aainstitutetext: Dipartimento di Fisica e Astronomia “Galileo Galilei”,
Università di Padova & INFN Padova, Via Marzolo 8, 35131 Padova,
Italybbinstitutetext: Dipartimento di Fisica, Università di Roma “Tor Vergata”
& INFN Roma 2,
Via della Ricerca Scientifica 1, 00133, Roma, Italyccinstitutetext: Instituto
de Física Teórica UAM/CSIC,
C/ Nicolás Cabrera, 13-15, C. U. Cantoblanco, E-28049 Madrid, Spain

# Revisiting $\alpha^{\prime}$ corrections to heterotic two-charge black holes

Stefano Massai b Alejandro Ruipérez c Matteo Zatti<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We find solutions of the heterotic string effective action describing the
first-order $\alpha^{\prime}$ corrections to two-charge black holes at finite
temperature. Making explicit use of these solutions, we compute the
corrections to the thermodynamic quantities: temperature, chemical potentials,
mass, charges and entropy. We check that the first law of black hole mechanics
is satisfied and that the thermodynamics agrees with the one extracted from
the Euclidean on-shell action. Finally, we show that our results are in
agreement with the corrections for the thermodynamics recently predicted by
Chen, Maldacena and Witten.

IFT-UAM/CSIC-23-143

##  1 Introduction

The study of two-charge black holes has attracted much attention since the
first investigations of black holes in string theory. This is mainly due to
the fact that they are supposed to describe perhaps the simplest configuration
in string theory which has a non-vanishing degeneracy of BPS states. This
microscopic system consists of a fundamental heterotic string with winding
$Q_{w}$ and momentum $Q_{p}$ charges along a compact direction
${\mathbb{S}}^{1}_{y}$. The degeneracy of BPS states of this system was
computed by Dabholkar and Harvey Dabholkar:1989jt ; Dabholkar:1990yf , and it
is given by

$S_{\rm{micro}}(Q_{p},Q_{w})=\log
d\left(Q_{p},Q_{w}\right)=4\pi\sqrt{Q_{p}Q_{w}}\,.$ (1.1)

Being a BPS degeneracy, it must be protected when extrapolating it to the
finite string-coupling regime where an effective black hole description is
expected to exist (a priori). In other words, it should be possible to match
this BPS degeneracy with the Bekenstein-Hawking entropy of the corresponding
black hole. However, when trying to do so one finds a puzzle: even though
there is a supergravity solution with the same charges and preserving the same
supersymmetries as the Dabholkar-Harvey states Sen:1994eb ; Cvetic:1995uj ;
Dabholkar:1995nc ; Callan:1995hn , it describes a singular black hole with
vanishing horizon area. Hence, the naive macroscopic entropy that can be
associated to the two-charge system vanishes.

In order to explain this mismatch, Sen proposed Sen:1995in  that two-charge
black holes have a small horizon of string size, and therefore it cannot be
resolved by supergravity unless the latter is supplemented with higher-
derivative terms capturing stringy $\alpha^{\prime}$ corrections.111For this
reason, these black holes are often referred to as _small_ black holes. Almost
ten years after this proposal, it was claimed in Dabholkar:2004yr ;
Dabholkar:2004dq  that four-derivative corrections in the context of type IIA
on ${\mathbb{K}}_{3}\times{\mathbb{T}}^{2}$ (which is dual to heterotic on
${\mathbb{T}}^{6}$) stretch the horizon of two-charge black holes (hiding the
singularity behind) and, what is even more remarkable, also give the precise
contribution to the black hole entropy so that it reproduces the microstate
counting of the two-charge system.

These results, however, have been recently questioned in a series of papers
Cano:2018hut ; Ruiperez:2020qda ; Cano:2021dyy  in which, working directly
within the heterotic theory, it has been shown that $\alpha^{\prime}$
corrections do not remove the singularity of BPS two-charge black holes.
Furthermore, it has been argued that the configuration studied in
Dabholkar:2004yr ; Dabholkar:2004dq  should correspond to a regular four-
dimensional black hole whose entropy accidentally matches the microscopic
degeneracy of the two-charge system, but which carries different charges and
preserves less supersymmetry. The fact that it preserves less supersymmetry is
indeed a smoking gun of the presence of additional sources (NS5 branes and
Kaluza-Klein monopoles), which would be the ultimate reason explaining why
this four-dimensional black hole has a regular horizon.

The fact that the two-charge system does not seem to admit a black hole
description in the BPS limit is something which appears rather natural from
the point of view of the correspondence between black holes and fundamental
strings Susskind:1993ws ; Horowitz:1996nw ; Horowitz:1997jc ; Damour:1999aw
(see also Chen:2021emg ; Chen:2021dsw ; Brustein:2021cza ; Matsuo:2022kvx ;
Balthazar:2022hno ; Ceplak:2023afb  for recent discussions). According to this
proposal, black holes should turn into highly-excited strings when their sizes
are of the order of the string scale. This has been recently discussed by
Chen, Maldacena and Witten Chen:2021dsw  precisely in the context of the two-
charge system. Let us consider a two-charge black hole at finite temperature.
This can be described in supergravity in terms of a solution with a large
(macroscopic) horizon. However, if the black hole starts losing its mass it
will reach the string size before reaching extremality, which would imply that
the right description of the system near extremality should be a sort of self-
gravitating string solution Horowitz:1997jc ; Damour:1999aw  rather than a
solution with a horizon Chen:2021dsw .

In this paper we will mainly focus on two-charge black holes at finite
temperature. More concretely, we consider two-charge black holes in heterotic
string theory and we study how the first-order $\alpha^{\prime}$ corrections
modify the solutions and their thermodynamic properties. The corrections to
the thermodynamics have been recently studied in Chen:2021dsw , exploiting the
fact that the two-charge solutions can be obtained by perfoming suitable
$O(2,2)$ transformations to the Schwarzschild-Tangherlini solution, whose
$\alpha^{\prime}$ corrections had been already studied in Callan:1988hs . In
principle this method can be used not only to obtain the corrections to the
thermodynamics but also the corrected solutions themselves, which were not
provided in Chen:2021dsw . This is just technically more involved, as one
would have to take into account that the $O(2,2)$ transformations receive
$\alpha^{\prime}$ corrections Bergshoeff:1995cg ; Kaloper:1997ux ;
Bedoya:2014pma ; Ortin:2020xdm ; Eloy:2020dko .222As explained in Chen:2021dsw
, one can ignore the explicit corrections to the $O(2,2)$ transformations if
the goal is just to obtain the corrected thermodynamics. This was precisely
the strategy followed in Giveon:2009da . However, as pointed out in
Chen:2021dsw , the corrected thermodynamics obtained in these two references
do not agree between each other. Our main motivation here is to perform an
independent “first principle” computation of the corrected solutions and their
thermodynamics; we are going to find the corrected solutions explicitly by
solving the $\alpha^{\prime}$-corrected equations of motion and then compute
the thermodynamic quantities with standard methods.

Anticipating our results, we are going to show that the $\alpha^{\prime}$
corrections to the thermodynamics that we compute fully agree with those of
Chen:2021dsw . This is a strong consistency check of both approaches, as well
as of the methods employed and of the results obtained in previous related
works by two of the authors and collaborators, see e.g. Cano:2018hut ;
Ruiperez:2020qda ; Cano:2021dyy ; Cano:2018qev ; Chimento:2018kop ;
Cano:2018brq ; RuiperezVicente:2020qfw ; Elgood:2020nls ; Elgood:2020xwu ;
Cano:2021nzo ; Ortin:2021win ; Cano:2022tmn ; Zatti:2023oiq  and references
therein. In particular, we want to emphasize that the (singular) solutions
found in Cano:2018hut ; Ruiperez:2020qda ; Cano:2021dyy  are properly
recovered from the non-extremal ones we have found in this paper after
suitably taking the extremal limit. This is discussed in subsection 2.4 and
further confirms the conclusions of Cano:2018hut ; Ruiperez:2020qda ;
Cano:2021dyy , yet from a different perspective.

The organization of the rest of the paper is the following. In section 2 we
review the two-derivative solutions describing heterotic two-charge black
holes in arbitrary dimension ($4\leq d\leq 9$) and then provide the details
about the $\alpha^{\prime}$-corrected solutions, focusing on the four- and
five-dimensional cases. In section 3 we compute the thermodynamic quantities
of the solutions and express them using two-different parametrizations: fixing
the value of the mass and charges (micro-canonical ensemble) and fixing the
inverse temperature and the chemical potentials (grand-canonical ensemble). We
show that the results we get are consistent with the first law of black hole
mechanics. Then in section 4 we corroborate the results of section 3 by
employing an alternative method to compute the corrected thermodynamics,
namely from the Euclidean on-shell action. Finally, in section 5 we compare
our results with those of Chen:2021dsw  finding that they are in perfect
agreement. The appendices contain additional information on the effective
action and equations of motion in app. A, on the procedure followed to find
the corrected solutions in app. B, and on the dimensional reduction of the
configurations on ${\mathbb{S}}^{1}_{y}$ in app. C.

Note on conventions. We adopt the conventions of Ortin:2015hya . In
particular, we use the mostly minus signature for the metric $(+-\dots-)$ and
the conventions for the Riemann tensor are such that

$[\nabla_{\mu},\nabla_{\nu}]\xi^{\sigma}=R_{\mu\nu\rho}{}^{\sigma}\xi^{\rho}\,,$
(1.2)

The conventions for the remaining fields are specified in appendix A. Hats
$\hat{\cdot}$ shall be used to denote both ten- and $(d+1)$-dimensional
fields, as they are identified with one another. Instead, the $d$-dimensional
fields which are obtained upon compactification on ${\mathbb{S}}^{1}_{y}$ will
not carry hats.

##  2 $\alpha^{\prime}$ corrections to heterotic two-charge black holes

###  2.1 Two-charge black holes at leading order in $\alpha^{\prime}$

Let us begin reviewing the two-derivative solution describing non-extremal
two-charge black holes in $d$ dimensions Horowitz:1996nw . Given that in
subsection 2.2 we will solve the corrected ten-dimensional equations of
motion, here we directly present the solution in its ten-dimensional form.
However, since the solutions have a ${\mathbb{T}}^{(9-d)}$ torus playing a
trivial role, we feel free to ignore these torus directions from now
on.333Taking them into account just amounts to add the flat metric on the
torus $-d{\vec{z}}^{2}_{(9-d)}$ to the $(d+1)$-dimensional metric (2.1). Doing
so, the resulting $(d+1)$-dimensional solution is given by

$\displaystyle{\mathrm{d}}{\hat{s}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f}{{f}_{p}{f}_{w}}\,{\mathrm{d}}t^{2}-f^{-1}{\mathrm{d}}\rho^{2}-\rho^{2}{\mathrm{d}}\Omega^{2}_{(d-2)}-k^{2}_{\infty}\frac{{f}_{p}}{{f}_{w}}\left({\mathrm{d}}y+\beta_{p}k^{-1}_{\infty}\left({f}_{p}^{-1}-1\right){\mathrm{d}}t\right)^{2}\,,$
(2.1) $\displaystyle{\hat{B}}$ $\displaystyle=$
$\displaystyle\beta_{w}k_{\infty}\left({f}_{w}^{-1}-1\right){\mathrm{d}}t\wedge{\mathrm{d}}y\,,$
(2.2) $\displaystyle e^{2\hat{\phi}}$ $\displaystyle=$ $\displaystyle
e^{2\hat{\phi}_{\infty}}{f}_{w}^{-1}\,,$ (2.3)

where $\mathrm{d}{\hat{s}}$ represents the line element in the string frame
and

${f}_{p}=1+\frac{q_{p}}{\rho^{d-3}}\,,\hskip
28.45274pt{f}_{w}=1+\frac{q_{w}}{\rho^{d-3}}\,,\hskip
28.45274ptf=1-\frac{\rho_{s}^{d-3}}{\rho^{d-3}}\,.$ (2.4)

The parameters $q_{p}$, $q_{w}$ and $\rho_{s}$ are related to the charges and
mass of the solutions. Together with the moduli ${\hat{\phi}}_{\infty}$ and
$k_{\infty}$ (representing the asymptotic values of the dilaton and the
Kaluza-Klein scalar), they constitute the set of independent parameters of the
solutions since $\beta_{p}$ and $\beta_{w}$ are subject to the following
constraints,

$\rho_{s}^{(d-3)}=q_{p}\left(\beta^{2}_{p}-1\right)=q_{w}\left(\beta^{2}_{w}-1\right)\,,$
(2.5)

implying that

$\beta_{i}=\epsilon_{i}\sqrt{1+\frac{\rho_{s}^{d-3}}{q_{i}}}\,,\hskip
28.45274pti=\left\\{p,w\right\\}\,.$ (2.6)

where $\epsilon^{2}_{i}=1$. These correspond to the signs of the winding and
momentum charges, respectively. In the BPS limit ($\rho_{s}\to 0$), the
solution with $\epsilon_{w}=\epsilon_{p}$ is supersymmetric, while the one
with $\epsilon_{w}=-\epsilon_{p}$ does not preserve any supersymmetry. The
analysis of the Killing spinor equations for these configurations can be found
for instance in Ruiperez:2020qda ; Cano:2021dyy ; Cano:2021nzo .

###  2.2 $\alpha^{\prime}$-corrected solutions

Our aim now is to compute the first-order $\alpha^{\prime}$ corrections to
these two-charge black holes. As usual, we treat the $\alpha^{\prime}$
corrections in a perturbative fashion and ignore ${\cal O}(\alpha^{\prime 2})$
terms. The first-order $\alpha^{\prime}$ corrections in the effective action
of the heterotic superstring were studied in Gross:1986mw ; Metsaev:1987zx ;
Bergshoeff:1989de . While different approaches were used, it was later shown
in Chemissany:2007he  that the resulting effective actions are equivalent up
to field redefinitions. Here we choose to work in the Bergshoeff-de Roo
scheme, Bergshoeff:1989de . In order to establish our conventions, we review
the effective action and equations of motion in appendix A.

Before entering into the details of the corrected solutions, let us briefly
explain the general strategy we have followed in order to find the corrected
solutions. The interested reader is referred to appendix B or to Cano:2022tmn
for more details. It turns out that an educated ansatz to solve the corrected
equations of motion is the following,

$\displaystyle{\mathrm{d}}{\hat{s}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f}{{f}_{p}{\tilde{f}}_{w}}{\mathrm{d}}t^{2}-{g}\left(f^{-1}{\mathrm{d}}\rho^{2}+\rho^{2}{\mathrm{d}}\Omega^{2}_{(d-2)}\right)-k^{2}_{\infty}\frac{{f}_{p}}{{\tilde{f}}_{w}}\left[{\mathrm{d}}y+\beta_{p}k^{-1}_{\infty}\left({f}_{p}^{-1}-1\right){\mathrm{d}}t\right]^{2}\,,$
(2.7) $\displaystyle{\hat{B}}$ $\displaystyle=$
$\displaystyle\beta_{w}k_{\infty}\left({f}_{w}^{-1}-1\right){\mathrm{d}}t\wedge{\mathrm{d}}y\,,$
(2.8)

where the functions $f,f_{p},f_{w},{\tilde{f}}_{w},g$ and the dilaton
$\hat{\phi}$ are assumed to depend only on the radial coordinate $\rho$. For
consistency with the perturbative approach, they must be of the form

$\displaystyle{f}_{p}=$ $\displaystyle
1+\frac{q_{p}}{\rho^{d-3}}+\alpha^{\prime}{\delta f_{p}}\,,\hskip
14.22636pt{\tilde{f}}_{w}=\,1+\frac{q_{w}}{\rho^{d-3}}+\alpha^{\prime}{\delta{\tilde{f}}_{w}}\,,\hskip
14.22636pt{g}=\,1+\alpha^{\prime}{\delta{g}}\,,$ (2.9) $\displaystyle{f}=$
$\displaystyle\,1-\frac{\rho_{s}^{d-3}}{\rho^{d-3}}+\alpha^{\prime}{\delta
f}\,,\hskip
14.22636pt{f}_{w}=\,1+\frac{q_{w}}{\rho^{d-3}}+\alpha^{\prime}{\delta{f}_{w}}\,.$

After linearization in $\alpha^{\prime}$, the equations of motion boil down to
a linear system of inhomogeneous second-order ODEs for the unknown functions
$\delta f_{p},\delta{\tilde{f}}_{w},\delta g,\delta f,\delta f_{w}$ and
$\hat{\phi}$. The strategy we are going to follow to solve them is the same as
in Cano:2022tmn , which consists of performing an asymptotic expansion (large
$\rho$) of the unknown functions and solve the equations of motion order by
order. Following this procedure, we can determine all the coefficients of the
asymptotic expansion except for a few of them which remain free, the
integration constants. Once the form of the asymptotic solution has been
found, we resum the asymptotic series with the help of Mathematica. The final
step is to fix the integration constants by imposing regularity at the horizon
and suitable boundary conditions. Our choice here will be such that we keep
the asymptotic charges and the mass fixed: i.e., we are going to give the form
of the corrected solution in the micro-canonical ensemble.

In what follows we give the corrected solutions in $d=5$ and $d=4$, as well as
its dimensional reduction on ${\mathbb{S}}^{1}_{y}$. Finally, we study their
BPS limits and check that they agree with the corrected solutions found in
Cano:2018hut ; Ruiperez:2020qda ; Cano:2021dyy .

####  2.2.1 Five-dimensional black holes

Let us first consider the $d=5$ case. The expression of the dilaton is found
by solving the equation of motion of the Kalb-Ramond 2-form $\hat{B}$. It is
given by

$\hat{\phi}=a_{\phi}+\frac{1}{2}\log\frac{-\rho^{3}{g}{\tilde{f}}_{w}{f}^{\prime}_{w}}{2q_{w}{f}^{2}_{w}}\,,$
(2.10)

where $a_{\phi}$ is an integration constant to be fixed by imposing the
asymptotic value of the string coupling is not renormalized, namely

$\lim_{\rho\to\infty}{\hat{\phi}}={\hat{\phi}}_{\infty}\,.$ (2.11)

After fixing the integration constants in the way we have explained, we find
the following solution:

$\displaystyle\delta{f}_{p}$ $\displaystyle=$
$\displaystyle-\frac{q_{p}\rho_{s}^{2}\left(1+\frac{\beta_{w}}{\beta_{p}}\right)\log\left(1+\frac{q_{w}}{\rho^{2}}\right)}{2q_{w}^{2}\rho^{2}}$
(2.12) $\displaystyle+$
$\displaystyle\frac{1}{32q_{w}\left(\rho^{2}+q_{w}\right)\rho^{6}}\left\\{16q_{p}\rho_{s}^{2}\rho^{4}-q_{w}q_{p}\rho^{2}\left(9\rho_{s}^{2}+32q_{w}\right)+7q_{w}^{2}q_{p}\rho_{s}^{2}\right.$
$\displaystyle+$
$\displaystyle\left.\frac{\beta_{w}}{\beta_{p}}\left[16q_{p}\rho_{s}^{2}\rho^{4}+8q_{w}\rho^{2}\left(q_{p}\rho_{s}^{2}-2q_{w}\left(\rho_{s}^{2}+2q_{p}\right)\right)+8q_{w}^{2}q_{p}\rho_{s}^{2}\right]\right\\}\,,$
$\displaystyle\delta{\tilde{f}}_{w}$ $\displaystyle=$
$\displaystyle-\frac{\rho_{s}^{2}\left(1+\frac{\beta_{w}}{\beta_{p}}\right)\log\left(1+\frac{q_{w}}{\rho^{2}}\right)}{2q_{w}\rho^{2}}+\frac{\beta_{w}\rho_{s}^{2}\left(q_{w}+2\rho^{2}\right)}{4\beta_{p}\rho^{6}}+\frac{7q_{w}\rho_{s}^{2}}{32\rho^{6}}+\frac{\rho_{s}^{2}}{2\rho^{4}}$
(2.13) $\displaystyle+$
$\displaystyle\frac{9\rho_{s}^{2}(q_{p}-q_{w})}{4\rho^{2}\left(4\rho_{s}^{2}(q_{w}+q_{p})+4q_{w}q_{p}+3\rho_{s}^{4}\right)}\,,$
$\displaystyle\delta{g}$ $\displaystyle=$
$\displaystyle\frac{\rho_{s}^{2}\left(1+\frac{\beta_{w}}{\beta_{p}}\right)\log\left(1+\frac{q_{w}}{\rho^{2}}\right)}{2q_{w}^{2}}+\frac{\beta_{w}\rho_{s}^{2}\left(q_{w}-2\rho^{2}\right)}{4\beta_{p}q_{w}\rho^{4}}-\frac{7\rho_{s}^{2}}{32\rho^{4}}$
(2.14) $\displaystyle+$
$\displaystyle\frac{1}{8\rho^{2}}\left(-\frac{4\rho_{s}^{2}}{q_{w}}+\frac{18q_{p}\left(\rho_{s}^{2}+2q_{w}\right)}{4\left(q_{w}+q_{p}\right)\rho_{s}^{2}+3\rho_{s}^{4}+4q_{w}q_{p}}-9\right)\,,$
$\displaystyle\delta f$ $\displaystyle=$
$\displaystyle\frac{\rho_{s}^{4}\left(1+\frac{\beta_{w}}{\beta_{p}}\right)\log\left(1+\frac{q_{w}}{\rho^{2}}\right)}{2q_{w}^{2}\rho^{2}}-\frac{\beta_{w}\rho_{s}^{2}\left(2\rho_{s}^{2}\rho^{4}+q_{w}\rho_{s}^{2}\rho^{2}+q_{w}^{2}\left(3\rho_{s}^{2}-4\rho^{2}\right)\right)}{4q_{w}\beta_{p}\rho^{6}\left(\rho^{2}+q_{w}\right)}$
(2.15) $\displaystyle+$
$\displaystyle\frac{\rho_{s}^{2}q_{w}}{\left(\rho^{2}+q_{w}\right)\rho^{4}}-\frac{3\left(\rho^{2}+q_{p}\right)}{4\rho^{4}}+\frac{\rho_{s}^{4}\left(-16\rho^{4}+9q_{w}\rho^{2}-7q_{w}^{2}\right)}{32q_{w}\rho^{6}\left(\rho^{2}+q_{w}\right)}$
$\displaystyle+$
$\displaystyle\frac{\left(\rho^{2}+q_{p}\right)\left(2q_{w}q_{p}-\left(q_{w}-2q_{p}\right)\rho_{s}^{2}\right)}{2\rho^{4}\left(4\left(q_{w}+q_{p}\right)\rho_{s}^{2}+3\rho_{s}^{4}+4q_{w}q_{p}\right)}\,,$
$\displaystyle\delta{f}_{w}$ $\displaystyle=$
$\displaystyle-\frac{\rho_{s}^{2}\left(1+\frac{\beta_{w}}{\beta_{p}}\right)\log\left(1+\frac{q_{w}}{\rho^{2}}\right)}{2q_{w}\rho^{2}}-\frac{\beta_{w}\rho_{s}^{2}\left(q_{w}-2\rho^{2}\right)}{4\beta_{p}\rho^{6}}-\frac{q_{w}\rho_{s}^{2}}{32\rho^{6}}$
(2.16) $\displaystyle+$
$\displaystyle\frac{\rho_{s}^{2}\left(4+\frac{18q_{w}\left(q_{w}-q_{p}\right)}{4\left(q_{w}+q_{p}\right)\rho_{s}^{2}+3\rho_{s}^{4}+4q_{w}q_{p}}\right)}{8\rho^{4}}\,,$

and $a_{\phi}={\hat{\phi}}_{\infty}$.

####  2.2.2 Four-dimensional black holes

We proceed as in the five-dimensional case. Solving the ten-dimensional Kalb-
Ramond equation of motion yields the expression for the dilaton,

$\hat{\phi}=\hat{\phi}_{\infty}+\frac{1}{2}\log\frac{-\rho^{2}{{g}}^{1/2}{\tilde{f}}_{w}{f}^{\prime}_{w}}{q_{w}{f}^{2}_{w}}\,,$
(2.17)

where we have already imposed that the asymptotic value of the string coupling
is not renormalized. Solving the remaining equations of motion as explained
before we obtain:

$\displaystyle\delta{f}_{p}$ $\displaystyle=$
$\displaystyle\frac{\beta_{p}\beta_{w}q_{p}\left[q_{p}\left(4q_{w}^{3}(\rho_{s}-3\rho)+\rho_{s}q_{w}^{2}\rho-3\rho_{s}q_{w}\rho^{2}-6\rho_{s}\rho^{3}\right)-6\rho_{s}q_{w}^{3}\rho\right]}{48q_{w}^{2}\rho^{4}(q_{p}+\rho_{s})(q_{w}+\rho)}$
(2.18) $\displaystyle-$
$\displaystyle\frac{q_{p}\left[q_{w}^{3}(39\rho-10\rho_{s})+q_{w}^{2}\rho(17\rho_{s}+3\rho)+9\rho_{s}q_{w}\rho^{2}+18\rho_{s}\rho^{3}\right]}{144q_{w}^{2}\rho^{4}(q_{w}+\rho)}$
$\displaystyle+$
$\displaystyle\log\left(1+\frac{q_{w}}{\rho}\right)\left(\frac{\beta_{p}\beta_{w}q_{p}^{2}\rho_{s}}{8q_{w}^{3}\rho(q_{p}+\rho_{s})}+\frac{q_{p}\rho_{s}}{8q_{w}^{3}\rho}\right)\,,$
$\displaystyle\delta{\tilde{f}}_{w}$ $\displaystyle=$
$\displaystyle\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}\left(4q_{w}^{2}+3q_{w}\rho-6\rho^{2}\right)}{48q_{w}\rho^{4}(q_{p}+\rho_{s})}+\frac{q_{w}^{2}(10\rho_{s}-3\rho)+9\rho_{s}q_{w}\rho-18\rho_{s}\rho^{2}}{144q_{w}\rho^{4}}$
(2.19) $\displaystyle+$
$\displaystyle\frac{q_{p}-q_{w}}{\rho\left(4q_{p}q_{w}+3q_{p}\rho_{s}+3\rho_{s}q_{w}+2\rho_{s}^{2}\right)}$
$\displaystyle+$
$\displaystyle\log\left(1+\frac{q_{w}}{\rho}\right)\left(\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}}{8q_{w}^{2}\rho(q_{p}+\rho_{s})}+\frac{\rho_{s}}{8q_{w}^{2}\rho}\right)\,,$
$\displaystyle\delta{g}$ $\displaystyle=$
$\displaystyle\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}\left(2q_{w}^{2}-3q_{w}\rho+6\rho^{2}\right)}{24q_{w}^{2}\rho^{3}(q_{p}+\rho_{s})}-\frac{\rho_{s}+3q_{w}}{8q_{w}\rho^{2}}-\frac{5\rho_{s}}{36\rho^{3}}$
(2.20) $\displaystyle-$
$\displaystyle\frac{(\rho_{s}+2q_{w})\left(-3q_{p}\rho_{s}+2q_{p}q_{w}-2\rho_{s}^{2}+6q_{w}^{2}+\rho_{s}q_{w}\right)}{4q_{w}^{2}\rho\left(3q_{p}\rho_{s}+4q_{p}q_{w}+2\rho_{s}^{2}+3\rho_{s}q_{w}\right)}$
$\displaystyle-$
$\displaystyle\log\left(1+\frac{q_{w}}{\rho}\right)\left(\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}}{4q_{w}^{3}(q_{p}+\rho_{s})}+\frac{\rho_{s}}{4q_{w}^{3}}\right)\,,$
$\displaystyle\delta f$ $\displaystyle=$
$\displaystyle\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}\left(2q_{w}^{3}(6\rho-5\rho_{s})-\rho_{s}q_{w}^{2}\rho+3\rho_{s}q_{w}\rho^{2}+6\rho_{s}\rho^{3}\right)}{48q_{w}^{2}\rho^{4}(q_{p}+\rho_{s})(q_{w}+\rho)}$
(2.21) $\displaystyle-$
$\displaystyle\frac{40q_{p}\rho_{s}^{2}q_{w}^{4}+30\rho_{s}^{3}q_{w}^{3}(q_{p}+q_{w})+20\rho_{s}^{4}q_{w}^{3}+\rho^{4}\left(288q_{w}^{3}+144\rho_{s}q_{w}^{2}\right)}{144q_{w}^{2}\rho^{4}(q_{w}+\rho)\left[3\rho_{s}(q_{p}+q_{w})+4q_{p}q_{w}+2\rho_{s}^{2}\right]}$
$\displaystyle+$
$\displaystyle\frac{\rho^{3}\left(-288q_{w}^{3}(q_{p}+q_{w})-144\rho_{s}q_{w}^{2}(q_{p}+q_{w})+54\rho_{s}^{3}(q_{p}+q_{w})+72q_{p}\rho_{s}^{2}q_{w}+36\rho_{s}^{4}\right)}{144q_{w}^{2}\rho^{4}(q_{w}+\rho)\left[3\rho_{s}(q_{p}+q_{w})+4q_{p}q_{w}+2\rho_{s}^{2}\right]}$
$\displaystyle+$
$\displaystyle\frac{\rho^{2}\left[9\rho_{s}^{2}q_{w}^{2}(5q_{p}+q_{w})+3\rho_{s}^{3}q_{w}(9q_{p}+11q_{w})+18\rho_{s}^{4}q_{w}-288q_{p}q_{w}^{4}-132q_{p}\rho_{s}q_{w}^{3}\right]}{144q_{w}^{2}\rho^{4}(q_{w}+\rho)\left[3\rho_{s}(q_{p}+q_{w})+4q_{p}q_{w}+2\rho_{s}^{2}\right]}$
$\displaystyle+$
$\displaystyle\frac{\rho\left[156q_{p}\rho_{s}q_{w}^{4}+\rho_{s}^{2}q_{w}^{3}(185q_{p}+117q_{w})+3\rho_{s}^{3}q_{w}^{2}(17q_{p}+43q_{w})+34\rho_{s}^{4}q_{w}^{2}\right]}{144q_{w}^{2}\rho^{4}(q_{w}+\rho)\left[3\rho_{s}(q_{p}+q_{w})+4q_{p}q_{w}+2\rho_{s}^{2}\right]}$
$\displaystyle-$
$\displaystyle\log\left(1+\frac{q_{w}}{\rho}\right)\left(\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}^{2}}{8q_{w}^{3}\rho(q_{p}+\rho_{s})}+\frac{\rho_{s}^{2}}{8q_{w}^{3}\rho}\right)\,,$
$\displaystyle\delta{f}_{w}$ $\displaystyle=$
$\displaystyle-\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}\left(2q_{w}^{2}-3q_{w}\rho+6\rho^{2}\right)}{48q_{w}\rho^{4}(q_{p}+\rho_{s})}+\frac{q_{w}(q_{w}-q_{p})}{\rho^{2}\left(3\rho_{s}q_{p}+3\rho_{s}q_{w}+4q_{p}q_{w}+2\rho_{s}^{2}\right)}$
(2.22) $\displaystyle-$
$\displaystyle\frac{\rho_{s}\left(8q_{w}^{2}-9q_{w}\rho+18\rho^{2}\right)}{144q_{w}\rho^{4}}+\frac{q_{w}}{48\rho^{3}}$
$\displaystyle+$
$\displaystyle\log\left(1+\frac{q_{w}}{\rho}\right)\left(\frac{\beta_{p}\beta_{w}q_{p}\rho_{s}}{8q_{w}^{2}\rho(q_{p}+\rho_{s})}+\frac{\rho_{s}}{8q_{w}^{2}\rho}\right)\,.$

###  2.3 Dimensional reduction on ${\mathbb{S}}^{1}_{y}$

The dimensional reduction of this solution to five dimensions can be carried
out using the formulae obtained in Elgood:2020xwu , which are collected in
appendix C. Applying them to the configuration at hands, we get

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=\,$
$\displaystyle\frac{f}{{f}_{p}{\tilde{f}}_{w}}{\mathrm{d}}t^{2}-{g}\left(f^{-1}{\mathrm{d}}\rho^{2}+\rho^{2}{\mathrm{d}}\Omega^{2}_{(d-2)}\right)\,,$
(2.23) $\displaystyle B$ $\displaystyle=\,$ $\displaystyle 0\,,$ (2.24)
$\displaystyle A$ $\displaystyle=\,$
$\displaystyle\beta_{p}\,k^{-1}_{\infty}\left({f}_{p}^{-1}-1\right){\mathrm{d}}t\,,\hskip
28.45274ptC=\,\beta_{w}\,k_{\infty}\left[{f}_{w}^{-1}\left(1+\alpha^{\prime}\beta_{w}^{-1}\Delta_{C}\right)-1\right]{\mathrm{d}}t\,,$
(2.25) $\displaystyle e^{2\phi}$ $\displaystyle=$ $\displaystyle
e^{2\hat{\phi}}\,k^{-1}_{\infty}\left(\frac{{\tilde{f}}_{w}}{f_{p}}\right)^{1/2}\,,\hskip
28.45274ptk=k_{\infty}\left(\frac{f_{p}}{{\tilde{f}}_{w}}\right)^{1/2}\,,$
(2.26)

where $\mathrm{d}s$ represents the line element in the string frame and

$\Delta_{C}=\frac{2\left(\beta_{p}+\beta_{w}\right)ff^{\prime}_{p}f^{\prime}_{w}-f^{\prime}\left(\beta_{w}f_{p}f^{\prime}_{w}+\beta_{p}f^{\prime}_{p}f_{w}\right)}{8f_{p}f_{w}}.$
(2.27)

###  2.4 Extremal limit

The extremal limit is implemented by setting $\rho_{s}\to 0$ while keeping the
charge parameters $q_{p}$ and $q_{w}$ fixed. The $\alpha^{\prime}$ corrections
in this limit have been already studied in the recent literature Cano:2018hut
; Ruiperez:2020qda ; Cano:2021dyy . The corrected solution in arbitrary
dimension is given by Ruiperez:2020qda ; Cano:2021dyy :444Here we are focusing
on the supersymmetric case $\epsilon_{p}=\epsilon_{w}$, which was the case
analyzed in Cano:2018hut ; Ruiperez:2020qda ; Cano:2021dyy . Surprisingly, in
the non-supersymmetric case $\epsilon_{p}=-\epsilon_{w}$ the corrections
simply vanish as the first-order correction in (2.29) is multiplied by
$1+\epsilon_{p}\epsilon_{w}$.

$\displaystyle f$ $\displaystyle=$ $\displaystyle g=1\,,$ (2.28)
$\displaystyle f_{p}$ $\displaystyle=$ $\displaystyle
1+\frac{q_{p}}{\rho^{d-3}}-\frac{(d-3)^{2}\alpha^{\prime}}{2}\frac{q_{p}q_{w}}{\rho^{d-1}\left(\rho^{d-3}+q_{w}\right)}\,,$
(2.29) $\displaystyle{\tilde{f}}_{w}$ $\displaystyle=$ $\displaystyle
f_{w}=1+\frac{q_{w}}{\rho^{d-3}}\,.$ (2.30)

We have checked that this solution is precisely recovered from the non-
extremal ones we have presented presented in subsection 2.2 upon taking
$\rho_{s}\to 0$.555While the limit is smooth in the five-dimensional case, in
the four-dimensional one the limit must be taken before fixing the integration
constants, as the expressions for the latter (which we have not provided
explicitly) diverge when $\rho_{s}\to 0$. This is an interesting consistency
check of our solutions.

##  3 Black hole thermodynamics

In this section we compute the thermodynamic quantities of the
$\alpha^{\prime}$-corrected solutions found in the previous section. Let us
first explain how to compute them in general. Then we apply the corresponding
formulae to the five- and four-dimensional solutions.

Temperature and chemical potentials. The inverse temperature $\beta$ is
obtained by demanding regularity in the Euclidean section Gibbons:1976ue . A
standard calculation shows that it is given by

$\beta=4\pi\frac{\sqrt{gf_{p}{\tilde{f}}_{w}}}{f^{\prime}}\Bigg{|}_{\rho=\rho_{h}}\,,$
(3.1)

where $\rho_{h}$ is the position of the outer horizon. The latter corresponds
to the (largest) root of the metric function $f$,

$f(\rho_{h})=0\,.$ (3.2)

As a consequence of our choice of boundary conditions the position of the
horizon $\rho_{h}$ is shifted by the $\alpha^{\prime}$ corrections.

The chemical potentials associated to the Kaluza-Klein and winding vectors are
obtained as usual,

$\displaystyle\Phi_{p}=\xi^{\mu}A_{\mu}|_{\infty}-\xi^{\mu}A_{\mu}|_{\rho=\rho_{h}}\,,\hskip
28.45274pt\Phi_{w}=\xi^{\mu}C_{\mu}|_{\infty}-\xi^{\mu}C_{\mu}|_{\rho=\rho_{h}}\,,$
(3.3)

where $\xi=\partial_{t}$ is the Killing vector that generates the horizon. As
it turns out, the explicit correction obtained in Elgood:2020xwu  for the
expression of the lower-dimensional winding vector $C_{\mu}$ (reported here in
(2.27)) plays a crucial role in order to obtain a result consistent with the
first law of black hole mechanics.

Electric charges and mass. The four-derivative corrections modify the two-
derivative Maxwell equations of the vector fields, giving rise to several
notions of charge Page:1983mke ; Marolf:2000cb . However, we expect on general
grounds that all of them should coincide when computed asymptotically,
provided the field strengths (and other quantities involved) decay fast
enough. Given this, we compute the so-called Maxwell charges Marolf:2000cb
:666Our conventions for the Hodge star operator are such that
$\star\left(dx^{\mu_{1}}\wedge\dots\wedge
dx^{\mu_{n}}\right)=\frac{1}{(d-n)!}\epsilon_{\nu_{1}\dots\nu_{d-n}}{}^{\mu_{1}\dots\mu_{n}}\,dx^{\nu_{1}}\wedge\dots\wedge
dx^{\nu_{d-n}}$ and $\epsilon_{01\dots d-1}=+\sqrt{|g|}$.

$Q_{p}=\frac{-1}{16\pi
G_{N}}\int_{{\mathbb{S}}^{(d-2)}_{\infty}}e^{-2\left(\phi-\phi_{\infty}\right)}k_{(1)}^{2}\star
F\,,\hskip 14.22636ptQ_{w}=\frac{-1}{16\pi
G_{N}}\int_{{\mathbb{S}}^{(d-2)}_{\infty}}e^{-2\left(\phi-\phi_{\infty}\right)}k^{-2}\star
G\,,$ (3.4)

where $F=\mathrm{d}A$, $G=\mathrm{d}C$, $k_{(1)}$ is the scalar combination
given in (C.5) and $G_{N}$ is the $d$-dimensional Newton constant,

$G_{N}=\frac{\hat{G}_{N}}{2\pi R_{y}}=\frac{\hat{G}_{N}}{2\pi
k_{\infty}\ell_{s}}\,,$ (3.5)

being $\hat{G}_{N}$ the $(d+1)$-dimensional one.

The expression for the mass $E$ can be obtained by applying the ADM formula.
In practice, we can just identify $E$ by looking at the asymptotic behavior of
the $tt$ component of the metric in the modified Einstein frame
Maldacena:1996ky ,

$g_{E}{}_{tt}=e^{-\frac{4}{d-2}(\phi-\phi_{\infty})}g_{tt}\approx
1-\frac{16\pi G_{N}E}{(d-2)\omega_{(d-2)}\rho^{d-3}}+\dots\,,$ (3.6)

where $\omega_{(d-2)}$ is the volume of the unit ${\mathbb{S}}^{(d-2)}$
sphere.

Black hole entropy. In higher-derivative theories the entropy follows from
Wald’s formula Wald:1993nt ; Iyer:1994ys . However, one of the key assumptions
in its derivation does not hold in presence of gravitational Chern-Simons
terms, such as the ones present in the heterotic theory.777Namely, that the
transformation of the $d$-form Lagrangian $\mathbf{L}$ under diffemorphisms is
$\delta_{\xi}{\mathbf{L}}={\cal L}_{\xi}{\mathbf{L}}$, being ${\cal L}_{\xi}$
the Lie derivative with respect to $\xi$. This property is however satisfied
by mixed Chern-Simons terms of the form $A_{(p)}\wedge{\rm{tr}}\left(R\wedge
R\right)$, where $A_{(p)}$ is some $p$-form potential. In this case one can
show Cassani:2023vsa  that Wald’s formula is still applicable if suitable
regularity conditions are imposed on $A_{(p)}$. As a consequence, different
strategies have been proposed in the literature in order to circumvent this
issue (see e.g. Sahoo:2006vz ; Sahoo:2006pm ; Faedo:2019xii ; Elgood:2020xwu ;
Ortin:2020xdm  for a limited list of references), which mainly involve a
convenient rewriting of the action. Nevertheless, it is also possible to
extend Wald’s formalism to properly account for gravitational Chern-Simons
terms. Doing so, general expressions for the black hole entropy were obtained
in Tachikawa:2006sz  and more recently in Elgood:2020nls . It is more
convenient for us to make use of the entropy formula given in Elgood:2020nls ,
as it has been derived precisely in the context of the heterotic effective
action. We report it here for completeness,

$S=\frac{(-1)^{d+1}{\hat{g}}_{s}^{2}}{8\hat{G}_{N}}\int_{\mathcal{BH}}e^{-2\hat{\phi}}\left\\{\left[\hat{\star}({\hat{e}}^{\hat{a}}\wedge{\hat{e}}^{\hat{b}})+\frac{\alpha^{\prime}}{2}\star\hat{R}_{(-)}{}^{\hat{a}\hat{b}}\right]{\hat{n}}_{\hat{a}\hat{b}}+(-1)^{d}\frac{\alpha^{\prime}}{2}\Pi_{n}\wedge\hat{\star}{\hat{H}}\right\\}\,,$
(3.7)

where $\mathcal{BH}$ stands for the bifurcation surface of the event horizon
and $\hat{R}_{(-)}{}^{\hat{a}\hat{b}}$ is the curvature two-form defined in
eq. (A.2). $\Pi_{n}$ is the vertical Lorentz momentum map associated to the
binormal to the Killing horizon, $\hat{n}^{\hat{a}\hat{b}}$, and it is defined
by the property

$d\Pi_{n}\stackrel{{\scriptstyle\mathcal{BH}}}{{=}}{\hat{R}}_{(-)}{}^{\hat{a}\hat{b}}{\hat{n}}_{\hat{a}\hat{b}}\,.$
(3.8)

The formula for the entropy is gauge-invariant and frame independent.
Performing a local Lorentz transformation we can always put the Vielbein
components $\hat{e}_{\hat{\mu}}{}^{\hat{a}}$ in an upper triangular form. In
such a frame and with our ansatz $\Pi_{n}$ has the explicit expression

$\Pi_{n}\stackrel{{\scriptstyle\mathcal{BH}}}{{=}}{\hat{\Omega}}_{(-)}{}^{\hat{a}\hat{b}}{\hat{n}}_{\hat{a}\hat{b}}\,,$
(3.9)

and (3.7) can be easily evaluated.

Having explained how to compute the different thermodynamic quantities, we now
apply the above formuale to find the corrected thermodynamics of two-charge
black holes. In order to facilitate the comparison with the previous
literature Giveon:2009da ; Chen:2021dsw  (which is something that we will do
later in section 5), we introduce the notation which is used in the
aforementioned references:

$q_{i}=\rho_{s}^{d-3}\sinh^{2}\gamma_{i}\,,\hskip
28.45274pti=\left\\{p,w\right\\}\,.$ (3.10)

In addition, we will write down the different expressions both in the micro-
and grand-canonical ensembles. By definition, the first is the one in which
the expressions for the mass $E$ and charges, $Q_{p}$ and $Q_{w}$, take the
same form as in the two-derivative solution (the solution in section 2.2 is
given using this parametrization). In turn, what is fixed in the latter
ensemble are the inverse temperature $\beta$ and the chemical potentials,
$\Phi_{p}$ and $\Phi_{w}$.

###  3.1 Thermodynamic quantities in the micro-canonical ensemble

Five-dimensional black holes. By definition, the expressions for the charges
are the same as in the two-derivative solution, namely

$\displaystyle Q_{p}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{p}k_{\infty}\pi}{8G_{N}}\rho_{s}^{2}\sinh\left(2\gamma_{p}\right)\,,$
(3.11) $\displaystyle Q_{w}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{w}\pi}{8G_{N}k_{\infty}}\rho_{s}^{2}\sinh\left(2\gamma_{w}\right)\,,$
(3.12) $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\pi\rho_{s}^{2}}{8G_{N}}\left[1+\cosh\left(2\gamma_{p}\right)+\cosh\left(2\gamma_{w}\right)\right]\,.$
(3.13)

Contrarily to the charges, the inverse temperature (3.1) and the chemical
potentials (3.3) receive $\alpha^{\prime}$ corrections. Parametrizing them as
follows,

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle
2\pi\cosh\gamma_{p}\cosh\gamma_{w}\rho_{s}\left(1+\frac{\alpha^{\prime}\Delta\beta}{\rho_{s}^{2}}\,\right)\,,$
(3.14) $\displaystyle\Phi_{p}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{p}\tanh\gamma_{p}}{k_{\infty}}\left(1+\frac{\alpha^{\prime}\Delta\Phi_{p}}{\rho_{s}^{2}}\right)\,,$
(3.15) $\displaystyle\Phi_{w}$ $\displaystyle=$
$\displaystyle\epsilon_{w}k_{\infty}\tanh\gamma_{w}\left(1+\frac{\alpha^{\prime}\Delta\Phi_{w}}{\rho_{s}^{2}}\right)\,,$
(3.16)

we get

$\displaystyle\Delta\beta$ $\displaystyle=\,$
$\displaystyle-\frac{\epsilon_{p}\epsilon_{w}}{2}\tanh\gamma_{p}\tanh\gamma_{w}-\,\frac{9\left(4\sinh^{2}\gamma_{p}\sinh^{2}\gamma_{w}-1\right)}{8\left(4\cosh^{2}\gamma_{p}\cosh^{2}\gamma_{w}-1\right)}\,,$
(3.17) $\displaystyle\Delta\Phi_{p}$ $\displaystyle=\,$
$\displaystyle-\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{w}}{\sinh\left(2\gamma_{p}\right)}-\frac{9\cosh\left(2\gamma_{w}\right)}{4\left(4\cosh^{2}\gamma_{p}\cosh^{2}\gamma_{w}-1\right)}\,,$
(3.18) $\displaystyle\Delta\Phi_{w}$ $\displaystyle=\,$
$\displaystyle-\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}}{\sinh\left(2\gamma_{w}\right)}-\frac{9\cosh\left(2\gamma_{p}\right)}{4\left(4\cosh^{2}\gamma_{p}\cosh^{2}\gamma_{w}-1\right)}\,.$
(3.19)

Finally, the result that we obtain for the black hole entropy is

$S=\frac{\pi^{2}\rho_{s}^{3}\cosh\gamma_{p}\cosh\gamma_{w}}{2G_{N}}\left[1+\frac{\alpha^{\prime}}{8\rho_{s}^{2}}\left(9+4\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}\right)\right]\,.$
(3.20)

These expressions pass several consistency checks. First, one can verify that
the first law of black hole mechanics,

$\mathrm{d}E=\beta^{-1}\mathrm{d}S+\Phi_{p}\,\mathrm{d}Q_{p}+\Phi_{w}\,\mathrm{d}Q_{w}\,,$
(3.21)

is obeyed. Second, the corrections agree with those of Cano:2022tmn , where
three-charge black holes were considered, in the limit in which the third
charge, associated to the presence of NS5 branes, goes to zero. Finally, the
expressions are consistent with T-duality, which exchanges
$\gamma_{p}\leftrightarrow\gamma_{w}$ and sends $k_{\infty}\to 1/k_{\infty}$.
One can see that the mass, entropy and temperature are left invariant, whereas
the chemical potentials and charges are interchanged, as expected.

Four-dimensional black holes. The expressions for the charges and mass read,

$\displaystyle Q_{p}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{p}k_{\infty}}{8G_{N}}\rho_{s}\sinh\left(2\gamma_{p}\right)\,,$
(3.22) $\displaystyle Q_{w}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{w}k_{\infty}^{-1}}{8G_{N}}\rho_{s}\sinh\left(2\gamma_{w}\right)\,,$
(3.23) $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{\rho_{s}}{8G_{N}}\left[2+\cosh\left(2\gamma_{p}\right)+\cosh\left(2\gamma_{w}\right)\right]\,.$
(3.24)

In turn, the inverse temperature and the chemical potential receive the
following $\alpha^{\prime}$ corrections,

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle
4\pi\cosh\gamma_{p}\cosh\gamma_{w}\rho_{s}\left(1+\frac{\alpha^{\prime}\Delta\beta}{\rho_{s}^{2}}\right)\,,$
(3.25) $\displaystyle\Phi_{p}$ $\displaystyle=$
$\displaystyle\epsilon_{p}k_{\infty}^{-1}\tanh\gamma_{p}\left(1+\frac{\alpha^{\prime}\Delta\Phi_{p}}{\rho_{s}^{2}}\right)\,,$
(3.26) $\displaystyle\Phi_{w}$ $\displaystyle=$
$\displaystyle\epsilon_{w}k_{\infty}\tanh\gamma_{w}\left(1+\frac{\alpha^{\prime}\Delta\Phi_{w}}{\rho_{s}^{2}}\right)\,,$
(3.27)

where

$\displaystyle\Delta\beta$ $\displaystyle=\,$
$\displaystyle\frac{\cosh(2\gamma_{p})\left[1-2\cosh(2\gamma_{w})\right]+\cosh(2\gamma_{w})}{2\cosh(2\gamma_{p})\left[1+2\cosh(2\gamma_{w})\right]+2\cosh(2\gamma_{w})}-\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}}{8}\,,$
(3.28) $\displaystyle\Delta\Phi_{p}$ $\displaystyle=\,$
$\displaystyle-\frac{2\cosh(2\gamma_{w})}{\cosh(2\gamma_{p})\left[1+2\cosh(2\gamma_{w})\right]+\cosh(2\gamma_{w})}-\frac{\epsilon_{p}\epsilon_{w}\tanh(\gamma_{w})}{4\sinh(2\gamma_{p})}\,,$
(3.29) $\displaystyle\Delta\Phi_{w}$ $\displaystyle=\,$
$\displaystyle-\frac{2\cosh(2\gamma_{p})}{\cosh(2\gamma_{p})\left[1+2\cosh(2\gamma_{w})\right]+2\cosh(2\gamma_{w})}-\frac{\epsilon_{p}\epsilon_{w}\tanh(\gamma_{p})}{4\sinh(2\gamma_{w})}\,.$
(3.30)

Finally, the expression for the entropy is

$S=\frac{\pi\rho_{s}^{2}\cosh\gamma_{p}\cosh\gamma_{w}}{G_{N}}\left[1+\frac{\alpha^{\prime}}{2\rho_{s}^{2}}\left(1+\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}}{4}\right)\right]\,.$
(3.31)

These corrections agree with those of Zatti:2023oiq , where the corrections to
a family of four-charge black holes have been computed. As in the five-
dimensional case, the thermodynamic quantities we have obtained transform as
expected under T-duality and obey the first law of black hole mechanics
(3.21).

###  3.2 Thermodynamic quantities in the grand-canonical ensemble

In order to obtain the thermodynamics in the grand-canonical ensemble, we must
consider a different choice of boundary conditions. This can be simply
implemented by considering a different parametrization of the solution,

$\rho_{s}\to\rho_{s}+\alpha^{\prime}\delta\rho_{s}\left(\rho_{s},\gamma_{i}\right)\,,\hskip
28.45274pt\gamma_{i}\to\gamma_{i}+\frac{\alpha^{\prime}\delta\gamma_{i}\left(\rho_{s},\gamma_{j}\right)}{\rho_{s}}\,,$
(3.32)

and fixing $\delta\rho_{s}$ and $\delta\gamma_{i}$ by imposing the vanishing
of the corrections to $\beta$ and the chemical potentials $\Phi_{i}$. The
resulting expressions for the thermodynamic quantities associated to the five-
and four-dimensional solutions are given below.

Five-dimensional black holes:

$\displaystyle\beta$ $\displaystyle=\,$ $\displaystyle
2\pi\rho_{s}\cosh\gamma_{p}\,\cosh\gamma_{w}\,,\hskip
14.22636pt\Phi_{p}=\,\frac{\epsilon_{p}\tanh\gamma_{p}}{k_{\infty}}\,,\hskip
14.22636pt\Phi_{w}=\,\epsilon_{w}k_{\infty}\tanh\gamma_{w}\,,$ (3.33)
$\displaystyle Q_{p}$ $\displaystyle=\,$
$\displaystyle\frac{\epsilon_{p}k_{\infty}\pi}{8G_{N}}\left[\rho_{s}^{2}\sinh\left(2\gamma_{p}\right)+\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\gamma_{w}\right]\,,$
(3.34) $\displaystyle Q_{w}$ $\displaystyle=\,$
$\displaystyle\frac{\epsilon_{w}\pi}{8G_{N}k_{\infty}}\left[\rho_{s}^{2}\sinh\left(2\gamma_{w}\right)+\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\right]\,,$
(3.35) $\displaystyle E$ $\displaystyle=\,$
$\displaystyle\frac{\pi\rho_{s}^{2}}{8G_{N}}\left[1+\cosh\left(2\gamma_{p}\right)+\cosh\left(2\gamma_{w}\right)+\frac{\alpha^{\prime}}{4\rho_{s}^{2}}\left(-9+4\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}\right)\right]\,,$
(3.36) $\displaystyle S$ $\displaystyle=\,$
$\displaystyle\frac{\pi^{2}\rho_{s}^{3}\cosh\gamma_{p}\cosh\gamma_{w}}{2G_{N}}\,.$
(3.37)

Four-dimensional black holes:

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle
4\pi\rho_{s}\cosh\gamma_{p}\cosh\gamma_{w}\,,\hskip
14.22636pt\Phi_{p}=\epsilon_{p}k_{\infty}^{-1}\tanh\gamma_{p}\,,\hskip
14.22636pt\Phi_{w}=\epsilon_{w}k_{\infty}\tanh\gamma_{w}\,,$ (3.38)
$\displaystyle Q_{p}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{p}k_{\infty}\rho_{s}\sinh\left(2\gamma_{p}\right)}{8G_{N}}\left[1+\frac{\alpha^{\prime}}{2\rho_{s}^{2}}\left(1+\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{w}}{4\tanh\gamma_{p}}\right)\right]\,,$
(3.39) $\displaystyle Q_{w}$ $\displaystyle=$
$\displaystyle\frac{\epsilon_{w}k_{\infty}^{-1}\rho_{s}\sinh\left(2\gamma_{w}\right)}{8G_{N}}\left[1+\frac{\alpha^{\prime}}{2\rho_{s}^{2}}\left(1+\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}}{4\tanh\gamma_{w}}\right)\right]\,,$
(3.40) $\displaystyle E\;$ $\displaystyle=$
$\displaystyle\;\;\frac{\rho_{s}\left(\cosh\left(2\gamma_{p}\right)+\cosh\left(2\gamma_{w}\right)+2\right)}{8G_{N}}\left[1+\frac{\alpha^{\prime}}{2\rho_{s}^{2}}\left(\frac{\cosh(2\gamma_{p})+\cosh(2\gamma_{w})-2}{\cosh(2\gamma_{p})+\cosh(2\gamma_{w})+2}\right.\right.$
(3.41)
$\displaystyle\left.\left.+\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}}{4}\right)\right]\,,$
$\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\pi\rho_{s}^{2}\cosh\gamma_{p}\cosh\gamma_{w}}{G_{N}}\left[1+\frac{\alpha^{\prime}}{2\rho_{s}^{2}}\left(1+\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}}{4}\right)\right]\,.$
(3.42)

##  4 Thermodynamics from the Euclidean on-shell action

In the saddle-point approximation the Euclidean on-shell action of the black
hole gives the dominant contribution to the grand-canonical partition function
Gibbons:1976ue . This leads to the so-called quantum statisical relation,

$I_{\infty}=\beta\,{\cal
G}=\beta\left(E-\Phi_{p}\,Q_{p}-\Phi_{w}\,Q_{w}\right)-S\,,$ (4.1)

where $I_{\infty}$ is the renormalized Euclidean on-shell action and ${\cal
G}$ is the grand-canonical potential, which is regarded as a function of the
(inverse) temperature and the chemical potentials. Knowing ${\cal G}={\cal
G}\left(\beta,\Phi_{p},\Phi_{w}\right)$ suffices to extract all the
thermodynamic quantities since the mass, charges and entropy can be obtained
(assuming the first law of black hole mechanics) as follows:

$Q_{p}=-\frac{\partial{\cal G}}{\partial{\Phi}_{p}}\,,\hskip
14.22636ptQ_{w}=-\frac{\partial{\cal G}}{\partial{\Phi}_{w}}\,,\hskip
14.22636ptS=-\frac{\partial{\cal G}}{\partial{\beta^{-1}}}\,,\hskip
14.22636ptE={\cal G}+\Phi_{p}\,Q_{p}+\Phi_{w}\,Q_{w}+\beta^{-1}S\,.$ (4.2)

As shown e.g. in Reall:2019sah ; Bobev:2022bjm ; Cassani:2022lrk , this method
to obtain the thermodynamics is particularly useful when dealing with higher-
derivative corrections.

The purpose of this section is to evaluate the Euclidean on-shell action of
the two-charge black holes at first order in $\alpha^{\prime}$ and check that
the thermodynamics that we get match the ones obtained in the previous
section. For simplicity, we are going to evaluate the $(d+1)$-dimensional
Euclidean effective action in the string frame, since its dimensional
reduction on ${\mathbb{S}}^{1}_{y}$ gives rise to much more terms
Baron:2017dvb ; Elgood:2020xwu ; Ortin:2020xdm ; Eloy:2020dko ; Liu:2023fqq .
Instead, the $(d+1)$-dimensional action coincides with the ten-dimensional one
(A.1) up to an overall factor which is absorbed in the $(d+1)$-dimensional
Newton constant ${\hat{G}}_{N}$.

Then, the heterotic Euclidean on-shell action $I$ for a manifold $M$ with
boundary $\partial M$ is given by

$I=-\frac{{\hat{g}}_{s}^{2}}{16\pi{\hat{G}}_{N}}\int_{M}\mathrm{d}^{d+1}x\sqrt{|\hat{g}|}\,{\cal
L}_{\rm{eff}}+\frac{{\hat{g}}_{s}^{2}}{8\pi{\hat{G}}_{N}}\int_{\partial
M}\mathrm{d}^{d}x\sqrt{|\hat{h}|}\,e^{-2\hat{\phi}}{\hat{K}}+\dots\,,$ (4.3)

where

${\cal
L}_{\rm{eff}}=e^{-2\hat{\phi}}\left[\hat{R}-4\,\partial^{\hat{\mu}}{\hat{\phi}}\,\partial_{\hat{\mu}}\hat{\phi}+\frac{1}{2\cdot
3!}{\hat{H}}^{2}+\frac{\alpha^{\prime}}{8}{\hat{R}}_{(-)}{}_{\hat{\mu}\hat{\nu}\hat{a}\hat{b}}{\hat{R}}_{(-)}{}^{\hat{\mu}\hat{\nu}\hat{a}\hat{b}}\right]\,$
(4.4)

is the effective Lagrangian of the heterotic superstring at first order in
$\alpha^{\prime}$ (see appendix A). The second term in (4.3) is the standard
Gibbons-Hawking-York (GHY) term written in the string frame:
${\hat{h}}_{\mu\nu}$ represents the metric induced at $\partial M$ and
$\hat{K}$ is the trace of the extrinsic curvature. Finally, the dots indicate
additional boundary terms associated to the higher-derivative corrections,
which on general grounds are expected to give a vanishing contribution for
asymptotically-flat solutions, Reall:2019sah  (hence, we shall ignore them
from now on). As observed in Tseytlin:1988tv ; Chen:2021dsw , the bulk
contribution reduces to a boundary term after using the equation of motion of
the dilaton (A.9), which implies

${\cal L}_{\rm{eff}}=-2{\hat{\nabla}}^{2}e^{-2\hat{\phi}}\,.$ (4.5)

Therefore, we have that (4.3) reduces to:

$I=\frac{{\hat{g}}^{2}_{s}}{8\pi{\hat{G}}_{N}}\int_{\partial
M}\mathrm{d}^{d}x\sqrt{|\hat{h}|}\,e^{-2\hat{\phi}}\left({\hat{K}}-2\,n^{\hat{\mu}}\,\partial_{\hat{\mu}}\hat{\phi}\right)\,,$
(4.6)

where $n^{\hat{\mu}}$ is the unit normal to the boundary.

Here we are interested in asymptotically-flat black holes whose boundary
${\partial M}$ has the topology of
${\mathbb{S}}^{1}_{\beta}\times{\mathbb{S}}^{d-2}$
($\times{\mathbb{S}}^{1}_{y}$). As it is well known, the GHY term diverges in
the limit in which the radius of the ${\mathbb{S}}^{d-2}$ goes to infinity,
just as in flat spacetime. In order to obtain a finite on-shell action, we
follow the prescription of Gibbons:1976ue . This amounts to first consider a
regulated spacetime $M_{R}$, where $R$ is a radial cutoff. The regulated
spacetime then corresponds to the region $\rho\leq R$, and its boundary
$\partial M_{R}$ is the hypersurface $\rho=R$. Second, we introduce an
auxiliary configuration with flat metric ${\hat{\delta}}_{R}$ and constant
dilaton ${\hat{\phi}}_{R}$ chosen so that the induced fields (metric and
dilaton) at $\rho=R$ coincide with the induced metric and dilaton of the black
hole solution, namely ${\hat{\delta}}_{R}|_{\rho=R}={\hat{g}}|_{\rho=R}$ and
$\hat{\phi}_{R}=\hat{\phi}|_{\rho=R}$. Once we have $\hat{\delta}_{R}$ and
$\hat{\phi}_{R}$, we substract the regulated action associated with the flat
spacetime $I_{R}[\hat{\delta}_{R},\hat{\phi}_{R}]$ to the one associated with
the black hole $I_{R}[\hat{g},\hat{\phi}]$ and only then take the $R\to\infty$
limit. Summarizing, the renormalized action $I_{\infty}$ is given by

$I_{\infty}=\lim_{R\to\infty}\left(I_{R}[\hat{g},\hat{\phi}]-I_{R}[\hat{\delta}_{R},\hat{\phi}_{R}]\right)\,,$
(4.7)

and, making use of (4.6), we get

$I_{\infty}=\lim_{R\to\infty}\left\\{\frac{{\hat{g}}_{s}^{2}}{8\pi{\hat{G}}_{N}}\int_{\rho=R}\mathrm{d}^{d}x\sqrt{|\hat{h}|}\,e^{-2\hat{\phi}}\left[\left({\hat{K}}-{\hat{K}}_{{\hat{\delta}}_{R}}\right)-2\,n^{\hat{\mu}}\,\partial_{\hat{\mu}}\hat{\phi}\right]\right\\}\,,$
(4.8)

where ${\hat{K}}_{{\hat{\delta}}_{R}}$ is the trace of the extrinsic curvature
associated to the metric ${\hat{\delta}}_{R}$. For the two-charge black holes
we are interested in, the auxiliary flat solution
$\\{{\hat{\delta}}_{R},\hat{\phi}_{R}\\}$ is given by

$\displaystyle-\hat{\delta}_{R}=$
$\displaystyle\frac{f(R)}{{f}_{p}(R){\tilde{f}}_{w}(R)}\mathrm{d}\tau^{2}+{g}(R)\left(\mathrm{d}\rho^{2}+\rho^{2}\mathrm{d}\Omega^{2}_{(d-2)}\right)+k^{2}_{R}\left[\mathrm{d}y+\beta_{p}k^{-1}_{\infty}\left({f}_{p}(R)^{-1}-1\right)\mathrm{d}t\right]^{2}\,,$
(4.9) $\displaystyle\hat{\phi}_{R}=$ $\displaystyle\hat{\phi}(R)\,,$

where $k_{R}^{2}=k^{2}_{\infty}\frac{{f}_{p}(R)}{{f}_{w}(R)}$. Now we have all
the ingredients to evaluate (4.8) using the corrected solutions found in the
previous section. Let us do this for the five- and four-dimensional solutions
separately.

Five-dimensional black holes. Expressing the result in the grand-canonical
ensemble, we get that the Euclidean on-shell action of the five-dimensional
two-charge black holes is given by

$I_{\infty}=\frac{\pi^{2}{\rho}_{s}^{3}\cosh\gamma_{p}\cosh\gamma_{w}}{4G_{N}}\left[1-\frac{9\alpha^{\prime}}{4{\rho}_{s}^{2}}-\frac{\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}}{{\rho}^{2}_{s}}\right]\,,$
(4.10)

and we recall that

$\beta=2\pi\rho_{s}\cosh\gamma_{p}\,\cosh\gamma_{w}\,,\hskip
14.22636pt\Phi_{p}=\,\frac{\epsilon_{p}\tanh\gamma_{p}}{k_{\infty}}\,,\hskip
14.22636pt\Phi_{w}=\,\epsilon_{w}k_{\infty}\tanh\gamma_{w}\,.$ (4.11)

It is a straightforward calculation to show that the corrected charges that
follow from the on-shell action (using (4.2)) are in perfect agreement with
the ones we computed in the previous section, namely with eqs. (3.34), (3.35),
(3.36) and (3.37).

Four-dimensional black holes. In the four-dimensional case the on-shell action
in the grand-canonical ensemble takes the form

$I_{\infty}=\frac{\pi\rho_{s}^{2}\cosh\gamma_{p}\cosh\gamma_{w}}{G_{N}}\left[1-\frac{\alpha^{\prime}}{2\rho_{s}^{2}}\left(1+\frac{\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}}{4}\right)\right]\,,$
(4.12)

with the inverse temperature and the chemical potentials given by

$\beta=4\pi\rho_{s}\cosh\gamma_{p}\,\cosh\gamma_{w}\,,\hskip
14.22636pt\Phi_{p}=\,\frac{\epsilon_{p}\tanh\gamma_{p}}{k_{\infty}}\,,\hskip
14.22636pt\Phi_{w}=\,\epsilon_{w}\,k_{\infty}\tanh\gamma_{w}\,.$ (4.13)

As before, the charges (3.39), (3.40), (3.41) and (3.42) are properly
recovered from (4.2).

##  5 Two-charge black holes from Schwarzshchild-Tangherlini

As already mentioned, the corrections to the thermodynamics of two-charge
black holes have been previously studied in Giveon:2009da ; Chen:2021dsw . The
strategy of these references is to find the $\alpha^{\prime}$ corrections by
performing a set of $O(2,2)$ transformations (boost with parameter
$\delta_{w}$ plus T-duality along $y$, followed by another boost with
parameter $\delta_{p}$) to the Schwarzschild-Tangherlini black hole, whose
$\alpha^{\prime}$ corrections had been already studied in Callan:1988hs . The
main difference between these two references is that Chen:2021dsw  just
focuses on the thermodynamic properties while in Giveon:2009da  the full
corrected solutions are obtained by means of this technique. This is
technically more complicated than just obtaining the thermodynamics, as one
has to take into account the explicit $\alpha^{\prime}$ corrections to the
$O(2,2)$ transformations. This might be the reason why the
$\alpha^{\prime}$-corrected thermodynamics obtained in these references do not
agree with one another.

The goal of this section is to show that our results for the
$\alpha^{\prime}$-corrected thermodynamics of heterotic two-charge black holes
are in agreement with those of Chen:2021dsw . To this aim, we find convenient
to review here their calculation. A key observation is that the Euclidean on-
shell action remains invariant after the $O(2,2)$ transformation. Therefore,

$I_{\infty}(\beta,\Phi_{p},\Phi_{w};\phi_{\infty},k_{\infty})={\tilde{I}}_{\infty}(\tilde{\beta};{\tilde{\phi}}_{\infty},{\tilde{k}}_{\infty})\,,$
(5.1)

where, following the conventions of Chen:2021dsw , we are using tildes for the
quantities associated to the Schwarzschild-Tangherlini solution.

The right-hand side of (5.1) is obtained from the $\alpha^{\prime}$
corrections to the Schwarzschild-Tangherlini solution Callan:1988hs . Focusing
just on the thermodynamic quantities, we have

${\tilde{E}}=\frac{d-2}{d-3}\frac{\gamma_{d}\,{{\tilde{R}}_{\beta}}^{d-3}}{8\pi{\tilde{G}}_{N}}\left(1-\frac{\epsilon_{d}\,\alpha^{\prime}}{4{\tilde{R}}_{\beta}^{2}}\right)\,,\hskip
14.22636pt{\tilde{S}}=\frac{\gamma_{d}\,{\tilde{R}}_{\beta}^{d-2}}{4{\tilde{G}}_{N}}\left(1-\frac{\sigma_{d}\,\alpha^{\prime}}{4{\tilde{R}}_{\beta}^{2}}\right)\,,$
(5.2)

where ${\tilde{R}}_{\beta}\equiv{\tilde{\beta}}/(2\pi)$ is the radius of the
thermal circle ${\mathbb{S}}^{1}_{\beta}$ and

$\gamma_{d}=\omega_{d-2}\left(\frac{d-3}{2}\right)^{d-2}\,,\hskip
14.22636pt\epsilon_{d}=\frac{2(d-4)(d-2)}{d-3}\,,\hskip
14.22636pt\sigma_{d}=\frac{2(d-5)(d-2)^{2}}{(d-3)^{2}}\,.$ (5.3)

Assuming the quantum statistical relation (4.1), we get that the Euclidean on-
shell action of the Schwarzschild-Tangherlini black hole is

${\tilde{I}}_{\infty}={\tilde{\beta}}{\tilde{E}}-{\tilde{S}}=\frac{\gamma_{d}\,{\tilde{R}}_{\beta}^{d-2}}{4{\tilde{G}}_{N}(d-3)}\left[1-\frac{(d-2)^{2}\alpha^{\prime}}{2(d-3){\tilde{R}}_{\beta}^{2}}\right]\,.$
(5.4)

Because of (5.1), the right-hand side of (5.4) computes the Euclidean on-shell
action of the two-charge black holes as well. This is however meaningless at
this stage, since we have not yet specified the expressions for $\beta$ and
the chemical potentials $\Phi_{p}$, $\Phi_{w}$ in terms of
${\tilde{R}}_{\beta}$ and the parameters of the $O(2,2)$ transformations. Such
expressions can be recovered from Chen:2021dsw . Taking into account all the
possibilities for the signs of the winding and momentum charges, we find

$\displaystyle R_{\beta}$ $\displaystyle=\,$
$\displaystyle{\tilde{R}}_{\beta}\cosh\delta_{p}\,\cosh\delta_{w}\left(1-\frac{\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\delta_{p}\tanh\delta_{w}}{2{\tilde{R}}_{\beta}^{2}}\right)\,,$
(5.5) $\displaystyle\Phi_{p}$ $\displaystyle=\,$
$\displaystyle\frac{\epsilon_{p}\tanh\delta_{p}}{k_{\infty}}\left(1-\frac{\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\delta_{w}}{{\tilde{R}}_{\beta}^{2}\sinh\left(2\delta_{p}\right)}\right)\,,$
(5.6) $\displaystyle\Phi_{w}$ $\displaystyle=\,$
$\displaystyle\epsilon_{w}k_{\infty}\tanh\delta_{w}\left(1-\frac{\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\delta_{p}}{{\tilde{R}}_{\beta}^{2}\sinh\left(2\delta_{w}\right)}\right)\,,$
(5.7)

where $\delta_{p,w}$ represent the parameters of the $O(2,2)$ transformations.
In addition to this, one must also bear in mind the relation between the
moduli of the solutions. In particular, we need the relation between the
asymptotic values of the $d$-dimensional dilaton $e^{\phi_{\infty}}=g_{s}$,
which is the following Chen:2021dsw

$g_{s}^{2}=\,{\tilde{g}}_{s}^{2}\cosh\delta_{p}\,\cosh\delta_{w}\left(1-\frac{\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\delta_{p}\tanh\delta_{w}}{2{\tilde{R}}_{\beta}^{2}}\right)\,.$
(5.8)

Taking into account that $G_{N}\propto g_{s}^{2}$, one gets that the Newton
constants are related by

$G_{N}={\tilde{G}}_{N}\cosh\delta_{p}\,\cosh\delta_{w}\left(1-\frac{\alpha^{\prime}\epsilon_{p}\epsilon_{w}\tanh\delta_{p}\tanh\delta_{w}}{2{\tilde{R}}_{\beta}^{2}}\right)\,.$
(5.9)

Using this in (5.4), we obtain

$I_{\infty}={\tilde{\beta}}{\tilde{E}}-{\tilde{S}}=\frac{\gamma_{d}\,{\tilde{R}}_{\beta}^{d-2}\cosh\delta_{p}\cosh\delta_{w}}{4{G}_{N}(d-3)}\left[1-\frac{\alpha^{\prime}}{2{\tilde{R}}_{\beta}^{2}}\left(\frac{(d-2)^{2}}{(d-3)}+\epsilon_{p}\epsilon_{w}\tanh\delta_{p}\tanh\delta_{w}\right)\right]\,.$
(5.10)

This already specifies the thermodynamics. However, the parametrization we are
using here differs from the one(s) used in the previous sections. It is not
difficult to find that the relation between
${\tilde{R}}_{\beta},\delta_{p},\delta_{w}$ and the parameters
$\rho_{s},\gamma_{p},\gamma_{w}$ used in the previous sections to express the
thermodynamics in the grand-canonical ensemble is given by

$\displaystyle{\tilde{R}}_{\beta}$ $\displaystyle=$
$\displaystyle\frac{2\rho_{s}}{d-3}\left(1-\frac{\epsilon_{p}\epsilon_{w}(d-3)^{2}\alpha^{\prime}\tanh\gamma_{p}\tanh\gamma_{w}}{8\rho_{s}^{2}}\right)\,,$
(5.11) $\displaystyle\delta_{p}$ $\displaystyle=$
$\displaystyle\gamma_{p}+\frac{\epsilon_{p}\epsilon_{w}(d-3)^{2}\alpha^{\prime}\tanh\gamma_{w}}{8\rho_{s}^{2}}\,,$
(5.12) $\displaystyle\delta_{w}$ $\displaystyle=$
$\displaystyle\gamma_{w}+\frac{\epsilon_{p}\epsilon_{w}(d-3)^{2}\alpha^{\prime}\tanh\gamma_{p}}{8\rho_{s}^{2}}\,.$
(5.13)

Making use of these relations, we can write the on-shell action of the two-
charge black holes in the grand-canonical ensemble:

$I_{\infty}=\frac{\omega_{d-2}\,\rho_{s}^{d-2}\,\cosh\gamma_{p}\cosh\gamma_{w}}{4(d-3)G_{N}}\left\\{1-\frac{(d-3)\alpha^{\prime}}{8\rho_{s}^{2}}\left[(d-2)^{2}+\epsilon_{p}\epsilon_{w}(d-3)^{2}\tanh\gamma_{p}\tanh\gamma_{w}\right]\right\\}\,.$
(5.14)

This properly reduces to (4.10) and to (4.12) when setting $d=5$ and $d=4$,
respectively. Given the grand-canonical potential ${{\cal
G}=\beta^{-1}I_{\infty}}$, we can obtain the charges, entropy and mass through
(4.2), as already discussed. Expressing them in the grand-canonical ensemble,
we obtain the following expressions

$\displaystyle\beta$ $\displaystyle=\,$
$\displaystyle\frac{4\pi\rho_{s}}{d-3}\cosh\gamma_{p}\,\cosh\gamma_{w}\,,\hskip
14.22636pt\Phi_{p}=\,\frac{\epsilon_{p}\tanh\gamma_{p}}{k_{\infty}}\,,\hskip
14.22636pt\Phi_{w}=\,{\epsilon}_{w}k_{\infty}\tanh\gamma_{w}\,,$ (5.15)
$\displaystyle Q_{p}$ $\displaystyle=$ $\displaystyle
Q^{(0)}_{p}\left[1-\frac{(d-3)^{2}\alpha^{\prime}}{16\rho_{s}^{2}}\left(\sigma_{d}-2\epsilon_{p}\epsilon_{w}(4-d+\text{coth}^{2}\gamma_{p})\tanh\gamma_{p}\tanh\gamma_{w}\right)\right]\,,$
(5.16) $\displaystyle Q_{w}$ $\displaystyle=$ $\displaystyle
Q^{(0)}_{w}\left[1-\frac{(d-3)^{2}\alpha^{\prime}}{16\rho_{s}^{2}}\left(\sigma_{d}-2\epsilon_{p}\epsilon_{w}(4-d+\text{coth}^{2}\gamma_{w})\tanh\gamma_{p}\tanh\gamma_{w}\right)\right]\,,$
(5.17) $\displaystyle S$ $\displaystyle=$ $\displaystyle
S^{(0)}\left[1-\frac{(d-3)^{2}\alpha^{\prime}}{16\rho_{s}^{2}}\left(\sigma_{d}+2(d-5)\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}\right)\right]\,,$
(5.18)

where

$\displaystyle Q^{(0)}_{p}=$
$\displaystyle\frac{(d-3)\epsilon_{p}k_{\infty}\omega_{d-2}\rho_{s}^{d-3}\sinh(2\gamma_{p})}{32\pi
G_{N}}\,,\hskip
14.22636ptQ^{(0)}_{w}=\,\frac{(d-3)\epsilon_{w}\omega_{d-2}\rho_{s}^{d-3}\sinh(2\gamma_{w})}{32\pi
G_{N}k_{\infty}}\,,$ (5.19) $\displaystyle S^{(0)}=$
$\displaystyle\frac{\gamma_{d-2}\rho_{s}^{d-2}\cosh\gamma_{p}\cosh\gamma_{w}}{32G_{N}}\,.$

Instead of the mass we provide the expression for the grand-canonical
potential ${\cal G}$, which is simpler

${\cal G}=\frac{\omega_{d-2}\,\rho_{s}^{d-3}}{16\pi
G_{N}}\left[1-\frac{(d-3)\alpha^{\prime}}{8\rho_{s}^{2}}\left((d-2)^{2}+(d-3)^{2}\,\epsilon_{p}\epsilon_{w}\tanh\gamma_{p}\tanh\gamma_{w}\right)\right]\,.$
(5.20)

The mass $E$ follows then from the last of (4.2). It is now straightforward to
compare these expressions with the ones we obtained in sections 3 and 4 and
see that they are in perfect agreement. Furthermore, we have also checked that
they agree with the corrected thermodynamics given in the appendix of
Chen:2021dsw , after using the map between the two parametrizations, provided
in (5.5), (5.6) and (5.7).

## Acknowledgements

We are grateful to Pablo A. Cano, Tomás Ortín and Pedro F. Ramírez for many
discussions and collaborations along these years on this and related topics.
We would like to further thank the FISPAC group at University of Murcia and
the group in Hamburg for their feedback while presenting this work and for
hospitality. AR is supported by a postdoctoral fellowship associated to the
MIUR-PRIN contract 2020KR4KN2, “String Theory as a bridge between Gauge
Theories and Quantum Gravity”. M.Z. has been supported by the fellowship
LCF/BQ/DI20/11780035 from “La Caixa” Foundation (ID100010434), by the MCI,
AEI, FEDER (UE) grants PID2021-125700NBC21 (“Gravity, Supergravity and
Superstrings” (GRASS)) and PID2021-123017NB-I00, and by the grant IFT Centro
de Excelencia Severo Ochoa CEX2020-001007-S.

##  Appendix A Effective action and equations of motion

The effective action of the heterotic string at first order in
$\alpha^{\prime}$ in the Bergshoeff-de Roo formulation Bergshoeff:1989de  is
given by,

$\displaystyle
S_{\rm{eff}}=\int\mathrm{d}^{10}x\sqrt{-{\hat{g}}}\,e^{-2\hat{\phi}}\left[\hat{R}-4\,\partial^{\hat{\mu}}{\hat{\phi}}\,\partial_{\hat{\mu}}\hat{\phi}+\frac{1}{2\cdot
3!}{\hat{H}}^{2}+\frac{\alpha^{\prime}}{8}{\hat{R}}_{(-)}{}_{\hat{\mu}\hat{\nu}\hat{a}\hat{b}}{\hat{R}}_{(-)}{}^{\hat{\mu}\hat{\nu}\hat{a}\hat{b}}\right]\,,$
(A.1)

where
${\hat{R}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}=\frac{1}{2}{\hat{R}}_{(-)}{}_{\hat{\mu}\hat{\nu}}{}^{\hat{a}}{}_{\hat{b}}\,dx^{\hat{\mu}}\wedge
dx^{\hat{\nu}}$ is defined as the curvature two-form associated to the
torsionful spin connection,

${\hat{R}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}=\mathrm{d}{\hat{\omega}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}-{\hat{\omega}}_{(-)}{}^{\hat{a}}{}_{\hat{c}}\wedge{\hat{\omega}}_{(-)}{}^{\hat{c}}{}_{\hat{b}}\,,$
(A.2)

where

${\hat{\omega}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}={\hat{\omega}}^{\hat{a}}{}_{\hat{b}}-\frac{1}{2}{\hat{H}}_{c}{}^{\hat{a}}{}_{\hat{b}}\,{\hat{e}}^{\hat{c}}\,.$
(A.3)

In the formula above, ${\hat{\omega}}^{\hat{a}}{}_{\hat{b}}$ represents the
standard Levi-Civita spin connection and the torsion is determined by the
three-form field strength ${\hat{H}}$, which satisfies the modified Bianchi
identity Green:1984sg ,

$\mathrm{d}{\hat{H}}-\frac{\alpha^{\prime}}{4}{\hat{R}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}\wedge{\hat{R}}_{(-)}{}^{\hat{b}}{}_{\hat{a}}=0\,.$
(A.4)

This implies that, locally, $\hat{H}$ is given by

${\hat{H}}=\mathrm{d}{\hat{B}}+\frac{\alpha^{\prime}}{4}{\hat{\Omega}}_{(-)}\,,$
(A.5)

where ${\hat{\Omega}}_{(-)}$ is the Lorentz Chern-Simons three-form, defined
as

${\hat{\Omega}}_{(-)}=\mathrm{d}{\hat{\omega}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}\wedge{\hat{\omega}}_{(-)}{}^{\hat{b}}{}_{\hat{a}}-\frac{2}{3}\,{\hat{\omega}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}\wedge{\hat{\omega}}_{(-)}{}^{\hat{b}}{}_{\hat{c}}\wedge{\hat{\omega}}_{(-)}{}^{\hat{c}}{}_{\hat{a}}\,.$
(A.6)

Let us note that the definition of the Kalb-Ramond two-form $\hat{B}$ in (A.5)
is a recursive one, since $\hat{\Omega}_{(-)}$ depends on $\hat{H}$ through
(A.3). Then, it should be implemented order by order in $\alpha^{\prime}$, as
follows:

${\hat{H}}^{(0)}=\mathrm{d}{\hat{B}}^{(0)}\,,\hskip
14.22636pt{\hat{H}}^{(1)}=\mathrm{d}{\hat{B}}^{(1)}+\frac{\alpha^{\prime}}{4}{\hat{\Omega}}^{(0)}_{(-)}\,,\hskip
14.22636pt...\hskip
14.22636pt{\hat{H}}^{(n)}=\mathrm{d}{\hat{B}}^{(n)}+\frac{\alpha^{\prime}}{4}{\hat{\Omega}}^{(n-1)}_{(-)}\,,$
(A.7)

where ${\hat{\Omega}}^{(n-1)}_{(-)}$ is computed using ${\hat{H}}^{(n-1)}$.
This implies that the term ${\hat{H}}^{2}$ in (A.1) actually contains and
infinite tower of $\alpha^{\prime}$ corrections. By consistency, we only keep
the first-order ones.

###  A.1 Equations of motion

The derivation of the equations of motion drastically simplifies when using a
lemma proven in Bergshoeff:1989de . It states that the variation of the action
with respect to ${\hat{\omega}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}$ produces terms
which are subleading in $\alpha^{\prime}$. Hence, we only need to vary the
terms where the fieds appear explicitly, ignoring implicit ocurrences of the
fields through ${\hat{\omega}}_{(-)}{}^{\hat{a}}{}_{\hat{b}}$. This leads to
the following equations of motion,

$\displaystyle{\hat{R}}_{\hat{\mu}\hat{\nu}}-2{\hat{\nabla}}_{\hat{\mu}}\partial_{\hat{\nu}}{\hat{\phi}}+\frac{1}{4}{\hat{H}}_{\hat{\mu}\hat{\rho}\hat{\sigma}}{\hat{H}}_{\hat{\nu}}{}^{\hat{\rho}\hat{\sigma}}+\frac{\alpha^{\prime}}{4}{\hat{R}}_{(-)}{}_{\hat{\mu}\hat{\rho}\hat{a}\hat{b}}{\hat{R}}_{(-)}{}_{\hat{\nu}}{}^{\hat{\rho}\hat{a}\hat{b}}$
$\displaystyle=$ $\displaystyle{\cal O}{(\alpha^{\prime 2})}\,,$ (A.8)
$\displaystyle
2{\hat{\nabla}}^{2}e^{-2\hat{\phi}}+e^{-2\hat{\phi}}\left[\hat{R}-4\,\partial^{\hat{\mu}}{\hat{\phi}}\,\partial_{\hat{\mu}}\hat{\phi}+\frac{1}{2\cdot
3!}{\hat{H}}^{2}+\frac{\alpha^{\prime}}{8}{\hat{R}}_{(-)}{}_{\hat{\mu}\hat{\nu}\hat{a}\hat{b}}{\hat{R}}_{(-)}{}^{\hat{\mu}\hat{\nu}\hat{a}\hat{b}}\right]$
$\displaystyle=$ $\displaystyle{\mathcal{O}}\left(\alpha^{\prime 2}\right)\,,$
(A.9) $\displaystyle\mathrm{d}\left(e^{-2\hat{\phi}}\hat{\star}\hat{H}\right)$
$\displaystyle=$ $\displaystyle{\mathcal{O}}\left(\alpha^{\prime 2}\right).$
(A.10)

As observed in Tseytlin:1988tv ; Chen:2021dsw , the equation of motion of the
dilaton tells us that

${\cal L}_{\rm{eff}}=-2{\hat{\nabla}}^{2}e^{-2\hat{\phi}}\,,$ (A.11)

where ${\cal L}_{\rm{eff}}$ is the effective Lagrangian. This observation is
really helpful to evaluate the Euclidean on-shell action, as it allows us to
reduce the problem to the evaluation of a boundary term.

##  Appendix B Additional details on the procedure to find the
$\alpha^{\prime}$-corrected solutions

The purpose of this appendix is to explain in more detail the procedure we
have followed to solve the $\alpha^{\prime}$-corrected equations motion (A.8),
(A.9) and (A.10).

### The ansatz

Let us begin motivating the ansatz we have used for the metric
$\hat{g}_{\mu\nu}$ and two-form $\hat{B}$. This is given in (2.7) and (2.8),
which we repeat here for convenience:

$\displaystyle\mathrm{d}{\hat{s}}^{2}$ $\displaystyle=$
$\displaystyle\frac{f}{{f}_{p}{\tilde{f}}_{w}}\mathrm{d}t^{2}-{g}\left(f^{-1}{\mathrm{d}}\rho^{2}+\rho^{2}{\mathrm{d}}\Omega^{2}_{(d-2)}\right)-k^{2}_{\infty}\frac{{f}_{p}}{{\tilde{f}}_{w}}\left[{\mathrm{d}}y+\beta_{p}k^{-1}_{\infty}\left({f}_{p}^{-1}-1\right){\mathrm{d}}t\right]^{2}\,,$
(B.1) $\displaystyle{\hat{B}}$ $\displaystyle=$
$\displaystyle\beta_{w}k_{\infty}\left({f}_{w}^{-1}-1\right){\mathrm{d}}t\wedge{\mathrm{d}}y\,.$
(B.2)

For the dilaton we only assume a dependence on just the radial coordinate
$\rho$.

The “recipe” followed to fix the ansatz is essentially to keep the same field
components active as in the two-derivative solution. Spherical symmetry
reduces the number of independent components of the metric to four:
${\hat{g}}_{tt},{\hat{g}}_{\rho\rho},{\hat{g}}_{yy}$ and ${\hat{g}}_{ty}$.
These are in one-to-one correspondence with the functions
$f,f_{p},{\tilde{f}}_{w}$ and $g$. The reason to choose this particular
parametrization is that we expect the form of these functions will be simpler,
just by experience with the two-derivative ones. The last function to be
considered is $f_{w}$, which is associated to the only non-vanishing component
of the two-form $\hat{B}$. Since we are going to treat the $\alpha^{\prime}$
corrections in a perturbative fashion, the form of these functions must be:

$\displaystyle{f}_{p}=$ $\displaystyle
1+\frac{q_{p}}{\rho^{d-3}}+\alpha^{\prime}{\delta f_{p}}\,,\hskip
14.22636pt{\tilde{f}}_{w}=\,1+\frac{q_{w}}{\rho^{d-3}}+\alpha^{\prime}{\delta{\tilde{f}}_{w}}\,,\hskip
14.22636pt{g}=\,1+\alpha^{\prime}{\delta{g}}\,,$ (B.3) $\displaystyle{f}=$
$\displaystyle\,1-\frac{\rho_{s}^{2}}{\rho^{d-3}}+\alpha^{\prime}{\delta
f}\,,\hskip
14.22636pt{f}_{w}=\,1+\frac{q_{w}}{\rho^{d-3}}+\alpha^{\prime}{\delta{f}_{w}}\,,$

so that the two-derivative solution is properly recovered in the
$\alpha^{\prime}\to 0$ limit.

When plugging the above ansatz in the corrected equations of motion, one gets
a coupled system of second-order differential equations for the unknown
functions. In what follows we describe the procedure that we have followed in
order to solve it, focusing on the five-dimensional case.

### The equation of motion of $\hat{B}$

The expression for the dilaton can be found by solving the equation of motion
of the two-form $\hat{B}$, (A.10). There is just one independent component
which is not trivially satisfied, and it yields the following equation (for
$d=5$):

$\frac{f^{\prime\prime}_{w}}{f^{\prime}_{w}}-\frac{2f^{\prime}_{w}}{f_{w}}+\frac{g^{\prime}}{g}+\frac{{\tilde{f}}^{\prime}_{w}}{{\tilde{f}}^{\prime}_{w}}+\frac{3}{\rho}-2\hat{\phi}^{\prime}=0\,,$
(B.4)

where primes denote derivatives with respect to $\rho$. This equation can be
integrated once to give

$\left[\hat{\phi}-\frac{1}{2}\log\frac{-\rho^{3}g{\tilde{f}}_{w}f^{\prime}_{w}}{f_{w}^{2}}\right]^{\prime}=0\,,$
(B.5)

which is solved by

$\hat{\phi}=a_{\phi}+\frac{1}{2}\log\frac{-\rho^{3}g{\tilde{f}}_{w}f^{\prime}_{w}}{2q_{w}f_{w}^{2}}\,,$
(B.6)

where $a_{\phi}$ is an appropriate integration constant which we are going to
fix imposing that the asymptotic value of the dilaton (string coupling) is not
renormalized. When expressing the solution in the microcanonical ensemble, the
term inside the logarithm goes to $1$ at infinity, which means that $a_{\phi}$
is identified with ${\hat{\phi}}_{\infty}$.

### Einstein and dilaton equations

As explained in the main text, the strategy to solve the corrected Einstein
and dilaton equations is to expand the unknown functions $\Psi=\\{\delta
f,\delta f_{p},\delta f_{w},\delta{\tilde{f}}_{w},\delta g\\}$ in a series in
$1/\rho^{2}$,

$\Psi=\frac{a_{\Psi}}{\rho^{2}}+\sum_{n>1}^{N}\frac{b^{(n)}_{\Psi}}{\rho^{2n}}\,,$
(B.7)

and then solve (A.8) and (A.9) order by order in $1/\rho^{2}$. This leads to a
set of algebraic equations that determine the values of the coefficients
$b_{\Psi}^{(n)}$ in terms of $a_{\Psi}$ and of the parameters of the two-
derivative solution, $\rho_{s},q_{p},q_{w}$. Two out of the five integration
constants $a_{\Psi}$ can be fixed right away. These are $a_{f_{p}}$ and
$a_{f_{w}}$, which are both set to zero by imposing that the charges of the
black hole $Q_{p}$ and $Q_{w}$ do not receive $\alpha^{\prime}$ corrections.
The next step is to find the generating functions that produce the asymptotic
expansions (B.7). This is done with the help of Mathematica.888For this
purpose, one has to compute the solution up to sufficiently high order in
$1/\rho^{2}$.  Finally, we must fix the three remaining integration constants
$a_{g}$, $a_{g}$ and $a_{\tilde{f}_{p}}$. Since we want to express the
solution in the microcanonical ensemble, we must fix one of these constants
(let us say, $a_{g}$) by imposing the mass does not receive $\alpha^{\prime}$
corrections. The resulting solution turns out to be singular at the horizon
for arbitrary values of the two remaining integration constants, $a_{f}$ and
$a_{\tilde{f}_{p}}$.999In particular, the dilaton and the Kaluza-Klein scalar
diverge when $\rho\to\rho_{H}$. Demanding regularity imposes two conditions
which fix both $a_{f}$ and $a_{\tilde{f}_{p}}$, leaving us with the solution
reported in section 2.2.

##  Appendix C Dimensional reduction on a circle

In this appendix we make use of the results of Elgood:2020xwu  in order to
find the dimensional reduction of the solutions to $d$ dimensions. The
$d$-dimensional fields are: the (string-frame) metric $g_{\mu\nu}$, the
dilaton $\phi$, the Kaluza-Klein scalar $k$ and vector $A_{\mu}$, the two-form
$B_{\mu\nu}$ and, finally, the winding vector $C_{\mu}$.101010Note that in
this section $\mu,\nu=0,\dots,d-1$, as opposed to the rest of the paper, where
$\mu,\nu=0,\dots,d$. These are given in terms of the higher-dimensional fields
by the following expressions Elgood:2020xwu :

$\displaystyle g_{\mu\nu}$ $\displaystyle=$
$\displaystyle{\hat{g}}_{\mu\nu}-\frac{{\hat{g}}_{\mu y}{\hat{g}}_{\nu
y}}{{\hat{g}}_{yy}}\,,\hskip 28.45274ptA_{\mu}=\frac{{\hat{g}}_{\mu
y}}{{\hat{g}}_{yy}}\,,\hskip 28.45274ptk^{2}=-{\hat{g}}_{yy}\,,$ (C.1)
$\displaystyle B_{\mu\nu}$ $\displaystyle=$
$\displaystyle\,{\hat{B}}_{\mu\nu}+{\hat{g}}_{y[\mu}{\hat{B}}_{\nu]y}+\frac{\alpha^{\prime}}{4}\,\frac{{\hat{g}}_{y[\mu}{\hat{\Omega}}_{(-)}{}_{\nu]}{}^{\hat{a}}{}_{\hat{b}}\,{\hat{\Omega}}_{(-)}{}_{y}{}^{\hat{b}}{}_{\hat{a}}}{{\hat{g}}_{yy}}\,,$
(C.2) $\displaystyle C_{\mu}$ $\displaystyle=$ $\displaystyle{\hat{B}}_{\mu
y}-\frac{\alpha^{\prime}}{4}\left({\hat{\Omega}}_{(-)}{}_{\mu}{}^{\hat{a}}{}_{\hat{b}}\,{\hat{\Omega}}_{(-)}{}_{y}{}^{\hat{b}}{}_{\hat{a}}-\frac{{\hat{g}}_{\mu
y}{\hat{\Omega}}_{(-)}{}_{y}{}^{\hat{a}}{}_{\hat{b}}\,{\hat{\Omega}}_{(-)}{}_{y}{}^{\hat{b}}{}_{\hat{a}}}{{\hat{g}}_{yy}}\right)\,,$
(C.3) $\displaystyle\phi$ $\displaystyle=$
$\displaystyle\hat{\phi}-\frac{1}{2}\log\sqrt{-{\hat{g}}_{yy}}\,.$ (C.4)

Finally, we define as in Elgood:2020xwu  the scalar combination

$k_{(1)}=k+\frac{\alpha^{\prime}}{4}k^{-1}{\hat{\Omega}}_{(-)}{}_{y}{}^{\hat{a}}{}_{\hat{b}}{\hat{\Omega}}{(-)}_{y}{}^{\hat{b}}{}_{\hat{a}}\,.$
(C.5)

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