UUITP-16/23
MIT-CTP/5574

The ABJM Hagedorn Temperature from Integrability
Simon Ekhammara,b, Joseph A. Minahana,c, and Charles Thulla

aDepartment of Physics and Astronomy, Uppsala University,
Box 516, SE-751 20 Uppsala, Sweden

and

b Mathematics Department, King’s College London,
The Strand, London WC2R 2LS, UK

and

cCenter for Theoretical Physics, Massachusetts Institute of Technology
Cambridge, MA 02139, USA

{simon.ekhammar, joseph.minahan<EMAIL_ADDRESS>
###### Abstract

We use the quantum spectral curve to compute the Hagedorn temperature for ABJM
theory in terms of the interpolating function $h(\lambda)$. At weak coupling
we compute this temperature up to eight-loop order, showing that it matches
the known tree-level and two-loop results. At strong coupling we compute the
dependence numerically, showing that it is consistent with expectations from
supergravity and the plane-wave limit for the four leading terms in the strong
coupling expansion, up to an overall shift of the zero-point energy for type
IIA string theory on AdS${}_{4}\times\mathbb{C}\textrm{P}^{3}$. We conjecture
an analytic form for this shift to leading order that is consistent with our
numerical results.

###### Contents

  1. 1 Introduction
  2. 2 The Hagedorn temperature at weak and strong coupling
    1. 2.1 Weak coupling
    2. 2.2 Strong coupling and supergravity
    3. 2.3 Strong coupling and the plane-wave limit
  3. 3 The ABJM QSC
    1. 3.1 The AdS4 Hagedorn QSC
    2. 3.2 Perturbative solution
      1. 3.2.1 Going to higher orders
      2. 3.2.2 The Hagedorn temperature to order $\mathcal{O}(h^{8})$
    3. 3.3 The numerical solution
      1. 3.3.1 The algorithm
      2. 3.3.2 The results
  4. 4 Discussion
  5. A ABJM Conventions
  6. B Tree level solution
  7. C $T_{H}^{(4)}$

##  1 Introduction

This is a companion paper to [1] where we conjectured the form of the sub-
leading contributions to the Hagedorn temperature $T_{H}$ for planar
$\mathcal{N}=4$ super Yang-Mills at large coupling $\lambda$. There we
improved upon the numerical accuracy in [2] of $T_{H}$ as a function of
$g\equiv\frac{\sqrt{\lambda}}{4\pi}$, and demonstrated the consistency of our
conjecture.

In this paper we generalize our analysis to the ABJM case [3] and make a
conjecture for the asymptotic dependence of the Hagedorn temperature. We do so
by twisting the AdS4/CFT3 quantum spectral curve (QSC), thus extending the
integrability formalism of [4, 5, 2] from $\mathcal{N}=4$ to ABJM. We then
carry out the numerical computation of the Hagedorn temperature in terms of
the interpolating function $h(\lambda)$, where $\lambda=N/k$ is the ’t Hooft
coupling. We show that these results are consistent with the conjecture.

The conjecture relies on the supergravity analysis of a light winding mode
wrapped around the time direction in Euclidean AdS4. The first order
correction to the Hagedorn temperature follows from the analysis in [6]. In
this paper we compute the next two orders in the expansion. The first such
correction also depends on the shift of the zero-point energy of the string on
the curved AdS${}_{4}\times\mathbb{C}\textrm{P}^{3}$ background. We take
inspiration from the plane-wave limit to motivate the temperature dependence
of this shift. The numerical results indicate that the shift on the full space
should be 12/5 the shift for the plane-wave. Interestingly, this shift is 3/4
of what is required for $\mathcal{N}=4$ super Yang-Mills. This motivates us to
conjecture a shift for general $AdS_{d+1}$ where the shift is proportional to
$d$, the dimension of the dual CFT.

We also study the Hagedorn temperature at weak coupling. Here we are able to
compute it up to the eight-loop order, which goes well beyond the previous
limits. Our result at tree-level agrees with the previously computed value in
[7]. Our result at two loops agrees with the value of [8], once a subtle
factor of two is taken into account.

The rest of this paper is structured as follows. In section 2.1 we discuss the
Hagedorn temperature at tree-level and at the two-loop level. In section 2.2
we compute the supergravity corrections to the Hagedorn temperature by mapping
the problem of a light winding mode on the compactified time direction in
AdSd+1 to a perturbed radially symmetric harmonic oscillator in $d$
dimensions. In section 2.3 we discuss the plane-wave limit of the type IIA
string theory and show that it gives a first order correction to the Hagedorn
temperature which matches the supergravity result. We also show that the shift
to the zero-point energy effects the next correction, but not the following
one. In section 3, we outline the general form of the twisted AdS4 QSC before
specializing for the ABJM Hagedorn temperature. In section 3.2 we give the
perturbative solution of the QSC to order $h^{8}$ and in 3.3 we discuss the
numerical solution for strong coupling.

Note added: As this paper was in the final stages of preparation [9] appeared
which also derives (2.22) and (2.27) in this paper. Furthermore, their paper
gives a world-sheet argument for the conjectured value of the zero-point shift
given in (3.53).

##  2 The Hagedorn temperature at weak and strong coupling

###  2.1 Weak coupling

At zero coupling the partition function of ABJM can be computed by stringing
beads together into a single-trace partition function and then exponentiating
to multi-trace contributions. In ABJM we have two different types of “half-
beads” coming from the two basic sets of fields, $(\phi_{a},\psi^{a})$ and
$(\overline{\phi}{}^{a},\overline{\psi}_{a})$, which sit in the
$(N,\overline{N})$ and $(\overline{N},N)$ of the gauge groups, respectively.
Here $\phi$’s are bosons, $\psi$’s fermions and the index $a$ labels the
(anti-)fundamental representation of $\mathfrak{su}_{4}$. Due to the structure
of the gauge groups this means that the chain for a single-trace operator has
to be alternating.

The tree-level Hagedorn temperature was first computed in [7]. We will here
briefly recall how this computation is carried out. Let us define

$Z=\textrm{tr}\,e^{-\beta H_{0}}=\textrm{tr}\,w^{2D_{0}}\,,$ (2.1)

where $w\equiv e^{-\frac{1}{2T}}$ and $D_{0}$ is the dilatation operator for
the free theory. Then the single-bead partition function is

$\displaystyle Z_{bead}(w)=(Z_{\phi}(w)+Z_{\psi}(w))^{2}\,,$ (2.2)

where

$\displaystyle Z_{\phi}=\frac{4w(1+w^{2})}{(1-w^{2})^{2}}\,\quad{\rm and}\quad
Z_{\psi}=\frac{8w^{2}}{(1-w^{2})^{2}}\,.$ (2.3)

Hence, it follows that

$\displaystyle Z_{bead}=\left(\frac{4w}{(1-w)^{2}}\right)^{2}\,.$ (2.4)

The tree level value of the Hagedorn temperature is then found by setting
$Z_{bead}(w_{H})=1$, from which it follows that

$w_{H}=3-2\sqrt{2}\,,$ (2.5)

and so

$\displaystyle T_{H}$ $\displaystyle=\frac{1}{2\log(3+2\sqrt{2})}\approx
0.283648\,.$ (2.6)

The leading correction to the Hagedorn temperature at weak coupling in ABJM
theory was computed in [8] from the two-loop dilatation operator of [10, 11].
They found that111We have included a subtle factor 2 compared to the result
stated in [8]. This factor arises because the “pendant” should include a
contribution from both $\langle D_{2}(x)\rangle$ and
$\langle\overline{D}_{2}(x)\rangle$, where $\langle
D_{2}(x)\rangle=\textrm{Tr}_{\mathcal{V}^{1}\otimes\overline{\mathcal{V}}^{1}\otimes\mathcal{V}^{1}}x^{\Delta}D_{2}$
and
$\langle\overline{D}_{2}(x)\rangle=\textrm{Tr}_{\overline{\mathcal{V}}^{1}\otimes\mathcal{V}^{1}\otimes\overline{\mathcal{V}}^{1}}x^{\Delta}D_{2}$.
See [8] for details. We thank G. Papathanasiou and M. Spradlin for
correspondence on this isssue.

$\frac{\delta T_{H}}{T_{H}}=\frac{\lambda^{2}}{\sqrt{2}}\langle
D_{2}(w_{H}^{-2})\rangle=4\lambda^{2}(\sqrt{2}-1).$ (2.7)

In section 3.2 we will verify the tree-level and two-loop results, and also
find the next three terms in the perturbative expansion using the QSC.

###  2.2 Strong coupling and supergravity

In this subsection we generalize the discussion in [1] to theories whose
supergravity dual is $AdS_{d+1}$. In this case the Euclidean $AdS$ metric is
given by

$\displaystyle
ds^{2}=(1+R^{2})d\tau^{2}+\frac{dR^{2}}{1+R^{2}}+R^{2}d\Omega_{d-1}^{2}\,,$
(2.8)

where we have set the radius to $1$ and made the identification
$\tau\equiv\tau+\beta$. We then assume that a string winds once around the
$\tau$ direction such that the world-sheet fermions have anti-periodic
boundary conditions. These boundary conditions shift the zero-point energy of
the string to a nonzero value, $C=C_{0}+\Delta C$ where
$C_{0}=-2/\alpha^{\prime}$ is the shift in flat space. If $\beta$ is tuned
properly this winding mode is very light and we can use the supergravity
approximation. The winding mode is a scalar field $\chi$ and its contribution
to the action is

$\displaystyle\int
d^{d+1}X\sqrt{g}\left(\nabla^{\mu}\chi\nabla_{\mu}\chi+m^{2}(R)\chi^{2}\right)\,,$
(2.9)

where $m^{2}(R)$ is the radial dependent mass-function

$\displaystyle
m^{2}(R)=(1+R^{2})\left(\frac{\beta}{2\pi\alpha^{\prime}}\right)^{2}+C\,.$
(2.10)

Choosing $\chi$ to be massless means finding a $\tau$ independent normalizable
solution to the equations of motion. Assuming that $\chi$ only has $R$
dependence then leads to the equation

$\displaystyle-\frac{1}{2}\frac{1}{R^{d-1}}\frac{d}{dR}R^{d-1}\frac{d}{dR}\chi(R)+\frac{1}{2}\left(\frac{\beta}{2\pi\alpha^{\prime}}\right)^{2}R^{2}\,\chi(R)+\Delta
H\,\chi(R)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad=-\frac{1}{2}\left(C+\left(\frac{\beta}{2\pi\alpha^{\prime}}\right)^{2}\right)\chi(R)\,,$
(2.11)

where

$\displaystyle\Delta
H=-\frac{1}{2}\frac{1}{R^{d-1}}\frac{d}{dR}R^{d+1}\frac{d}{dR}\,.$ (2.12)

We want a solution to (2.2) that is normalizable and hence falls off as
$R\to\infty$. Hence, solving (2.2) is equivalent to solving the radial
eigenvalue equation for a perturbed $d$-dimensional harmonic oscillator with
frequency $\omega=\frac{\beta}{2\pi\alpha^{\prime}}$ and energy

$\displaystyle
E=-\frac{1}{2}\left(C+\left(\frac{\beta}{2\pi\alpha^{\prime}}\right)^{2}\right)\,.$
(2.13)

To lowest order we drop $\Delta H$ and set $C=-2/\alpha^{\prime}$. The ground-
state energy for the harmonic oscillator then satisfies

$\displaystyle
E=\frac{1}{2}\left(\frac{2}{\alpha^{\prime}}-\left(\frac{\beta}{2\pi\alpha^{\prime}}\right)^{2}\right)=\frac{d}{2}\,\omega=\frac{d}{2}\,\frac{\beta}{2\pi\alpha^{\prime}}\,,$
(2.14)

which has the solution

$\displaystyle\beta=\pi\alpha^{\prime}\left(\sqrt{\frac{8}{\alpha^{\prime}}+d^{2}}-d\right)\,.$
(2.15)

Hence, to leading order we have that

$\displaystyle
T_{H}=\frac{1}{\beta}=\frac{1}{2\sqrt{2\alpha^{\prime}}}+\frac{d}{8\pi}+\dots\,.$
(2.16)

The first order correction matches that in [6].

For the next two orders we use first and second order perturbation theory for
the perturbed oscillator, as well as the correction $\Delta C$. As described
in [1] and in the next subsection, an analysis of the plane-wave Hagedorn
temperature suggests that

$C=-\frac{2}{\alpha^{\prime}}+\frac{\beta^{2}}{2\pi^{2}\alpha^{\prime}}\Delta
c+\mathcal{O}(\alpha^{\prime}).$ (2.17)

Furthermore, the correction to the energy is

$\displaystyle\Delta E$ $\displaystyle=\langle\psi_{0}|\Delta
H|\psi_{0}\rangle+\frac{\langle\psi_{0}|\Delta
H|\psi_{2}\rangle\langle\psi_{2}|\Delta
H|\psi_{0}\rangle}{-2\omega}+\frac{\langle\psi_{0}|\Delta
H|\psi_{4}\rangle\langle\psi_{4}|\Delta H|\psi_{0}\rangle}{-4\omega}$ (2.18)
$\displaystyle=\frac{d(d+2)}{8}+0-\frac{d(d+2)}{32\omega}\,,$ (2.19)

where the relevant normalized wave-functions are

$\displaystyle\psi_{0}(R)$
$\displaystyle=\sqrt{\frac{2}{\Gamma(d/2)}}\omega^{d/4}e^{-\frac{1}{2}\omega
R^{2}}\,,$ $\displaystyle\psi_{2}(R)$
$\displaystyle=\sqrt{\frac{2}{\Gamma(d/2+1)}}\omega^{d/4+1}\left(2R^{2}-\frac{d}{\omega}\right)e^{-\frac{1}{2}\omega
R^{2}}\,,$ $\displaystyle\psi_{4}(R)$
$\displaystyle=\frac{1}{\sqrt{\Gamma(d/2+2)}}\omega^{d/4+2}\left(R^{4}-\frac{d+2}{\omega}R^{2}+\frac{d(d+2)}{4\omega^{2}}\right)e^{-\frac{1}{2}\omega
R^{2}}\,.$

Hence the equation in (2.14) becomes

$\frac{1}{2}\left(\frac{2}{\alpha^{\prime}}-\left(\frac{\beta}{2\pi\alpha^{\prime}}\right)^{2}-\frac{\beta^{2}}{2\pi^{2}\alpha^{\prime}}\Delta
c\right)=\frac{d}{2}\frac{\beta}{2\pi\alpha^{\prime}}+\frac{d(d+2)}{8}-\frac{d(d+2)}{32}\frac{2\pi\alpha^{\prime}}{\beta}\,,$
(2.20)

which we can rearrange into the form

$\displaystyle\frac{\beta^{2}}{4\pi\alpha^{\prime}}=2\pi-\frac{d}{2}\beta-\frac{\beta^{2}\Delta
c}{2\pi}-\frac{d(d+2)}{4}\pi\alpha^{\prime}+\frac{d(d+2)\pi^{2}(\alpha^{\prime})^{2}}{8\,\beta}\,.$
(2.21)

From this we find the Hagedorn temperature

$T_{H}(\alpha^{\prime})=\frac{1}{2\pi\sqrt{2\alpha^{\prime}}}+\frac{d}{8\pi}+\frac{d(d+1)+8\Delta
c}{16\sqrt{2}\pi}\sqrt{\alpha^{\prime}}+\frac{d(d+2)(4d-1)}{256\pi}\alpha^{\prime}+\mathcal{O}\left((\alpha^{\prime})^{3/2)}\right).$
(2.22)

Based on the numerical results in [1] and in section 3.3.2 we conjecture the
value of $\Delta c$ in (3.53).

Applying (2.22) to ABJM, we set $d=3$ and use the dictionary
$\alpha^{\prime}=\frac{1}{\pi\sqrt{2\hat{\lambda}}}$, where
$\hat{\lambda}\equiv\lambda-\frac{1}{24}$ is the shifted ’t Hooft parameter
[12] 222At the highest order shown in (2.22) it does not matter if we use
$\lambda$ or $\hat{\lambda}$. However, later when we discuss numerical results
we will go to higher order in $\hat{\lambda}^{-1}$, where $\hat{\lambda}$ is
the more natural parameter. We thank N. Bobev for comments about this.. Hence,
we find

$\displaystyle T_{H}(\hat{\lambda})$ $\displaystyle=$
$\displaystyle\frac{(\hat{\lambda}/2)^{1/4}}{2\sqrt{\pi}}+\frac{3}{8\pi}+\frac{3+2\Delta
c}{8\pi^{3/2}}\left(\frac{\hat{\lambda}}{2}\right)^{-1/4}+\frac{165}{512\pi^{2}}\left(\frac{\hat{\lambda}}{2}\right)^{-1/2}+\mathcal{O}(\hat{\lambda}^{-3/4})$
(2.23) $\displaystyle=$ $\displaystyle
0.237212\hat{\lambda}^{1/4}+0.119366+(0.0800874+0.0533916\Delta
c)\hat{\lambda}^{-1/4}$
$\displaystyle\qquad\qquad\qquad\qquad+0.0461774\hat{\lambda}^{-1/2}+\mathcal{O}(\hat{\lambda}^{-3/4})\,.$

Significantly, the structure of the proposed zero-point shift in (2.17) means
that the $\lambda^{-1/2}$ coefficient is independent of the shift.

###  2.3 Strong coupling and the plane-wave limit

In [1] we used the plane-wave limit of type IIB string theory to gain some
insight on the structure of the zero-point energy corrections. We also argued
that the first correction to the Hagedorn temperature can be extracted from
the growth of states in the plane-wave limit. In this subsection we will find
a similar structure when taking the plane-wave limit of type IIA string theory
on AdS${}_{4}\times\mathbb{C}\textrm{P}^{3}$.

The type IIA plane-wave metric and flux are given by [13, 14]

$\displaystyle
ds^{2}=-2dx^{+}dx^{-}A(x^{I})dx^{+}dx^{+}+\sum_{I=1}^{8}dx^{I}dx^{I}\,,$
$\displaystyle\qquad\qquad F_{+123}=\mu\,,\qquad F_{+4}=\frac{\mu}{3}\,,$
(2.24)

where the prefactor $A(x^{I})$ is given by

$\displaystyle
A(x^{I})=\left(\frac{\mu}{3}\right)^{2}\sum_{I=1}^{4}x^{I}x^{I}+\left(\frac{\mu}{6}\right)^{2}\sum_{I=5}^{8}x^{I}x^{I}\,.$
(2.25)

For the plane-wave reduction from AdS${}_{4}\times\mathbb{C}\textrm{P}^{3}$
one has that $\mu=3/\sqrt{2}$.

The free energy for the non-interacting type IIA plane-wave string theory was
computed in [15]. By looking for a divergence they found that the Hagedorn
temperature is found by solving the equation

$\frac{\beta^{2}}{4\pi\alpha^{\prime}}=2\pi-\frac{\beta\mu}{\sqrt{2}}+\frac{5\beta^{2}\mu^{2}}{36\pi}\log(2)\\\
+\sum_{k=2}^{\infty}\frac{(-1)^{k}\Gamma(k-\frac{1}{2})}{\Gamma(-\frac{1}{2})\Gamma(k+1)}16\pi(2^{2k-2}-1)(1+2^{-2k})\zeta(2k-1)\left(\frac{\beta\mu}{6\sqrt{2}\pi}\right)^{2k}\,.$
(2.26)

Note that the shift in the string zero-point energy has the form in (2.17).

If we use the flux for the AdS${}_{4}\times\mathbb{C}\textrm{P}^{3}$ reduction
[7], we find

$\displaystyle\frac{\beta^{2}}{4\pi\alpha^{\prime}}=2\pi-\frac{3}{2}\,\beta+\frac{5\beta^{2}}{8\pi}\log(2)+\dots\,.$
(2.27)

The term linear in $\beta$ matches the corresponding term in (2.21), hence the
growth of states in the plane-wave gives the same first-order correction to
the flat space Hagedorn temperature as computed from supergravity. The same
result can be found by generalizing the discussion in [16]. If we compare the
next term in (2.27) with (2.21) then we see that the plane wave gives $\Delta
c=-\frac{5}{4}\log(2)$.

##  3 The ABJM QSC

The ABJM QSC was first formulated in [17], building on the AdS4/CFT3 TBA [18,
19]. The full structure of the QSC is explained in detail in [20]. The main
object for us will be the “spinorial” Q-functions $Q_{a|i},\ Q^{a}{}_{|i},\
a,i=1,2,3,4$, constrained to satisfy $\det Q^{a}{}_{|i}=\det Q_{a|i}=-1$. As
opposed to the Q-functions that appear in the study of spin chains, the
$Q_{a|i}$ and $Q^{a}{}_{|i}$ are functions with branch points at $\pm
2h-\frac{i}{2}-in$, $n\in\mathbb{Z}_{\geq 0}$. Here $h(\lambda)$ is the
integrability coupling constant whose conjectured relation to $\lambda$ is
given by [21]

$\lambda=\frac{\sinh(2\pi
h)}{2\pi}{}_{3}F_{2}\left(\frac{1}{2},\frac{1}{2},\frac{1}{2};1,\frac{3}{2};-\sinh^{2}(2\pi
h)\right)\,.$ (3.1)

In particular, for $\lambda\ll 1$ we have that

$h(\lambda)=\lambda-\frac{\pi^{2}}{3}\lambda^{3}+\frac{5\pi^{4}}{12}\lambda^{5}-\frac{893\pi^{6}}{1260}\lambda^{7}+{\mathcal{O}}(\lambda^{9})$
(3.2)

while for $\lambda\gg 1$ the behavior is

$\displaystyle h(\lambda)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}\left(\lambda-\frac{1}{24}\right)}-\frac{\log(2)}{2\pi}+{\mathcal{O}}\left(e^{-\pi\sqrt{8\lambda}}\right)$
(3.3) $\displaystyle=$
$\displaystyle\sqrt{\frac{1}{2}\,\hat{\lambda}}-\frac{\log(2)}{2\pi}+{{\mathcal{O}}}\left(e^{-\pi\sqrt{8\hat{\lambda}}}\right)\,.$

At strong coupling the shifted ’t Hooft parameter $\hat{\lambda}$ is the more
natural parameter [12], as can be seen for example when localizing ABJM [22,
23].

$Q_{a|i}$ and $Q^{a}{}_{|i}$ are not independent but are related through

$\displaystyle-\kappa^{ij}Q^{a}{}_{|j}Q_{a|k}=\delta^{i}_{j}\,,$
$\displaystyle\kappa^{ij}Q^{a}{}_{|i}Q_{b|j}=\delta^{a}_{b}\,,$ (3.4)

with $\kappa^{ij}=(-1)^{i}\delta^{i+j,5}$. Two more important bilinear
expressions are

$\displaystyle\mathbf{P}_{A}=-\frac{1}{2}Q^{+}_{a|i}\kappa^{ij}\bar{\sigma}^{ab}_{A}Q_{b|j}^{-}\,,$
(3.5)
$\displaystyle\mathbf{Q}_{I}=-\frac{1}{2}(Q^{a}{}_{|i})^{+}\,\overline{\Sigma}_{I}^{ij}\,Q_{a|j}^{-}\,,$
(3.6)

where $A=1,\dots,6$ and $I=1\dots,5$. Explicit expressions for
$\bar{\sigma}^{ab}_{A}$, $\overline{\Sigma}_{I}^{ij}$, as well as other
matrices that appear throughout the main text are collected in Appendix A. The
$\mathbf{P}_{A}$ and $\mathbf{Q}_{I}$ have particularly simple analytic
properties. $\mathbf{P}_{A}$ must be a function with a short cut while
$\mathbf{Q}_{I}$ must be a function with a long cut. We show these cuts in
figure 1.

Figure 1: Cut structure of the $\mathbf{P}$-, $\mathbf{Q}$\- and
$Q_{a|i}$-functions.

To study the Hagedorn temperature we need to twist the QSC. This implies that
the Q-functions will have exponential asymptotics, as opposed to the
polynomial asymptotic featured in the spectral problem. We can turn on a total
of $3+2=5$ twists for $\mathfrak{osp}(6|4)$. We will use the notation
$\mathbf{x}=\\{x_{1},x_{2},x_{5}\\}$ for the $\mathfrak{so}_{6}$ twists and
$\mathbf{y}=\\{y_{1},y_{2}\\}$ for the
$\mathfrak{sp}_{4}\simeq\mathfrak{so}_{5}$ twists. In the fully twisted case
the asymptotics of $Q_{a|i}$ for large $u$, up to constants prefactors, are
given by

$\displaystyle
Q_{a|i}\sim\mathbf{x}^{iu\omega_{a}}\,\mathbf{y}^{-iu\nu_{i}}\,,$
$\displaystyle
Q^{a}{}_{|i}\sim\mathbf{x}^{-iu\omega_{a}}\,\mathbf{y}^{-iu\nu_{i}}\,,$
$\displaystyle\omega=\begin{pmatrix}+++\\\ +--\\\ -+-\\\
\--+\end{pmatrix}\,,\quad\nu=\begin{pmatrix}++\\\ +-\\\ -+\\\
\--\end{pmatrix}\,,$ (3.7)

where $\pm=\pm\frac{1}{2}$ so that, for example,
$\mathbf{x}^{+++}=x^{\frac{1}{2}}_{1}x_{2}^{\frac{1}{2}}x_{5}^{\frac{1}{2}}$.
Using (3.5) this translates in the fully twisted case into

$\displaystyle\mathbf{P}_{A}\sim x_{A}^{iu}\,,$
$\displaystyle\mathbf{Q}_{I}\sim y^{-iu}_{I}\,,$ (3.8)

with the identifications
$x_{3}=\frac{1}{x_{2}},x_{4}=\frac{1}{x_{1}},x_{6}=\frac{1}{x_{5}},y_{3}=\frac{1}{y_{2}}$
and $y_{4}=\frac{1}{y_{1}}$. These pairings are a consequence of our choice of
basis. We emphasize that (3.8) is valid only if we assume that all $x_{a}$ and
$y_{i}$ are independent, otherwise one should include additional powers of
$u$.

Throughout this article, we will work in the symmetric sector. In this case we
can identify

$\displaystyle Q^{a}{}_{|i}=-\kappa^{ab}Q_{b|j}\mathds{K}^{j}_{i}\,,$ (3.9)

which enforces $x_{5}=1$. Furthermore, in this sector we also have
$\mathbf{P}_{5}=\mathbf{P}_{6}$ and $\mathbf{Q}_{5}=0$.

###  3.1 The AdS4 Hagedorn QSC

We will from now on turn off all R-symmetry twists, thus setting $x_{a}=1$. We
fix the remaining twists to

$y_{1}^{-iu}=y_{2}^{-iu}\equiv y^{-iu}=e^{-\pi u}e^{-\frac{iu}{2\,T_{H}}}\,.$
(3.10)

The QSC naturally computes the Witten index. The factor of $e^{\pi u}$ twists
the trace in such a way to give instead the standard thermal partition
function. Since there is no twisting of the $R$-symmetry, the asymptotics of
(3.7) picks up additional powers of $u$, resulting in

$Q_{a|i}\sim\begin{pmatrix}y^{-iu}&u&1&y^{iu}\\\
y^{-iu}u^{2}&u^{3}&u^{2}&y^{iu}u^{2}\\\ y^{-iu}u&u^{2}&u&y^{iu}u\\\
y^{-iu}u^{3}&u^{4}&u^{3}&y^{iu}u^{3}\end{pmatrix}_{a|i}\,,$ (3.11)

and

$\displaystyle\mathbf{Q}_{I}\simeq\begin{pmatrix}B_{1}\,y^{-iu}\,u\\\
B_{2}\,y^{-iu}\\\ B_{3}\,y^{iu}\,u\\\ B_{4}\,y^{iu}\\\ \
0\end{pmatrix}_{I}\,,$
$\displaystyle\mathbf{P}_{A}\simeq\begin{pmatrix}A_{1}\,u\\\ A_{2}\\\
A_{3}\,u^{4}\\\ A_{4}\,u^{3}\\\ A_{5}\,u^{2}\\\
A_{5}\,u^{2}\end{pmatrix}_{A}\,.$ (3.12)

The prefactors $A$ and $B$ satisfy the relations

$\displaystyle-
A_{2}A_{3}=-\frac{1}{4}A_{1}A_{4}=\frac{1}{3}A_{5}A_{6}=\frac{4}{3}\cosh^{4}{\frac{1}{4T_{H}}}\,,$
$\displaystyle B_{1}B_{4}=B_{2}B_{3}=i\coth^{3}{\frac{1}{4T_{H}}}\,.$ (3.13)

Q-functions are not uniquely specified, rather we can transform our entire
Q-system by acting on $Q_{a|i}$ with linear transformations acting on either
the $a$ or $i$ index. These transformations are usually called $H$-rotations.
These transformations are further restricted by 3.9 and by (3.11). We found it
convenient to fix part of this freedom by imposing that

$\displaystyle A_{1}=A_{2}=1\,,$ $\displaystyle
A_{5}=A_{6}=2\cosh^{2}{\frac{1}{4T_{H}}}\,,$ $\displaystyle
B_{1}=B_{2}=8\cosh^{3}{\frac{1}{4T}}\,.$ (3.14)

The Q-system also exibits parity symmetry, that is, after sending
$u\rightarrow-u$ the Q-system is equivalent to the original one up to symmetry
transformations. Explicitly

$\displaystyle\underline{Q}_{a|i}=\mathbb{g}_{a}{}^{b}\,Q_{b|j}\,(\Omega^{j}{}_{i})^{+}$
$\displaystyle\underline{\mathbf{P}}_{A}=\textrm{diag}(-1,1,1,-1,1,1)_{A}{}^{B}\mathbf{P}_{B}\,,$
(3.15)

where we have used the notation $f(-u)=\underline{f}(u)$. Here $\mathbb{g}$ is
a constant matrix while $\Omega^{i}{}_{j}=(\Omega^{i}{}_{j})^{[4]}$ is in
general a more complicated function with branch cuts. However, at large $u$ it
is simply a constant. With our specific choice of gauge

$\displaystyle\mathbb{g}=\begin{pmatrix}i&0&0&0\\\ 0&i&0&0\\\ 0&0&-i&0\\\
0&0&0&-i\end{pmatrix}\,,$
$\displaystyle\Omega\simeq\begin{pmatrix}0&0&0&-\frac{1}{8\sinh^{3}{\frac{1}{2T_{H}}}}\\\
0&i&0&0\\\ 0&0&-i&0\\\ 8\sinh^{3}{\frac{1}{2T_{H}}}&0&0&0\\\ \end{pmatrix}\,.$
(3.16)

The interesting part of these matrices is the distribution of non-zero entries
rather than their exact value, which is dependent on normalization.

Since $\mathbf{P}_{A}$ is a function with one short cut, we can parameterize
it as

$\mathbf{P}_{A}=\sum_{m=n_{A}}^{\infty}\frac{c_{A,m}}{x(u)^{m}}\,,$ (3.17)

where $x(u)$ is the Zhukovsky variable

$\displaystyle x(u)=\frac{1}{2h}\left(u+\sqrt{u+2h}\sqrt{u-2h}\right)\,,$
(3.18)

and $n_{A}$ is an $A$ dependent integer. Using the parity symmetry and the
asymptotic behavior described in (3.12) and (3.13), our ansatz in (3.17) takes
the form

$\displaystyle\mathbf{P}_{1}$
$\displaystyle=xh+\sum_{n=1}^{\infty}c_{1,2n-1}\frac{h^{2n-1}}{x^{2n-1}}\,,$
(3.19a) $\displaystyle\mathbf{P}_{2}$
$\displaystyle=1+\sum_{n=1}^{\infty}c_{2,2n}\frac{h^{2n}}{x^{2n}}\,,$ (3.19b)
$\displaystyle\mathbf{P}_{3}$
$\displaystyle=-\frac{4\cosh^{4}\left(\frac{1}{4T_{H}}\right)}{3}(xh)^{4}\left(1+\sum_{n=2}^{\infty}c_{3,2n-4}\frac{h^{2(n-4)}}{x^{2n}}\right)\,,$
(3.19c) $\displaystyle\mathbf{P}_{4}$
$\displaystyle=-\frac{16\cosh^{4}\left(\frac{1}{4T_{H}}\right)}{3}(xh)^{3}\left(1+c_{4,-1}\frac{h^{-2}}{x^{2}}+\sum_{n=2}^{\infty}c_{4,2n-3}\frac{h^{2(n-3)}}{x^{2n}}\right)\,,$
(3.19d) $\displaystyle\mathbf{P}_{5}=\mathbf{P}_{6}$
$\displaystyle=2\cosh^{2}\left(\frac{1}{4T_{H}}\right)(xh)^{2}\left(1+c_{5,0}\frac{h^{-2}}{x^{2}}+\sum_{n=2}^{\infty}c_{5,2n-2}\frac{h^{2(n-2)}}{x^{2n}}\right)\,,$
(3.19e)

where powers of $h$ have been inserted so that
$c_{A,n}\simeq\mathcal{O}(h^{0})$. We have also used the $H$-symmetry to
eliminate a possible $x^{2}$ term in $\mathbf{P}_{4}$.

Finally, to close the QSC we supplement it with the so-called gluing
conditions. As a function with short cuts $\mathbf{Q}_{I}$ is an upper half-
plane analytic function with a ladder of cuts
$(-2h-in,2h-in),n\in\mathbb{Z}_{\geq 0}$ in the lower half-plane. In order to
restore the long cut structure described previously, the analytic continuation
of $\mathbf{Q}_{I}$ around the branch-points on the real axis, denoted by
$\widetilde{\mathbf{Q}}_{I}$, must be a lower half-plane analytic function
with a ladder of cuts in the upper half-plane. Luckily, we can easily
construct such a function; namely
$\mathbf{Q}_{I}(-u)=\underline{\mathbf{Q}}_{I}$. We demand that
$\widetilde{\mathbf{Q}}_{I}=\mathcal{L}_{I}{}^{J}\underline{\mathbf{Q}}_{J}$
where $\mathcal{L}_{I}^{J}$ is a matrix with periodicity
${\mathcal{L}_{I}{}^{J}}^{[2]}=\mathcal{L}_{I}{}^{J}$. To fix the form of
$\mathcal{L}_{I}{}^{J}$ one should utilize the so-called
$\mathbf{Q}\,\omega$-system as done in [20] for the spectral problem. We will
not perform this analysis but rather argue the general form of
$\mathcal{L}_{I}{}^{J}$ from symmetry.

To constrain $\mathcal{L}_{I}{}^{J}$ we use the asymptotic properties of
$\mathbf{Q}_{I}$. At $u=i\infty$ we have that
$\mathbf{Q}_{1}>\mathbf{Q}_{2}>\mathbf{Q}_{3}>\mathbf{Q}_{4}$, and demand that
the same ordering is true for $\widetilde{\mathbf{Q}}_{I}$ at $-i\infty$. This
means that $\mathcal{L}_{I}{}^{J}$ is upper-triangular. We also require that
the ordering is preserved at $u=-\infty$ which determines the exponential
terms on the diagonal. The square-root cuts further add the restriction
$\mathcal{L}_{I}{}^{J}\underline{\mathcal{L}}_{J}{}^{K}=\delta_{I}^{K}$
constraining the coefficients on the diagonal. Finally, we require that we do
not add terms with different powers of $u$ in their asymptotics. Supplementing
these restrictions with the weak coupling algorithm in the next section we
found the matrix relation

$\widetilde{\mathbf{Q}}_{I}={\begin{pmatrix}e^{-2\pi u}&0&l_{13}&0&0\\\
0&-e^{-2\pi u}&0&l_{24}&0\\\ 0&0&-e^{2\pi u}&0&0\\\ 0&0&0&e^{2\pi u}&0\\\
0&0&0&0&\bullet\end{pmatrix}_{I}}^{\ J}\,\underline{\mathbf{Q}}_{J}\,.$ (3.20)

The explicit form of $\mathcal{L}$ depends on $H$-rotations. For the choices
described in this section we find that
$l_{13}=-l_{24}=16i\sinh^{2}{\frac{1}{2T_{H}}}$.

###  3.2 Perturbative solution

We now turn to the question of solving the constraints described in the
previous section to obtain $T_{H}$. In this section we consider the case at
weak coupling, while in section 3.3 we consider strong coupling.

The first step in solving the ABJM QSC at weak coupling is to find the
Q-system at zero coupling. At $h=0$ we expect that the Q-system should be
fully regular, that is, $Q_{a|i}$ should be polynomials in $u$ with
exponential prefactors. To emphasize that we are now working at $h=0$ we write
$y_{0}^{-iu}=e^{-\pi u}\,w_{H}^{iu},\ w_{H}=\exp(-\frac{1}{2T_{H}^{(0)}})$.
The polynomial structure of the Q-functions at $h=0$ can be anticipated both
from the knowledge of how to treat a twisted spin-chain using Q-functions [24]
and from the Hagedorn QSC for AdS${}_{5}\times$S5 [2]. To fix the exact form
of the Q-functions we use that the ABJM QSC requires that [20]

$\displaystyle Q_{A|IJ}$
$\displaystyle=(\overline{\sigma}_{A})^{ab}Q_{a|i}^{+}\kappa^{ij}(\Sigma_{IJ})_{j}{}^{k}Q_{b|k}^{-}=-(\sigma_{A})_{ab}(Q^{b}{}_{|i}{})^{-}\kappa^{ij}(\Sigma_{IJ})_{j}{}^{k}(Q^{a}{}_{|k}{})^{+}\,,$
$\displaystyle Q_{AB|I}$
$\displaystyle=(\sigma_{AB})_{a}{}^{b}(Q^{a}{}_{|i}{})^{+}(\overline{\Sigma}_{I})^{ij}Q_{b|j}^{-}=(\sigma_{AB})_{a}{}^{b}(Q^{a}{}_{|i}{})^{-}(\overline{\Sigma}_{I})^{ij}Q_{b|j}^{+}\,.$

as well as (3.4). In addition, we use the last $H$-symmetry to impose the
condition $\mathbf{Q}_{1}\propto u$. This fixes all constants in the ansatz,
leaving only the Hagedorn temperature to be fixed. We collect the explicit
expressions for $\mathbf{P}$ and $\mathbf{Q}$ in Appendix B.

To fix $T_{H}$ we compute $\widetilde{\mathbf{Q}}_{I}$ using (3.20). Since
$\widetilde{\mathbf{Q}}_{I}+\mathbf{Q}_{I}$ and
$\frac{\widetilde{\mathbf{Q}}_{I}-\mathbf{Q}_{I}}{\sqrt{u-2h}\sqrt{u+2h}}$ do
not have cuts on the real axis, they must be regular functions around $u\simeq
0$, that is

$\displaystyle\widetilde{\mathbf{Q}}_{I}+\mathbf{Q}_{I}\bigg{|}_{u\simeq
0}=\text{reg}\,,$
$\displaystyle\frac{\widetilde{\mathbf{Q}}_{I}-\mathbf{Q}_{I}}{\sqrt{u-2h}\sqrt{u+2h}}\bigg{|}_{u\simeq
0}=\text{reg}\,.$ (3.21)

Imposing these relations at $h=0$ implies that
$T_{H}^{(0)}=\frac{1}{2\log(3+2\sqrt{2})}$, which we recognize as a solution
to $Z_{bead}=1$ 333The other solution to $Z_{bead}=1$ results in a negative
temperature which we discard.. This is how the QSC finds the Hagedorn
temperature at $h=0$.

####  3.2.1 Going to higher orders

Techniques to solve the AdS4 QSC perturbatively at small coupling were
developed in [25, 26], see also [27, 28, 29]. We will introduce a slightly
different algorithm inspired by [30].

To describe our perturbative algorithm we start by rewriting (3.5) as

$Q_{a|i}^{+}+\mathbf{P}_{ab}\kappa^{bc}Q^{-}_{c|j}\mathds{K}^{j}_{i}=0\,,$
(3.22)

where $\mathbf{P}_{ab}=\mathbf{P}_{A}\,\sigma^{A}_{ab}$. Let $Q^{(m)}_{a|i}$
solve (3.22) to order $\mathcal{O}(h^{2m})$ and write

$\displaystyle
Q_{a|i}=Q_{a|i}^{(m)}+h^{2m+2}Q_{a|j}^{(0)}(b^{j}_{i})^{+}+\mathcal{O}(h^{2m+4})\,.$
(3.23)

Using (3.4) we can solve for $b^{i}_{j}$ as

$b^{i}_{j}-(-1)^{\frac{i(i-1)+j(j-1)}{2}}(b^{[2]})^{i}_{j}=(dS^{(m+1)})^{i}_{j}\,,$
(3.24)

where

$\displaystyle(dS^{(m+1)})^{i}_{j}=-\mathds{K}_{k}^{i}\kappa^{kl}((Q^{a}{}_{|l})^{(0)})^{+}\mathbf{P}_{ab}\kappa^{bc}(Q_{c|j}^{(m)})^{-}\bigg{|}_{h^{2m+2}}\,.$
(3.25)

We then notice that
$(dS^{(m)})^{k}_{l}=y_{0}^{-iu\lfloor\frac{k}{2}\rfloor+iu\lfloor\frac{l}{2}\rfloor}(ds^{(m)})^{k}_{l}$
, where $ds^{(m)}$ does not contain any exponential prefactors. If we now
choose the parameterization
$b^{k}_{l}=y_{0}^{-iu\lfloor\frac{k}{2}\rfloor+iu\lfloor\frac{l}{2}\rfloor}p^{k}_{l}$
, then (3.24) becomes

$\displaystyle
p^{k}_{l}-z\left(p^{k}_{l}\right)^{[2]}=(ds^{(m+1)})^{k}_{l}\,,$
$\displaystyle
z\in\\{1,w_{H},w_{H}^{2},\frac{1}{w_{H}},\frac{1}{w_{H}^{2}}\\}\,,$ (3.26)

where $w_{H}$ is given in (2.5). It is interesting to note that the additional
signs appearing in (3.24) cancel against signs coming from $e^{\pm\pi u}$.

The technology for solving equations of the type appearing in (3.26) was
developed in [31, 32, 33]. Let us briefly recall the procedure. In the first
iteration the source-term is $ds^{(1)}$, which from our parameterization is a
Laurent polynomial in $u$. For positive powers of $u$ it is straightforward to
find $p$ as a polynomial in $u$. For negative powers we need the twisted
$\eta$-function $\eta^{z}_{s}$ [30], which are functions without poles in the
upper half-plane that solve

$\eta^{z}_{s}-z\,(\eta^{z}_{s})^{[2]}=\frac{1}{u^{s}}\,.$ (3.27)

A formal expression is given by

$\eta^{z}_{s}=\sum_{n=0}^{\infty}\frac{z^{n}}{(u+in)^{s}}\,.$ (3.28)

Going to $\mathcal{O}(h^{4})$ will require $\eta^{z}_{s}$ appearing on the
right-hand side of (3.26). To solve equations of this type we need generalized
twisted $\eta$-functions, which are defined to satisfy

$\eta_{s,S}^{z,Z}-zZ\left(\eta_{s,S}^{z,Z}\right)^{[2]}=\frac{Z}{u^{s}}(\eta^{Z}_{S})^{[2]}\,,$
(3.29)

without poles in the upper half-plane and with $S$ and $Z$ multi-indices. A
formal expression for
$\eta^{z_{1},z_{2},\dots,z_{k}}_{s_{1},s_{2},\dots,s_{k}}$ is

$\eta_{s_{1},s_{2},\dots,s_{k}}^{z_{1}\,z_{2},\dots z_{k}}=\sum_{0\leq
n_{1}<n_{2}<\dots<n_{k}<\infty}\frac{z_{1}^{n_{1}}}{(u+in)^{s_{1}}}\frac{z_{2}^{n_{2}}}{(u+in_{2})^{s_{2}}}\dots\frac{z_{k}^{n_{k}}}{(u+in_{k})^{s_{k}}}\,.$
(3.30)

There is no need to introduce any more functions.

We note that there are also homogeneous solutions to (3.26). For example, when
$z=1$ we can always add an $i$-periodic function to $p$ and still solve
$\eqref{eq:finiteDiffP}$. However, since by construction $p$ must have
polynomial asymptotic behavior and cannot have poles in the upper-half plane,
we can discard all homogeneous solutions except for constants. We thus write

$p_{l}{}^{k}=\Psi_{z}\left((ds^{(m)})^{k}_{l}\right)+\delta_{\lfloor\frac{k}{2}\rfloor,\lfloor\frac{l}{2}\rfloor}(p_{H})^{k}_{l}\,,$
(3.31)

with $(p_{H})^{k}_{l}$ constants and $\Psi_{z}$ a particular solution of
(3.26) analytic in the upper half-plane, i.e. written in terms of
$\eta$-functions.

Thus, at each order in $h$ we can find $Q_{a|i}$ as a function of
$\eta$-functions and $u$ with undetermined coefficients $c_{A,m}$ and $p_{H}$.
To fix these coefficients we follow the strategy in [2]. The first step is to
construct $Q^{a}{}_{|i}$ using (3.9) and then impose (3.4). The second step is
to verify the asymptotics (3.12) for $\mathbf{Q}_{I}$. In our weak-coupling
expansion

$T_{H}=\sum_{m=0}^{\infty}T_{H}^{(m)}\,h^{2m}\,,$ (3.32)

so that

$y^{iu}=e^{\pi
u}e^{\frac{iu}{2T_{H}^{(0)}}}\left(1-iu\,\frac{T_{H}^{(1)}}{2(T_{H}^{(0)})^{2}}\,h^{2}\right)+\mathcal{O}(h^{4})\,.$
(3.33)

It follows that $\mathbf{Q}_{I}$, and all other functions containing
$y^{-iu}$, when expanded first in $h$, will look like they contain higher
powers of $u$ compared to (3.12). To make sure we still have the correct
asymptotic behavior we solve [2]

$\frac{\mathbf{Q}_{1}}{\mathbf{Q}_{2}}\simeq u\,.$ (3.34)

The final step in the algorithm is to impose (3.21). When doing so it is
necessary to expand the $\eta$-functions close to $u=0$. As can be seen from
(3.30), this will introduce the polylogarithms

$\displaystyle\text{Li}_{s_{1},\dots,s_{k}}(z_{1},z_{2},\dots,z_{k})=\sum_{1\leq
n_{1}<n_{2}<\dots<n_{k}}\frac{z_{1}^{n_{1}}\dots
z_{k}^{n_{k}}}{n_{1}^{s_{1}}\dots n_{k}^{s_{k}}}\,,$ (3.35)

with $z_{i}\in\\{1,w_{H},w_{H}^{2},w^{-1}_{H},w^{-2}_{H}\\}$. After ensuring
that (3.21) is satisfied we can read off $T_{H}$ from (3.13).

####  3.2.2 The Hagedorn temperature to order $\mathcal{O}(h^{8})$

Using the algorithm described in section 3.2.1 we computed the Hagedorn
temperature up to $\mathcal{O}(h^{8})$. The results are (using
$w_{H}=3-2\sqrt{2}$)

$\displaystyle T_{H}^{(0)}=$
$\displaystyle\frac{1}{2\log\left(3+2\sqrt{2}\right)}\approx
0.283648164276628\,,$ (3.36) $\displaystyle T_{H}^{(1)}=$
$\displaystyle\frac{\sqrt{2}-1}{\log\left(1+\sqrt{2}\right)}\approx
0.469963666342443\,,$ (3.37)
$\begin{split}T_{H}^{(2)}=&7\sqrt{2}-8-4\left(1+2\sqrt{2}\right)\text{Li}_{1}\left(\frac{1}{\left(1+\sqrt{2}\right)^{2}}\right)-\frac{2\left(1+2\sqrt{2}\right)\text{Li}_{2}\left(\frac{1}{\left(1+\sqrt{2}\right)^{2}}\right)}{\log\left(1+\sqrt{2}\right)}\\\
\approx&-2.54281120753405\,,\end{split}$ (3.38)
$\displaystyle\begin{split}T_{H}^{(3)}=&\frac{4}{3}\left(48\text{Li}_{1,1}\left(w_{H},w_{H}\right)-48\text{Li}_{1,1}\left(w_{H}^{-1},w_{H}\right)\right.\\\
&+12\sqrt{2}\left(\text{Li}_{2}\left(w_{H}^{2}\right)-2\log\left(w_{H}\right)\text{Li}_{1}\left(w_{H}^{2}\right)\right)\\\
&+(72\sqrt{2}+48)\left(-\frac{\log\left(w_{H}\right)}{6}\text{Li}_{2}\left(w_{H}\right)+\frac{1}{2}\text{Li}_{3}\left(w_{H}\right)-\frac{\text{Li}_{4}\left(w_{H}\right)}{2\log\left(w_{H}\right)}\right)\\\
&+\text{Li}_{1}\left(w_{H}\right)\left(48\sqrt{2}\text{Li}_{2}\left(w_{H}\right)-9\left(4-5\sqrt{2}\right)\log\left(w_{H}\right)\right)\\\
&-24\sqrt{2}\log\left(w_{H}\right)\left(\text{Li}_{1}\left(w_{H}\right)\right)^{2}-\frac{6\sqrt{2}\left(\text{Li}_{2}\left(w_{H}\right)\right)^{2}}{\log\left(w_{H}\right)}\\\
&\left.+(-45\sqrt{2}+84)\text{Li}_{2}\left(w_{H}\right)-\frac{1}{2}(35\sqrt{2}-52)\log\left(w_{H}\right)+45\sqrt{2}-66\right)\end{split}$
(3.39) $\displaystyle\approx$ $\displaystyle 21.7782105898884\,,$
$\displaystyle T_{H}^{(4)}\approx$ $\displaystyle-222.299692062791\,.$ (3.40)

The analytic expression for $T^{(4)}_{H}$ is given in appendix C. All
numerical estimates have been produced using DiffExp [34]. We see that our
result for $T_{H}^{(0)}$ and $T_{H}^{(1)}$ agrees with the tree-level result
of [7] and with the 2-loop calculation in [8], up to the aforementioned factor
of $2$. The results for $T_{H}^{(2)},T_{H}^{(3)}$ and $T_{H}^{(4)}$ are new.

When the spectral problem for AdS4 was explored at finite coupling [25, 26], a
Padé approximation computed from the perturbative data was used to get a
better agreement with the numerical results. For the Hagedron temperature we
can do the same and find that the $[4/4]$ Padé approximation is

$T_{H}^{\textrm{Padé}[4/4]}=\frac{0.283648164276628+4.16468030174843h^{2}+10.4253195236392h^{4}}{1+13.0257026158742h^{2}+24.1373808661550h^{4}}.$
(3.41)

###  3.3 The numerical solution

To go to strong coupling we will resort to numerics. We will use standard
methods first introduced for $\mathcal{N}=4$ in [35] and implemented for the
spectral problem of ABJM in [26] (see [36] for earlier numerical results from
TBA). However, due to our exponential asymptotics we are forced to work
slightly harder compared to the spectral problem, just as in [2].

####  3.3.1 The algorithm

We can also use the ansatz in (3.19) for the numerical solution. We only need
to truncate all sums at $n=K$ so as to work with a finite number of unknowns.
We then assume that for large $u$ that $Q_{a,i}$ takes the form

$Q_{a|i}=y^{s_{i}iu}u^{p_{a|i}}\sum_{n=0}^{N}b_{a|i}^{(n)}u^{-n}\,,$ (3.42)

where

$\displaystyle s_{i}=\begin{cases}-1,\ i=1\\\ 1,\ i=4\\\ 0,\ i=2,3\end{cases}$
$\displaystyle p_{a|i}=\delta_{i,2}+\begin{cases}0&a=1\\\ 2&a=2\\\ 1&a=3\\\
3&a=4\end{cases}\,,$

and $N$ is an integer cut-off that satisfies $N=2K-2$. The large $u$ expansion
of $Q_{a|i}$ is constrained by the parity condition (3.15). Parity implies
that $b_{a|2}^{(n)},b_{a|3}^{(n)}$ are zero for $n$ odd and

$b_{a|4}^{(n)}=\frac{i(-1)^{n}}{8\sinh^{3}\frac{1}{2T_{H}}}b_{a|1}^{(n)}$
(3.43)

for all $n$. Note that the asymptotic form of $\Omega$ in (3.16) is exact only
up to exponentially suppressed terms at large u, so we can impose these parity
constraints only on the large-$u$ expansion. As a gauge choice we also set

$\displaystyle b_{1|1}^{(0)}$
$\displaystyle=2\cosh\left(\frac{1}{4T_{H}}\right),$ $\displaystyle
b_{1|2}^{(0)}$
$\displaystyle=\frac{1}{2\sinh\left(\frac{1}{4T_{H}}\right)}\,.$ (3.44)

With these simplifications we can start solving order by order in $u^{-1}$ the
equation

$\displaystyle
Q_{a|i}^{+}+\mathbf{P}_{ab}\kappa^{bc}Q_{c|j}^{-}\mathds{K}^{j}_{i}=0\,.$
(3.45)

At the leading five orders we solve for the following coefficients:

$\displaystyle u^{6}$ $\displaystyle:$ $\displaystyle b_{4|2}^{(0)},$
$\displaystyle u^{5}$ $\displaystyle:$ $\displaystyle
b_{4|3}^{(0)},b_{3|2}^{(0)},b_{2|2}^{(0)},b_{4|1}^{(0)},$ $\displaystyle
u^{4}$ $\displaystyle:$ $\displaystyle
b_{4|2}^{(2)},b_{3|3}^{(0)},b_{2|3}^{(0)},b_{4|1}^{(1)},b_{2|1}^{(0)},b_{3|1}^{(0)},$
$\displaystyle u^{3}$ $\displaystyle:$ $\displaystyle
b_{4|3}^{(2)},b_{3|2}^{(2)},b_{2|2}^{(2)},b_{4|1}^{(2)},b_{2|1}^{(1)},b_{3|1}^{(1)},$
$\displaystyle u^{2}$ $\displaystyle:$ $\displaystyle
b_{4|2}^{(4)},b_{3|3}^{(2)},b_{2|3}^{(2)},b_{4|1}^{(3)},b_{2|1}^{(2)},b_{3|1}^{(2)},c_{5,0},c_{4,-1}.$

At this point it is convenient to solve a constraint coming from the
restriction to the symmetric sector. Combining (3.9) and (3.4) leads to

$\displaystyle\kappa^{ij}Q_{a|j}(-\kappa^{bc}Q_{c|k}\mathds{K}^{k}_{i})=\delta_{a}^{b}\,.$
(3.46)

Expanding this to order $u^{4}$ then fixes $b_{1|3}^{(0)}$. Now we can go back
to the main equation where we solve the next three orders:

$\displaystyle u^{1}$ $\displaystyle:$ $\displaystyle
b_{4|3}^{(4)},b_{3|2}^{(4)},b_{2|2}^{(4)},b_{4|1}^{(4)},b_{2|1}^{(3)},b_{3|1}^{(3)},$
$\displaystyle u^{0}$ $\displaystyle:$ $\displaystyle
b_{4|2}^{(6)},b_{3|3}^{(4)},b_{2|3}^{(4)},b_{4|1}^{(5)},b_{2|1}^{(4)},b_{3|1}^{(4)},c_{4,1},c_{3,0},$
$\displaystyle u^{-1}$ $\displaystyle:$ $\displaystyle
b_{4|3}^{(6)},b_{3|2}^{(6)},b_{2|2}^{(6)},b_{4|1}^{(6)},b_{2|1}^{(5)},b_{3|1}^{(5)},b_{1|1}^{(1)}.$

After this, for all $k\geq 1$ we can solve in order for:

$\displaystyle u^{-2k}$ $\displaystyle:$ $\displaystyle
b_{4|2}^{(6+2k)},b_{3|3}^{(4+2k)},b_{2|3}^{(4+2k)},b_{4|1}^{(5+2k)},b_{2|1}^{(4+2k)},b_{3|1}^{(4+2k)},b_{1|1}^{(2k)},b_{1|2}^{(2k)},c_{3,2k},$
$\displaystyle u^{-(2k+1)}$ $\displaystyle:$ $\displaystyle
b_{4|3}^{(6+2k)},b_{3|2}^{(6+2k)},b_{2|2}^{(6+2k)},b_{4|1}^{(6+2k)},b_{2|1}^{(5+2k)},b_{3|1}^{(5+2k)},b_{1|1}^{(1+2k)},b_{1|3}^{(2k)}.$

The aim now is to check that $\mathbf{Q}_{I}$ glues correctly on the cut along
the real line at the points

$I={\left\\{-2h\cos\left(\frac{\pi}{I_{P}}\left(n-\frac{1}{2}\right)\right)\right\\}_{n=1,\dots,I_{P}}}\,.$

To get there we can start for each $v\in I$ with $\mathbf{Q}_{a|i}$ from
$u=v+i\frac{U}{2}$ for $U$ a large odd positive integer and iterate using the
equation

$\displaystyle Q_{a|i}^{-}$
$\displaystyle=-\mathbf{P}_{ab}\kappa^{bc}Q_{c|j}^{+}\mathds{K}^{j}_{i}$
(3.47)

to get $\mathbf{Q}_{a|i}^{+}(v)$. Then we can compute $\mathbf{Q}_{I}$ and
$\widetilde{\mathbf{Q}}_{I}$ on opposite sides of the cut from

$\displaystyle\mathbf{Q}_{I}(v+i0)$
$\displaystyle=-\frac{1}{2}\left(\overline{\Sigma}_{I}\right)^{ij}\kappa^{ab}Q_{b|k}^{+}(v)\mathds{K}^{k}_{i}\mathbf{P}_{ac}(v+i0)\kappa^{cd}Q_{d|l}^{+}(v)\mathds{K}^{l}_{j}$
(3.48) $\displaystyle\widetilde{\mathbf{Q}}_{I}(v-i0)$
$\displaystyle=\mathcal{L}_{I}{}^{J}(x)\left(-\frac{1}{2}\left(\overline{\Sigma}_{J}\right)^{ij}\kappa^{ab}Q_{b|k}^{+}(-v)\mathds{K}^{k}_{i}\mathbf{P}_{ac}(-v+i0)\kappa^{cd}Q_{d|l}^{+}(-v)\mathds{K}^{l}_{j},\right)$
(3.49)

with $\mathcal{L}$ the gluing matrix from (3.20). We can also get
$\widetilde{\mathbf{P}}$ from $\mathbf{P}$ by sending the Zhukovsky variable
$x$ to $\frac{1}{x}$ and thus get tilded versions of the above expressions by
replacing $\mathbf{P}$ with $\widetilde{\mathbf{P}}$. Continuity through the
cut implies that for the exact solution the function

$F(\\{c_{A,n}\\},y)=\sum_{v\in
I}\sum_{J=3}^{4}\left(\left|\frac{\mathbf{Q}_{J}(v+i0)}{\widetilde{\mathbf{Q}}_{J}(v-i0)}-1\right|^{2}+\left|\frac{\widetilde{\mathbf{Q}}_{J}(v+i0)}{{\mathbf{Q}}_{J}(v-i0)}-1\right|^{2}\right)$
(3.50)

is zero. We can then use the Levenberg-Marquardt algorithm [35] to find an
approximate numerical solution.

####  3.3.2 The results

For weak coupling in the range $0<h^{2}\leq 0.1$ we use the values $K=15$,
$N=28$, $U=91$ and $I_{P}=64$. For strong coupling with $h\leq 1.99$ we use
the same values. For $1.99<h\leq 3.99$ we use $K=20$, $N=38$, $U=91$ and
$I_{P}=84$, for $4\leq h\leq 4.44$ we use $K=21$, $N=40$, $U=121$ and
$I_{P}=88$, while for $4.48\leq h$ we increase $K$ and $N$ to $K=25$, $N=48$.
We provide all results for the Hagedorn temperature at the above range of
couplings in the ancillary files weak_h_squared.csv and strong_h.csv.

To check the numerical accuracy, we perturbed the result at $h^{2}=0.1$ and
reran the algorithm 21 times at this same coupling to find that the Hagedorn
temperature converged to the same value with an accuracy of 41 digits on all
21 runs. At the strong coupling point $h=3.99$ we did the same for 10 runs
with perturbations of order $10^{-6}$ and two runs at order $10^{-5}$ to find
results agreeing with 12 digit accuracy.

At weak $h$ the numerics agrees well with the perturbative solution of the
QSC, as shown in figure 2. By fitting the numerical data with an even
polynomial of order $h^{98}$ we match the first five terms with 15 digit
precision against their exact values (3.40). In the plot we can observe the
break down of the perturbative expansion in $h^{2}$ before $h^{2}=0.1$.
However the Padé approximation 3.41 fits well on the whole range of the weak
coupling data.

Figure 2: The perturbative solution of the QSC compared to the numerical
solution at weak coupling. In the legend we indicate the order in $h$ of the
perturbative expansion. The Padé approximation (3.41) fits the numerical data
to a relative difference of $4\times 10^{-4}$ across the full range shown in
the plot.

At strong coupling we convert from $h$ to $\hat{\lambda}=\lambda-\frac{1}{24}$
using (3.3) and fit against a series in negative powers in
$\hat{\lambda}^{1/4}$. To mitigate the errors choosing the correct range of
data to fit on, we average over different values for the lowest value of
$\hat{\lambda}$ and choose the range which gives the smallest estimated
standard deviation for the leading fitted coefficient. We also vary how many
orders we include in the series. This way we can check that the large
$\hat{\lambda}$ asymptotics agrees with the string theory prediction at orders
$\hat{\lambda}^{1/4}$ and $\hat{\lambda}^{0}$. To get better numerical
estimates for the subleading terms, we then subtract the exact string theory
prediction for the two leading terms from our numerical data. Taking the mean
of the resulting coefficients and estimating the standard deviation we find

$T_{H}(\hat{\lambda})=\frac{\hat{\lambda}^{1/4}}{2^{5/4}\sqrt{\pi}}+\frac{3}{8\pi}\\\
-(0.0308\pm 0.0004)\hat{\lambda}^{-1/4}+(0.046\pm
0.003)\hat{\lambda}^{-1/2}-(0.017\pm 0.006)\hat{\lambda}^{-3/4}\\\ +(0.005\pm
0.004)\hat{\lambda}^{-1}+(0.0004\pm
0.0006)\hat{\lambda}^{-5/4}+\mathcal{O}(\hat{\lambda}^{-3/2}),$ (3.51)

We plot the numerical data and the fitted function in figure 3. We observe
that the $\hat{\lambda}^{-1/2}$ term matches the prediction in (2.23) within
the error margins. If we now assume that the predicted value for this
coefficient is correct, we can make a more precise estimate for the
$\hat{\lambda}^{-1/4}$ coefficient, and hence the value for $\Delta c$, by
refitting the curve. Our result is

$\Delta c=-2.0782\pm 0.0016\approx-3\log(2)=2.07944\dots\,.$ (3.52)

In [1] we found that $\Delta c=-4\log(2)$ for AdS5 with slightly better
numerical accuracy. This along with (3.52) leads us to conjecture that for
general AdSd+1 the contribution from the zero-point shift is

$\Delta c=-d\log(2).$ (3.53)

Figure 3: We show (half of) the numeric results for the Hagedorn temperature
up to $\hat{\lambda}^{1/4}\approx 2.74$. The strong coupling fit matches
within a relative error of less than one tenth of a percent all the way down
to $\hat{\lambda}^{1/4}\approx 0.83$, while the Padé approximation (3.41) fits
within the same error margins all the away up to $\hat{\lambda}^{1/4}\approx
0.88$.

##  4 Discussion

In this paper we computed the Hagedorn temperature at weak and strong coupling
for ABJM using the QSC. At weak coupling we found the result up to eight
loops, although there is theoretically no limit as to how high we could go. At
strong coupling we found the Hagedorn temperature numerically in inverse
powers of $\lambda^{1/4}$ to four nontrivial orders in the expansion, showing
that the last two terms are consistent with a conjectured form for the
coefficients in the expansion.

Putting equations (2.22) and (3.52) together we observe a linear dependence on
the dimension of the CFT for $\Delta c$. However one could twist the
$R$-symmetries by turning on fugacities and explore how deformations related
to the internal space affect the asymptotic behavior. In this case there is
less symmetry in the QSC so the numerical calculations are slower. Perhaps
recent work on numerical solutions of the QSC would be of use [37]. This
investigation is in progress.

It would also be interesting to study the AdS3 case. The first correction in
the expansion was already given in [38] and verified on the world-sheet in
[16], with the next two terms given in (2.23) and (3.53). For AdS3 there is a
conjectured form for the QSC [39, 40] and a TBA [41]. However, even for the
spectral problem, it is not known how to take the AdS3 QSC to strong coupling
with the present technology due to the novel analytic properties of the curve.

## Acknowledgements

We thank Martijn Hidding for help with the numerical evaluation of
polylogarithms. We also thank N. Bobev, N. Gromov, R. Tateo, G. Papathanasiou
and M. Spradlin for discussions. This research is supported in part by the
Swedish Research Council under grant #2020-03339 and by the National Science
Foundation under Grant No. NSF PHY-1748958. Computations were done on a
cluster provided by the National Academic Infrastructure for Supercomputing in
Sweden (NAISS) at UPPMAX, partially funded by the Swedish Research Council
under grant #2022-06725. S.E was partially supported by the European Research
Council (ERC) under the European Union’s Horizon 2020 research and innovation
programme (grant agreement No. 865075) EXACTC. J.A.M. thanks the KITP for
hospitality during the beginning of this work.

##  Appendix A ABJM Conventions

We follow the conventions of [20]. We use $A,B=1,\dots,6$ for $SO(6)$ vector
indices, and $a,b=1,\dots,4$ for the corresponding spinor indices. Relevant
matrices pertaining to the $SO(6)$ are

$\displaystyle\eta_{AB}=\begin{pmatrix}0&0&0&1&0&0\\\ 0&0&-1&0&0&0\\\
0&-1&0&0&0&0\\\ 1&0&0&0&0&0\\\ 0&0&0&0&0&1\\\ 0&0&0&0&1&0\end{pmatrix}\,,$
$\displaystyle\kappa_{ab}=\begin{pmatrix}0&0&0&1\\\ 0&0&-1&0\\\ 0&1&0&0\\\
-1&0&0&0\end{pmatrix}\,,$ (A.1) $\displaystyle
V_{A}\sigma^{A}_{ab}=\begin{pmatrix}0&-V_{1}&-V_{2}&-V_{5}\\\
-V_{1}&0&-V_{6}&-V_{3}\\\ V_{2}&V_{6}&0&-V_{4}\\\
V_{5}&V_{3}&V_{4}&0\end{pmatrix}\,,$ $\displaystyle
V_{A}(\bar{\sigma}^{A})^{ab}=\begin{pmatrix}0&V_{4}&-V_{3}&V_{6}\\\
-V_{4}&0&V_{5}&-V_{2}\\\ V_{3}&-V_{5}&0&V_{1}\\\
V_{5}&V_{3}&-V_{1}&0\end{pmatrix}\,.$ (A.2)

The $SO(5)$ conventions are derived from the above $SO(6)$ conventions by
identifying components 5 and 6. We use $I,J=1,\dots 5$ for the vector and
$i,j=1,\dots 4$ for the spinor indices of the $SO(5)$. Relevant matrices are

$\displaystyle\rho_{IJ}=\begin{pmatrix}0&0&0&1&0\\\ 0&0&-1&0&0\\\
0&-1&0&0&0\\\ 1&0&0&0&0\\\ 0&0&0&0&\frac{1}{2}\end{pmatrix}\,,$ (A.3)
$\displaystyle\Sigma^{I}=(\sigma^{1},\sigma^{2},\sigma^{3},\sigma^{4},\sigma^{5}+\sigma^{6}),$
$\displaystyle\overline{\Sigma}^{I}=(\overline{\sigma}^{1},\overline{\sigma}^{2},\overline{\sigma}^{3},\overline{\sigma}^{4},\overline{\sigma}^{5}+\overline{\sigma}^{6})\,,$
(A.4) $\displaystyle\kappa_{ij}=\begin{pmatrix}0&0&0&1\\\ 0&0&-1&0\\\
0&1&0&0\\\ -1&0&0&0\end{pmatrix}\,,$
$\displaystyle\mathds{K}^{i}_{j}=\begin{pmatrix}1&0&0&0\\\ 0&-1&0&0\\\
0&0&-1&0\\\ 0&0&0&1\end{pmatrix}\,.$ (A.5)

##  Appendix B Tree level solution

For the tree-level solution, $h=0$, we find the following expressions for
$\mathbf{P}$ and $\mathbf{Q}$

$\displaystyle\mathbf{P}_{A}$ $\displaystyle=\begin{pmatrix}u\\\ 1\\\
-\frac{4}{3}u^{4}\cosh^{4}\frac{1}{4T}+\frac{\left(\cosh^{2}\frac{1}{4T}-6\right)^{2}}{9}-1\\\
-\frac{16}{3}u^{3}\cosh^{4}\frac{1}{4T}-\frac{4}{3}\cosh^{2}\frac{1}{4T}\left(\cosh^{2}\frac{1}{4T}-6\right)u\\\
2u^{2}\cosh^{2}\frac{1}{4T}+\frac{1}{3}\left(\cosh^{2}\frac{1}{4T}-6\right)\\\
2u^{2}\cosh^{2}\frac{1}{4T}+\frac{1}{3}\left(\cosh^{2}\frac{1}{4T}-6\right)\end{pmatrix}\,,$
(B.1) $\displaystyle\mathbf{Q}_{I}$
$\displaystyle=\begin{pmatrix}8y^{-iu}u\cosh^{3}\frac{1}{4T}\\\
8y^{-iu}\cosh^{3}\frac{1}{4T}\\\
y^{iu}\frac{1}{8\sinh^{3}\frac{1}{4T}}\left(iu+\frac{\sinh^{2}\frac{1}{4T}-1}{\sinh\left(\frac{1}{4T}\right)\cosh\left(\frac{1}{4T}\right)}\right)\\\
y^{iu}\frac{i}{8\sinh^{3}\frac{1}{4T}}\\\ 0\end{pmatrix}\,,$ (B.2)

##  Appendix C $T_{H}^{(4)}$

Using $w_{H}=3-2\sqrt{2}$ from section 2.1 we find:

$\begin{split}T_{H}^{(4)}=&56-296w_{H}+4\left(104w_{H}-\frac{49}{3}\right)\log\left(w_{H}\right)+\frac{1}{4}\left(-331w_{H}-31\right)\log^{2}\left(w_{H}\right)\\\
+&64\left(\log^{2}\left(w_{H}\right)\left(w_{H}-3\right)-3\log\left(w_{H}\right)\right)\text{Li}_{1}\left(w_{H}\right){}^{3}\\\
+&8\left(3\left(13-7w_{H}\right)\log^{2}\left(w_{H}\right)+\left(9w_{H}+5\right)\log\left(w_{H}\right)-4\left(w_{H}-1\right)\right.\\\
&\left.+24\left(1-\log\left(w_{H}\right)\left(w_{H}-3\right)\right)\text{Li}_{2}\left(w_{H}\right)\right)\text{Li}_{1}\left(w_{H}\right){}^{2}\\\
+&4\left(48\left(w_{H}-3\right)\text{Li}_{2}\left(w_{H}\right){}^{2}+4\left(w_{H}-11\right)\text{Li}_{1}\left(w_{H}^{2}\right)\log^{2}\left(w_{H}\right)\right.\\\
&+\text{Li}_{2}\left(w_{H}\right)\left(75w_{H}+4\left(4w_{H}-11\right)\log\left(w_{H}\right)-97\right)\log\left(w_{H}\right)\\\
&\left.-8\left(w_{H}-1\right)\text{Li}_{2}\left(w_{H}^{2}\right)\log\left(w_{H}\right)\right)\text{Li}_{1}\left(w_{H}\right)-8\left(23w_{H}+27\right)\text{Li}_{4}\left(w_{H}\right)\\\
-&\frac{64\left(w_{H}-3\right)\text{Li}_{2}\left(w_{H}\right){}^{3}}{\log\left(w_{H}\right)}+2\left(-93w_{H}+8\left(11-4w_{H}\right)\log\left(w_{H}\right)+311\right)\text{Li}_{2}\left(w_{H}\right){}^{2}\\\
+&16\left(\left(3w_{H}-5\right)\text{Li}_{2}\left(w_{H}^{2}\right)+\left(5-\frac{13w_{H}}{3}\right)\log^{2}\left(w_{H}\right)\right)\text{Li}_{2}\left(w_{H}\right)\\\
+&4\left(5\left(\frac{2w_{H}}{3}-3\right)\log\left(w_{H}\right)+2\left(27w_{H}-17\right)\right)\log\left(w_{H}\right)\text{Li}_{3}\left(w_{H}\right)\\\
+&100\left(9-2w_{H}\right)\left(\frac{2}{5}\log\left(w_{H}\right)\text{Li}_{4}\left(w_{H}\right)-\text{Li}_{5}\left(w_{H}\right)+\frac{\text{Li}_{6}\left(w_{H}\right)}{\log\left(w_{H}\right)}\right)\\\
+&\left(\text{Li}_{1}\left(w_{H}\right)\log\left(w_{H}\right)-\text{Li}_{2}\left(w_{H}\right)\right)\left(384\text{Li}_{1,1}\left(w_{H},w_{H}\right)+\frac{32\left(w_{H}+1\right)\text{Li}_{3}\left(w_{H}^{2}\right)}{\log\left(w_{H}\right)}\right.\\\
&+48\left(4w_{H}-11\right)\left(\frac{\text{Li}_{4}\left(w_{H}\right)}{\log\left(w_{H}\right)}-\text{Li}_{3}\left(w_{H}\right)\right)-4\left(4\zeta(2)\left(w_{H}+5\right)+63w_{H}+19\right)\\\
&\left.-\frac{32\zeta(3)\left(w_{H}+1\right)}{\log\left(w_{H}\right)}+\frac{16}{3}\left(14w_{H}+19\right)\log\left(w_{H}\right)\right)\\\
+&\left(16\left(17-3w_{H}\right)\text{Li}_{2}\left(w_{H}\right)+32\left(7-5w_{H}\right)\log\left(w_{H}\right)-64\left(w_{H}-3\right)\right)\log\left(w_{H}\right)\text{Li}_{1}\left(w_{H}^{2}\right)\\\
+&2\left(-32w_{H}+\left(27w_{H}-145\right)\log^{2}\left(w_{H}\right)+4\left(11w_{H}-1\right)\log\left(w_{H}\right)\right)\text{Li}_{2}\left(w_{H}^{2}\right)\\\
+&16\left(5-3w_{H}\right)\text{Li}_{3}\left(w_{H}^{2}\right)\log\left(w_{H}\right)+4\left(19w_{H}-121\right)\text{Li}_{4}\left(w_{H}^{2}\right)\\\
+&\left(64\left(w_{H}-1\right)+4\left(11w_{H}-65\right)\log^{2}\left(w_{H}\right)-16\left(9w_{H}+5\right)\log\left(w_{H}\right)\right)\text{Li}_{1,1}\left(w_{H},w_{H}\right)\\\
+&8\left(w_{H}-3\right)\log\left(w_{H}\right)\text{Li}_{1,2}\left(\frac{1}{w_{H}},w_{H}\right)-128\log\left(w_{H}\right)\text{Li}_{1,2}\left(\frac{1}{w_{H}},w_{H}^{2}\right)\\\
+&8\left(3w_{H}+7\right)\log\left(w_{H}\right)\text{Li}_{2,1}\left(\frac{1}{w_{H}},w_{H}\right)+32\left(w_{H}-3\right)\log\left(w_{H}\right)\text{Li}_{2,1}\left(w_{H},w_{H}\right)\\\
+&32\left(w_{H}-11\right)\left(\text{Li}_{1,3}\left(w_{H},w_{H}\right)-\text{Li}_{1,3}\left(\frac{1}{w_{H}},w_{H}\right)+\frac{\log\left(w_{H}\right)}{2}\text{Li}_{2,1}\left(\frac{1}{w_{H}},w_{H}^{2}\right)\right)\\\
+&8\left(41-3w_{H}\right)\left(\text{Li}_{2,2}\left(\frac{1}{w_{H}},w_{H}\right)-\text{Li}_{2,2}\left(w_{H},w_{H}\right)\right)\\\
+&16\left(w_{H}+5\right)\log^{2}\left(w_{H}\right)\text{Li}_{1,1}\left(\frac{1}{w_{H}},w_{H}^{2}\right)+384\left(\text{Li}_{3,1}\left(\frac{1}{w_{H}},w_{H}\right)-\text{Li}_{3,1}\left(w_{H},w_{H}\right)\right).\end{split}$

## References

  * [1] S. Ekhammar, J.A. Minahan and C. Thull, _The asymptotic form of the Hagedorn temperature in planar $\mathcal{N}=4$ super Yang-Mills_, 2306.09883.
  * [2] T. Harmark and M. Wilhelm, _Solving the Hagedorn temperature of AdS 5/CFT4 via the Quantum Spectral Curve: chemical potentials and deformations_, _JHEP_ 07 (2022) 136 [2109.09761].
  * [3] O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, _N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_ , _JHEP_ 10 (2008) 091 [0806.1218].
  * [4] T. Harmark and M. Wilhelm, _Hagedorn Temperature of AdS 5/CFT4 via Integrability_, _Phys. Rev. Lett._ 120 (2018) 071605 [1706.03074].
  * [5] T. Harmark and M. Wilhelm, _The Hagedorn temperature of AdS 5/CFT4 at finite coupling via the Quantum Spectral Curve_, _Phys. Lett. B_ 786 (2018) 53 [1803.04416].
  * [6] E.Y. Urbach, _String Stars in Anti De Sitter Space_ , _JHEP_ 04 (2022) 072 [2202.06966].
  * [7] T. Nishioka and T. Takayanagi, _On Type IIA Penrose Limit and N=6 Chern-Simons Theories_ , _JHEP_ 08 (2008) 001 [0806.3391].
  * [8] G. Papathanasiou and M. Spradlin, _The Morphology of N=6 Chern-Simons Theory_ , _JHEP_ 07 (2009) 036 [0903.2548].
  * [9] F. Bigazzi, T. Canneti and A.L. Cotrone, _Higher Order Corrections to the Hagedorn Temperature at Strong Coupling_ , 2306.17126.
  * [10] B.I. Zwiebel, _Two-loop Integrability of Planar N=6 Superconformal Chern-Simons Theory_ , _J. Phys. A_ 42 (2009) 495402 [0901.0411].
  * [11] J.A. Minahan, W. Schulgin and K. Zarembo, _Two loop integrability for Chern-Simons theories with N=6 supersymmetry_ , _JHEP_ 03 (2009) 057 [0901.1142].
  * [12] O. Bergman and S. Hirano, _Anomalous radius shift in AdS(4)/CFT(3)_ , _JHEP_ 07 (2009) 016 [0902.1743].
  * [13] K. Sugiyama and K. Yoshida, _Type IIA string and matrix string on PP wave_ , _Nucl. Phys. B_ 644 (2002) 128 [hep-th/0208029].
  * [14] S.-j. Hyun and H.-j. Shin, _N=(4,4) type 2A string theory on PP wave background_ , _JHEP_ 10 (2002) 070 [hep-th/0208074].
  * [15] S.-j. Hyun, J.-D. Park and S.-H. Yi, _Thermodynamic behavior of IIA string theory on a pp wave_ , _JHEP_ 11 (2003) 006 [hep-th/0304239].
  * [16] F. Bigazzi, T. Canneti and W. Mück, _Semiclassical Quantization of the Superstring and Hagedorn Temperature_ , 2306.00588.
  * [17] A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, _Quantum Spectral Curve of the $\mathcal{N}=$ 6 Supersymmetric Chern-Simons Theory_, _Phys. Rev. Lett._ 113 (2014) 021601 [1403.1859].
  * [18] D. Bombardelli, D. Fioravanti and R. Tateo, _TBA and Y-system for planar AdS(4)/CFT(3)_ , _Nucl. Phys. B_ 834 (2010) 543 [0912.4715].
  * [19] N. Gromov and F. Levkovich-Maslyuk, _Y-system, TBA and Quasi-Classical strings in AdS(4) x CP3_ , _JHEP_ 06 (2010) 088 [0912.4911].
  * [20] D. Bombardelli, A. Cavaglià, D. Fioravanti, N. Gromov and R. Tateo, _The full Quantum Spectral Curve for $AdS_{4}/CFT_{3}$_, _JHEP_ 09 (2017) 140 [1701.00473].
  * [21] N. Gromov and G. Sizov, _Exact Slope and Interpolating Functions in N=6 Supersymmetric Chern-Simons Theory_ , _Phys. Rev. Lett._ 113 (2014) 121601 [1403.1894].
  * [22] N. Bobev, J. Hong and V. Reys, _Large N partition functions of the ABJM theory_ , _JHEP_ 02 (2023) 020 [2210.09318].
  * [23] N. Bobev, J. Hong and V. Reys, _Large N Partition Functions, Holography, and Black Holes_ , _Phys. Rev. Lett._ 129 (2022) 041602 [2203.14981].
  * [24] V. Kazakov, S. Leurent and D. Volin, _T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve_ , _JHEP_ 12 (2016) 044 [1510.02100].
  * [25] L. Anselmetti, D. Bombardelli, A. Cavaglià and R. Tateo, _12 loops and triple wrapping in ABJM theory from integrability_ , _JHEP_ 10 (2015) 117 [1506.09089].
  * [26] D. Bombardelli, A. Cavaglià, R. Conti and R. Tateo, _Exploring the spectrum of planar AdS 4/CFT3 at finite coupling_, _JHEP_ 04 (2018) 117 [1803.04748].
  * [27] R.N. Lee and A.I. Onishchenko, _ABJM quantum spectral curve and Mellin transform_ , _JHEP_ 05 (2018) 179 [1712.00412].
  * [28] R.N. Lee and A.I. Onishchenko, _Toward an analytic perturbative solution for the ABJM quantum spectral curve_ , _Teor. Mat. Fiz._ 198 (2019) 292 [1807.06267].
  * [29] R.N. Lee and A.I. Onishchenka, _ABJM quantum spectral curve at twist 1: algorithmic perturbative solution_ , _JHEP_ 11 (2019) 018 [1905.03116].
  * [30] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, _Pomeron Eigenvalue at Three Loops in $\mathcal{N}=$ 4 Supersymmetric Yang-Mills Theory_, _Phys. Rev. Lett._ 115 (2015) 251601 [1507.04010].
  * [31] S. Leurent and D. Volin, _Multiple zeta functions and double wrapping in planar $N=4$ SYM_, _Nucl. Phys. B_ 875 (2013) 757 [1302.1135].
  * [32] C. Marboe and D. Volin, _Quantum spectral curve as a tool for a perturbative quantum field theory_ , _Nucl. Phys. B_ 899 (2015) 810 [1411.4758].
  * [33] N. Gromov and F. Levkovich-Maslyuk, _Quantum Spectral Curve for a cusped Wilson line in $\mathcal{N}=4$ SYM_, _JHEP_ 04 (2016) 134 [1510.02098].
  * [34] M. Hidding, _DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions_ , _Comput. Phys. Commun._ 269 (2021) 108125 [2006.05510].
  * [35] N. Gromov, F. Levkovich-Maslyuk and G. Sizov, _Quantum Spectral Curve and the Numerical Solution of the Spectral Problem in AdS5/CFT4_ , _JHEP_ 06 (2016) 036 [1504.06640].
  * [36] F. Levkovich-Maslyuk, _Numerical results for the exact spectrum of planar AdS4/CFT3_ , _JHEP_ 05 (2012) 142 [1110.5869].
  * [37] N. Gromov, A. Hegedus, J. Julius and N. Sokolova, _Fast QSC Solver: tool for systematic study of N=4 Super-Yang-Mills spectrum_ , 2306.12379.
  * [38] E.Y. Urbach, _The black hole/string transition in AdS 3 and confining backgrounds_, 2303.09567.
  * [39] S. Ekhammar and D. Volin, _Monodromy bootstrap for SU(2—2) quantum spectral curves: from Hubbard model to AdS 3/CFT2_, _JHEP_ 03 (2022) 192 [2109.06164].
  * [40] A. Cavaglià, N. Gromov, B. Stefański, Jr., Jr. and A. Torrielli, _Quantum Spectral Curve for AdS 3/CFT2: a proposal_, _JHEP_ 12 (2021) 048 [2109.05500].
  * [41] S. Frolov and A. Sfondrini, _Mirror thermodynamic Bethe ansatz for AdS3/CFT2_ , _JHEP_ 03 (2022) 138 [2112.08898].