††institutetext: Department of Particle Physics and Astrophysics, Weizmann
Institute of Science, Rehovot, Israel

# The black hole/string transition in AdS3 and confining backgrounds

Erez Y. Urbach<EMAIL_ADDRESS>
###### Abstract

String stars, or Horowitz-Polchinski solutions, are Euclidean string theory
saddles with a normalizable condensate of thermal winding strings. String
stars were suggested as a possible description of stringy (Euclidean) black
holes close to the Hagedorn temperature. In this work, we continue the study
initiated in Urbach:2022xzw  by investigating the thermodynamic properties of
string stars in asymptotically (thermal) anti-de Sitter backgrounds. First, we
discuss the case of AdS3 with mixed RR and NS-NS fluxes (including the pure
NS-NS system) and comment on a possible BTZ/string transition unique to AdS3.
Second, we present new “winding-string gas” saddles for confining holographic
backgrounds such as the Witten model and determine the subleading correction
to their Hagedorn temperature. We speculate a black brane/string transition in
these models and argue for a possible relation to the deconfined phase of 3+1
dimensional pure Yang-Mills.

##  1 Introduction

In string theory, black holes are solutions of the low energy effective
action, usually valid only for horizon size larger than the string scale
$r_{h}\gg l_{s}$ (and small string coupling $g_{s}$). What is the correct
description of a black hole microstate when it is adiabatically shrunk to the
string scale? The black hole/string transition or the correspondence principle
Veneziano:1986zf ; Susskind:1993ws ; Horowitz:1996nw  is the idea that such
stringy black holes are indistinguishable from high energy stringy bound
states. In Horowitz:1997jc  a canonical ensemble version of this idea was
explored (see also Damour:1999aw ; Khuri:1999ez ; Kutasov:2005rr ;
Giveon:2005jv ; Chen:2021emg ; Brustein:2021cza ; Chen:2021dsw ;
Balthazar:2022szl ; Bedroya:2022twb ; Balthazar:2022hno ). As a thermodynamic
phase of quantum gravity, the Euclidean black hole is a saddle of the string
theory thermal partition on asymptotically $R^{d}\times S^{1}$.111Assuming the
asymptotic circumference of the circle can be consistently frozen at the value
$\beta$ Atick:1988si . In terms of the temperature, this saddle is reliable
for $\beta\gg l_{s}$. Slightly below the Hagedorn temperature of string theory
$\beta_{H}$, Horowitz and Polchinski found Horowitz:1997jc  a new saddle, a
condensate of strings winding around the thermal $S^{1}$ bound by
gravitational interactions. We term such a saddle a “string star”, imagining
it describes a gravitational bound state of hot strings. The black hole and
the string star saddle are reliable at different regimes of temperatures, but
a crude extrapolation to intermediate temperatures shows a qualitative
agreement of their thermodynamic properties. The black hole/string transition
can now be phrased as a precise conjecture: there exists a line of string
theory backgrounds (well-defined only for small $g_{s}$) connecting the
Euclidean black hole to the string star, and eventually (at the Hagedorn
temperature $\beta=\beta_{H}$) to thermal $R^{d}\times S^{1}$. Technically,
this line of saddles can be interpreted as a line of worldsheet CFTs
connecting the Euclidean black hole to $R^{d}\times S^{1}$ through winding
condensates. In Chen:2021dsw  an argument was presented against the existence
of such a line for $d>2$ type II string theory, via a worldsheet index
argument. A refined version of the conjecture thus allows for the possibility
of phase transitions along the line.

In Urbach:2022xzw  this discussion was extended to anti-de Sitter (AdS)
spaces, by studying the properties of AdSd+1 string stars for $d>2$. An AdS
string star is a thermal winding condensate in the middle of thermal AdSd+1.
The saddle is reliable near the AdS Hagedorn temperature $\beta_{c}$,
$\beta-\beta_{c}\ll\beta_{c}$. As the temperature increases the solution’s
size grows, with a maximal size of $L\sim\sqrt{l_{ads}l_{s}}$. Its amplitude
and free energy decrease to zero at the Hagedorn temperature, where it merges
with thermal AdSd+1. At lower temperatures, the solution size becomes small
$L\sim l_{s}$, at which point a correspondence principle to Euclidean AdS
black holes can be drawn similar to flat space: a conjectured line of string
theory backgrounds connecting the Euclidean AdS black hole solution and
thermal AdS at the Hagedorn temperature via thermal winding-string
condensate.222The definition of this line should be understood only in the
$g_{s}\rightarrow 0$ limit. At finite but small $g_{s}$ the 1-loop corrections
become leading close enough to the Hagedorn temperature. As the tree level
result gives an on-shell action $I\sim
g_{s}^{-2}\left(\frac{\beta-\beta_{c}}{\beta_{c}}\right)^{2}$, we find the
solution is trustworthy only for $(\beta-\beta_{c})/\beta_{c}\gg g_{s}$.
Unlike flat space, string theory on AdSd+1 also has a holographic dual in
terms of $d$-dimensional CFT. The black hole/string transition in terms of the
holographic CFT would be a conjectured line of large $N$ saddles in the
thermal partition function on $S^{d-1}$, connecting the deconfined phase to
the confined phase at the Hagedorn temperature (defined by the $N=\infty$
spectrum Aharony:2003sx ; Harmark:2021qma ) Sundborg:1999ue ; Aharony:2003sx ;
Alvarez-Gaume:2005dvb .

In this work, we continue our study of AdS string stars. The first extension
is the study of AdS3 string stars ($d=2$), presented in section 2. This is an
interesting test for the correspondence principle that is inherently different
from the flat space case. The reason is that both black holes and string stars
do not exist in asymptotically flat $R^{2}\times S^{1}_{\beta}$. However, in
AdS3 black hole solutions do exist and are known by the name BTZ
Banados:1992wn . Similarly, string star solutions don’t exist on flat
$R^{2}\times S^{1}_{\beta}$. But in thermal AdS3 we find that the AdS3
curvature stabilizes the potential for the winding condensate. As a result,
thermal AdS3 allows for reliable string star solutions close to the Hagedorn
temperature. This is an interesting check that such a correspondence is
possible for a more general setting and may exist whenever black holes exist
in a weakly coupled string theory. However, contrary to higher dimensional
small AdS black holes, a BTZ with small horizon radius $r_{h}\ll l_{ads}$ has
very low temperature $\beta\gg l_{ads}\gg\beta_{c}$. As the BTZ gets smaller,
its temperature continues to decrease further away from the Hagedorn
temperature, and a simple black hole/string transition between them is
impossible.

It is still possible to conjecture a different kind of correspondence. Below
temperature $\beta_{c}^{\prime}=4\pi^{2}l_{ads}^{2}/\beta_{c}$ the BTZ
solution has a tachyonic instability from strings winding the angular circle
Berkooz:2007fe ; Lin:2007gi ; Rangamani:2007fz . This is exactly the Hagedorn
instability of thermal AdS3, after switching the two asymptotic circles. The
natural candidate for a “BTZ/string transition” in AdS3 would be conjectured
line of saddles connecting the BTZ at $\beta_{c}^{\prime}$ to thermal AdS3 at
$\beta_{c}$, with winding condensates at each end of the line Jafferis:2021ywg
; Halder:2022ykw . See figure 1. Notice that unlike at higher dimensions, in
AdS3 there exists a large temperature difference between the regimes of
validity of the two saddles $\beta_{c}\ll\beta_{c}^{\prime}$. As a result, no
qualitative match can be extrapolated between the AdS3 string star and small
BTZs as a sanity check. We admit that the evidence for such a line is much
weaker compared to higher dimensions.

Another interesting feature of AdS3 string stars is the possibility in AdS3 to
turn on NS-NS flux Maldacena:1997re ; Giveon:1998ns ; Kutasov:1999xu ;
David:2002wn ; Cho:2018nfn . As the string is charged under the Kalb-Ramond
field, the winding string field $\chi(x)$ is electrically charged in the
effective two-dimensional description. Non-zero flux changes the solution at
leading order. We study the properties of AdS3 string stars for mixed RR and
NS-NS fluxes and for the pure NS-NS system. Denoting the fraction of the NS-NS
flux by $0\leq\lambda\leq 1$, the size of the RR flux is proportional to
$\sqrt{1-\lambda^{2}}$. In section 3.2 we study string star solutions for
finite RR flux. We find that the solutions are similar to those found in
higher dimensions Urbach:2022xzw . The NS-NS flux only contributes to the
effective mass of the winding mode (at leading order in $\alpha^{\prime}$),
and the size of the solution is
$L\sim\sqrt{l_{s}l_{ads}}\cdot(1-\lambda^{2})^{-1/4}$. In section 4 we study
solutions for small RR flux $\sqrt{1-\lambda^{2}}\sim l_{s}/l_{ads}$ and for
the pure NS-NS system. In this regime, the background NS-NS flux is high
enough to (almost) flatten the potential for the winding mode, similar to long
strings in AdS3 with NS-NS flux Maldacena:1998uz ; Seiberg:1999xz . As a
result, the size of the solution increases to the AdS scale $L\sim l_{ads}$.
Moreover, the back-reacted NS-NS flux from the winding mode now appears at
leading order, along with the metric and the dilaton.

In section 3 we also discuss winding string condensates in a family of
confining backgrounds which includes the D3 and D4 Witten models Witten:1998qj
, the Maldacena-Nuñez Maldacena:2000yy  and the Klebanov-Strassler
Klebanov:2000hb  backgrounds. These gravitational backgrounds are
holographically dual to theories believed to be continuously related (at low
energies) to confining field theories on flat space $R^{p+1}$ ($p=2,3$). The
thermodynamics of these models was studied using a semiclassical bulk analysis
Freedman:2000xb ; Gubser:2001ri ; Buchel:2001gw ; Buchel:2001qi ;
Gubser:2001eg ; Buchel:2001dg ; Buchel:2003ah ; Aharony:2006da ;
Aharony:2007vg ; Buchel:2007vy ; Buchel:2018bzp . The confined phase has a
“thermal” background dual given by analytically continuing the boundary to
$R^{p}\times S^{1}_{\beta}$. The deconfined phase has a black-brane dual. In
this work, we find a new saddle with a thermal winding string condensate on
top of the thermal background. The solution is homogeneous parallel to the
spatial boundary directions $R^{p}$, and it is localized to the bulk IR
region. We term this saddle a ‘winding-string gas’. As before, it is reliable
only close to the Hagedorn temperature of the holographic confining string
$\beta-\beta_{c}\ll\beta_{c}$, and it is highly subdominant. We describe the
properties of the solution and find the first correction to the Hagedorn
temperature in these models due to the bulk curvature. It is possible to
conjecture a black brane/string transition for these models, connecting the
black brane saddle to the winding-string gas saddle. The phase diagram for the
Witten models is similar to AdS3, discussed above. In this case, it is also
natural to conjecture that the string winding gas is continuously related to
the deconfined phase of pure $3+1$ Yang-Mills.

Several future directions are possible. The AdS3 calculation we performed was
only at leading order in the winding condensate and at leading order in
$\alpha^{\prime}$. The worldsheet theory in the pure NS-NS case ($\lambda=1$)
is known exactly Giveon:1998ns ; Kutasov:1999xu ; Maldacena:2000hw , and it is
exciting to find the string star profile exactly in $\alpha^{\prime}$
Jafferis:2021ywg ; Halder:2022ykw ; halder_private . Of course, ultimately one
would like to find (for the pure NS-NS background) the entire line of
worldsheet CFTs interpolating between the thermal AdS3 and BTZ. At the moment
it sounds improbable, as such a deformation away from
$\beta-\beta_{c}\ll\beta_{c}$ will require the entire string spectrum that
couples to the winding mode, and all the orders of their interactions.
Finally, the confining backgrounds solution gives interesting predictions on
its holographic dual. It would be interesting to compare these predictions to
the weakly coupled regime.

##  2 General properties of the AdS3 string star

###  2.1 Winding-string condensates in type II string theory

In this section, we will describe the low energy effective field theory (EFT)
for type II string theory Euclidean saddles with a perturbative (normalizable)
winding-string condensate. This is an extension of the EFT for the winding
mode together with the metric around flat space Atick:1988si ; Horowitz:1997jc
; Chen:2021dsw  to more general cases that include curved metric and
(possibly) background fluxes such as the Kalb-Ramond flux $H_{3}\neq 0$.

By “perturbative saddle” we mean that we start with a type II “background
saddle” without a winding-string condensate, and will consider a small
deformation of it that will include such a condensate. We will assume the
undeformed metric is of the form $M_{D}\times X_{10-D}$, with $X_{10-D}$ being
a $10-D$ dimensional compact space. We denote by $t\sim t+\beta$ the compact
’temporal’ direction in $M_{D}$ on which we will reduce the theory, and the
rest of the directions by $x^{i}$, $i=1,...,d=D-1$. We will assume the
undeformed metric on $M_{D}$ has the form

$G=g_{tt}(x)dt^{2}+g_{ij}(x)dx^{i}dx^{j}.$ (1)

Besides the metric, the background saddle can also include non-zero NS-NS and
RR gauge fields. All the background fields are assumed to be in a gauge where
they are independent of $t$, making $t$ translations a symmetry. We set the
units such that the curvature scale of the background fields is set to one
$l_{curv}=1$, while keeping the $\alpha^{\prime}$ explicit, and assume the
background is weakly-curved $\alpha^{\prime}\ll 1$.

Following Atick:1988si  we would like to consistently add an action for
$\chi$, the string mode that winds once around the temporal circle $t$. We
take $\chi(x)$ to be homogeneous on the compact $X_{10-D}$ directions and to
depend only on $x$. Denoting the size of the normalizable $\chi(x)$ profile
$L$, we will assume that it is much larger than the string scale $L\gg l_{s}$.
Assuming the entire deformation from the saddle has $L\gg l_{s}$, higher order
terms in the $\alpha^{\prime}$ expansion of the action will be suppressed by
derivatives. At leading order in $\alpha^{\prime}$ the total action is

$I=I_{\text{SUGRA}}+I_{\chi}.$ (2)

The type II supergravity (SUGRA) action (at leading order in
$\alpha^{\prime}$) is Polchinski:1998rr

$I_{\text{SUGRA}}=\frac{1}{16\pi G_{N}^{(10)}}\int
d^{10}x\sqrt{G}\left[e^{-2\Phi}\left(-R-4\partial_{\mu}\Phi\partial^{\mu}\Phi+\frac{1}{2}|H_{3}|^{2}\right)+\frac{1}{2}|F_{p}|^{2}\right],$
(3)

with $H_{3}=dB_{2}$, $F_{p+1}=dC_{p}$ and
$|F_{p}|^{2}=\frac{1}{p!}F_{\mu_{1}...\mu_{p}}F^{\mu_{1}...\mu_{p}}$.
$G_{N}^{(10)}$ is the $10$-dimensional gravitational constant. The last term
stands for the possible RR $F_{p}$ fluxes, and we omitted the possible Chern-
Simons terms for brevity.

Considering the action for the winding mode, the new ingredient compared to
Urbach:2022xzw  is that here we need to describe the coupling of the winding
mode to the Kalb-Ramond field. The leading interaction comes from the electric
coupling of the fundamental string to the Kalb-Ramond field along the $t$
direction. It is convenient to define the 1-form gauge field

$A_{i}=B_{t\ i},\text{ and }F=dA.$ (4)

In these conventions for $A_{i}$, $\chi$ is electrically charged under the
$U(1)$ winding with charge $g=\frac{\beta}{2\pi\alpha^{\prime}}$.333To see
this, notice the worldsheet action of the configuration
$t(\sigma,\tau)=\frac{\beta}{2\pi}\sigma,x^{i}(\sigma,\tau)=x^{i}(\tau)$ is
$S_{B}=\frac{i\beta}{2\pi\alpha^{\prime}}\int
d\tau\dot{x}^{i}(\tau)B_{ti}(x^{i}).$ (5) Up to $O(\alpha^{\prime})$ terms,
the $\chi$ $d$-dimensional action takes the form

$\begin{split}I_{\chi}&=\frac{\beta}{16\pi G_{N}}\int d^{d}x\ \text{vol}\
e^{-2\Phi}\Bigg{(}|D\chi|^{2}\\\
&+m^{2}\left(R(x)\right)|\chi|^{2}+O(\alpha^{\prime-1}|\chi|^{4})\\\
&+\left(g_{1}\mathcal{R}_{D}[g]+g_{2}\frac{g^{tt}}{2}F_{ij}F^{ij}+\frac{g_{3}}{12}H_{ijk}H^{ijk}\right)|\chi|^{2}\Bigg{)},\end{split}$
(6)

with $R(x)\equiv\frac{\beta}{2\pi}\sqrt{g_{tt}}$,
$D_{j}=\partial_{j}-igA_{j}$, and $g_{1}$, $g_{2}$, $g_{3}$ are some unknown
$O(1)$ couplings. $G_{N}$ and

$\text{vol}=\sqrt{g}\sqrt{g_{tt}},$ (7)

are the $D$-dimensional gravitational constant and volume respectively. For
type II string theory the winding mode mass and the flat space Hagedorn
temperature $R_{H}=\frac{1}{2\pi}\beta_{H}$ are

$\begin{split}m^{2}(R)=\frac{R^{2}-R^{2}_{H}}{\alpha^{\prime
2}},\qquad&R_{H}^{2}/\alpha^{\prime}=2.\end{split}$ (8)

In these notations, we also define the effective mass for the winding mode as

$m^{2}_{\text{eff}}(x)=m^{2}(R(x))+g^{2}A_{i}A^{i}.$ (9)

In (2) we added the first winding mode and the supergravity fields but
neglected the rest of the higher winding modes and the rest of the massive
string spectrum. This is consistent only when the mass of the winding mode is
parametrically below the string scale $\alpha^{\prime}m^{2}_{\text{eff}}\ll 1$
at the support of $\chi$. In practice, we need to tune the temperature of the
background to be close to the Hagedorn temperature $R-R_{H}\ll R_{H}$. The
second line of (6) includes all the $\alpha^{\prime-1}$ terms. After fine-
tuning the quadratic mass term to be small, we still have higher order
interaction terms starting at $\alpha^{\prime-1}|\chi|^{4}$. We will neglect
these terms below by our assumption that the deformation is perturbative and
the profile of the winding mode is parametrically small $|\chi|\ll 1$.

In the third line of (6) we listed the first $\alpha^{\prime 0}$ terms in the
action. These terms correct the mass term by a curvature scale (which also
controls the Kalb-Ramond flux) amount $\Delta m^{2}\sim O(1)$.444One can also
write $\alpha^{\prime 0}$ terms that correct the $\varphi|\chi|^{2}$ coupling,
which are also subleading. These terms are naively leading whenever the size
of the winding mode profile is comparable to the curvature scale. Indeed some
of the solutions we will find below will satisfy this condition. Nevertheless,
we will argue that these terms are negligible for these cases.

Imagining a general deformation of the supergravity fields, what are the
leading interaction terms? Expanding the total action around the background,
the quadratic order (as in any Euclidean theory) is trivial. The leading
$\alpha^{\prime-1}$ cubic interaction comes by expanding the kinetic term for
$\chi$ to linear order around the background. Parametrizing the deformation of
the $tt$ component of the metric by $G_{tt}=g_{tt}e^{2\varphi(x)}$, there is a
universal cubic interaction by expanding the mass term

$m^{2}(x)|\chi|^{2}=m^{2}(R(x))|\chi|^{2}+\frac{2R(x)^{2}}{\alpha^{\prime
2}}\varphi|\chi|^{2}+O(\varphi^{2}|\chi|^{2}),$ (10)

where on the RHS $R(x)$ is of the (undeformed) background metric. Close to the
Hagedorn temperature $R\sim l_{s}$ the interaction term is of leading
$\alpha^{\prime-1}$ order. Whenever the background solution has a non-trivial
Kalb-Ramond flux around the temporal circle $F_{ij}\neq 0$ there is also a
leading cubic term from the $A_{i}A^{i}|\chi|^{2}$ term, of the schematic form
$\frac{1}{\alpha^{\prime}}\delta A_{i}|\chi|^{2}$ (we will deal with this term
in the next section). However, the dilaton $\Phi$ cubic coupling to $\chi$ is
of higher $\alpha^{\prime}$ order due to our assumption $L\gg l_{s}$ and is
negligible.555We assume $\Phi=0$ for the undeformed background. Considering a
general deformation of the supergravity fields, the leading cubic interactions
of $I_{\text{SUGRA}}$ are all with $O(1)$ (the curvature scale) couplings and
can be neglected as well. Only the quadratic interactions in
$I_{\text{SUGRA}}$ around the saddle are leading.

It is, therefore, crucial to find out which fields couple to $\varphi$ (and
$\delta A_{i}$) at quadratic order in $I_{\text{SUGRA}}$. The Einstein-Hilbert
term by itself gives only a kinetic term for $\varphi$, but the linear term
does give a quadratic coupling to the dilaton. To see this we proceed as
follows. The overall volume in (3) is $\sqrt{G}\exp(\varphi-2\Phi)$. It is
therefore useful to redefine the dilaton as

$\phi=\Phi-\frac{1}{2}\varphi.$ (11)

In this form, the quadratic order of the metric + dilaton action is given by

$\begin{split}\frac{1}{16\pi G_{N}^{(10)}}&\int
d^{10}x\sqrt{G}e^{-2\Phi}\left[-R-4\partial_{\mu}\Phi\partial^{\mu}\Phi\right]\\\
&=\frac{\beta}{16\pi G_{N}}\int
d^{d}x\text{vol}\left[\partial_{i}\varphi\partial^{i}\varphi-4\partial_{i}\phi\partial^{i}\phi-2\phi\frac{\partial_{i}g_{tt}\partial^{i}\varphi}{g_{tt}}+(...)\right].\end{split}$
(12)

On the RHS we integrated over the other $10-d$ directions (including $t$) and
kept only quadratic terms in $\varphi,\phi$. The last term gives a leading
quadratic $\phi\partial\varphi$ interaction, that generically appears whenever
$g_{tt}$ is not flat.666This is the reason this term is absent in Atick:1988si
; Horowitz:1997jc ; Chen:2021dsw . The dilaton however is expected to couple
quadratically in a similar way to any supergravity field with a non-trivial
profile along the $x^{i}$ directions. To find a saddle with a winding
condensate, it is necessary to consider a general deformation of the metric
and the dilaton (and other fields present in the background).

In many cases, however, there is a significant simplification that can be
considered assuming the solution’s length scale is also much smaller than the
curvature scale $L\ll 1$. For those cases, we can further expand in orders of
$L$ and neglect lower derivatives or higher powers of $x$. Looking back at
(12), the kinetic terms are of order $L^{-2}$ while the interaction term is of
$L^{-1}$ subleading order. In fact, in the gauge we will choose $A_{i}\sim
O(L)$, and it is possible to show that the $\delta B|\chi|^{2}$ is also
subleading (we will explain it for the AdS3 case below). As a result, at
leading order in $L$ the dilaton $\phi$ and the Kalb-Ramond field decouple,
and we can consistently consider a perturbative deformation of the winding
mode $\chi$ and the temporal metric $\varphi$ alone. The effective action in
this limit will be reviewed in section 3.1. Whenever the size of the profile
reaches the curvature scale $L\sim 1$, this approximation breaks. For those
cases a general supergravity deformation is necessary. We will describe such a
case in section 4 below.

###  2.2 The mixed AdS3 background

In this section, we begin the discussion of the AdS3 string star. We work in
units of $l_{ads}=1$. The set of backgrounds we will consider are Euclidean
10d solutions of type IIB string theory with both NS-NS and RR fluxes, and the
geometry of (thermal) $AdS_{3}\times S^{3}\times M_{4}$, where $M_{4}$ can be
a four torus $T^{4}$ or a $K3$ manifold Maldacena:1997re ; David:2002wn ;
Cho:2018nfn . The thermal AdS3 component is given by the following background

$\begin{split}ds^{2}&=d\rho^{2}+\sinh^{2}\rho\ d\phi^{2}+\cosh^{2}\rho\
dt^{2},\\\ B_{2}&=i\lambda\sinh^{2}\rho\ d\phi\wedge dt,\\\
C_{2}&=i\sqrt{1-\lambda^{2}}\sinh^{2}\rho\ d\phi\wedge dt,\end{split}$ (13)

with $t\sim t+\beta$ and $0\leq\lambda\leq 1$.777The quantization of the
fluxes (on $S^{3}$) requires both $\lambda/\alpha^{\prime}$,
$\sqrt{1-\lambda^{2}}/\alpha^{\prime}$ to be integers. In this work, we expand
at leading order in $\alpha^{\prime}$, and therefore this quantization is
invisible Maldacena:1997re ; Cho:2018nfn . The gauge fields $B,C$ also have
fluxes from the $S^{3}$ component, which we omitted. $\lambda=0$ is the pure
RR solution, for which string stars solutions were studied in Urbach:2022xzw ,
and $\lambda=1$ is the pure NS-NS solution. We note that the latter can be
studied not only for type IIB but in any closed string theory, and it is also
known exactly in $\alpha^{\prime}$ via an $\text{SL}(2,R)$ WZW worldsheet
description Giveon:1998ns ; Kutasov:1999xu ; Maldacena:2000hw . As in
Urbach:2022xzw  $\beta$ is also the holographic temperature in the dual CFT on
$S^{1}_{2\pi}\times S^{1}_{\beta}$.

In these conventions the background gauge field (4) is
$A_{\phi}=i\lambda\sinh^{2}\rho$ and $R(\rho)=R_{0}\cosh(\rho)$, with
$R_{0}=\frac{\beta}{2\pi}$. The (background) effective mass (9) is given by

$m^{2}_{\text{eff}}(\rho)=\frac{1}{\alpha^{\prime
2}}\left(R_{0}^{2}\left(\cosh^{2}\rho-\lambda^{2}\sinh^{2}\rho\right)-R_{H}^{2}\right).$
(14)

For self consistency we assumed scale separation
$\alpha^{\prime}m^{2}_{\text{eff}}(\rho)\ll 1$, which in this case gives
$R_{0}-R_{H}\ll R_{H}$ for $\rho\sim O(1)$. The EFT is therefore applicable
only for temperatures close to the Hagedorn temperature.

Following the previous section, to write the two-dimensional EFT we need to
expand $I_{\text{SUGRA}}$ to quadratic order around a general supergravity
deformation. Assuming the spherical symmetry of the solution, we are
instructed to consider a deformation for every field with a nontrivial
background profile along the $\rho$ direction. A general spherically symmetric
deformation of the AdS3 components of (13) has the form888We gauge fixed the
field potential $B_{2}$, $C_{2}$ to also depend solely on $\rho$, which fixes
the allowed components.

$\begin{split}\Phi&=\phi+\frac{1}{2}(\varphi+\psi),\\\
G&=d\rho^{2}+e^{2\varphi}\cosh^{2}\rho\ dt^{2}+e^{2\psi}\sinh^{2}\rho\
d\phi^{2},\\\ B_{2}&=i\lambda e^{\zeta}\sinh^{2}\rho\ dt\wedge d\phi,\\\
C_{2}&=i\sqrt{1-\lambda^{2}}e^{\eta}\sinh^{2}\rho\ dt\wedge d\phi,\end{split}$
(15)

with $\phi,\varphi,\zeta$ and $\eta$ all function of $\rho$. In particular, we
don’t need to consider deformations of other RR fluxes. Similar to (11), we
redefined $\Phi$ in such a way to satisfy

$\sqrt{G}e^{-2\Phi}=\text{vol}\ e^{-2\phi},$ (16)

with $\text{vol}=\cosh\rho\ \sinh\rho$ the volume form of the background
metric.

Following the discussion in the previous section, we assume small back-
reaction $|\phi|,|\varphi|,|\psi|,|\zeta|,|\eta|\ll 1$ and consider only its
leading coupling to $\chi$. Expanding the winding mode action (6) to
($\alpha^{\prime-1}$) cubic order, and integrating over the angular direction,
gives

$\begin{split}I_{\chi}&=\frac{2\pi\beta}{16\pi G_{N}}\int d\rho\
\text{vol}\left[|\chi^{\prime}|^{2}+m^{2}_{\text{eff}}(\rho)|\chi|^{2}+\frac{2R_{0}^{2}}{\alpha^{\prime
2}}\left(\cosh^{2}\rho\ \varphi-\sinh^{2}\rho\
(\zeta-\psi)\right)|\chi|^{2}\right],\end{split}$ (17)

with $G_{N}$ being the 3-dimensional gravitational constant, and the effective
mass (14). Here and below, primes are derivatives with respect to $\rho$. In
particular, we find both $\varphi|\chi|^{2}$ and $(\zeta-\psi)|\chi|^{2}$
terms of the same leading $\alpha^{\prime-1}$ order. Substituting the
deformation (15) in the SUGRA action (3) gives to quadratic order

$\begin{split}I_{\text{SUGRA}}^{(2)}&=\frac{2\pi\beta}{16\pi G_{N}^{(3)}}\int
d\rho\ \text{vol}\
\Bigg{[}-4(\phi^{\prime})^{2}+(\varphi^{\prime})^{2}+(\psi^{\prime})^{2}\\\
&-\lambda^{2}\left(\frac{1}{2}\tanh^{2}\rho\
(\zeta^{\prime})^{2}+4(\psi+\varphi-\zeta)^{2}-4\tanh\rho\
\zeta^{\prime}(\psi+\varphi-\zeta)\right)\\\
&-(1-\lambda^{2})\left(\frac{1}{2}\tanh^{2}\rho\
(\eta^{\prime})^{2}+4\eta^{2}-4\eta(\varphi+\psi)-2\tanh\rho\
\eta^{\prime}(\psi+\varphi-2\eta)\right)\\\ &-4\phi\left(\tanh\rho\
\varphi^{\prime}+\coth\rho\
\psi^{\prime}+\lambda^{2}(2\varphi+2\psi-2\zeta-\tanh\rho\
\zeta^{\prime})\right)\Bigg{]}.\end{split}$ (18)

To avoid a curvature singularity at $\rho=0$ we need to maintain $\psi(0)=0$.
Regularity of the fields at $\rho=0$ also sets
$\chi^{\prime}(0)=\phi^{\prime}(0)=\varphi^{\prime}(0)=\psi^{\prime}(0)=\zeta^{\prime}(0)=\eta^{\prime}(0)=0$.
At $\rho=\infty$ all the fields should vanish for normalizability.

###  2.3 Thermodynamics and the BTZ/string transition

The total EFT action compared to that of thermal AdS3 is the sum of (17) and
(18)

$I_{\text{String star}}-I_{\text{TAdS}_{3}}=I^{(2)}_{\text{SUGRA}}+I_{\chi}.$
(19)

The AdS3 string star is the bound state saddle of (19). Being a solution of
the action, it is a Euclidean saddle of the (Euclidean) string theory
partition function with asymptotic thermal AdS3 boundary condition. As a
Euclidean saddle, it has free energy of order $G_{N}^{-1}$, just like black
holes. It is applicable only near the Hagedorn temperature $R_{0}-R_{H}\ll
R_{H}$.

In $d=2$ the AdS black hole solution is known by the BTZ solution
Banados:1992wn . It’s on-shell action, compared to the thermal AdS3, is (in
units $l_{ads}=1$)

$I_{\text{BTZ}}-I_{\text{TAdS}_{3}}=-\frac{\pi}{4G_{N}}\left(R_{0}^{-1}-R_{0}\right).$
(20)

Thus, thermal AdS3 is canonically dominant for low temperatures $R_{0}>1$,
while the BTZ is dominant for high temperatures $R_{0}<1$. Whenever the string
star saddle is reliable $R\sim R_{H}\ll 1$ the dominant saddle is the BTZ. The
string star saddle is therefore non-perturbatively unstable, and exponentially
subdominant canonically. As we will see numerically in figure 7 below (and
also in Urbach:2022xzw ) the string star free energy (19) is positive (note
that the expression is not clearly positive due to the $\zeta,\eta$ kinetic
terms (18)). Therefore it is subdominant also compared to the thermal AdS3
solution.

For higher dimensional AdSd+1 $d>2$ an AdS black hole/string transition was
suggested between the AdS string star saddle and AdS black holes
Urbach:2022xzw . The argument was carried from the black hole/string
transition previously suggested between the flat-space string star saddle (or
the Horowitz-Polchinski solution) and the (Euclidean) Schwarzschild saddle
Horowitz:1997jc ; Chen:2021dsw . The reason is that at the two ends of the
correspondence point around $R_{0}\sim l_{s}$, both the AdS black hole and the
AdS string star are of size much smaller than the AdS scale and can be well
approximated by their asymptotically flat counterparts.

Figure 1: A conjectured phase diagram for AdS3 (for $l_{ads}\gg l_{s}$). The
Euclidean action as a function of the inverse temperature $R_{0}$ is sketched.
The gray and black lines describe the thermal AdS3 and the BTZ respectively.
The string star saddle is the blue line close to the Hagedorn temperature
$R_{c}$. Its dual, a BTZ winding condensate, is the blue line connected to the
BTZ saddle at $R_{c}^{\prime}$. A BTZ/string transition in AdS3 would be a
hypothetical line of saddles connecting the two (dashed blue line). We note
that the sketch should be understood in the limit $g_{s}\rightarrow 0$. At
finite but small $g_{s}$ the string star saddle can’t be trusted arbitrarily
close to $R_{c}$ whenever the 1-loop effect is leading.

In $d=2$ however asymptotically flat Schwarzschild solutions simply don’t
exist, so a similar argument can’t be drawn. Going back to AdSd+1, AdS black
holes exist only for $R_{0}<l_{ads}/\sqrt{d(d-2)}$. For small enough horizon
radius $r_{h}/l_{ads}<\sqrt{\frac{d-2}{d}}$ the temperature grows as the
radius shrinks, just like in flat space. This is not the case for $d=2$. The
temperature of the BTZ solution always grows with the radius $R_{0}=1/r_{h}$.
As a result, any attempt to connect the AdS3 string star to a small black hole
close to the Hagedorn temperature is impossible, because at high temperatures
only large black hole solutions exist. A clue comes when we consider the
asymptotic symmetries of the question. Any worldsheet with asymptotic
$S^{1}_{\phi}\times S^{1}_{\beta}$ has two $U(1)$ windings. The thermal AdS3
saddle breaks the $S^{1}_{\phi}$ winding spontaneously, while the BTZ saddle
breaks the $S^{1}_{\beta}$ winding. The AdS3 string star saddle on the other
hand (spontaneously) breaks both winding symmetries. To connect the AdS3
string star saddle to the BTZ we need to consider deformations of the BTZ by
$S^{1}_{\phi}$ winding strings.

Denoting the AdS3 Hagedorn temperature by $R_{c}$, the BTZ solution has a
known instability at $R_{c}^{\prime}=1/R_{c}$ Berkooz:2007fe ; Lin:2007gi ;
Rangamani:2007fz . This instability is exactly the tachyon instability of the
thermal AdS3 at the AdS Hagedorn temperature after we switch the roles of the
two asymptotic circles ($R_{0}\mapsto 1/R_{0}$). In other words, below
$R_{0}>R_{c}^{\prime}$ strings winding around the $\phi$-circle are tachyonic.
Considering the AdS3 string star saddle after switching the circles we get a
new saddle that describes a normalizable $\phi$-winding condensate in the BTZ
background. This saddle also breaks both winding symmetries spontaneously, but
it is connected (in its regime of validity $R_{c}^{\prime}-R_{0}\ll
R_{c}^{\prime}$) to the BTZ instead of the thermal AdS3. In terms of global
symmetries pattern, it is therefore natural to speculate these two branches
meet around $R_{0}\sim 1$.999By ‘meet’ we mean through a line of string theory
backgrounds labeled by the asymptotic temperature $\beta=2\pi R_{0}$.  This
phase diagram for AdS3 was also suggested in Halder:2022ykw . A schematic plot
of this suggested picture is drawn in figure 1.

##  3 Sub-AdS solutions and confining strings

###  3.1 General discussion

This section studies general properties of holographic thermal confining
theories, in cases where the “IR wall” is geometrically realized by a
contractible (non-thermal) cycle. Thermal $AdS_{d+1}$ is one such example
Urbach:2022xzw , in which an $S^{d-1}$ closes in the IR region. We start by
showing that around such a background there exists a saddle of thermal winding
string condensate that depends (to leading order) only on the curvature scale
at the IR point at which the cycle closes. The reason would be that for this
background, the size of the solution would be much smaller than the
characteristic curvature scale. In terms of AdSd+1 these solutions satisfy
$L\ll l_{ads}$ and were reviewed in Urbach:2022xzw . The analysis allows us to
find the characteristic behavior of the solution and the leading correction to
the Hagedorn temperature in the dual holographic theory.

The topology of the geometries we will study is
$S^{1}_{\beta}\times\mathbb{R}^{d}\times\mathcal{M}$. $S^{1}_{\beta}$
asymptotes to the thermal cycle of the holographic theory, $\mathbb{R}^{d}$ is
made of a radial direction $r$ and a shrinking $S^{d-1}$, and $\mathcal{M}$ is
some (perhaps non-compact) manifold. We further assume the solution preserves
$SO(2)\times SO(d)$ symmetry. The string metric close to the shrinking point
$r=0$ of the $S^{d-1}$ sphere is, to subleading order in $r^{2}$,

$ds^{2}=dr^{2}+\left(1+\frac{r^{2}}{l^{2}}\right)dt^{2}+r^{2}d\Omega^{2}_{d-1}+ds^{2}_{\mathcal{M}}$
(21)

The Euclidean time direction satisfies $t\sim t+2\pi R_{0}$. Higher orders of
the metric are assumed to be down by higher orders of $r^{2}/l^{2}$ with $l$
being the curvature scale of the (IR) geometry. As a result, we implicitly
assume $r\ll l$ and $l_{s}\ll l$. On general grounds the parameter $R_{0}$ is
related linearly to the holographic theory temperature $\beta$. We also assume
that for $r\ll l$ the internal space geometry $ds^{2}_{\mathcal{M}}$ is
independent of $r$. In this geometry, and for temperatures close to the
Hagedorn temperature, we look for non-trivial winding solutions that are
homogeneous both on $\mathcal{M}$ and on the closing $S^{d-1}$ sphere. In
cases that $\mathcal{M}$ is non-compact these solutions are dual (in the
holographic theory) to a gas of interacting (confining) strings, and so we
term them “winding-string gas” solutions.

In section 2.1 we explained that whenever the winding mode profile size is
parametrically smaller than the curvature scale $L\ll l$ we can expand the EFT
in orders of $L/l$ and consider only the winding mode $\chi$ and the temporal
metric deformation $\varphi$. In practice, we take only the highest number of
derivatives (within the $O(\alpha^{\prime})$ action) and fewer powers of $r$.
Assuming both $\chi(r),\varphi(r)$ depend only on $r$, the leading
$O(l^{2}/L^{2})$ terms in (2) are

$I_{L\ll l}=\frac{\beta V_{d-1}}{16\pi G_{N}}\int
r^{d-1}dr\left[(\varphi^{\prime})^{2}+|\chi^{\prime}|^{2}+m^{2}_{\text{eff}}(r)|\chi|^{2}+\frac{2R_{0}^{2}}{\alpha^{\prime
2}}\varphi|\chi|^{2}\right],$ (22)

with $V_{d-1}$ the volume of a unit $S^{d-1}$ sphere. At leading order for
small $r$ it is enough to expand the effective mass to quadratic order in $r$.
Assuming no Kalb-Ramond flux in the $t$ direction the effective mass (9) is

$m^{2}_{\text{eff}}(r)=\frac{1}{\alpha^{\prime
2}}\left(R_{0}^{2}-R_{H}^{2}+\frac{R_{0}^{2}r^{2}}{l^{2}}\right).$ (23)

Notice that the only $l$ dependence in the action (22) is through (23).
Whenever a (non-singular) Kalb-Ramond flux exists it only corrects the $r^{2}$
term and can be swallowed into a redefinition of $l$. Naively the second term
is subleading for $L\ll l$, but for temperatures close enough to the Hagedorn
temperature $(R_{0}^{2}-R_{H}^{2})/R_{0}^{2}\sim L^{2}/l^{2}$ it is of the
same order and should be added (while higher orders in $r$ are still
subleading). The resulting equations for $\chi(r),\varphi(r)$ are

$\begin{split}\chi^{\prime\prime}+\frac{d-1}{r}\,\chi^{\prime}-\left(m^{2}_{\text{eff}}(r)+\frac{4}{\alpha^{\prime}}\varphi\right)\chi&=0,\\\
\varphi^{\prime\prime}+\frac{d-1}{r}\,\varphi^{\prime}-\frac{2}{\alpha^{\prime}}|\chi|^{2}&=0.\end{split}$
(24)

These equations are of the same form found in the near-Hagedorn limit at
Urbach:2022xzw .

The winding EFT regime of validity for the winding EFT is
$(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\ll 1$. In this section, we further restrict
ourselves to $(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\sim l_{s}/l$. We will denote
this regime of temperatures as “near-Hagedorn” and parameterize it explicitly
by defining

$R_{0}^{2}\equiv R_{H}^{2}+\delta\cdot\frac{l_{s}^{3}}{l}.$ (25)

We can further reparameterize

$\chi(r)=\frac{l_{s}}{l}\cdot\hat{\chi}(r/L),\quad\varphi(r)=\frac{l_{s}}{l}\cdot\
\hat{\varphi}(r/L),$ (26)

with $L^{2}\equiv l_{s}l$, to get ($x=r/L$)

$\begin{split}\hat{\chi}^{\prime\prime}+\frac{d-1}{x}\,\hat{\chi}^{\prime}-\left(\delta+2x^{2}+4\hat{\varphi}\right)\hat{\chi}&=0,\\\
\hat{\varphi}^{\prime\prime}+\frac{d-1}{x}\,\hat{\varphi}^{\prime}-2|\hat{\chi}|^{2}&=0,\end{split}$
(27)

with ′ denote an $x$ derivative. These equations were studied numerically in
Urbach:2022xzw . Because (27) is dimensionless we find $|\chi|,|\varphi|\sim
l_{s}/l$ and solution’s size $L\sim\sqrt{l_{s}l}$. Because (for the original
metric to be weakly curved) $l_{s}/l\ll 1$ the solutions for (27) are all
self-consistent. Specifically note that $l_{s}\ll L\ll l$, justifying the
approximation (22). Numerically it was found that the solution’s size
increases with the temperature Urbach:2022xzw .

The Hagedorn temperature of the theory is defined as the temperature where the
linearized equations have a normalizable solution, giving rise to a
normalizable zero mode in the partition function. At the order we work with,
this is given by $\delta_{c}=-\sqrt{2}d$, and the corresponding normalizable
mode is $\chi(r)=\exp(-r^{2}/(\sqrt{2}L^{2}))$. In other words, the leading
curvature correction to the Hagedorn value for $R_{0}$ is

$R_{c}^{2}=2\alpha^{\prime}\left(1-\frac{d}{\sqrt{2}}\frac{l_{s}}{l}+O(l_{s}^{2}/l^{2})\right).$
(28)

However, $R_{0}$ is not directly related to the holographic theory temperature
$\beta$. In the examples below the geometry is given by an analytic
continuation from a Lorentzian manifold with a boundary of $R^{p,1}$ for some
$p$. The Lorentzian solutions take the general form

$ds^{2}=dr^{2}+f(r)\left(-(dx^{0})^{2}+\sum_{i=1}^{p}(dx^{i})^{2}\right)+(...),$
(29)

where $r$ is again the radial bulk direction, $x^{0},...,x^{p}$ are parallel
to the boundary $R^{p,1}$, and in the bracket we ignored the rest of the
coordinates. The Euclidean solution is then taken by analytically continuing
$x_{0}=it$, with the same $t\sim t+\beta$ as in (21). Comparing to (21) gives
$f(r=0)=4\pi^{2}R_{0}^{2}/\beta^{2}$ at $r=0$. The confining string tension
$T_{st}$ is given by calculating the bulk worldsheet action in the $t,x^{i}$
IR plane ($r=0$) measured in units of the holographic boundary lengths given
by $\beta$:

$T_{st}=\frac{1}{2\pi\alpha^{\prime}}\sqrt{g_{tt}g_{xx}}\mid_{r=0}=\frac{f(0)}{2\pi\alpha^{\prime}}=\frac{2\pi
R_{0}^{2}}{\alpha^{\prime}\beta^{2}}.$ (30)

Together with (28), the leading curvature correction to the holographic
Hagedorn temperature is

$\beta_{c}=\sqrt{\frac{4\pi}{T_{st}}}\left(1-\frac{d}{2\sqrt{2}}\frac{l_{s}}{l}+O(l_{s}^{2}/l^{2})\right).$
(31)

###  3.2 AdS3 with finite RR flux

We now go back to study the AdS3 string star solutions in the background (13).
Following the previous section we will when small solutions $L\ll l_{ads}$
exist. Assuming $\rho\ll l_{ads}$ we can directly expand (18) and (17) to
$L^{-2}$ order and see that all the fields besides $\varphi$ decouple, leaving
us with (22) (at $d=2$). In particular, notice that while both $\varphi$ and
$\zeta$ couple to $\chi$ at the same $\alpha^{\prime-1}$ order (17), the
$\zeta$ coupling is smaller by a factor of $\rho^{2}/l_{ads}^{2}\ll 1$.
Expanding the effective mass (14) to leading order gives (bringing back the
$l_{ads}$ units)101010The discussion here is for $\lambda\sim O(1)$. We note
that for small enough $\lambda\sim l_{s}/l_{ads}$ the $\lambda$ (and the
entire NS-NS flux) dependence is absent at leading order in $\alpha^{\prime}$,
and appears together with the suppressed higher $\alpha^{\prime}$ terms in
(6).

$m^{2}_{\text{eff}}(\rho)=\frac{1}{\alpha^{\prime
2}}\left(R_{0}^{2}+\frac{(1-\lambda^{2})R_{0}^{2}}{l_{ads}^{2}}\rho^{2}-R_{H}^{2}\right).$
(32)

This is the same form as in (23) only with the effective scale
$l=l_{ads}/\sqrt{1-\lambda^{2}}$. We can follow the same steps as in the
previous section and assume a small solution $l_{s}\ll L\ll l$. We also note
that as the winding mass of order $m^{2}\sim l_{ads}/l_{s}^{3}$, we can
consistently neglect higher-curvature contributions to the mass which are of
subleading order $\Delta m\sim l_{ads}^{-2}$.

Following the previous section, for near-Hagedorn temperatures
$(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\sim l_{s}/l_{ads}$ the AdS3 string star
solution has an amplitude of

$|\chi|,|\varphi|\sim\frac{l_{s}}{l_{ads}}\sqrt{1-\lambda^{2}},$ (33)

and a characteristic size

$L\sim\sqrt{l_{s}l_{ads}}(1-\lambda^{2})^{-1/4}.$ (34)

In this geometry $\beta=2\pi R_{0}$ and therefore, following (31), the leading
correction to the AdS3 Hagedorn temperature is

$R_{c}^{2}=2\alpha^{\prime}-2^{3/2}\sqrt{1-\lambda^{2}}\frac{l_{s}^{3}}{l_{ads}}+O(\alpha^{\prime
2}/l_{ads}^{2}).$ (35)

In Urbach:2022xzw  the leading correction for the Hagedorn temperature in
$AdS_{5}\times S^{5}$ was matched with an integrability analysis of
$\mathcal{N}=4$ SYM Harmark:2021qma . It would be interesting to find a way to
similarly compare (35) to the recent integrability results for the pure RR
AdS3 case ($\lambda=0$) Cavaglia:2021eqr ; Ekhammar:2021pys ; Cavaglia:2022xld
; Frolov:2021bwp .

The thermodynamic stability of these solutions was studied in Urbach:2022xzw .
$\zeta$ is negligible in (7) and the string star free energy can be shown to
be positive. This saddle is therefore non-perturbatively unstable compared to
the thermal AdS3 saddle (and to the BTZ saddle).

This discussion is self-consistent only when the size of the solution is
smaller than the AdS curvature scale, $L\ll l_{ads}$, or
$1-\lambda^{2}\gg\alpha^{\prime}/l_{ads}^{2}$. Note that
$\sqrt{1-\lambda^{2}}$ is the magnitude of the RR flux $F_{3}$ (13). We,
therefore, conclude that for near-Hagedorn temperatures
$(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\sim l_{s}/l_{ads}$ and large enough RR-flux
$1-\lambda^{2}\gg\alpha^{\prime}/l_{ads}^{2}$ the leading behavior of the mass
around $\rho=0$ solely controls the profile of the string star. As we lower
the RR flux the profile size is larger and the amplitude is smaller. Around
$1-\lambda^{2}\sim\alpha^{\prime}/l_{ads}^{2}$ the solution size reaches the
AdS scale and the approximation we took breaks. We will discuss this case in
section 4 below.

For lower temperatures, the solution size turns shorter. Below the near-
Hagedorn regime, but for temperatures in which the EFT is still valid
$l_{s}/l_{ads}\ll(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\ll 1$ one might expect
reliable solutions that are arbitrarily short (although with $L\gg l_{s}$).
This is not the case and for a good reason. If arbitrarily small solutions
would exist, they would approximately solve the flat space equations for
$d=2$, and it is known that no such solutions exist Horowitz:1997jc ;
Chen:2021dsw . Instead, whenever the solution is reliable
($|\chi|,|\varphi|\ll 1$) its size remains comparable to
$L\sim\sqrt{l_{s}l_{ads}}$. In other words, this section describes the
behavior of AdS3 string stars whenever they exist for finite RR flux
$1-\lambda^{2}\gg\alpha^{\prime}/l_{ads}^{2}$.

###  3.3 The D4 Witten model

Consider $4+1$ dimensional maximally symmetric $SU(N)$ Yang-Mills theory (with
coupling $g_{5}$) compactified on a circle of radius $R$ with anti-periodic
boundary condition for the fermions. We denote the $4+1$ ’t Hooft coupling by
$\lambda_{5}=g_{5}^{2}N$. It is holographically dual to the near horizon limit
of $N$ D4 branes Witten:1998zw , with the $9+1$ (type IIA) metric

$\begin{split}ds^{2}&=\left(\frac{u}{R_{D4}}\right)^{3/2}\left[-(dx^{0})^{2}+\sum_{i=1}^{3}(dx^{i})^{2}+f(u)(dx^{4})^{2}\right]+\left(\frac{R_{D4}}{u}\right)^{3/2}\left[\frac{du^{2}}{f(u)}+u^{2}d\Omega_{4}^{2}\right]\\\
F_{(4)}&=\frac{2\pi N}{V_{4}}\epsilon_{4},\quad
e^{\phi}=g_{s}\left(\frac{u}{R_{D4}}\right)^{3/4},\quad R_{D4}^{3}\equiv\pi
g_{s}Nl_{s}^{3},\quad f(u)\equiv
1-\left(\frac{u_{\Lambda}}{u}\right)^{3},\end{split}$ (36)

The string coupling $g_{s}$ is related to the $4+1$ Yang-Mills coupling by
$4\pi^{2}g_{s}l_{s}=g_{5}^{2}$. Here $x_{4}\sim x_{4}+2\pi R$ with
$R=\frac{2}{3}\left(\frac{R_{D4}^{3}}{u_{\Lambda}}\right)^{1/2}$ (by demanding
smooth solution where the $x_{4}$ circle shrinks at $u=u_{\Lambda}$).

The thermodynamics of this system was studied in Aharony:2006da . The thermal
solution is produced by analytically continuing the time direction to a circle
of size $\beta$, $x_{0}=it$ $t\sim t+\beta$. On the gravity side, the
analytically continued solution is the dominating solution for low
temperatures. For high temperatures, one can take the same solutions, only
with the $R$ and $\beta$ circles switched, so that now the temporal circle
shrinks in the bulk. This last solution is a Euclidean black brane solution.
The phase transition between the two happens at $\beta=2\pi R$ Aharony:2006da
.

Here we are interested in winding condensate solutions around the
(analytically continued) thermal saddle (36). The solutions extend
homogeneously on the $x^{i}$ directions but localize around $u=u_{\Lambda}$.
Defining
$r^{2}=\frac{4}{3}\left(\frac{R_{D4}^{3}}{u_{\Lambda}}\right)^{1/2}(u-u_{\Lambda})$,
$x_{0}=it$ and $\phi=\frac{x_{4}}{R}$ we have to leading order in $r^{2}$

$\begin{split}ds^{2}&=\left(\frac{u_{\Lambda}}{R_{D4}}\right)^{3/2}\left[dt^{2}+\sum_{i=1}^{3}(dx^{i})^{2}\right]+r^{2}d\phi^{2}+dr^{2}+\left(\frac{R_{D4}^{3}}{u_{\Lambda}}\right)^{1/2}d\Omega_{4}^{2}.\end{split}$
(37)

And at the next order in $r^{2}$

$G_{tt}=\left(\frac{u_{\Lambda}}{R_{D4}}\right)^{3/2}\left(1+\frac{3}{2}\frac{u-u_{\Lambda}}{u_{\Lambda}}\right)=\left(\frac{u_{\Lambda}}{R_{D4}}\right)^{3/2}\left(1+\frac{27R}{16R_{D4}^{3}}r^{2}\right).$
(38)

Comparing with (21) we find the curvature scale
$l^{2}=\frac{16}{27}\frac{R_{D4}^{3}}{R}$. In this model, the (confining)
string tension is given by
$T_{st}=\frac{1}{2\pi\alpha^{\prime}}\left(\frac{u_{\Lambda}}{R_{D4}}\right)^{3/2}=\frac{4}{27\pi\alpha^{\prime}}(R_{D4}/R)^{3}$
and the mass gap (called the glueball mass) is of order $M_{gb}=1/R$. Plugging
inside (31) with $d=2$ gives the leading correction to the holographic
Hagedorn temperature

$\begin{split}\beta_{c}=\sqrt{\frac{4\pi}{T_{st}}}-\frac{1}{\sqrt{2}}\frac{M_{gb}}{T_{st}}+O(M_{gb}^{2}T_{st}^{-3/2}),\end{split}$
(39)

or in terms of the dimensionless ratio $\lambda_{5}/R$:

$\beta_{c}/R=\sqrt{108\pi^{3}}\cdot\left(\lambda_{5}/R\right)^{-1/2}-\frac{27\pi^{2}}{\sqrt{2}\left(\lambda_{5}/R\right)^{-1}}+O((\lambda_{5}/R)^{-3/2}).$
(40)

For a worldsheet derivation of (39), see Bigazzi:2022gal .

We found that this model admits a ‘winding-string gas’ saddle for near-
Hagedorn temperatures: a normalizable condensate of winding strings around
Euclidean time $t$ that extends homogeneously on the $R^{3}$ directions (and
$x^{4}$). The condensate is localized in the bulk around $u=u_{\Lambda}$ with
a size of $L\sim\sqrt{l_{s}l}\sim l_{s}\cdot(T_{st}/M_{gb}^{2})^{1/4}$.

The winding-string gas saddle is highly metastable, as it has free energy
higher than both the thermal solution and (because we are above the phase
transition) the black brane solution. Note that the phase diagram is very
similar to that of the AdS3, due to $2\pi R\iff\beta$ symmetry. Just like for
the BTZ solution, the black brane solution has a tachyonic instability below
the temperature

$\beta^{\prime}_{c}=\frac{4\pi}{3}R_{D4}\left(\frac{R^{2}}{\alpha^{\prime}}\right)^{1/3}=2^{4/3}\pi\left(T_{st}M_{gb}^{-5}\right)^{1/3}.$
(41)

For temperatures slightly above it, there exists a condensate of winding
$x^{4}$ strings homogeneous on $R^{3}\times S^{1}_{\beta}$, obtaining by
interchanging the asymptotic circles for the winding-string gas saddle. It is
possible to speculate a new kind of correspondence principle or a black
brane/string transition. The black brane with $x^{4}$ winding condensate and
the thermal solution with $t$-winding condensate are connected via a line of
metastable saddles.

To understand the relation to pure Yang-Mills, it is useful to write the
critical temperatures in terms of $R$ and the $4+1$ Yang-Mills coupling
$\lambda_{5}$

$\beta_{c}/R=\sqrt{108\pi^{3}\frac{R}{\lambda_{5}}},\quad\beta_{HP}/R=2\pi,\quad\beta_{c}^{\prime}/R=\frac{2}{3}\left(\pi^{2}\frac{\lambda_{5}}{R}\right)^{1/3}.$
(42)

The semiclassical approximation is valid when $l\gg l_{s}\gg l_{pl}$ or
$1\ll\lambda_{5}/R\ll N$. In this regime $\beta_{c}/R\ll
1\ll\beta_{c}^{\prime}/R$. In the opposite limit, $\lambda_{5}/R\ll 1$ the
holographic theory at energies below $\sim 1/R$ is that of pure $3+1$ Yang-
Mills. In Aharony:2006da  the phase diagram of the system was suggested for
any $\lambda_{5},R$. At $\lambda_{5}/R\ll 1$ and for low temperatures
$\beta\gg R$ the theory is described by $3+1$ SYM. This theory has a first-
order confinement/deconfinement close to its Hagedorn temperature
$\beta_{c}\gg R$. As a result, we also expect a phase transition at high
temperatures $\beta_{c}^{\prime}\ll R$. The simplest scenario suggested in
Aharony:2006da  was a single new dominating phase for temperatures
$\beta_{c}^{\prime}<\beta<\beta_{c}$. In such a scenario, this phase should
have the unique property that it maps to itself when interchanging
$\beta,R$.111111In other words, the line of solutions is smooth at $\beta=2\pi
R$ and the saddle there is mapped to itself under switching the circles. In
more complicated scenarios, there are further phase transitions and phases
that are mapped to each other under interchanging $\beta,R$.

Going back to the gravity limit $1\ll\lambda_{5}/R\ll N$. If indeed the
winding-string gas saddle is the one that connects the two saddles between
$\beta_{c}$ and $\beta_{c}^{\prime}$, the simplest option would be that its
continuation to the $\lambda_{5}/R\ll 1$ is the dominant phase for
temperatures $\beta_{c}^{\prime}<\beta<\beta_{c}$, see figure 2. Notice that
because the winding-string gas phase breaks the winding of both circles, it is
indeed invariant to switching $\beta,R$. This picture is also supported by the
analysis of Aharony:2005ew  which studied $2d$ SYM on a torus. In the weakly
coupled regime, the authors found a dominating phase with non-zero holonomies
on both circles.

Figure 2: The phase diagram of the D4 Witten model as suggested by the
analysis of Aharony:2006da . Left: The phase diagram when the bulk geometry is
weakly curved $R\ll\lambda_{5}$. The two black lines represent the thermal and
the black brane geometry, with a Hawking-Page transition at $\beta=2\pi R$.
The two red dots represent the two critical temperatures $\beta_{c}$,
$\beta_{c}^{\prime}$ above (below) which the thermal (black brane) saddle
suffers from tachyonic instability. The blue line represents the ‘winding-
string gas’ saddle, which we suggest connects the two winding condensate
solutions of each saddle. Right: In the regime $R\gg\lambda_{5}$ the low
energy behavior is that of $3+1$ Yang-Mills, with a confinement/deconfinement
transition at $\beta_{c}\gg R$. A similar transition happens for the ‘black
brane’ below some $\beta_{c}^{\prime}\ll R$. We suggest that the dominating
phase for $\beta_{c}^{\prime}<\beta<\beta_{c}$ is nothing but the ‘winding-
string gas’ saddle.

###  3.4 The D3 Witten model

It is possible to repeat the previous construction also for D3s compactified
on a circle Witten:1998qj ; Aharony:1999ti ; Csaki:1998qr . The holographic
theory is now $3+1$ $\mathcal{N}=4$ $SU(N)$ Yang-Mills on $R^{3}\times S^{1}$
with anti-periodic boundary conditions on the fermions. The gravity dual is
the near-Horizon limit of $N$ D3 branes on $S^{1}$. After analytically
continuing $x^{0}=it$ we find the thermal solution

$\begin{split}ds^{2}/l_{ads}^{2}&=f(u)d\tau^{2}+f^{-1}(u)du^{2}+u^{2}\left[dt^{2}+\sum_{i=1}^{2}(dx^{i})^{2}\right]+d\Omega^{2}_{5},\\\
f(u)&=u^{2}-\frac{u_{0}^{4}}{u^{2}},\quad F_{5}=16\pi N\alpha^{\prime
2}\omega_{5},\end{split}$ (43)

with $l_{ads}^{2}=\sqrt{4\pi g_{s}N}\alpha^{\prime}$, $\tau\sim\tau+\pi/u_{0}$
(the $\tau$-circle closes at $u=u_{0}$) and the length of the asymptotic
circle is $R=1/(2u_{0})$. There is a non-zero RR flux through the $S^{5}$,
$\int_{S^{5}}\omega_{5}=\pi^{3}$. Close to $u=u_{0}$ the radial coordinate is
$r=R((u-u_{0})/u_{0})^{1/2}$ and

$G_{tt}=R^{2}u_{0}^{2}\left(1+2\frac{r^{2}}{l_{ads}^{2}}+...\right).$ (44)

Comparing with (21) we find the curvature scale $l^{2}=l_{ads}^{2}/2$ with
$d=2$ (shrinking $S^{1}$). Following (31) the leading correction to the
holographic Hagedorn temperature is

$\beta_{c}=\sqrt{\frac{4\pi}{T_{st}}}\left(1-(4\pi g_{s}N)^{-1/4}+O\left((4\pi
g_{s}N)^{-1/2}\right)\right),$ (45)

with $T_{st}=R^{2}u_{0}^{2}/(2\pi\alpha^{\prime})$. The phase diagram of this
model (when the gravity approximation is valid) is similar to the D4 case. A
black brane solution exists by replacing the two Euclidean circles and
dominates the canonical ensemble for high temperatures $\beta>2\pi R$. The
black brane is tachyonic for temperatures below (at leading $g_{s}N$ order)

$\beta_{c}^{\prime}=\frac{\pi}{2^{3/2}}\frac{R}{l_{s}u_{0}}=\frac{\pi}{4}\sqrt{\frac{4\pi
g_{s}N}{T_{st}}}.$ (46)

The same line of argument used for the D4 suggests a similar black
brane/string transition. The black brane and the thermal saddle are connected
between $\beta_{c}\leq\beta\leq\beta_{c}^{\prime}$, with perturbative winding
condensates being the near-critical description at each side.

###  3.5 The Maldacena-Nuñez background

In Maldacena:2000yy  Maldacena and Nuñez (interpreting the solutions of
Chamseddine:1997nm ) considered $N$ D5 branes on $R^{4}\times S^{2}$, with a
twisted normal bundle on $S^{3}$. The holographic effective description is
$5+1$ maximally supersymmetric YM on $R^{4}\times S^{2}$ with twisting of the
$S^{2}$ using a gauge background for a $U(1)_{R}\subset SU(2)_{R}\subset
SO(4)$ global symmetry. In terms of $R^{4}$ the twist preserves
$\mathcal{N}=1$ supersymmetry. We denote the $5+1$ ’t Hooft coupling
$\lambda_{6}$ and the radius of the $S^{2}$ by $R$. The near horizon 10d
metric and RR flux of the D5 system is given by

$\begin{split}ds^{2}_{str}/\alpha^{\prime}&=e^{\phi_{D}}\left(-(dx^{0})^{2}+\sum_{i=1}^{3}(dx^{i})^{2}+N\left[d\rho^{2}+e^{2g(\rho)}d\Omega_{2}^{2}+\frac{1}{4}\sum_{a=1}^{3}(w_{a}-A_{a})^{2}\right]\right)\\\
F_{3}&=N\left[-\frac{1}{4}(w^{1}-A^{1})\wedge(w^{2}-A^{2})\wedge(w^{3}-A^{3})+\frac{1}{4}\sum_{a=1}{3}F^{a}\wedge(w^{a}-A^{a})\right],\end{split}$
(47)

with

$\begin{split}e^{2\phi_{D}}&=g_{s}^{2}\frac{\sinh
2\rho}{2e^{g}}=g_{s}^{2}\left(1+\frac{8}{9}\rho^{2}+O(\rho^{4})\right),\\\
e^{2g}&=\rho\coth(2\rho)-\frac{\rho^{2}}{\sinh^{2}(2\rho)}-\frac{1}{4}.\end{split}$
(48)

here $w_{a}$ label 1-forms on $S^{3}$, and $A_{a}$ is a smooth solution that
asymptotes at large $\rho$ to the prescribed $S^{2}$ twist, and $F=dA$. In
this geometry, the asymptotic $S^{2}$ shrinks at $\rho=0$. This solution is
weakly coupled and curved for $1\ll g_{s}N\ll N$, or $1\ll\lambda_{6}/R^{2}\ll
N$. For $\lambda_{6}/R^{2}\ll 1$ (in which gravity is highly curved) on the
other hand the four-dimensional $\mathcal{N}=1$ theory decouples at energies
below $1/R$ Maldacena:2000yy .

The thermal ensemble is produced by analytically continuing $x_{0}=it/l_{s}$
($t\sim t+\beta$). Close to $\rho=0$ the radial coordinate is
$r=(\alpha^{\prime}g_{s}N)^{1/2}\rho$, and the temporal (string) metric
component is

$G_{tt}=e^{\phi_{D}}=g_{s}\left(1+\frac{4}{9}\frac{r^{2}}{\alpha^{\prime}g_{s}N}+O\left(\frac{r^{4}}{(\alpha^{\prime}g_{s}N)^{2}}\right)\right).$
(49)

At $r=0$ the $S^{3}$ has no twist ($A_{i}$ is flat), and as long as the
solution’s size is smaller than $L\ll(\alpha^{\prime}g_{s}N)^{1/2}$ we can
approximate the geometry by $S_{1}\times R^{3}\times R^{3}\times S^{3}$ ($r$
being the radial direction in $R^{3}$). Comparing with (21), we find ‘winding-
string gas’ saddles $\chi(r),\varphi(r)$ homogeneous on $R^{3}\times S^{3}$
for near-Hagedorn temperatures. In terms of section 3.1
$l=\frac{3}{2}(\alpha^{\prime}g_{s}N)^{1/2}$ and $d=3$ (as we have a shrinking
$S^{2}$), and the first correction to the Hagedorn temperature is (31)

$\beta_{c}=\sqrt{\frac{4\pi}{T_{st}}}\left(1-(2g_{s}N)^{-1/2}+O(N^{-1})\right).$
(50)

with $T_{st}=\frac{1}{2\pi\alpha^{\prime}}g_{s}$.121212The high energy
description of this model is in terms of little string theory, which has its
own, different, Hagedorn temperature Kutasov:2000jp . The latter appear at
much higher temperatures
$\beta_{H,LST}=2\pi(\alpha^{\prime}N)^{1/2}\ll\beta_{c}$ (in the case we are
considering $g_{s}N\ll 1$).

Close to the Hagedorn temperature $\beta-\beta_{c}\ll\beta_{c}$ there exists a
reliable winding-string gas saddle homogeneous on the $R^{3}$ (and the rest of
the compact directions) and localize at the IR $\rho=0$ region with size
$L\sim l_{s}(g_{s}N)^{1/4}$. The black brane-like solution, in which the
thermal circle closes, and its thermodynamics were studied in Buchel:2001qi ;
Gubser:2001eg ; Buchel:2001dg .131313For the thermodynamics of other
supergravity duals of 4d confining gauge theories see Freedman:2000xb ;
Buchel:2003ah ; Buchel:2007vy  A black brane/string transition for this model
suggests the black brane (with a small horizon radius) is connected to the
winding-string gas saddle via a line of worldsheet CFTs.

###  3.6 The Klebanov-Strassler conifold background

The Klebanov-Strassler geometry is given by Klebanov:2000hb

$ds^{2}=h^{-1/2}\left(-(dx^{0})^{2}+\sum_{i=1}^{3}(dx^{i})^{2}\right)+h^{1/2}\left(\frac{\varepsilon^{4/3}K(\tau)}{6K^{3}(\tau)}d\tau^{2}+(...)\right)$
(51)

In $(...)$ we omitted the remaining 5 dimensions of the conifold geometry. For
small $\tau$

$\begin{split}K(\tau)&=(2/3)^{1/3}-\tau^{2}/(5\cdot 2^{2/3}\cdot
3^{1/3})+O(\tau^{4})\\\
h(\tau)&=(g_{s}M\alpha^{\prime})^{2}2^{2/3}\varepsilon^{-8/3}(a_{0}-2^{-1/3}3^{-4/3}\tau^{2}+O(\tau^{4})),\end{split}$
(52)

and $a_{0}\approx 0.71805$ PandoZayas:2003yb . Here $M$ is the number of
fractional $D3$ branes and $\varepsilon$ is the deformation parameter. The
background also includes RR and NS-NS fluxes, which are immaterial in the
discussion of winding condensate.

The thermodynamics of Klebanov-Strassler was studied in Gubser:2001ri ;
Buchel:2001gw ; Aharony:2007vg ; Buchel:2018bzp . Here we are interested only
in the thermal solution, given by analytically continuing (51) by
$x^{0}=il_{s}t$ ($t\sim t+\beta$). At $\tau=0$ an $S^{2}$ of the conifold
geometry closes, and the geometry is approximately $R^{4}\times R^{3}\times
S^{3}$. Defining the radial coordinate $r=G_{\tau\tau}^{1/2}\tau$ we find
141414here the $x_{i}$ have units of length squared. When we write $G_{tt}$ we
already multiplying by $\alpha^{\prime}$ to make contact with (21).

$G_{tt}=\frac{\varepsilon^{4/3}}{2^{1/3}a_{0}^{1/2}\
g_{s}M}\left(1+\frac{1}{3a_{0}^{3/2}}\frac{r^{2}}{g_{s}M\alpha^{\prime}}\right).$
(53)

In terms of (21) $d=3$ and a curvature scale
$l^{2}=3a_{0}^{3/2}g_{s}M\alpha^{\prime}$. As a result, the leading correction
to the Hagedorn temperature in this background is

$\beta_{c}=\sqrt{\frac{4\pi}{T_{st}}}\left(1-\frac{3^{1/2}}{2^{3/2}a_{0}^{3/4}}\left(g_{s}M\right)^{-1/2}+O\left((g_{s}M)^{-1}\right)\right),$
(54)

they with the confining string tension
$T_{st}=\frac{\varepsilon^{4/3}}{2^{4/3}a_{0}^{1/2}\pi}(\alpha^{\prime}g_{s}M)^{-1}$.

##  4 AdS-sized solutions for AdS3 with small RR flux

###  4.1 The effective action

In section 3.2 we concluded that the analysis for sub-AdS solutions breaks for
small RR fluxes $1-\lambda^{2}\sim\alpha^{\prime}/l_{ads}^{2}$, where the
solution’s size reaches the AdS scale $L\sim l_{ads}$. For those cases, we
can’t neglect the backreaction of the winding mode on the supergravity fields
(15). In this section, we turn to study the behavior of AdS3 string stars for
small RR fluxes, as well as the pure NS-NS case. To that end we define

$1-\lambda^{2}\equiv a\cdot\frac{\alpha^{\prime}}{l_{ads}^{2}},$ (55)

and assume $0\leq a\ll l_{ads}^{2}/\alpha^{\prime}$.151515Notice that the
quantization of the RR charge requires
$\sqrt{1-\lambda^{2}}l_{ads}^{2}/\alpha^{\prime}=\sqrt{a}l_{ads}/l_{s}$ to be
integers. In our $l_{ads}\gg l_{s}$ limit it gives no restriction on the range
of $a$. Section 3.2 considered only near-Hagedorn temperatures
$(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\sim l_{s}/l_{ads}$. For small RR flux, the
important regime is the tighter
$(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\sim\alpha^{\prime}/l_{ads}^{2}$ (which we can
call the very-near-Hagedorn regime). We therefore define

$R_{0}^{2}=R_{H}^{2}+\hat{m}^{2}\cdot\frac{\alpha^{\prime 2}}{l_{ads}^{2}}.$
(56)

Solutions are expected to be valid as long as $R_{0}^{2}-R_{H}^{2}\ll
R_{H}^{2}$, or $\hat{m}^{2}\ll l_{ads}^{2}/\alpha^{\prime}$, which we will
therefore assume.

In the rest of the section, we will set the units to $l_{ads}=1$ for brevity.
In terms of the new variables and at leading order in $l_{s}/l_{ads}$ the
effective mass (14) is

$m^{2}_{\text{eff}}=\hat{m}^{2}+2a\ \sinh^{2}\rho.$ (57)

In the limit $\lambda\approx 1$ the RR flux terms in the action (18) are
subleading and can be neglected $\eta=0$. For the rest of the fields, it is
useful to redefine by (in $l_{ads}=1$ units)

$\chi=\alpha^{\prime}\hat{\chi},\quad\phi=\alpha^{\prime}\hat{\phi},\quad\varphi=\alpha^{\prime}\hat{\varphi},\quad\psi=\alpha^{\prime}\hat{\psi},\quad\zeta=\alpha^{\prime}\hat{\zeta}.$
(58)

At leading order, we get the action

$\displaystyle I_{\text{String star}}-I_{\text{TAdS}_{3}}$
$\displaystyle=\frac{\beta\alpha^{\prime 2}}{16\pi G_{N}}\cdot F,$ (59)

with, upon integrating by parts, the normalized free energy (compared to that
of thermal AdS3)

$\begin{split}F=\int d\rho\ \text{vol}\
&\Bigg{[}(\hat{\psi}^{\prime})^{2}+(\hat{\varphi}^{\prime})^{2}-4(\hat{\phi}^{\prime})^{2}\\\
&-\frac{1}{2}\tanh^{2}\rho\
(\hat{\zeta}^{\prime})^{2}-4(\hat{\psi}+\hat{\varphi}-\hat{\zeta})^{2}+4\tanh\rho\
\hat{\zeta}^{\prime}(\hat{\psi}+\hat{\varphi}-\hat{\zeta})\\\
&+4\hat{\phi}^{\prime}\left(\tanh\rho\ \hat{\varphi}+\coth\rho\
\hat{\psi}-\tanh\rho\ \hat{\zeta}\right)\\\
&+|\hat{\chi}^{\prime}|^{2}+m^{2}_{\text{eff}}\rho\
|\hat{\chi}|^{2}+4\left(\cosh^{2}\rho\ \hat{\varphi}-\sinh^{2}\rho\
(\hat{\zeta}-\hat{\psi})\right)|\hat{\chi}|^{2}\Bigg{]},\end{split}$ (60)

The equation of motion for the dilaton gives the first-order constraint (upon
using the boundary conditions for $\hat{\psi}$)

$2\hat{\phi}^{\prime}=\tanh\rho\ \hat{\varphi}+\coth\rho\
\hat{\psi}-\tanh\rho\ \hat{\zeta}.$ (61)

Substituting (61) gives the equations of motion for the rest of the fields

$\begin{split}\hat{\chi}^{\prime\prime}+v(\rho)\hat{\chi}^{\prime}&=(\hat{m}^{2}+2a\
\sinh^{2}\rho)\hat{\chi}+4\left(\cosh^{2}\rho\ \hat{\varphi}+\sinh^{2}\rho\
(\hat{\psi}-\hat{\zeta})\right)\hat{\chi},\\\
\hat{\varphi}^{\prime\prime}+v(\rho)\hat{\varphi}^{\prime}&=2\tanh\rho\
\hat{\zeta}^{\prime}+\tanh^{2}\rho\
(\hat{\varphi}-\hat{\zeta})+\hat{\psi}-4(\hat{\varphi}+\hat{\psi}-\hat{\zeta})+2\cosh^{2}\rho\
|\hat{\chi}|^{2},\\\
\hat{\psi}^{\prime\prime}+v(\rho)\hat{\psi}^{\prime}&=2\tanh\rho\
\hat{\zeta}^{\prime}+\hat{\varphi}-\hat{\zeta}+\coth^{2}\rho\
\hat{\psi}-4(\hat{\varphi}+\hat{\psi}-\hat{\zeta})+2\sinh^{2}\rho\
|\hat{\chi}|^{2},\\\
\hat{\zeta}^{\prime\prime}+u(\rho)\hat{\zeta}^{\prime}&=4\coth\rho\
(\hat{\varphi}^{\prime}+\hat{\psi}^{\prime})+2(\hat{\varphi}-\hat{\zeta}+\coth^{2}\rho\
\hat{\psi})+4\cosh^{2}\rho\ |\hat{\chi}|^{2},\end{split}$ (62)

with

$v(\rho)=\tanh\rho+\coth\rho,\quad u(\rho)=3\coth\rho-\tanh\rho.$ (63)

$a$ adds a potential to the linear equation for $\hat{\chi}$ and therefore for
$a>0$ we expect a normalizable solution with size $L\sim 1$. Assuming an
$O(1)$ result (for $\hat{m}^{2}\sim O(1)$) gives
$|\chi|,|\varphi|,|\zeta|\sim\alpha^{\prime}\ll 1$ which is self-consistent
with our assumptions. The reader may worry that as the effective mass is now
$m^{2}_{\text{eff}}(\rho)\sim O(1)$ (in $l_{ads}$ units), higher curvature
corrections are now leading. In section 4.3 below we argue that the resulting
mass shift is proportional to $\Delta m^{2}\propto(1-\lambda^{2})$. For finite
$a$ this is a $O(\alpha^{\prime})$ correction to the mass and therefore
subleading.

In the extreme case $a=0$ ($\lambda=1$) we find the pure NS-NS background. In
this case, the $\rho$ dependence in $m^{2}_{\text{eff}}$ is exactly canceled
between the metric and the NS-NS flux, which results in a constant mass

$m^{2}_{\text{eff}}(\rho)=\hat{m}^{2}=\frac{1}{\alpha^{\prime
2}}(R^{2}_{0}-R_{H}^{2}),$ (64)

equals to the flat space result (8). This property means that very close to
the Hagedorn temperature strings winding around the thermal circle are light
arbitrarily close to the boundary $\rho\gg 1$, although the proper length of
the thermal circle diverges exponentially. Physically, this is due to the
cancellation between the gravitational attraction (of AdS space) and the
electric repulsion (or the non-zero $H_{3}$). This is related to the well-
studied pure NS-NS AdS3 long strings Maldacena:1998uz ; Seiberg:1999xz . These
strings wind around the angular circle of Lorentzian AdS3 with finite energy
arbitrarily close to the boundary. Because asymptotically there is no
difference between the two circles, the cancellation of the energy divergence
at large $\rho$ is of the same origin. As a result, for $a=0$ the potential in
(62) is quadratic and not exponential. Nevertheless, we expect a normalizable
solution for $\chi,\varphi,\zeta$ also in this case for high enough
$\hat{m}^{2}$, with a generic $L\sim 1$. We present numerical solutions to
this equation in section 4.4 below.

###  4.2 The AdS3 Hagedorn temperature

To find the leading correction to the AdS3 Hagedorn temperature, we should
look for the maximal value of $\hat{m}^{2}$ (as a function of $a\geq 0$) for
which there’s a solution to the linearized (62) equation for
$\hat{\chi}$161616At the linearized order the other fields in (62) have no
non-trivial solutions.

$\hat{\chi}^{\prime\prime}+v(\rho)\,\hat{\chi}^{\prime}-\left(\hat{m}^{2}+2a\
\sinh^{2}\rho\right)\hat{\chi}=0.$ (65)

Denoting this critical value by $\hat{m}^{2}_{c}$, the first correction to the
AdS3 Hagedorn temperature is

$R_{c}^{2}(\lambda)=\frac{2}{k}+\frac{\hat{m}^{2}_{c}}{k^{2}}+O(k^{-3}),$ (66)

with $k=l_{ads}^{2}/\alpha^{\prime}$. For $a=0$ the (smooth) linear solution
can be found analytically to be

$\hat{\chi}_{\text{lin}}(\rho)=P_{\frac{-1+\sqrt{1+\hat{m}^{2}}}{2}}(\cosh(2\rho)),$
(67)

with $P_{n}(x)$ being the Legendre function Berkooz:2007fe . For $\rho\gg 1$
and $\hat{m}^{2}>-1$ we have
$\hat{\chi}_{\text{lin}}\sim\exp(-\Delta_{\chi}\rho)$ with
$\Delta_{\chi}=\real(1-\sqrt{1+\hat{m}^{2}})$, and as
$\hat{\chi}_{\text{lin}}\sim\rho\exp(-\rho)$ for $\hat{m}^{2}=-1$. The
solution is thus non-normalizable for any $\hat{m}^{2}$, but it turns delta-
function normalizable at $\hat{m}^{2}=-1$ (at the Breitenlohner-Freedman
bound). Therefore for $a=0$ the critical value is $\hat{m}^{2}_{c}=-1$, and
the first correction to the pure NS-NS AdS3 Hagedorn temperature is

$R_{c}^{2}(\lambda=1)=\frac{2}{k}-\frac{1}{k^{2}}+O(k^{-3}).$ (68)

Interestingly, for the type II pure NS-NS this is also the exact result
$R_{c}^{2}=2/k-1/k^{2}$ Berkooz:2007fe ; Mertens:2014nca .171717I thank David
Kutasov for pointing out that the first correction here is $O(k^{-2})$, and
not $O(k^{-3/2})$ as found above (35) and in Urbach:2022xzw . It is his
comment that led me to study the AdS3 case. The fact that the zero-mode (67)
is only delta-function normalizable is similar to flat space Atick:1988si ,
and probably corresponds to the fact the Lorentzian spectrum, in this case, is
continuous.

For $a>0$ the linear equation has a potential and so we expect only discrete
values of $\hat{m}^{2}$ for which exist normalizable solutions. We are
interested in the ‘ground state’ solution, with the largest $\hat{m}^{2}$. We
weren’t able to find an analytical solution in this case, but it is possible
to find it numerically. In figure 3 we plotted $\hat{m}_{c}^{2}$ as a function
of $a$. It can be seen from the figure that $\hat{m}^{2}$ decreases as $a$
grows. This is to be expected, as when $a\gg 1$ we expect from section 3.2 to
find $\hat{m}^{2}_{c}=-2^{3/2}a^{1/2}$ (by translating (35) in terms of
$\hat{m}^{2}$). This behavior is shown in the figure by a black dashed line.

Figure 3: The value of the leading correction to the Hagedorn temperature
$\hat{m}^{2}_{c}$ as a function of $a$ (blue). The $a\gg 1$ behavior from
section 3.2 is plotted in a black dashed line.

###  4.3 An argument against higher curvature corrections

Higher curvature corrections always appear with non-negative powers of the
string scale and therefore are naively subleading. For $d=2$, two operators
still contribute at zeroth order in the string scale:
$\mathcal{R}_{2}|\chi|^{2}$ and $g^{tt}F_{ij}F^{ij}|\chi|^{2}$. For the mixed
background (13) their value is exactly a constant

$\mathcal{R}_{2}=-2,\quad g^{tt}F_{ij}F^{ij}=-4\lambda^{2}.$ (69)

We can write these terms in the Lagrangian as

$\mathcal{L}_{\text{hige-
curv}}=g_{1}\cdot\mathcal{R}_{2}|\chi|^{2}+g_{2}\cdot\frac{1}{2}g^{tt}F^{ij}|\chi|^{2}=-2(g_{1}+g_{2}\lambda^{2})|\chi|^{2},$
(70)

for some unknown couplings $g_{1},g_{2}\sim O(1)$. Therefore the overall
effect of these two terms is to shift the effective mass of $\chi$ by a
constant $\Delta m^{2}=-2(g_{1}+g_{2}\lambda^{2})$.181818Of course, both terms
also induce a correction to the coupling of $\varphi$, $\zeta$ to
$|\chi|^{2}$. Whenever the constant shift is subleading, so is the coupling.
As a result, the effective mass for the pure NS-NS case is
($k=l_{ads}^{2}/\alpha^{\prime}$)

$m^{2}_{\text{eff}}(\rho)=k^{2}(R_{0}^{2}-R_{H}^{2})+\Delta m^{2}+O(1/k).$
(71)

We now turn specifically to the pure NS-NS theory $\lambda=1$, in which the
worldsheet theory on thermal AdS is known Berkooz:2007fe ; Rangamani:2007fz ;
Mertens:2014nca . The mass of the particle dictates the asymptotic behavior of
its mode close to the boundary. As such, it should match the AdS3 $SL(2,R)$
Casimir of the corresponding worldsheet vertex operator. To find the exact
worldsheet mass, we follow the discussion of Mertens:2014nca , which uses the
description of thermal AdS3 as a specific coset of $SL(2,C)$ Hemming:2002kd .
For a given $\beta$, a bosonic $SL(2,C)$ WZW primary with winding $\pm 1$ and
$SL(2,C)$ Casimir $J^{2}=s^{2}+\frac{1}{4}$ (and $h_{int}=0$) has dimension

$h=\bar{h}=\frac{s^{2}+\frac{1}{4}}{k-2}+\frac{k\beta^{2}}{16\pi^{2}}$ (72)

We change variables to $R_{0}=\frac{\beta}{2\pi}$, $-4m^{2}=s^{2}+\frac{1}{4}$
to get

$h=\bar{h}=-\frac{1}{4}\frac{m^{2}}{k-2}+\frac{kR_{0}^{2}}{4}.$ (73)

To write the type II expression we shift $k\mapsto k+2$ (only for the first
term, as we keep the temperature in string units $kR_{0}^{2}$ constant) and
find

$h=\bar{h}=-\frac{1}{4}\frac{m^{2}}{k}+\frac{kR_{0}^{2}}{4}=\frac{1}{2}.$ (74)

As a result, the mass of the winding vertex operator as a function of $R_{0}$
is

$m^{2}=k^{2}\left(R_{0}^{2}-\frac{2}{k}\right).$ (75)

Comparing with (71) we find that the higher curvature correction to the mass
vanishes $\Delta m^{2}=0$, for $\lambda=1$. In other words $g_{1}=-g_{2}$, and
(for any $\lambda$)

$\Delta m^{2}=-2g_{1}(1-\lambda^{2}).$ (76)

Because in this section we consider only $1-\lambda^{2}\sim 1/k$, this is a
subleading effect.

###  4.4 Numerical results

Figure 4: The two profiles of $\hat{\chi}$, $\hat{\varphi}$ and $\hat{\zeta}$
for temperature $\hat{m}^{2}=0$ in the pure NS-NS theory $a=0$ (left) and for
$a=0.5$ (left).

Figure 5: The value of $\hat{\chi}(0)$, $\hat{\varphi}(0)$ and
$\hat{\zeta}(0)$ as a function of $\hat{m}^{2}$, for $a=0.1$ (left) and
$a=0.5$ (right). The horizontal dashed green line indicates the AdS Hagedorn
temperature $\hat{m}_{c}^{2}$.

Figure 6: The size of $\hat{\chi}(\rho)$ as a function of $\hat{m}^{2}$, for
$a=0.1$ (left) and $a=0.5$ (right). The size is defined as the radius $\rho$
in which the value of $\hat{\chi}(\rho)$ is decreased by a factor of $e$
compared to $\hat{\chi}(0)$. The horizontal dashed green line indicates the
AdS Hagedorn temperature $\hat{m}_{c}^{2}$.

Figure 7: The string star normalized free energy as a function of
$\hat{m}^{2}$, in the pure NS-NS theory $a=0.1$ (left) and $a=0.5$ (right).
The horizontal dashed green line indicates the AdS Hagedorn temperature
$\hat{m}_{c}^{2}$.

To further study the profile of the AdS-sized string star solution we used
numerical analysis to find the non-linear solutions for (62) of
$\hat{\chi},\hat{\varphi},\hat{\psi}$ and $\hat{\zeta}$ as a function of $a$
and $\hat{m}^{2}$. We set the boundary condition $\hat{\psi}(0)=0$ to avoid
curvature singularity and for a smooth and normalizable solution

$\hat{\chi}^{\prime}(0)=\hat{\varphi}^{\prime}(0)=\hat{\psi}^{\prime}(0)=\hat{\zeta}^{\prime}(0)=0,\quad\hat{\chi}(\infty)=\hat{\varphi}(\infty)=\hat{\psi}(\infty)=\hat{\zeta}(\infty)=0.$
(77)

In practice, we chose $\rho_{\text{min}}=0.00001$ and $\rho_{\text{max}}=3$,
and tried to satisfy $\psi(\rho_{\text{min}})=0$ with

$\hat{\chi}^{\prime}(\rho_{\text{min}})=\hat{\varphi}^{\prime}(\rho_{\text{min}})=\hat{\psi}^{\prime}(\rho_{\text{min}})=\hat{\zeta}^{\prime}(\rho_{\text{min}})=0,\quad\hat{\chi}(\rho_{\text{max}})=\hat{\varphi}(\rho_{\text{max}})=\hat{\psi}(\rho_{\text{max}})=\hat{\zeta}(\rho_{\text{max}})=0,$
(78)

using the shooting method. That is, we optimized the initial values at
$\rho_{\text{min}}$ to satisfy the boundary condition at $\rho_{\text{max}}$.
Notice that the variable $\rho$ is already the dimensionless length in which
the characteristic size of the solution should be $O(1)$. Figure 4 shows the
profile of the solution at $\hat{m}^{2}=0$ for the pure NS-NS system $a=0$ and
for $a=0.5$. We found that the near-Hagedorn solutions require higher and
higher accuracy for smaller values of $a$. The reason is that $a$ controls the
size of the potential for $\hat{\chi}$. In the extreme case of $a=0$, the
linear equation for $\hat{\chi}$ has a continuous normalizable spectrum above
the Hagedorn temperature. As a result, below we treat $a$ as a regulator for
the simulation and plot $a=0.1$ instead of $a=0$.

As the solutions are monotonically decreasing functions of $x$, the initial
value at $\rho=0$ sets the amplitude of the solution. Figure 5 shows the
initial values $\hat{\chi}(0)$ (black), $\hat{\varphi}(0)$ (red) and
$\hat{\zeta}(0)$ (blue) as a function of the temperature $\hat{m}^{2}$. For
large $\hat{m}^{2}\gg 1$ (lower temperatures) the solution amplitude seems to
grow linearly with $\hat{m}^{2}$, or
$|\chi|\sim(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}$. This is the same scaling found
for (higher dimensional) flat space solutions Chen:2021dsw . As in flat space,
the solution is no longer reliable at
$|\chi|\sim(R_{0}^{2}-R_{H}^{2})/R_{H}^{2}\sim 1$ or for $\hat{m}^{2}\sim
l_{ads}^{2}/\alpha^{\prime}$. This is self-consistent, as we assumed
$\hat{m}^{2}\ll l_{ads}^{2}/\alpha^{\prime}$ to derive (62). Notice that the
temperature regime considered for finite RR flux $\hat{m}^{2}\sim
l_{ads}/l_{s}$ is inside the regime of validity.

As expected, the solution merges with the trivial solution
$\hat{\chi}=\hat{\varphi}=\hat{\psi}=\hat{\zeta}=0$ at the Hagedorn
temperature $\hat{m}^{2}_{c}$ (drawn in green dashed line). Close to
$\hat{m}^{2}_{c}$ the winding mode gets similar to the linear mode
$\hat{\chi}_{\text{lin}}$ found numerically above, while
$|\hat{\varphi}|,|\hat{\psi}|,|\hat{\zeta}|\ll|\hat{\chi}|$ are much smaller.
This behavior is similar to the one found for higher dimensional AdS string
stars Urbach:2022xzw  as well as for finite RR flux above (which is the large
$a\gg 1$ limit of the simulation).

Figure 6 shows the size $L$ of the winding mode profile (in AdS units) as a
function of the temperature. $L$ is defined as the value of $\rho$ at which
the amplitude of $\hat{\chi}$ reaches $e^{-1}\hat{\chi}(0)$. The length grows
with the temperature, as was found for finite $RR$ flux (including higher
dimensions Urbach:2022xzw ). For finite $\hat{m}^{2}$ the size is at the AdS
scale and bounded from above by the size of the linear solution at the
Hagedorn temperature $\hat{m}^{2}=\hat{m}^{2}_{c}$. At lower temperatures,
$\hat{m}^{2}\gg 1$ the solution size reaches the string scale and the
solutions are unreliable.

The normalized AdS3 string star free energy $F$ (60) is presented in figure 7
as a function of the temperature. The free energy turns out to always be
positive, which is not immediately clear from its form (60). The AdS3 string
star is therefore thermodynamically disfavored compared to the thermal AdS3
solution. As we explain above, the solution is also non-perturbatively
metastable due to the dominant BTZ solution. The free energy decreases with
the temperature and goes to zero at the Hagedorn temperature when the solution
becomes trivial. General EFT arguments predict that close enough to the
critical value $|\hat{\chi}|\sim(\hat{m}^{2}-\hat{m}^{2}_{c})^{1/2}$ (and
$|\hat{\varphi}|,|\hat{\psi}|,|\hat{\zeta}|\sim(\hat{m}^{2}-\hat{m}^{2}_{c})^{1}$).
As a result, we expect the Euclidean action to scale as

$I_{\text{String star}}-I_{\text{TAdS}_{3}}\sim\frac{\alpha^{\prime
2}}{G_{N}}(\hat{m}^{2}-\hat{m}^{2}_{c})^{2}\sim\frac{l_{s}}{G_{N}}\left(\frac{R_{0}-R_{c}}{R_{c}}\right)^{2},$
(79)

where in the last line we moved back to general units for brevity.

## Acknowledgments

I would like to thank Micha Berkooz, Shai Chester, Rajesh Gopakumar, Indranil
Halder, Daniel Jafferis, Yiyang Jia, Zohar Komargodski, Suman Kundu, Ohad
Mamroud, Adar Sharon, Tal Sheaffer, Gabriel Wong and Xi Yin for useful
discussions and comments. I especially thank Ofer Aharony and David Kutasov
for their guidance and comments on the manuscript.
This work was partly funded by an Israel Science Foundation center for
excellence grant (grant number 2289/18), by grant no. 2018068 from the United
States-Israel Binational Science Foundation (BSF), by the Minerva foundation
with funding from the Federal German Ministry for Education and Research, by
the German Research Foundation through a German-Israeli Project Cooperation
(DIP) grant “Holography and the Swampland”, and by a research grant from
Martin Eisenstein.

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