{\prime}}{\tilde{R}}+2\pi\tilde{\Omega}\tilde{J}^{\prime}\right).$
(147)

The point of (143) - (147) is that once we know
$\\{\tilde{M}^{\prime},\tilde{S}^{\prime},\tilde{J}^{\prime}\\}$ of the
uncharged solution, the thermodynamic quantities of the charged solution will
be completely determined. For example, a consistency check of these
expressions is that if we plug in the
$\\{\tilde{M}^{\prime},\tilde{S}^{\prime},\tilde{J}^{\prime}\\}$ for a four
dimensional Kerr black hole, we correctly reproduce the thermodynamics of the
two-charge black hole in Sen:1994eb .

As an aside, it is interesting to note from the expressions that the
combination of
$\frac{\tilde{S}^{\prime}}{\tilde{R}}+2\pi\tilde{\Omega}\tilde{J}^{\prime}$ is
the “charge” being transformed.

###  A.1 Generating $\alpha^{\prime}$ corrections in various cases

A practical application of (143) - (147) is that one could use them to work
out the $\alpha^{\prime}$ correction to the rotating two-charge system using
the knowledge of the $\alpha^{\prime}$ correction to the seed solution,
without needing to find the explicit corrected solution. Of course, this is
assuming that we don’t encounter any singular solutions along the solution
generating transformation, so perturbative $\alpha^{\prime}$ corrections are
under control. For this reason, one cannot apply the results here directly to
the solution we studied in the main text, which is singular at the two
derivative level. Nonetheless, the formulas would apply when we are far away
from the singular limit $\Omega=2\pi i/\beta$.

The starting point of the derivation is the $\alpha^{\prime}$ correction to