#  $AdS_{2}$ holography and ModMax

Hemant Rathi and Dibakar Roychowdhury
Department of Physics, Indian Institute of Technology Roorkee,
Roorkee 247667, Uttarakhand, India
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a JT gravity set up in the presence of projected ModMax corrections
in two dimensions. Our starting point is the Einstein’s gravity in four
dimensions accompanied by the ModMax Lagrangian. The 2D gravity action is
obtained following a suitable dimensional reduction which contains a 2D image
of the 4D ModMax Lagrangian. We carry out a perturbative analysis to find out
the vacuum structure of the theory which asymptotes to $AdS_{2}$ in the
absence of $U(1)$ gauge fields. We estimate the holographic central charge and
obtain corrections perturbatively upto quadratic order in the ModMax and the
$U(1)$ coupling. We also find out ModMax corrected 2D black hole solutions and
discuss their extremal limits.

##  1 Overview and motivation

The non-linear generalisations of Maxwell electrodynamics [1]-[2] in four
dimensions play a pivotal role in understanding the dynamics of charged
particles in the strong field regime. For example, the Born-Infeld (BI) theory
[1] was proposed in order to obtain the finite self energy corrections for a
charged particle in an electromagnetic field. On the other hand, the
Heisenberg-Euler-Kockel (HEK) model [2] describes the vacuum polarization
effects of Quantum Electrodynamics111See a recent review [3] for different
versions of the non-linear modified theories of Maxwell electrodynamics..
However, both of these (non-linear) theories meet the standard Maxwell
electrodynamics in the limit of “weak” field approximations.

Generally, the non-linear generalizations of Maxwell electrodynamics (NLE) is
characterised by an action that contains a Lorentz scalar and a pseudo scalar
which are quadratic in the field strength ($F^{\mu\nu}$) [3]-[4]

$\displaystyle S=\frac{1}{2}F_{\mu\nu}F^{\mu\nu}\hskip 2.84526pt,\hskip
5.69054ptP=\frac{1}{2}F_{\mu\nu}\tilde{F}^{\mu\nu},$ (1)

where $\tilde{F}^{\mu\nu}$ is the Hodge dual of $F^{\mu\nu}$.

For instance, the BI electrodynamics is described by the following Lagrangian
density [1]

$\displaystyle\mathcal{L}_{BI}=T-\sqrt{T^{2}+\frac{T}{2}F_{\mu\nu}F^{\mu\nu}-\frac{1}{16}\big{(}F_{\mu\nu}\tilde{F}^{\mu\nu}\big{)}^{2}},$
(2)

where $T$ is the coupling parameter having the dimension of energy density.
Clearly, in the weak field limit ($T\rightarrow\infty$), the Lagrangian
density (2) reduces to the standard Maxwell electrodynamics.

Unlike the standard Maxwell electrodynamics, its non-linear modifications are
generally not invariant under the $SO(2)$ duality transformations and in fact
break the conformal symmetry in four dimensions. For instance, the HEK theory
[2] is not invariant under the electromagnetic duality and does not have a
conformal symmetry. However, the BI electrodynamics is invariant under the
$SO(2)$ duality [5] although it is not conformal invariant due to the presence
of the dimensionful coupling ($T$) in the theory (2).

Recently, there has been a radical proposal [6]-[7] to (non-linearly)
generalize the Maxwell electrodynamics which retains its conformal invariance
(in four dimensions) as well as preserves the $SO(2)$ duality symmetry. This
goes under the name of the “ModMax” electrodynamics222For details, see the
recent review [3]..

The ModMax electrodynamics is a 1-parameter deformation of the Maxwell
electrodynamics in four dimensions that is described by the following
Lagrangian density333In the limit $\gamma\rightarrow 0$, the ModMax
electrodynamics reduces to the standard Maxwell electrodynamics. [6]-[7]

$\displaystyle\mathcal{L}_{MM}$
$\displaystyle=\frac{1}{2}\Big{(}S\cosh{\gamma}-\sqrt{S^{2}+P^{2}}\sinh{\gamma}\Big{)},$
(3)

where $\gamma$ is the dimensionless coupling constant that measures the
strength of the electromagnetic self interaction.

The physical requirements that the theory must be unitary and preserves the
causality restrict the ModMax parameter $(\gamma)$ to take only positive
values ($\gamma>0$) [6]. The above restriction guarantees that the Lagrangian
density (3) is a convex function of the electric field strength $E^{i}$.

There have been some further modifications to the ModMax electrodynamics in
the literature which include the 1-parameter generalisation of the BI
theory444In the weak field limit, the ($\gamma$BI) theory reduces to the
standard ModMax electrodynamics (3). ($\gamma$BI) [8] and $\mathcal{N}=1$
supersymmetric extension of the ModMax electrodynamics555See [10]-[12] for
further details.  [9]. The supersymmetric version of the ModMax
electrodynamics is invariant under the electromagnetic duality as well as
posses the superconformal symmetry [9].

The ModMax electrodynamics finds an extensive application in theories of
gravity [13]-[17] as well. In fact, a large number of solutions have been
obtained down the line. For instance, accelerated black holes [13], the Taub-
NUT [15]-[16] and Reissner-Nordstorm solutions [17] in diverse spacetime
dimensions have been constructed in the presence of ModMax interactions and
the effects of non-linearity were explored on their thermal properties.
Recently, the non-linear models of electrodynamics have also found their
applications in the context of strongly correlated systems [18]-[20] by means
of the celebrated $AdS_{d+1}/CFT_{d}$ correspondence [21]-[23].

Despite of several notable applications those are alluded to the above, ModMax
theories are least explored in $AdS_{2}$ holography and in particular in the
context of the JT/SYK correspondence [24]-[46]. The purpose of the present
paper is to fill up some of these gaps in the literature and find out an
interpretation for the projected ModMax interactions within the realm of 2D
gravity theories.

The pure Jackiw-Teitelboim (JT) gravity [24]-[25] is the two dimensional
theory of Einstein-dilaton gravity in the presence of a negative cosmological
constant. Under certain special circumstances, this theory is conjectured to
be the dual description of the Sachdev-Ye-Kitaev (SYK) model [26]-[46] which
is a quantum mechanical theory of N interacting (Majorana) fermions in one
dimension666The bulk dual of the pure SYK model contains an infinite tower of
massive particles which are dual to primary $O(N)$ singlet operators [29],
[45]. However, the pure JT gravity does not contain the tower of such massive
particles. Therefore, the SYK/JT correspondence should make sense only in the
soft/Schwarzian limit. . Interestingly, this model can be solved exactly at
strong coupling and in the Large N limit. The generalisation of the JT/SYK
correspondence in the presence of $U(1)$ gauge fields and SU(2) Yang-Mills
fields have been carried out in a series of papers [47]-[53].

In the present paper, we cook up a theory of JT gravity in the presence of 2D
“projected” ModMax interactions and compute various physical entities
associated with the boundary theory. For instance, we construct the
holographic stress-energy tensor [48], [51], [54]-[55] and compute the
associated central charge [48], [51], [54], [56] for the boundary theory.
Finally, we construct black hole solutions in two dimensions and explore the
effects of projected ModMax interactions on their thermal behaviour.

The organisation for the rest of the paper is as follows :

$\bullet$ In Section 2, we follow suitable dimensional reduction procedure
[47]-[48], [53] to construct a model for JT gravity in the presence of 2D
projected ModMax interactions. We also clarify the meaning of projected ModMax
interactions in 2D and in particular present a detail comparison with the 4D
ModMax interactions.

$\bullet$ In Section 3, we calculate the conformal dimensions of different
scalar operator in deep IR limit and make a comparative analysis between them.
We further explore the vacuum structure of the theory using the Fefferman-
Graham gauge [48], [57] by treating the non-linear $U(1)$ gauge interactions
as “perturbations” over the pure JT gravity solutions. We estimate these
solutions upto quadratic order in the gauge and ModMax couplings.

$\bullet$ In Section 4, we construct the “renormalised” boundary stress tensor
and investigate its transformation properties under the combined action of the
diffeomorphism and the $U(1)$ gauge transformations [51]. We compute the
central charge $(c_{M})$ associated with the boundary theory [51] up to
quadratic order in the (ModMax and $U(1)$) couplings.

$\bullet$ In Section 5, we construct the black hole solutions upto quadratic
order in the couplings. We observe that the non-linear interactions (or the
projected ModMax interactions) play a crucial role in obtaining a finite value
for the background fields at the horizon.

Furthermore, we compute the Hawking temperature for 2D black holes [58] and
calculate the associated Wald entropy [59]-[61]. We also investigate the
“extremal” limit associated with these 2D black hole solutions and calculate
the corresponding Wald entropy.

$\bullet$ We draw our conclusion in Section 6, along with some future remarks.

##  2 JT gravity and 2D projected ModMax

The ModMax theory coupled to Einstein gravity in four dimensions is defined as
[6]-[7], [13]

$\displaystyle I^{(4)}=\frac{1}{16\pi G_{4}}\int
d^{4}x\sqrt{-g_{(4)}}\Big{(}R^{(4)}-2\Lambda-4\kappa\mathcal{L}^{(4)}_{\text{MM}}\Big{)},$
(4)

where $R^{(4)}$ is the Ricci scalar in 4 dimensions, $\Lambda=-3$ is the
cosmological constant777Here, we set the $AdS$ length $l=1$., $G_{4}$ is the
Newton’s constant in four dimensions, $\kappa$ is the coupling constant and
$\mathcal{L}^{(4)}_{\text{MM}}$ is the ModMax Lagrangian density in four
dimensions [6]-[7], [13]

$\displaystyle\mathcal{L}^{(4)}_{\text{MM}}$
$\displaystyle=\frac{1}{2}\Big{(}S\cosh{\gamma}-\sqrt{S^{2}+P^{2}}\sinh{\gamma}\Big{)},$
$\displaystyle\hskip 5.69054ptS$ $\displaystyle=\frac{1}{2}F_{MN}F^{MN}\hskip
2.84526pt,\hskip 5.69054ptP=\frac{1}{2}F_{MN}\tilde{F}^{MN}\hskip
2.84526pt,\hskip 5.69054pt\tilde{F}^{MN}=\frac{1}{2}\epsilon^{MNUV}F_{UV}.$
(5)

Here, $\gamma$ is the ModMax parameter and $(M,N)$ are the 4 dimensional
space-time indices. Clearly, the standard Maxwell electrodynamics is recovered
in the limit $\gamma\rightarrow 0$ [6]-[7], [13].

The imprint of the ModMax theory (2) in two dimensions can be obtained via
dimensional reduction [47]-[48], [53] of the following form

$\displaystyle ds_{(4)}^{2}$
$\displaystyle=ds_{(2)}^{2}+\Phi(x^{\mu})dx_{i}^{2}\hskip 2.84526pt,\hskip
5.69054ptds_{(2)}^{2}=g_{\mu\nu}(x^{\alpha})dx^{\mu}dx^{\nu},$ $\displaystyle
A_{\mu}$ $\displaystyle\equiv A_{\mu}(x^{\nu}),\hskip 2.84526ptA_{i}\equiv
A_{i}(x^{\mu}),$ (6)

where $(\mu,\nu)$ are the two dimensional indices and $(i,j)$ are the indices
of the compact dimensions.

Substituting (2) into (4) and integrating over the compact directions, one
finds888The Newton’s constant in two and four dimensions are related by
$G_{2}=\frac{G_{4}}{V_{2}}$, where $V_{2}$ is the volume of the compact space.

$\displaystyle I_{\text{bulk}}=\frac{1}{16\pi G_{2}}\int
d^{2}x\sqrt{-g_{(2)}}\Big{(}\Phi
R^{(2)}-2\Lambda\Phi-4\kappa\Phi\mathcal{L}^{(2)}_{\text{(}MM)}\Big{)},$ (7)

where $R^{(2)}$ is the Ricci scalar in two dimensions, $G_{2}$ is the Newton’s
constant in two dimensions and

$\displaystyle\mathcal{L}^{(2)}_{\text{MM}}$
$\displaystyle=\frac{1}{2}\Big{(}s\cosh{\gamma}-\sqrt{s^{2}+p^{2}}\sinh{\gamma}\Big{)},$
$\displaystyle s$
$\displaystyle=\frac{1}{2}F_{\mu\nu}F^{\mu\nu}+\Phi^{-1}\Big{(}(\partial\chi)^{2}+(\partial\xi)^{2}\Big{)},\hskip
2.84526ptp=-2\Phi^{-1}\epsilon^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\xi$
(8)

is what we define as the Lagrangian density of the projected ModMax theory in
two dimensions. Here, we denote $A_{2}=\chi(x^{\mu}),\hskip
2.84526ptA_{3}=\xi(x^{\mu})$ and introduce
$\epsilon^{\mu\nu}=\frac{\varepsilon^{\mu\nu}}{\sqrt{-g_{(2)}}}$ as the Levi-
Civita tensor in two dimensions.

Notice that, in the limit $\gamma\rightarrow 0$, we do not recover the
standard Maxwell electrodynamics in two dimensions [48], [51]-[53]. On
contrary, we do have additional contributions coming from non-vanishing scalar
fields $\xi$ and $\chi$ which arise by virtue of the dimensional reduction
procedure. This turns out to be the unique feature of the projected ModMax
interactions in two dimensions. The $\gamma\rightarrow 0$ limit is what we
refer as the 2D Maxwell interaction in this paper.

$\bullet$ A comparative study of 4D ModMax and the 2D projected ModMax:

Below, we draw a comparative analysis between 4D ModMax [6]-[7] and its 2D
projection which plays the central role in what follows. 4D ModMax preserves
the conformal invariance in its usual sense which is also evident from the
generic structure of the associated stress-energy tensor

$\displaystyle T^{(4)}_{MN}\sim
f(\gamma)\Bigg{(}-\frac{1}{2}F^{2}g_{MN}+2g^{QP}F_{QM}F_{PN}\Bigg{)},$ (9)

where we define the function

$\displaystyle
f(\gamma)=\Bigg{(}\cosh{\gamma}-\frac{F^{2}\sinh{\gamma}}{\sqrt{\big{(}F_{RS}F^{RS}\big{)}^{2}+\big{(}F_{RS}\tilde{F}^{RS}\big{)}^{2}}}\Bigg{)}.$
(10)

Clearly, the trace $T^{M(4)}_{M}$ vanishes identically in four dimensions. On
the other hand, the trace of the projected ModMax in two dimensions turns out
to be

$\displaystyle T^{\mu(2)}_{\mu}=g^{\mu\nu}T_{\mu\nu}^{(2)}=\frac{\Phi
F^{2}}{2}\Bigg{(}\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^{2}+p^{2}}}\Bigg{)},$
(11)

which is a non-vanishing entity.

This reflects to the fact that the projected theory losses its conformal
invariance in two dimensions. Furthermore, the absence of the (Hodge) dual two
form ($\tilde{F}^{\mu\nu}$) in two dimensions spoils the electromagentic SO(2)
duality invariance of the 2D projected theory in comparison to its 4D cousin.
However, it is noteworthy to mention that the ModMax coupling $(\gamma)$ that
appears in the 2D projected version is same as that of the 4D parent theory.

The equations of motion corresponding to different field contents can be
obtained by varying the action (7)

$\displaystyle\delta I_{\text{bulk}}=\frac{1}{16\pi G_{2}}\int
d^{2}x\sqrt{-g}\Big{(}\mathcal{H}_{\mu\nu}\delta
g^{\mu\nu}+\mathcal{H}_{\Phi}\delta\Phi+\mathcal{H}_{\mu}\delta
A^{\mu}+\mathcal{H}_{\chi}\delta\chi+\mathcal{H}_{\xi}\delta\xi\Big{)},$ (12)

where we define individual entities as

$\displaystyle\mathcal{H}_{\Phi}=\hskip 2.84526pt$ $\displaystyle
R-2\Lambda-4\kappa\mathcal{L}^{(2)}_{\text{MM}}+2\kappa\Phi^{-1}\Bigg{[}\Big{(}(\partial\xi)^{2}+(\partial\chi)^{2}\Big{)}\cosh{\gamma}-\Bigg{\\{}\frac{s\Big{(}(\partial\xi)^{2}+(\partial\chi)^{2}\Big{)}}{\sqrt{s^{2}+p^{2}}}-$
$\displaystyle\frac{2p\epsilon^{\mu\nu}\nabla_{\mu}\chi\nabla_{\nu}\xi}{\sqrt{s^{2}+p^{2}}}\Bigg{\\}}\sinh{\gamma}\Bigg{]}=0,$
(13) $\displaystyle\mathcal{H}_{\mu\nu}=\hskip 2.84526pt$
$\displaystyle\square\Phi g_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}\Phi+\Lambda\Phi
g_{\mu\nu}-2\kappa\Phi\Bigg{[}\mathcal{F_{\mu\nu}}\cosh{\gamma}-\frac{\sinh{\gamma}}{\sqrt{s^{2}+p^{2}}}\Bigg{(}s\mathcal{F_{\mu\nu}}-\frac{1}{2}s^{2}g_{\mu\nu}\Bigg{)}$
$\displaystyle-\frac{s}{2}g_{\mu\nu}\cosh{\gamma}\Bigg{]}=0,$ (14)
$\displaystyle\mathcal{H}_{\chi}=\hskip 2.84526pt$
$\displaystyle\kappa\nabla_{\mu}\Bigg{[}\nabla^{\mu}\chi\cosh{\gamma}-\frac{s\nabla^{\mu}\chi-p\epsilon^{\mu\nu}\nabla_{\nu}\xi}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{]}=0,$
(15) $\displaystyle\mathcal{H}_{\xi}=\hskip 2.84526pt$
$\displaystyle\kappa\nabla_{\mu}\Bigg{[}\nabla^{\mu}\xi\cosh{\gamma}-\frac{s\nabla^{\mu}\xi+p\epsilon^{\mu\nu}\nabla_{\nu}\chi}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{]}=0,$
(16) $\displaystyle\mathcal{H}_{\mu}=\hskip 2.84526pt$
$\displaystyle\kappa\nabla_{\mu}\Bigg{[}\Phi\Bigg{(}\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^{2}+p^{2}}}\Bigg{)}F^{\mu\nu}\Bigg{]}=0,$
(17)

along with the function

$\displaystyle\mathcal{F_{\mu\nu}}=$ $\displaystyle\hskip
2.84526ptF_{\mu\alpha}F_{\nu\beta}g^{\alpha\beta}+\Phi^{-1}\Big{(}\partial_{\mu}\xi\partial_{\nu}\xi+\partial_{\mu}\chi\partial_{\nu}\chi\Big{)}.$
(18)

##  3 General solution with 2D projected ModMax

The purpose of this Section is to obtain the most general solutions of
(13)-(17) in the Fefferman-Graham gauge999The explicit form of these equations
(13)-(17) have been provided in the Appendix A. [48], [57]

$\displaystyle ds^{2}=\hskip 2.84526pt$ $\displaystyle
d\eta^{2}+h_{tt}(t,\eta)dt^{2},\hskip 2.84526ptA_{\mu}dx^{\mu}=\hskip
2.84526ptA_{t}(t,\eta)dt,$ $\displaystyle\Phi=\hskip 2.84526pt$
$\displaystyle\Phi(t,\eta),\hskip 2.84526pt\chi=\chi(t,\eta),\hskip
2.84526pt\xi=\xi(t,\eta).$ (19)

$\bullet$ A note on conformal dimensions:

Here, we present a calculation on the conformal dimensions of the dual
operators $\Delta_{\chi}$, $\Delta_{\xi}$ and $\Delta_{\Phi}$ corresponding to
the bulk scalar fields $\chi$, $\xi$ and $\Phi$ respectively. This allows us
to make a comparative study between various operator dimensions in the deep IR
limit.

The IR fixed point [62]-[64] is defined as the set of solutions to the
equations of motion (13)-(17) for constant values of the scalar fields

$\displaystyle\chi(t,\eta)=\chi^{*}\hskip 2.84526pt,\hskip
5.69054pt\xi(t,\eta)=\xi^{*}\hskip 2.84526pt,\hskip
5.69054pt\Phi(t,\eta)=\Phi^{*},$ (20)

where the superscript ‘*’ denotes the values of the background scalars at the
IR fixed point.

Using (20), one can solve the above set of equations (13)-(17) in the
Fefferman-Graham gauge (3) to obtain

$\displaystyle\omega^{*}=$ $\displaystyle\hskip
2.84526pt\alpha(t)e^{\sqrt{2}\eta\lambda}+\beta(t)e^{-\sqrt{2}\eta\lambda},$
(21) $\displaystyle A_{t}^{*}=$ $\displaystyle\hskip
2.84526pt\mu(t)+\frac{c}{\sqrt{2}\lambda}\Big{(}\alpha(t)e^{\sqrt{2}\eta\lambda}-\beta(t)e^{-\sqrt{2}\eta\lambda}\Big{)},$
(22)

where we define $\lambda=\sqrt{-\Lambda}=\sqrt{3}$, $\omega=\sqrt{-h_{tt}}$
and $c$ is the integration constant. Here, $\alpha(t)$, $\beta(t)$ and
$\mu(t)$ are some arbitrary functions of time.

In order to compute the conformal dimensions of the dual operators, we expand
the scalar fields ($\chi$, $\xi$ and $\Phi$) around the fixed point (20) and
retain the equations of motion (14)-(16) upto linear order in scalar
fluctuations which yields

$\displaystyle\Bigg{[}\partial^{2}_{\eta}+\frac{1}{\omega^{*}}\Big{(}\partial_{\eta}\omega^{*}\Big{)}\partial_{\eta}-\frac{1}{\omega^{*}}\partial_{t}\Bigg{(}\frac{1}{\omega^{*}}\partial_{t}\Bigg{)}-m^{2}\Bigg{]}\tilde{\Phi}=$
$\displaystyle\hskip 2.84526pt0,$ (23)
$\displaystyle\Bigg{[}\partial_{\eta}\Big{(}\omega^{*}\partial_{\eta}\Big{)}-\partial_{t}\Bigg{(}\frac{1}{\omega^{*}}\partial_{t}\Bigg{)}\Bigg{]}\tilde{\chi}=$
$\displaystyle\hskip 2.84526pt0,$ (24)
$\displaystyle\Bigg{[}\partial_{\eta}\Big{(}\omega^{*}\partial_{\eta}\Big{)}-\partial_{t}\Bigg{(}\frac{1}{\omega^{*}}\partial_{t}\Bigg{)}\Bigg{]}\tilde{\xi}=$
$\displaystyle\hskip 2.84526pt0,$ (25)

where we define $m^{2}=\big{(}6-2c^{2}\kappa e^{-\gamma}\big{)}$ and scalar
fluctuations $\tilde{\mathcal{Y}}=\mathcal{Y}-\mathcal{Y}^{*}$, where
$\mathcal{Y}$ collectively denotes the scalar fields ($\Phi$, $\chi$ and
$\xi$).

It should be noted that, the mass-squared term ($m^{2}$) defined above must
satisfy the Breitenlohner-Freedman (BF) bound101010In ($d+1$) spacetime
dimensions, the BF bound is defined as
$m^{2}\geq-\Big{(}\frac{d^{2}}{2L}\Big{)}^{2},$ where $L$ is the AdS length.
[65], which for the present example sets a constraint of the form
$c\leq\sqrt{\frac{25e^{\gamma}}{8\kappa}}$. Notice that, unlike (23), the
equations of motion for scalar fluctuations $\tilde{\chi}$ (24) and
$\tilde{\xi}$ (25) do not contain any mass-squared term. This indicates that
these scalar fields ($\chi$ and $\xi$) are massless. This is consistent with
the fact that these scalar fields ($\chi$ and $\xi$) carry only kinetic terms
in the Lagrangian (2).

From the above set of equations (23)-(25), one could finally decode the
conformal dimensions111111The conformal dimension of the dual operator
($\Delta$) is defined as $\Delta(\Delta-1)=m^{2}$ [43], [62], [65], where $m$
represents the mass of the scalar field. of the dual operators as

$\displaystyle\Delta_{\Phi\pm}=\hskip
2.84526pt\frac{1}{2}\left(1\pm\sqrt{25-8e^{-\gamma}c^{2}\kappa}\right)\hskip
2.84526pt,\hskip 5.69054pt\Delta_{\chi}=\hskip 2.84526pt\Delta_{\xi}=\hskip
2.84526pt1,$ (26)

where the subscript ‘$\pm$’ denotes the two possible values of
$\Delta_{\Phi}$.

It is interesting to notice that the conformal dimension,
$(\Delta_{\chi}=\Delta_{\xi})>\Delta_{\Phi+}$ for the range of constant,
$\sqrt{\frac{3e^{\gamma}}{\kappa}}<c\leq\sqrt{\frac{25e^{\gamma}}{8\kappa}}$.
This indicates that the dynamics of the dilaton fluctuation $\tilde{\Phi}$
dominates [62]–[64] over the scalar fluctuations $\tilde{\chi}$ and
$\tilde{\xi}$ in the deep IR.

On the other hand, one could set the conformal dimension,
$(\Delta_{\chi}=\Delta_{\xi})<\Delta_{\Phi+}$ given the range $0\leq
c<\sqrt{\frac{3e^{\gamma}}{\kappa}}$, which suggests that the IR dynamics is
dominated by the scalar fluctuation $\tilde{\chi}$ and $\tilde{\xi}$. However,
for a particular choice of constant $c=\sqrt{\frac{3e^{\gamma}}{\kappa}}$, the
conformal dimensions, $\Delta_{\chi}=\Delta_{\xi}=\Delta_{\Phi+}=1$. In this
case, the dynamics of all scalar fluctuations $\tilde{\mathcal{Y}}$ are
equally important in the deep IR.

On a similar note, one finds that the maximum value of the conformal
dimension121212In this case, the constant $c$ is restricted to the range
$\sqrt{\frac{3e^{\gamma}}{\kappa}}\leq
c\leq\sqrt{\frac{25e^{\gamma}}{8\kappa}}$. If
$c<\sqrt{\frac{3e^{\gamma}}{\kappa}}$, then the conformal dimension
$\Delta_{\Phi-}$ become negative. $\Delta_{\Phi-}$ is $1/2$. Therefore, in
this case, the dilaton fluctuation $(\tilde{\Phi})$ always dominates over the
scalar fluctuations. Therefore, to summarise, one could conjecture that the
dilaton fluctuation always dominates over scalar fluctuation if the constant
falls in the range
$\sqrt{\frac{3e^{\gamma}}{\kappa}}<c\leq\sqrt{\frac{25e^{\gamma}}{8\kappa}}$.

Finally, it is noteworthy to compare our results with the existing literature
[62]-[63]. The authors in [62], construct a 2D theory of gravity in the
presence of a dilaton ($e^{-2\psi}$), scalar field ($\chi$) and a $U(1)$ gauge
field following a consistent reduction of Einstein gravity in five dimensions.
Unlike the present example, the authors in [62] obtained a mass-squared term
for the scalar field ($\chi$) which is thereby used to calculate the conformal
dimension of the dual operator. Interestingly, they found that the dual
operator is always irrelevant compared to the dilaton operator in the IR. In
other words, the dilaton fluctuation always dominates over the scalar
fluctuations in the deep IR.

$\bullet$ Remarks about perturbative solutions:

Now, we compute the most general solutions of (13)-(17) in the Fefferman-
Graham gauge. Clearly, these equations (13)-(17) are quite difficult to solve
exactly in the ModMax coupling $\gamma$. Therefore, to proceed further, we
simplify the fields as $\Phi\equiv\Phi(\eta),\hskip 2.84526pth_{tt}\equiv
h_{tt}(\eta),\hskip 2.84526ptA_{t}\equiv A_{t}(\eta),\hskip
2.84526pt\xi\equiv\xi(\eta)\hskip 2.84526pt\text{and}\hskip
2.84526pt\chi\equiv\chi(t)$ and solve them “perturbatively” treating the 2D
Maxwell coupling ($\kappa$) and the 2D ModMax coupling $(\gamma)$ as expansion
parameters.

One can systematically expand these fields using the expansion parameters
($\kappa$ and $\gamma$) as

$\displaystyle\mathcal{A}=\mathcal{A}_{0}+\kappa\mathcal{A}_{1}+\gamma\kappa\mathcal{A}_{2}+\kappa^{2}\mathcal{A}_{3}+...,$
(27)
$\displaystyle\mathcal{B}=\mathcal{B}_{1}+\gamma\mathcal{B}_{2}+\kappa\mathcal{B}_{3}+...,\hskip
5.69054pt|\kappa|<<1,\hskip 4.2679pt|\gamma|<<1,$ (28)

where $\mathcal{A}$ collectively denotes the fields ($\Phi$, $\omega$) and
$\mathcal{B}$ denotes the remaining fields ($A_{t}$, $\chi$, $\xi$). Here, the
subscript ‘0’ denotes the pure JT gravity solution. On the other hand,
subscripts ‘$1$’ and ‘$2$’ denote the leading order corrections due to the 2D
Maxwell term and the 2D projected ModMax interaction respectively.
Furthermore, the subscript ‘$3$’ stands for the quadratic order corrections
due to the 2D Maxwell term alone.

Notice that, the $\mathcal{B}$ fields (28) are expanded differently from that
of the $\mathcal{A}$ fields (27). This is due to the fact that the
$\mathcal{B}$ fields are coupled with an overall 2D Maxwell coefficient,
$\kappa$ in the Lagrangian (7). Therefore, one should think of the expansion
(28) to be multiplied with an overall factor of $\kappa$. On the other hand,
the effects of the 2D projected ModMax comes into the picture at the quadratic
level ($\gamma\kappa$). To summarise, we solve the equations of motion
(13)-(17) up to quadratic order ($\gamma\kappa$ and $\kappa^{2}$) in the
couplings and ignore all the higher order corrections.

###  3.1 Zeroth order solution

In order to obtain the pure JT gravity solutions, one has to take the limits
$\kappa\rightarrow 0$ and $\gamma\rightarrow 0$ in the equations (13)-(17),
which yields

$\displaystyle\omega_{0}^{\prime\prime}+\Lambda\omega_{0}=$
$\displaystyle\hskip 2.84526pt0,$ (29)
$\displaystyle\Phi_{0}^{\prime\prime}+\Lambda\Phi_{0}=$ $\displaystyle\hskip
2.84526pt0,$ (30)
$\displaystyle\frac{\Phi_{0}^{\prime}\omega_{0}^{\prime}}{\omega_{0}}+\Lambda\Phi_{0}=$
$\displaystyle\hskip 2.84526pt0.$ (31)

On solving (29)-(31), one finds

$\displaystyle\omega_{0}=$ $\displaystyle\hskip
2.84526pta_{1}e^{\eta\lambda}+a_{2}e^{-\eta\lambda},$ (32)
$\displaystyle\Phi_{0}=$ $\displaystyle\hskip
2.84526pt\frac{b_{1}}{a_{1}\lambda}e^{-\eta\lambda}\left(a_{1}e^{2\eta\lambda}-a_{2}\right),$
(33)

where $a_{1},a_{2}$ and $b_{1}$ are the integration constants.

Equations (32)-(33) are the zeroth order solutions of the theory (7). In the
following Sections, we will be using these solutions to obtain the next to
leading order corrections for $\mathcal{A}$ and $\mathcal{B}$.

###  3.2  Order $\kappa$ solution

The leading order corrections to the fields $\mathcal{A}$ and $\mathcal{B}$
are due to the presence of the Maxwell interactions in (7),

$\displaystyle\mathcal{L}_{\text{Maxwell}}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}\Phi^{-1}\Big{(}(\partial\chi)^{2}+(\partial\xi)^{2}\Big{)}.$
(34)

On comparing the coefficients of $\kappa$ in the equations (13)-(17), we
obtain

$\displaystyle\omega_{0}\Phi_{1}^{\prime\prime}+\omega_{1}\Phi_{0}^{\prime\prime}-\Big{(}\Phi_{0}^{\prime}\omega_{1}^{\prime}+\Phi_{1}^{\prime}\omega_{0}^{\prime}\Big{)}+2\omega_{0}\Bigg{(}\xi_{1}^{\prime
2}+\frac{\dot{\chi_{1}}^{2}}{\omega_{0}^{2}}\Bigg{)}=$ $\displaystyle\hskip
2.84526pt0,$ (35)
$\displaystyle\omega_{1}^{\prime\prime}+\Lambda\omega_{1}-\frac{A_{t1}^{\prime
2}}{\omega_{0}}=$ $\displaystyle\hskip 2.84526pt0,$ (36)
$\displaystyle\partial_{\eta}\Bigg{(}\frac{\Phi_{0}}{\omega_{0}}A_{t1}^{\prime}\Bigg{)}=$
$\displaystyle\hskip 2.84526pt0,$ (37)
$\displaystyle\partial_{\eta}\Big{(}\omega_{0}\xi_{1}^{\prime}\Big{)}=$
$\displaystyle\hskip 2.84526pt0,$ (38) $\displaystyle\ddot{\chi}_{1}=$
$\displaystyle\hskip 2.84526pt0.$ (39)

Using the zeroth order solutions (32)-(33), one can solve the above set of
equations to yield

$\displaystyle\Phi_{1}=$ $\displaystyle\hskip
2.84526pt\frac{e^{-\eta\lambda}}{4\lambda^{2}}\Bigg{(}\frac{4\lambda}{a_{1}}\left(a_{3}b_{1}e^{2\eta\lambda}+a_{2}\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\right)+4\lambda
e^{2\eta\lambda}\Big{(}2\eta\lambda-$
$\displaystyle\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\Big{)}+\tan^{-1}\left(\frac{\sqrt{a_{1}}e^{\eta\lambda}}{\sqrt{a_{2}}}\right)\Bigg{(}\frac{1}{a_{1}^{3/2}\sqrt{a_{2}}}-\frac{e^{2\eta\lambda}}{\sqrt{a_{1}}a_{2}^{3/2}}\Bigg{)}\Bigg{)},$
(40) $\displaystyle\omega_{1}=$ $\displaystyle\hskip
2.84526pt\frac{c_{1}^{2}}{4a_{2}}e^{-\eta\lambda}\left(2\eta\lambda
e^{2\eta\lambda}-\frac{1}{a_{1}}\left(a_{1}e^{2\eta\lambda}+a_{2}\right)\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\right)+a_{3}e^{\eta\lambda},$
(41) $\displaystyle A_{t1}=$ $\displaystyle\hskip
2.84526ptc_{1}\Bigg{[}\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)-\eta\lambda\Bigg{]}+c_{2},$
(42) $\displaystyle\xi_{1}=$ $\displaystyle\hskip
2.84526pt\frac{e_{1}}{\lambda}\tan^{-1}\left(\frac{\sqrt{a_{1}}e^{\eta\lambda}}{\sqrt{a_{2}}}\right)+e_{2},$
(43) $\displaystyle\chi_{1}=$ $\displaystyle\hskip 2.84526ptd_{1}t+d_{2},$
(44)

where $a_{3},c_{i},d_{i}$ and $e_{i}$, $(i=1,2)$ are the integration
constants.

Equations (40)-(44) represent the leading order corrections to the fields
$\mathcal{A}$ and $\mathcal{B}$ in the presence of the 2D Maxwell interactions
(34).

###  3.3  Order $\gamma\kappa$ solution

Next, we take into account the projected ModMax interactions and their imprint
on the background fields $\mathcal{A}$ (27) and $\mathcal{B}$ (28).

A straight forward analysis reveals the following set of equations at order
$\gamma\kappa$

$\displaystyle\omega_{0}\Phi_{2}^{\prime\prime}+\omega_{2}\Phi_{0}^{\prime\prime}-\Phi_{0}^{\prime}\omega_{2}^{\prime}-\Phi_{2}^{\prime}\omega_{0}^{\prime}+4\omega_{0}\xi_{1}^{\prime}\xi_{2}^{\prime}+\frac{4}{\omega_{0}}\dot{\chi_{1}}\dot{\chi_{2}}-f_{0}=$
$\displaystyle\hskip 2.84526pt0,$ (45)
$\displaystyle\omega_{2}^{\prime\prime}-\omega_{2}\lambda^{2}-\frac{2}{\omega_{0}}A_{t1}^{\prime}A_{t2}^{\prime}+\frac{s_{0}A_{t1}^{\prime
2}}{\omega_{0}\sqrt{s_{0}^{2}+p_{0}^{2}}}=$ $\displaystyle\hskip 2.84526pt0,$
(46)
$\displaystyle\partial_{\eta}\Bigg{[}\frac{\Phi_{0}}{\omega_{0}}\Bigg{(}A_{t2}^{\prime}-\frac{s_{0}}{\sqrt{s_{0}^{2}+p_{0}^{2}}}A_{t1}^{\prime}\Bigg{)}\Bigg{]}=$
$\displaystyle\hskip 2.84526pt0,$ (47)
$\displaystyle\partial_{\eta}\Bigg{[}\omega_{0}\xi_{2}^{\prime}-\frac{\Big{(}s_{0}\xi_{1}^{\prime}\omega_{0}-p_{0}\dot{\chi_{1}}\Big{)}}{\sqrt{s_{0}^{2}+p_{0}^{2}}}\Bigg{]}=$
$\displaystyle\hskip 2.84526pt0,$ (48) $\displaystyle\ddot{\chi_{2}}=$
$\displaystyle\hskip 2.84526pt0,$ (49)

where we identify the above functions as

$\displaystyle s_{0}=$ $\displaystyle\hskip
2.84526pt-\frac{1}{\omega_{0}^{2}}A_{t1}^{\prime
2}+\frac{1}{\Phi_{0}}\Bigg{(}-\frac{\dot{\chi_{1}^{2}}}{\omega_{0}^{2}}+\xi_{1}^{\prime
2}\Bigg{)}\hskip 2.84526pt,\hskip
5.69054ptp_{0}=-\frac{2}{\Phi_{0}\omega_{0}}\dot{\chi_{1}}\xi_{1}^{\prime},$
(50) $\displaystyle f_{0}=$ $\displaystyle\hskip
2.84526pt\frac{2\omega_{0}s_{0}}{\sqrt{s_{0}^{2}+p_{0}^{2}}}\Bigg{(}\xi_{1}^{\prime
2}+\frac{\dot{\chi_{1}}^{2}}{\omega_{0}^{2}}\Bigg{)}.$ (51)

The above set of equations (45)-(49) are difficult to solve for generic values
of $\eta$. However, for our present purpose, it will be sufficient to solve
them near the asymptotic limit ($\eta\rightarrow\infty$) of the space-time
which yields

$\displaystyle\Phi_{2}=$ $\displaystyle\hskip
2.84526pt\frac{1}{\lambda}\Big{(}b_{2}e^{\eta\lambda}-b_{3}\lambda+e^{-\eta\lambda}\Big{)}-\frac{b_{1}}{a_{1}}\eta
e^{-\eta\lambda},$ (52) $\displaystyle\omega_{2}=$ $\displaystyle\hskip
2.84526pte^{-\eta\lambda}\left(a_{4}e^{2\eta\lambda}+a_{5}+\eta\lambda\right),$
(53) $\displaystyle\xi_{2}=$ $\displaystyle\hskip
2.84526pt\frac{e_{3}}{\lambda}e^{-\eta\lambda}+e_{4},$ (54) $\displaystyle
A_{t2}=$ $\displaystyle\hskip 2.84526ptc_{3}\eta\lambda+c_{4},$ (55)
$\displaystyle\chi_{2}=$ $\displaystyle\hskip 2.84526ptd_{3}t+d_{4}.$ (56)

where $a_{i},b_{j},c_{k},d_{k}$ and $e_{k}$, $(i=4,5,\hskip
2.84526ptj=2,3,\hskip 2.84526ptk=3,4)$ are the integration constants.

As we show below, not all of these integration constants are actually
important for our analysis. In fact, a few of them finally survive which can
be fixed by making use of the residual gauge freedom [51] in the Fefferman-
Graham gauge (3). In particular, the re-scaling of the time coordinate
$t\rightarrow a_{1}t$ preserves the gauge condition $g_{\eta t}=0$ and
$g_{\eta\eta}=1$. Therefore, we can use this freedom to fix the
constant131313Interestingly, with this particular choice of the integration
constant $a_{1}$ (57), the final expression of the central charge (96) appears
to be independent of all the remaining integration constants.

$\displaystyle a_{1}=\frac{1}{a_{2}b_{3}}.$ (57)

###  3.4 Order $\kappa^{2}$ solution

Finally, we estimate the quadratic order ($\kappa^{2}$) corrections due to the
Maxwell (34) term alone.

The resulting equations of motion (13)-(17) can be expressed as

$\displaystyle\omega_{0}\partial_{\eta}\Bigg{(}\frac{\Phi_{0}A_{t3}^{\prime}}{\omega_{0}}-\frac{\Phi_{0}\omega_{1}A_{t1}^{\prime}}{\omega_{0}^{2}}+\frac{\Phi_{1}A_{t1}^{\prime}}{\omega_{0}}\Bigg{)}-\omega_{1}\partial_{\eta}\Bigg{(}\frac{\Phi_{0}A_{t1}^{\prime}}{\omega_{0}}\Bigg{)}=$
$\displaystyle\hskip 2.84526pt0,$ (58)
$\displaystyle\omega_{0}\Phi_{3}^{\prime\prime}+\omega_{1}\Phi_{1}^{\prime\prime}+\omega_{3}\Phi_{0}^{\prime\prime}-\big{(}\Phi_{0}^{\prime}\omega_{3}^{\prime}+\Phi_{1}^{\prime}\omega_{1}^{\prime}+\Phi_{3}^{\prime}\omega_{0}^{\prime}\big{)}+f_{2}=$
$\displaystyle\hskip 2.84526pt0,$ (59)
$\displaystyle\omega_{3}^{\prime\prime}-\lambda^{2}\omega_{3}-\frac{1}{\omega_{0}}\Bigg{(}2A_{t1}^{\prime}A_{t3}^{\prime}-\frac{\omega_{1}}{\omega_{0}}A_{t1}^{\prime
2}\Bigg{)}=$ $\displaystyle\hskip 2.84526pt0,$ (60)
$\displaystyle\partial_{\eta}\big{(}\omega_{0}\xi_{3}^{\prime}+\omega_{1}\xi_{1}^{\prime}\big{)}-\frac{\omega_{1}}{\omega_{0}}\partial_{\eta}\big{(}\omega_{0}\xi_{1}^{\prime}\big{)}=$
$\displaystyle\hskip 2.84526pt0,$ (61) $\displaystyle\ddot{\chi}_{3}=$
$\displaystyle\hskip 2.84526pt0,$ (62)

where

$\displaystyle
f_{2}=2\big{(}2\omega_{0}\xi_{1}^{\prime}\xi_{3}^{\prime}+\omega_{1}\xi_{1}^{\prime
2}\big{)}+\frac{2}{\omega_{0}}\Bigg{(}2\dot{\chi}_{1}\dot{\chi}_{3}-\dot{\chi}_{1}^{2}\frac{\omega_{1}}{\omega_{0}}\Bigg{)}.$
(63)

The above set of equations (58)-(62) could be solved near the asymptotics
($\eta\rightarrow\infty$) of the spacetime which yield

$\displaystyle\Phi_{3}=$ $\displaystyle\hskip
2.84526pt-\frac{1}{2a_{1}^{2}\lambda^{2}}\Big{(}e^{-\eta\lambda}(2\eta\lambda+3)\left(2a_{2}a_{3}\lambda-
b_{1}c_{1}c_{5}\right)\Big{)}+\frac{b_{4}e^{\eta\lambda}}{\lambda}+\frac{a_{1}\left(a_{1}^{2}+1\right)+a_{3}}{6a_{1}^{2}\left(a_{1}^{2}+1\right)a_{2}},$
(64) $\displaystyle\omega_{3}=$ $\displaystyle\hskip
2.84526pt\frac{1}{4}e^{-\eta\lambda}\left(\frac{c_{1}(2\eta\lambda+1)\left(a_{3}c_{1}\lambda-2a_{1}c_{5}\right)}{a_{1}^{2}\lambda}+4a_{6}e^{2\eta\lambda}+4a_{7}\right),$
(65) $\displaystyle\xi_{3}=$ $\displaystyle\hskip
2.84526pt\frac{1}{36a_{1}^{5/2}\lambda}\Big{(}\sqrt{a_{2}}e_{1}e^{-3\eta\lambda}\left(c_{1}^{2}(1-6\eta\lambda)-12a_{2}a_{3}\right)\Big{)}-\frac{e_{5}e^{-\eta\lambda}}{\lambda}+e_{6},$
(66) $\displaystyle A_{t3}=$ $\displaystyle\hskip
2.84526pt\frac{c_{1}e^{-\eta\lambda}}{4a_{1}a_{2}b_{1}\lambda}+c_{6}\eta+c_{5},$
(67) $\displaystyle\chi_{3}=$ $\displaystyle\hskip 2.84526ptd_{5}t+d_{6},$
(68)

where $a_{i},b_{j},c_{k},d_{k}$ and $e_{k}$ ($i=6,7,\hskip
2.84526ptj=4,5,\hskip 2.84526ptk=5,6$) are the integration constants.

##  4 Boundary stress tensor and central charge

In this Section, we work out the “renormalised” boundary stress tensor [48],
[51], [54]-[56] and study its transformation properties under both the
diffeomorphism and the $U(1)$ gauge transformations. In particular, we examine
the effects of the projected ModMax interactions on the central charge of the
boundary theory.

To begin with, we workout the boundary terms141414The boundary in the
Ferrerman-Graham gauge is located near $\eta\rightarrow\infty$. for the action
(7). This is required in order to implement a consistent variational principle
[48], [51]. Systematically, one can decompose the boundary terms into
following two pieces,

$\displaystyle I_{\text{boundary}}$ $\displaystyle\hskip
2.84526pt=I_{\text{GHY}}+I_{\text{counter}},$ (69)

where $I_{\text{GHY}}$ is the standard Gibbons-Hawking-York boundary term and
$I_{\text{counter}}$ represents the boundary counter terms.

The Gibbons-Hawking-York boundary term [48], [51], [66] in 2D gravity is given
by

$\displaystyle I_{\text{GHY}}=\frac{1}{8\pi
G_{2}}\int_{0}^{\beta}dt\sqrt{-h}\Phi K\hskip 2.84526pt,\hskip
5.69054ptK=\frac{1}{2}h^{tt}\partial_{\eta}h_{tt},$ (70)

where $K$ is the trace of extrinsic curvature, $\beta$ is the inverse
temperature and $h_{tt}$ is the induced metric on the boundary.

On the other hand, the counter term that is required to absorb all the near
boundary divergences of the on-shell action can be expressed as

$\displaystyle I_{\text{counter}}=-\frac{1}{8\pi
G_{2}}\int_{0}^{\beta}dt\sqrt{-h}\Bigg{(}\lambda\Phi+2\kappa\frac{b_{1}}{c_{1}}\sqrt{-h^{ab}A_{a}A_{b}}\Bigg{)},$
(71)

where $(a,b)$ are the one dimensional boundary indices151515Here, we set the
constant $c_{6}=-\frac{\pi
c_{1}^{3}}{16\sqrt{a_{1}a_{2}^{3}}b_{1}(a_{1}-c_{1}^{2})}$ in order to cancel
the boundary divergences up to quadratic order ($\gamma\kappa$ and
$\kappa^{2}$) in the couplings..

Finally, the complete renormalised action is given by

$\displaystyle I_{\text{renormalised}}=I_{\text{bulk}}+I_{\text{boundary}},$
(72)

where $I_{\text{bulk}}$ and $I_{\text{boundary}}$ are given in (7) and (69)
respectively.

Notice that, the combination of the $U(1)$ gauge field in the
$I_{\text{counter}}$ (71) seems to break the gauge invariance under the
transformation

$\displaystyle A_{\alpha}\rightarrow A_{\alpha}+\partial_{\alpha}\Sigma,$ (73)

which yields the following extra piece under the $U(1)$ gauge (73)

$\displaystyle
I_{\text{counter}}\sim\int_{0}^{\beta}dt\sqrt{-h}\Big{(}\sqrt{-h^{ab}A_{a}A_{b}}\Big{)}\rightarrow\int_{0}^{\beta}dt(A_{t}+\partial_{t}\Sigma).$
(74)

However, one can preserve the gauge invariance by imposing the condition that
$\partial_{t}\Sigma$ (see (4)) must vanish near the boundary,
$\eta\rightarrow\infty$ [51].

Using the renormalised action (72), it is now straightforward to calculate the
variation $\delta I_{\text{boundary}}$ under the combined action of the
diffeomorphism and the $U(1)$ gauge, where $\delta I_{\text{boundary}}$ can be
systematically expressed as161616$\delta I_{\text{boundary}}$ already
incorporates the bulk contributions $(\delta I_{\text{bulk}})$ near the
asymptotic limit, $\eta\rightarrow\infty$.

$\displaystyle\delta I_{\text{boundary}}=\frac{1}{16\pi G_{2}}\int
dt\sqrt{-h}\Big{(}\mathcal{G}^{ab}\delta
h_{ab}+\mathcal{G}_{\Phi}\delta\Phi+\mathcal{G}^{a}\delta
A_{a}+\mathcal{G}_{\chi}\delta\chi+\mathcal{G}_{\xi}\delta\xi\Big{)}.$ (75)

Here, the boundary contributions can be expressed as

$\displaystyle\mathcal{G}^{ab}=$ $\displaystyle\hskip
2.84526ptn_{\mu}\nabla^{\mu}\Phi
h^{ab}+n^{\mu}\frac{\Phi}{\sqrt{-h}}\Big{(}\partial_{\mu}\sqrt{-h}\Big{)}h^{ab}-\lambda\Phi
h^{ab}-2\kappa\frac{b_{1}}{c_{1}}h^{ab}\sqrt{-h^{cd}A_{c}A_{d}}$
$\displaystyle+2\kappa\frac{b_{1}}{c_{1}}\frac{A^{a}A^{b}}{\sqrt{-h^{cd}A_{c}A_{d}}},$
(76) $\displaystyle\mathcal{G}^{a}=$ $\displaystyle\hskip 2.84526pt-4\kappa
n_{\mu}\Phi\Bigg{(}\cosh{\gamma}-\frac{s}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{)}F^{\mu
a}+4\kappa\frac{b_{1}}{c_{1}}\frac{h^{ab}A_{b}}{\sqrt{-h^{cd}A_{c}A_{d}}},$
(77) $\displaystyle\mathcal{G}_{\chi}=$ $\displaystyle\hskip 2.84526pt-4\kappa
n_{\mu}\Bigg{(}\nabla^{\mu}\chi\cosh{\gamma}-\frac{s\nabla^{\mu}\chi-p\epsilon^{\mu
a}\nabla_{a}\xi}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{)},$ (78)
$\displaystyle\mathcal{G}_{\xi}=$ $\displaystyle\hskip 2.84526pt-4\kappa
n_{\mu}\Bigg{(}\nabla^{\mu}\xi\cosh{\gamma}-\frac{s\nabla^{\mu}\xi+p\epsilon^{\mu
a}\nabla_{a}\chi}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{)},$ (79)
$\displaystyle\mathcal{G}_{\Phi}=$ $\displaystyle\hskip 2.84526pt2K-2\lambda,$
(80)

where $n^{\mu}=\delta^{\mu}_{\eta}$ is the unit normal vector at the boundary.

With all these preliminaries, we now introduce the boundary stress tensor
[48], [51] corresponding to the action (72)

$\displaystyle T^{ab}=\frac{2}{\sqrt{-h}}\frac{\delta
I_{\text{boundary}}}{\delta h_{ab}}=\frac{\mathcal{G}^{ab}}{8\pi G_{2}},$ (81)

where $\mathcal{G}^{ab}$ is given in (76).

Our next task is to explore the transformation properties of the background
fields (27)-(28) and hence the boundary stress tensor (81) under the combined
effects of the diffeomorphism and the $U(1)$ gauge transformation.

Under the diffeomorphism,

$\displaystyle x^{\mu}\rightarrow x^{\mu}+\epsilon^{\mu}(x),$ (82)

the background fields (27)-(28) transform as

$\displaystyle\delta_{\epsilon}A_{\mu}=$ $\displaystyle\hskip
2.84526pt\epsilon^{\nu}\nabla_{\nu}A_{\mu}+A_{\nu}\nabla_{\mu}\epsilon^{\nu},$
(83) $\displaystyle\delta_{\epsilon}g_{\mu\nu}=$ $\displaystyle\hskip
2.84526pt\nabla_{\mu}\epsilon_{\nu}+\nabla_{\nu}\epsilon_{\mu},$ (84)
$\displaystyle\delta_{\epsilon}\mathcal{S}=$ $\displaystyle\hskip
2.84526pt\epsilon^{\mu}\nabla_{\mu}\mathcal{S},$ (85)

where $\mathcal{S}$ collectively denotes the scalar fields $\Phi,\hskip
2.84526pt\xi$ and $\chi$.

The diffeomorphism parameter, $\epsilon^{\mu}(x)$ can be obtained using (84)
and the space-time metric (3), which yields the following

$\displaystyle\epsilon_{t}=e^{2\eta\lambda}f(t)+\frac{1}{2\lambda}\Bigg{(}\frac{2\left(a_{1}^{2}+1\right)}{3a_{1}^{2}\lambda}-\frac{4a_{3}\kappa}{3a_{1}^{3}\lambda}\Bigg{)}\partial^{2}_{t}f(t)\hskip
2.84526pt,\hskip
5.69054pt\epsilon_{\eta}=\Bigg{(}\frac{2\left(a_{1}^{2}+1\right)}{3a_{1}^{2}\lambda}-\frac{4a_{3}\kappa}{3a_{1}^{3}\lambda}\Bigg{)}\partial_{t}f(t),$
(86)

where $f(t)$ is some function171717In the Fefferman-Graham gauge [48], [57],
the variation of the space-time metric (under diffeomorphism (84)) yields a
set of coupled differential equations that contain the derivatives of the
diffeomorphism parameters $\epsilon_{t}$ and $\epsilon_{\eta}$ with respect to
the variable “$\eta$”. Therefore, the function $f(t)$ in these equations
appears as an integration constant. However, one can further compute the
function $f(t)$ using suitable boundary conditions for the background fields
$\mathcal{A}$ (27) and $\mathcal{B}$ (28). of time [51].

It should be noted that, we perform all the analysis in a gauge in which one
of the components of the $U(1)$ gauge field, $A_{\eta}$ is set to be zero (3).
On the other hand, under the diffeomorphism (82), $A_{\eta}$ transforms as

$\displaystyle\delta_{\epsilon}A_{\eta}=A_{t}\partial_{\eta}\Bigg{(}\frac{\epsilon_{t}}{h_{tt}}\Bigg{)}\neq
0,$ (87)

which breaks the gauge condition $A_{\eta}=0$.

In order to restore this gauge condition, we employ the $U(1)$ gauge
transformation, $A_{\alpha}\rightarrow A_{\alpha}+\partial_{\alpha}\Sigma$ and
compute the $U(1)$ gauge parameter $\Sigma$ such that
$(\delta_{\epsilon}+\delta_{\Sigma})A_{\eta}=0$, which yields the following

$\displaystyle\Sigma=\hskip 2.84526pt-\int d\eta
A_{t}\partial_{\eta}\Bigg{(}\frac{\epsilon_{t}}{h_{tt}}\Bigg{)},$ (88)

where we have used the variation (87).

Now, one can perform the above integration (88) using the background fields
(27)-(28) and the diffeomorphism parameter (86), which yields

$\displaystyle\Sigma=$ $\displaystyle\hskip
2.84526pt\frac{e^{-2\eta\lambda}}{12a_{1}^{5}a_{2}\lambda^{3}}\Bigg{(}f^{\prime\prime}(t)\big{(}2\left(a_{1}^{2}+1\right)a_{2}a_{1}\big{(}c_{1}\lambda\left(2\log\left(a_{1}\right)-(\gamma-1)(2\eta\lambda+1)\right)+2\gamma
c_{3}\lambda+$ $\displaystyle\hskip
2.84526pt\kappa\left(2\lambda\left(c_{5}\eta+c_{4}\right)+c_{5}\right)\big{)}+a_{1}^{2}c_{1}\kappa\lambda\left(c_{1}^{2}\log\left(a_{1}\right)-4a_{2}a_{3}\right)\left(2\log\left(a_{1}\right)+2\eta\lambda+1\right)+$
$\displaystyle\hskip
2.84526ptc_{1}\kappa\lambda\left(c_{1}^{2}\log\left(a_{1}\right)-8a_{2}a_{3}\right)\left(2\log\left(a_{1}\right)+2\eta\lambda+1\right)\big{)}-3a_{1}a_{2}c_{1}^{3}\kappa\lambda^{3}f(t)\big{(}2\log\left(a_{1}\right)+$
$\displaystyle\hskip 2.84526pt2\eta\lambda+1\big{)}\Bigg{)}.$ (89)

It is interesting to notice that the $U(1)$ gauge parameter $\Sigma$ vanishes
naturally in the asymptotic limit ($\eta\rightarrow\infty$), which is
consistent with the gauge preserving condition (74).

Finally, we note down the transformation of the boundary stress tensor (81)
under the combined action of the diffeomorphism (82) and the $U(1)$ gauge
transformation which yields

$\displaystyle(\delta_{\epsilon}+\delta_{\Sigma})T_{tt}=$ $\displaystyle\hskip
2.84526pt\frac{1}{8\pi
G_{2}}\Bigg{[}\Bigg{(}\partial_{\eta}\Phi-\lambda\Phi-2\kappa\frac{b_{1}}{c_{1}}\frac{A_{t}}{\omega}\Bigg{)}(\delta_{\epsilon}h_{tt})+4\kappa\frac{b_{1}\omega}{c_{1}}\big{(}(\delta_{\epsilon}+\delta_{\Sigma})A_{t}\big{)}$
$\displaystyle+\frac{\Phi}{2}\partial_{\eta}(\delta_{\epsilon}h_{tt})-\frac{1}{2}\Big{(}\partial_{\eta}\omega^{2}-2\lambda\omega^{2}\Big{)}(\delta_{\epsilon}\Phi)-\partial_{\eta}(\delta_{\epsilon}\Phi)\omega^{2}\Bigg{]}.$
(90)

The variations of the background fields $h_{tt},\hskip 2.84526ptA_{t}$ and
$\Phi$ can be obtained using (83)-(86) and (4), which yields the following

$\displaystyle(\delta_{\epsilon}+\delta_{\Sigma})A_{t}=$ $\displaystyle\hskip
2.84526pt\mathcal{H}_{1}(\eta)\partial_{t}f(t)+\mathcal{H}_{2}(\eta)\partial_{t}^{3}f(t),$
(91) $\displaystyle\delta_{\epsilon}h_{tt}=$ $\displaystyle\hskip
2.84526pt\mathcal{H}_{3}(\eta)\partial_{t}f(t)+\mathcal{H}_{4}(\eta)\partial_{t}^{3}f(t),$
(92) $\displaystyle\delta_{\epsilon}\Phi=$ $\displaystyle\hskip
2.84526pt\mathcal{H}_{5}(\eta)\partial_{t}f(t),$ (93)

where the explicit form of the functions $\mathcal{H}_{i}(\eta)$,
$(i=1,2...5)$ are given in the Appendix B.

Using these variations (91)-(93), the transformation of the boundary stress
tensor (90) can be expressed in a more elegant way

$\displaystyle(\delta_{\epsilon}+\delta_{\Sigma})\tilde{T}_{tt}=2\tilde{T}_{tt}\partial_{t}f(t)+f(t)\partial_{t}\tilde{T}_{tt}-c_{M}\partial^{3}_{t}f(t).$
(94)

Here, we define the re-scaled stress tensor as

$\displaystyle\tilde{T}_{tt}=\frac{T_{tt}}{b_{3}(1+a_{2}^{2}b_{3}^{2})},$ (95)

and identify the coefficient “$c_{M}$” (coefficient of $\partial_{t}^{3}f(t)$)
as being the central charge [48], [51] of the boundary theory,

$\displaystyle c_{M}=\frac{1}{144\sqrt{3}\pi
G_{2}}\Big{(}\kappa-12\gamma\kappa+2\kappa^{2}\Big{)},$ (96)

where we substitute $\lambda=\sqrt{3}$.

It should be noted that the above expression of the central charge (96) is a
perturbative result up to quadratic order in the ModMax coupling ($\gamma$)
and the $U(1)$ gauge coupling ($\kappa$). Clearly, in the limit
$\gamma\rightarrow 0$, the central charge (96) reduces to
$\sim\frac{1}{G_{2}}$ which is consistent with the existing result in the
literature [51].

##  5 Black holes and 2D projected ModMax

We now construct the 2D black hole solutions and investigate their thermal
properties in the presence of 2D projected ModMax interactions (7). In
particular, we emphasise on the role played by the ModMax parameter, that is
required to set all the fields “finite” near the horizon. These solutions are
further used to compute the Wald entropy [59]-[61] associated with these 2D
black holes. Finally, we also comment on the possibilities for extremal black
hole solutions in two dimensions.

###  5.1 Black hole solutions

We estimate the 2D black hole solutions of (7) by means of perturbative
techniques up to quadratic order in the ModMax parameter $(\gamma)$ and the
Maxwell’s coupling ($\kappa$). Technically speaking, it is not convenient to
determine the black hole horizon in the Ferrferman-Graham gauge due to the
presence of the non-trivial couplings in $U(1)$ gauge fields (7). However, one
can perform an elegant calculation using the light cone gauge. In this gauge,
the space-time metric can be expressed as

$\displaystyle ds^{2}=$ $\displaystyle\hskip
2.84526pte^{2\omega(z)}\big{(}-dt^{2}+dz^{2}\big{)},\hskip
2.84526ptA_{\mu}dx^{\mu}=\hskip 2.84526ptA_{t}(z)dt,$
$\displaystyle\Phi=\hskip 2.84526pt$ $\displaystyle\Phi(z),\hskip
2.84526pt\chi=\chi(t),\hskip 2.84526pt\xi=\xi(z).$ (97)

Like before as in (27)-(28), one can systematically expand the background
fields in the couplings $\kappa$ and $\gamma$ as

$\displaystyle\mathcal{A}^{(bh)}=\mathcal{A}_{0}^{(bh)}+\kappa\mathcal{A}_{1}^{(bh)}+\gamma\kappa\mathcal{A}_{2}^{(bh)}+\kappa^{2}\mathcal{A}_{3}^{(bh)}...,$
(98)
$\displaystyle\mathcal{B}^{(bh)}=\mathcal{B}_{1}^{(bh)}+\gamma\mathcal{B}_{2}^{(bh)}+\kappa\mathcal{B}_{3}^{(bh)}...,\hskip
5.69054pt|\kappa|<<1,\hskip 4.2679pt|\gamma|<<1,$ (99)

where $\mathcal{A}^{(bh)}$ collectively represents the fields ($\Phi$,
$\omega$) and $\mathcal{B}^{(bh)}$ represents the remaining fields ($A_{t}$,
$\chi$, $\xi$). Furthermore, the superscript “${bh}$” in $\mathcal{A}^{(bh)}$
and $\mathcal{B}^{(bh)}$ denote the black hole solution.

####  5.1.1 Zeroth order solution

In order to calculate black hole solutions at zeroth order, we switch off the
$U(1)$ gauge couplings ($\kappa\rightarrow 0$, $\gamma\rightarrow 0$) in the
equations of motion (13)-(17), which yields the following set of equations

$\displaystyle\Phi_{0}^{\prime\prime}-\omega_{0}^{\prime}\Phi_{0}^{\prime}+\Lambda
e^{2\omega_{0}}\Phi_{0}=$ $\displaystyle\hskip 2.84526pt0,$ (100)
$\displaystyle\omega_{0}^{\prime}\Phi_{0}^{\prime}+\Lambda
e^{2\omega_{0}}\Phi_{0}=$ $\displaystyle\hskip 2.84526pt0,$ (101)
$\displaystyle\omega_{0}^{\prime\prime}+e^{2\omega_{0}}\Lambda=$
$\displaystyle\hskip 2.84526pt0,$ (102)

where ′ denotes the derivative with respect to $z$.

On solving the equations (100)-(102), one finds

$\displaystyle
e^{2\omega_{0}^{(bh)}}=-\frac{4\mu}{\Lambda\sinh^{2}{(2\sqrt{\mu}}z)}\hskip
2.84526pt,\hskip 5.69054pt\Phi_{0}^{(bh)}=\phi_{0},$ (103)

where $\phi_{0}$ is a constant.

It should be noted that we treat the dilaton ($\Phi$) as constant while taking
the limits $\kappa\rightarrow 0$ and $\gamma\rightarrow 0$. However, it
possesses a non-trivial profile in the presence of $U(1)$ gauge fields (see
Section (5.1.2) and (5.1.3)).

####  5.1.2 Order $\kappa$ solution

The leading order corrections to $\mathcal{A}^{(bh)}$ and $\mathcal{B}^{(bh)}$
could be estimated by solving the equations of motion (13)-(17) at order
$\kappa$

$\displaystyle\Phi_{1}^{\prime\prime}-2\big{(}\omega_{0}^{\prime}\Phi_{1}^{\prime}+\omega_{1}^{\prime}\Phi_{0}^{\prime}\big{)}+2\big{(}\dot{\chi}_{1}^{2}+\xi_{1}^{\prime
2}\big{)}=$ $\displaystyle\hskip 2.84526pt0,$ (104)
$\displaystyle\omega_{1}^{\prime\prime}+2\Lambda\omega_{1}e^{2\omega_{0}}-e^{-2\omega_{0}}A_{t1}^{\prime
2}=$ $\displaystyle\hskip 2.84526pt0,$ (105)
$\displaystyle\partial_{z}\Big{(}\Phi_{0}e^{-2\omega_{0}}A_{t1}^{\prime}\Big{)}=$
$\displaystyle\hskip 2.84526pt0,$ (106) $\displaystyle\xi_{1}^{\prime\prime}=$
$\displaystyle\hskip 2.84526pt0,$ (107) $\displaystyle\ddot{\chi}_{1}=$
$\displaystyle\hskip 2.84526pt0,$ (108)

where . and ′ denote the derivatives with respect to $t$ and $z$ respectively.

In order to solve the above differential equations (104)-(108), we adopt the
following change in coordinates

$\displaystyle\rho=\sqrt{\mu}\coth{(2\sqrt{\mu}z)}.$ (109)

Using the zeroth order solutions (103) together with (109), one finds

$\displaystyle\omega_{1}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{q_{2}}{\sqrt{\mu}}\rho\tanh^{-1}{\Bigg{(}\frac{\rho}{\sqrt{\mu}}\Bigg{)}}+\frac{q_{1}\rho}{\sqrt{\mu}}+\frac{m_{1}^{2}}{2\Lambda\phi_{0}^{2}}-q_{2},$
(110) $\displaystyle\Phi_{1}^{(bh)}=$ $\displaystyle\hskip
2.84526pt-\frac{(n_{1}^{2}+l_{1}^{2})\rho}{4\mu^{\frac{3}{2}}}\tanh^{-1}{\Bigg{(}\frac{\rho}{\sqrt{\mu}}\Bigg{)}}+g_{1}\rho,$
(111) $\displaystyle\xi_{1}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{l_{1}}{2\sqrt{\mu}}\coth^{-1}{\Bigg{(}\frac{\rho}{\sqrt{\mu}}\Bigg{)}}+l_{2},$
(112) $\displaystyle A_{t1}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{2m_{1}\rho}{\Lambda\phi_{0}}+m_{2},$ (113)
$\displaystyle\chi_{1}^{(bh)}=$ $\displaystyle\hskip 2.84526ptn_{1}t+n_{2},$
(114)

where $m_{i},n_{i},l_{i},q_{i}$ and $g_{1}$, $(i=1,2)$ are the integration
constants.

####  5.1.3 Order $\gamma\kappa$ solution

The contributions due to the projected ModMax interactions could be estimated
by solving the equations of motion (13)-(17) at order $\gamma\kappa$

$\displaystyle\Phi_{2}^{\prime\prime}-2\big{(}\omega_{0}^{\prime}\Phi_{2}^{\prime}+\omega_{2}^{\prime}\Phi_{0}^{\prime}\big{)}+4\big{(}\dot{\chi}_{1}\dot{\chi}_{2}+\xi_{1}^{\prime}\xi_{2}^{\prime}\big{)}-\frac{2s_{0}}{\sqrt{s_{0}^{2}+p_{0}^{2}}}\big{(}\dot{\chi}_{1}^{2}+\xi_{1}^{\prime
2}\big{)}=$ $\displaystyle\hskip 2.84526pt0,$ (115)
$\displaystyle\omega_{2}^{\prime\prime}+2\Lambda\omega_{2}e^{2\omega_{0}}-2e^{-2\omega_{0}}A_{t1}^{\prime}A_{t2}^{\prime}-e^{2\omega_{0}}\sqrt{s_{0}^{2}+p_{0}^{2}}+f_{1}=$
$\displaystyle\hskip 2.84526pt0,$ (116)
$\displaystyle\partial_{z}\Bigg{[}\xi_{2}^{\prime}-\frac{1}{\sqrt{s_{0}^{2}+p_{0}^{2}}}\Big{(}s_{0}\xi_{1}^{\prime}-p_{0}\dot{\chi}_{1}\Big{)}\Bigg{]}=$
$\displaystyle\hskip 2.84526pt0,$ (117)
$\displaystyle\partial_{z}\Bigg{[}e^{-2\omega_{0}}\Phi_{0}\Bigg{(}A_{t2}^{\prime}-\frac{A_{t1}^{\prime}s_{0}}{\sqrt{s_{0}^{2}+p_{0}^{2}}}\Bigg{)}\Bigg{]}=$
$\displaystyle\hskip 2.84526pt0,$ (118) $\displaystyle\ddot{\chi}_{2}=$
$\displaystyle\hskip 2.84526pt0,$ (119)

where we define the above quantities as

$\displaystyle f_{1}=$ $\displaystyle\hskip
2.84526pt\frac{1}{\Phi_{0}\sqrt{s_{0}^{2}+p_{0}^{2}}}\Big{(}s_{0}\big{(}-\dot{\chi}_{1}^{2}+\xi_{1}^{\prime
2}\big{)}-2p_{0}\dot{\chi}_{1}\xi_{1}^{\prime}\Big{)}\hskip 2.84526pt,\hskip
5.69054ptp_{0}=-2\Phi_{0}^{-1}e^{-2\omega_{0}}\dot{\chi}_{1}\xi_{1}^{\prime},$
$\displaystyle s_{0}=$ $\displaystyle\hskip
2.84526pt-e^{-4\omega_{0}}A_{t1}^{\prime
2}+\Phi_{0}^{-1}e^{-2\omega_{0}}\big{(}-\dot{\chi}_{1}^{2}+\xi_{1}^{\prime
2}\big{)}.$ (120)

Clearly, the above differential equations (115)-(119) are quite non trivial to
solve exactly in the radial variable ($z$). However, for the purpose of our
present analysis, it is sufficient to solve them near the black hole horizon.

Using (109), the location of the horizon $(\rho_{H})$ can be determined by
noting the spacetime metric (5.1)

$\displaystyle
ds^{2}_{(bh)}\approx\frac{4(\mu-\rho^{2})}{\Lambda}\Big{(}1+2\kappa\omega_{1}^{(bh)}+2\gamma\kappa\omega_{2}^{(bh)}\Big{)}\Bigg{(}-dt^{2}+\frac{d\rho^{2}}{4(\mu-\rho^{2})^{2}}\Bigg{)},$
(121)

which yields $\rho=\rho_{H}=\sqrt{\mu}$.

Finally, the near horizon solutions of the equations of motion (115)-(119)
could be listed as

$\displaystyle\Phi_{2}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{\rho}{192\mu^{5/2}}\Bigg{(}64\mu\rho\left(\rho-6\sqrt{\mu}\right)+n_{1}^{2}\rho\Bigg{(}\frac{64\mu^{3/2}}{n_{1}^{8}l_{1}^{2}}\Bigg{(}8\sqrt{\mu}(n_{1}^{2}-l_{1}^{2})^{2}\left(n_{1}^{2}+l_{1}^{2}\right)\left(\rho-9\sqrt{\mu}\right)$
$\displaystyle\hskip
2.84526pt-3n_{1}^{4}l_{1}^{4}\Bigg{)}-15\sqrt{\mu}+2\rho\Bigg{)}+2n_{1}n_{3}\rho\left(2\rho-15\sqrt{\mu}\right)+\rho
l_{1}\Big{(}l_{1}\left(\rho-12\sqrt{\mu}\right)+$ $\displaystyle
3l_{3}\left(\rho-9\sqrt{\mu}\right)\Big{)}+192\mu^{5/2}g_{2}\Bigg{)}+\frac{\rho\left(n_{1}^{2}+2n_{3}n_{1}+l_{1}\left(l_{1}+2l_{3}\right)\right)\log\left(\rho-\sqrt{\mu}\right)}{8\mu^{3/2}},$
(122) $\displaystyle\xi_{2}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{1}{16\mu^{3/2}}\Bigg{(}-\frac{256m_{1}^{4}n_{1}^{2}l_{1}\mu^{2}\rho\left(\rho-2\sqrt{\mu}\right)}{\Lambda^{2}\text{$\phi_{0}$}^{2}(n_{1}^{2}+l_{1}^{2})^{3}}-\frac{32\mu\rho\left(2\sqrt{\mu}\left(n_{1}^{2}-2l_{1}^{2}\right)+l_{1}^{2}\rho\right)}{n_{1}^{2}l_{1}}+$
$\displaystyle\hskip
2.84526pt(l_{3}+l_{1})\Bigg{(}-\frac{1}{2}\rho\left(\rho-6\sqrt{\mu}\right)-4\mu\log\left(\rho-\sqrt{\mu}\right)\Bigg{)}\Bigg{)}+l_{4},$
(123) $\displaystyle A_{t2}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{32\sqrt{\mu}m_{1}^{3}n_{1}^{2}l_{1}^{2}\rho\left(\rho-2\sqrt{\mu}\right)}{\Lambda^{2}\text{$\phi_{0}$}^{2}(n_{1}^{2}+l_{1}^{2})^{3}}-m_{3}\rho+m_{4},$
(124) $\displaystyle\omega_{2}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{m_{1}}{2\text{$\phi_{0}$}^{2}}\left(\frac{m_{1}\left(n_{1}^{2}-l_{1}^{2}\right)}{\Lambda(n_{1}^{2}+l_{1}^{2})}-m_{3}\text{$\phi_{0}$}\right)+q_{3}I_{0}\left(\tilde{\rho}\right)+q_{4}K_{0}\left(\tilde{\rho}\right),$
(125) $\displaystyle\chi_{2}^{(bh)}=$ $\displaystyle\hskip
2.84526ptn_{3}t+n_{4},$ (126)

where we define $\tilde{\rho}=2\sqrt{\frac{\rho}{\sqrt{\mu}}-1}$ and
$m_{i},n_{i},l_{i},q_{i}$, $g_{2}$, $(i=3,4)$ are the integration constants.
Furthermore, here $I_{0}\left(\tilde{\rho}\right)$ and
$K_{0}\left(\tilde{\rho}\right)$ are respectively the modified Bessel
functions [67] of the first $(I_{n}(\tilde{\rho}))$ and the second kind
$(K_{n}(\tilde{\rho}))$.

####  5.1.4 Order $\kappa^{2}$ solution

The contribution due to the Maxwell (34) term alone at quadratic
($\kappa^{2}$) could be estimated by solving the equations (13)-(17) at order
$\kappa^{2}$

$\displaystyle-2\omega_{1}\partial_{z}\Big{(}\Phi_{0}e^{-2\omega_{0}}A_{t1}^{\prime}\Big{)}+\partial_{z}\Big{[}e^{-2\omega_{0}}\Big{(}-2\omega_{1}\Phi_{0}A_{t1}^{\prime}+\Phi_{1}A{t1}^{\prime}+\Phi_{0}A_{t3}^{\prime}\Big{)}\Big{]}$
$\displaystyle=0,$ (127) $\displaystyle\omega_{3}^{\prime\prime}+2\Lambda
e^{2\omega_{0}}\big{(}\omega_{1}^{2}+\omega_{3}\big{)}-2e^{-2\omega_{0}}\big{(}-\omega_{1}A_{t1}^{\prime
2}+A_{t1}^{\prime}A_{t3}^{\prime}\big{)}$ $\displaystyle=0,$ (128)
$\displaystyle\Phi_{3}^{\prime\prime}-2\big{(}\omega_{3}^{\prime}\Phi_{0}^{\prime}+\omega_{0}^{\prime}\Phi_{3}^{\prime}+\omega_{1}^{\prime}\Phi_{1}^{\prime}\big{)}+4(\dot{\chi}_{1}\dot{\chi}_{3}+\xi_{1}^{\prime}\xi_{3}^{\prime})$
$\displaystyle=0,$ (129) $\displaystyle\xi_{3}^{\prime\prime}$
$\displaystyle=0,$ (130) $\displaystyle\ddot{\chi}_{3}$ $\displaystyle=0.$
(131)

The solutions of the above equations (127)-(131) are quite complicated,
therefore we mention them in the Appendix C. Like before, one can further
simplify these solutions (122)-(126) and (158)-(162) by making use of the
residual gauge freedom in the light cone gauge (121). In particular, the re-
scaling of the time coordinate, $t\rightarrow n_{1}t$ does not affect the
gauge condition $g_{t\rho}=0$. Therefore, one can use this freedom to fix the
constant $n_{1}=\sqrt{1-l_{1}^{2}}$.

It is evident from (111), (112), (159) and (161) that the leading order
($\kappa$) corrections as well as the quadratic order $(\kappa^{2})$
corrections diverge as we move closer towards the black hole horizon
($\rho\sim\rho_{H}=\sqrt{\mu}$). Similar divergences persist even at quadratic
order ($\gamma\kappa$) (see (122) and (123)). However, for a particular choice
of constants

$\displaystyle n_{3}=$ $\displaystyle\hskip
2.84526pt\frac{1}{2\gamma\sqrt{1-l_{1}^{2}}}\Big{(}(2l_{1}^{2}-1)(1+\gamma)-2\kappa
n_{5}\sqrt{1-l_{1}^{2}}-\kappa q_{2}\Big{)},$ (132) $\displaystyle l_{3}=$
$\displaystyle\hskip 2.84526pt\frac{1}{\gamma
l_{1}}\Big{(}-l_{1}^{2}(1+\gamma)+\kappa n_{5}\sqrt{1-l_{1}^{2}}\Big{)},$
(133)

the divergences at order $\kappa$ and $\kappa^{2}$ cancel with those at the
quadratic order $(\gamma\kappa)$ thereby resulting in a finite expression for
$\xi^{(bh)}$ and $\Phi^{(bh)}$ near the horizon $(\rho\sim\sqrt{\mu})$. This
turns out to be a unique feature of projected ModMax interactions in two
dimensions.

###  5.2 2D Black hole thermodynamics

With the above solutions at hand, we now explore the thermal properties of 2D
black holes in the presence of projected ModMax interactions. In particular,
we compute the Wald entropy [59]-[61] for 2D black holes. Finally, we also
comment on the Wald entropy associated with the _extremal_ black holes in two
dimensions.

To begin with, we compute the Hawking temperature [58] for the 2D black holes
which receives quadratic order corrections due to $U(1)$ gauge and ModMax
couplings

$\displaystyle
T_{H}=\frac{1}{2\pi}\sqrt{-\frac{1}{4}g^{tt}g^{\rho\rho}\big{(}\partial_{\rho}g_{tt}\big{)}^{2}}\Bigg{|}_{\rho\rightarrow\sqrt{\mu}}=\frac{\sqrt{\mu}}{\pi}\Bigg{[}1-\Big{(}\kappa
q+\gamma\kappa q+\kappa^{2}p\Big{)}\Bigg{]},$ (134)

where we set the constants $q_{4}=q_{2}=q$ and $p$ is defined as

$\displaystyle
p=\frac{m_{1}^{2}}{12\Lambda\mu\phi_{0}^{3}}\Big{(}1+24q\mu\phi_{0}+\log(4)-8\mu^{\frac{3}{2}}g_{1}\Big{)}.$
(135)

The Wald entropy [59]-[61] is defined as

$\displaystyle S_{W}=-2\pi\frac{\delta\mathcal{L}}{\delta
R_{\mu\nu\alpha\beta}}\epsilon_{\mu\nu}\epsilon_{\alpha\beta},$ (136)

where $R_{\mu\nu\alpha\beta}$ is the Riemann curvature tensor, $\mathcal{L}$
is the Lagrangian density181818Here we used the convention, $I=\int
d^{2}x\sqrt{-g}\mathcal{L}$.  in two dimensions and $\epsilon_{\mu\nu}$ is the
anti-symmetric rank two tensor having the normalization condition,
$\epsilon^{\mu\nu}\epsilon_{\mu\nu}=-2$.

Using (72), the Wald entropy (136) for 2D black holes turns out to
be191919Here, the entities $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ are
respectively the values of $\Phi_{1}^{(bh)}$ (111), $\Phi_{2}^{(bh)}$ (122)
and $\Phi_{3}^{(bh)}$ (159) at the horizon $\rho=\rho_{H}=\sqrt{\mu}$.

$\displaystyle
S_{W}=\frac{\Phi^{(bh)}}{4G_{2}}\Bigg{|}_{\rho\rightarrow\sqrt{\mu}}=\hskip
2.84526pt\frac{1}{4G_{2}}\Big{(}\phi_{0}+\kappa\phi_{1}+\gamma\kappa\phi_{2}+\kappa^{2}\phi_{3}\Big{)},$
(137)

where we denote the above entities as

$\displaystyle\phi_{1}=$ $\displaystyle\hskip
2.84526pt\sqrt{\mu}g_{1}-\frac{1}{192\mu}\Big{(}12\log(4\mu)+2l_{1}^{2}-13\Big{)},$
(138) $\displaystyle\phi_{2}=$ $\displaystyle\hskip
2.84526pt\frac{64\mu}{3l_{1}^{2}}\frac{\left(1-2l_{1}^{2}\right){}^{2}}{\left(l_{1}^{2}-1\right){}^{3}}+\frac{l_{1}^{2}}{l_{1}^{2}-1}+\sqrt{\mu}g_{2}-\frac{5}{3},$
(139) $\displaystyle\phi_{3}=$ $\displaystyle\hskip
2.84526pt\frac{1}{192\mu}\Bigg{(}192\mu^{3/2}\left(g_{1}\left(q_{1}+q\right)+g_{3}\right)+2\sqrt{1-l_{1}^{2}}n_{5}+q(36\log(\mu)$
$\displaystyle+13-24\log(2))-48q_{1}\Bigg{)},$ (140)

and $\phi_{0}$ is the usual constant dilaton solution in the limit
$\kappa\rightarrow 0$ and $\gamma\rightarrow 0$ (103).

###  5.3 A special case : Extremal 2D black holes

As a special case, we study the extremal 2D black hole solutions and compute
the associated Wald entropy. Extremal black holes correspond to the vanishing
of the Hawking temperature (134)

$\displaystyle\kappa q+\gamma\kappa q+\kappa^{2}p=1,$ (141)

which for the present example stands as an extremality condition in two
dimensions.

Using (141) and (137), the Wald entropy for 2D extremal black holes
$\big{(}S_{W}^{\text{(ext)}}\big{)}$ turns out to be

$\displaystyle
S_{W}^{\text{(ext)}}=\frac{1}{4G_{2}}\Bigg{[}\phi_{0}+\frac{\phi_{2}}{q}+\kappa\Big{(}\phi_{1}-\phi_{2}\Big{)}+\kappa^{2}\Bigg{(}\phi_{3}-\frac{p}{q}\phi_{2}\Bigg{)}\Bigg{]},$
(142)

where the entities $p$, $\phi_{1}$, $\phi_{2}$ and $\phi_{3}$ are respectively
given in (135), (138), (139) and (140).

##  6 Concluding remarks

To summarise, in the present paper, we construct the 2D analogue of the four
dimensional ModMax electrodynamics (coupled with Einstein gravity) using the
notion of dimensional reduction. We investigate the effects of projected
ModMax interactions on various physical entities associated with the boundary
theory in one dimension. Finally, we construct the associated 2D black hole
solutions and explore their thermal properties.

Below, we outline some of the future extensions of the present work.

$\bullet$ In the literature, there exists an alternative way to derive the
thermodynamic entropy of 2D black holes by noting the asymptotic growth of the
physical states of a CFT by means of the Cardy formula ($S_{C}$) [51],
[68]-[69]

$\displaystyle S_{C}=2\pi\sqrt{\frac{c_{M}\Delta}{6}},$ (143)

where $\Delta$ is the eigen value of the associated Virasoro generator
$L_{0}$.

The authors in [51] establish a 2D/3D dictionary which by virtue of the Cardy
formula (143) predicts the correct Bekenstein-Hawking entropy for 2D black
holes. Therefore, it would be indeed an interesting project to uplift the 2D
black hole solutions (98)-(99) into three dimensions and establish a suitable
2D/3D mapping in the presence of 2D projected ModMax interactions.

$\bullet$ It would be an interesting project to add SU(2) Yang-Mills
interactions and investigate their imprints on various physical observables
associated with the boundary theory. In particular, the authors in [53]
observe that the SU(2) Yang-Mills field play an important role in obtaining
the Hawking-Page transition in the context of JT gravity. Therefore, one can
investigate similar effects and/or possible deviations in the presence of
projected ModMax interactions in two dimensions.

$\bullet$ Finally, it would be nice to construct the 2D wormhole solutions
[43], [52] and explore their thermal stability for the ModMax corrected JT
gravity models.

We would like to address some of the above issues in the near future.

## Acknowledgments

The authors are indebted to the authorities of Indian Institute of Technology,
Roorkee for their unconditional support towards researches in basic sciences.
DR would like to acknowledge The Royal Society, UK for financial assistance.

##  Appendix A Equations of motion

In this Appendix, we note down the most general form of the equations of
motion (13)-(17) in the Fefferman-Graham gauge (3),

$\displaystyle A_{t}\hskip 2.84526pt:\hskip 2.84526pt$
$\displaystyle\frac{\kappa}{\sqrt{-h_{tt}}}\partial_{\eta}\Bigg{[}\frac{\Phi
A_{t}^{\prime}}{\sqrt{-h_{tt}}}\Bigg{(}\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^{2}+p^{2}}}\Bigg{)}\Bigg{]}=0,$
(144) $\displaystyle\chi\hskip 2.84526pt:\hskip 2.84526pt$
$\displaystyle\frac{\kappa}{\sqrt{-h_{tt}}}\partial_{t}\Bigg{(}-\frac{\dot{\chi}}{\sqrt{-h_{tt}}}\cosh{\gamma}+\frac{s\dot{\chi}+p\xi^{\prime}\sqrt{-h_{tt}}}{\sqrt{-h_{tt}}\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{)}+\frac{\kappa}{\sqrt{-h_{tt}}}\partial_{\eta}\Bigg{(}\sqrt{-h_{tt}}\times$
$\displaystyle\chi^{\prime}\cosh{\gamma}-\frac{s\chi^{\prime}\sqrt{-h_{tt}}+p\dot{\xi}}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{)}=0,$
(145) $\displaystyle\xi\hskip 2.84526pt:\hskip 2.84526pt$
$\displaystyle\frac{\kappa}{\sqrt{-h_{tt}}}\partial_{t}\Bigg{(}-\frac{\dot{\xi}}{\sqrt{-h_{tt}}}\cosh{\gamma}-\frac{-s\dot{\xi}+p\chi^{\prime}\sqrt{-h_{tt}}}{\sqrt{-h_{tt}}\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{)}+\frac{\kappa}{\sqrt{-h_{tt}}}\partial_{\eta}\Bigg{(}\sqrt{-h_{tt}}\times$
$\displaystyle\xi^{\prime}\cosh{\gamma}-\frac{s\xi^{\prime}\sqrt{-h_{tt}}-p\dot{\chi}}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}\Bigg{)}=0,$
(146)

$\displaystyle\Phi\hskip 2.84526pt:\hskip 2.84526pt$
$\displaystyle\sqrt{-h_{tt}}(\sqrt{-h_{tt}})^{\prime\prime}+(\sqrt{-h_{tt}})^{2}\Lambda-\kappa
A_{t}^{\prime 2}\cosh{\gamma}+\frac{\kappa sA_{t}^{\prime
2}}{\sqrt{s^{2}+p^{2}}}\sinh{\gamma}=0,$ (147) $\displaystyle g_{tt}\hskip
2.84526pt:\hskip 2.84526pt$
$\displaystyle\Phi^{\prime\prime}+\Lambda\Phi+2\kappa\Phi\Bigg{(}\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^{2}+p^{2}}}\Bigg{)}\Bigg{(}\Phi^{-1}(\chi^{\prime
2}+\xi^{\prime 2})-\frac{s}{2}\Bigg{)}=0,$ (148) $\displaystyle
g_{\eta\eta}\hskip 2.84526pt:\hskip 2.84526pt$
$\displaystyle-\frac{1}{\sqrt{-h_{tt}}}\partial_{t}\Bigg{(}\frac{\dot{\Phi}}{\sqrt{-h_{tt}}}\Bigg{)}+\frac{(\sqrt{-h_{tt}})^{\prime}\Phi^{\prime}}{\sqrt{-h_{tt}}}+\Lambda\Phi-2\kappa\Phi\Bigg{(}\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^{2}+p^{2}}}\Bigg{)}\times$
$\displaystyle\Bigg{(}\frac{\Phi^{-1}(\dot{\chi}^{2}+\dot{\xi}^{2})}{(\sqrt{-h_{tt}})^{2}}+\frac{s}{2}\Bigg{)}=0,$
(149) $\displaystyle g_{\eta t}\hskip 2.84526pt:\hskip 2.84526pt$
$\displaystyle-(\dot{\Phi})^{\prime}+\frac{\dot{\Phi}(\sqrt{-h_{tt}})^{\prime}}{\sqrt{-h_{tt}}}-2\kappa\Bigg{(}\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^{2}+p^{2}}}\Bigg{)}(\dot{\xi}\xi^{\prime}+\dot{\chi}\chi^{\prime})=0,$
(150)

along with the functions

$\displaystyle s=$ $\displaystyle-\frac{A_{t}^{\prime
2}}{(\sqrt{-h_{tt}})^{2}}+\Phi^{-1}\Bigg{(}-\frac{1}{(\sqrt{-h_{tt}})^{2}}(\dot{\chi}^{2}+\dot{\xi}^{2})+\chi^{\prime
2}+\xi^{\prime 2}\Bigg{)},$ (151) $\displaystyle p=$
$\displaystyle-2\frac{\Phi^{-1}}{\sqrt{-h_{tt}}}\big{(}\dot{\chi}\xi^{\prime}-\chi^{\prime}\dot{\xi}\big{)},$
(152)

where . and ′ denote the derivatives with respect to $t$ and $\eta$
respectively.

##  Appendix B Details of the functions $\mathcal{H}_{i}$’s

In this Appendix, we present the explicit details of the functions
$\mathcal{H}_{i}$, $(i=1,2,..5)$

$\displaystyle\mathcal{H}_{1}(\eta)=$ $\displaystyle\hskip
2.84526pt\frac{1}{4}\Bigg{[}c_{1}\Bigg{(}-\frac{a\kappa
e^{-\eta\lambda}}{a_{1}a_{2}b_{1}}+\frac{\kappa}{a_{1}^{3}}\left(a_{3}\left(8\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)-8\eta\lambda\right)-\frac{e^{-\eta\lambda}}{a_{2}b_{1}\lambda}\right)+$
$\displaystyle\hskip
2.84526pt\frac{1}{a_{1}^{2}}\Big{(}4(\gamma+1)\eta\lambda-4\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\Big{)}-4a(\gamma+1)\lambda+\frac{8aa_{1}\lambda
e^{2\eta\lambda}}{a_{1}e^{2\eta\lambda}-a_{2}}\Bigg{)}-$ $\displaystyle\hskip
2.84526pt\frac{4}{a_{1}^{2}}\left(\kappa\left(c_{5}\left(\eta-
aa_{1}^{2}\right)+c_{4}\right)+\gamma
c_{3}\right)-\frac{1}{a_{1}^{4}a_{2}}\Big{(}c_{1}^{3}\kappa
e^{-2\eta\lambda}\Big{(}2a_{1}e^{2\eta\lambda}\Big{(}2\eta^{2}\lambda^{2}+$
$\displaystyle\hskip
2.84526pt\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)^{2}-3\eta\lambda\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\Big{)}+a_{2}\left(2\log\left(a_{1}\right)+2\eta\lambda+1\right)\Big{)}\Big{)}\Bigg{]},$
(153) $\displaystyle\mathcal{H}_{2}(\eta)=$ $\displaystyle\hskip
2.84526pt-\frac{1}{8a_{1}^{3}a_{2}b_{1}\lambda^{2}}\Bigg{(}ae^{-3\eta\lambda}\Big{(}c_{1}\Big{(}-2a_{1}a_{2}b_{1}\lambda
e^{\eta\lambda}\Big{(}-2\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)+2\log\left(a_{1}\right)$
$\displaystyle\hskip
2.84526pt-\gamma+4\eta\lambda+1\Big{)}+4a_{2}a_{3}b_{1}\kappa\lambda
e^{\eta\lambda}\Big{(}-2\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)+2\log\left(a_{1}\right)+4\eta\lambda$
$\displaystyle\hskip
2.84526pt+1\Big{)}+\kappa\Big{)}+b_{1}c_{1}^{3}\kappa\lambda
e^{\eta\lambda}\Big{(}2\Big{(}\log^{2}\left(a_{2}-a_{1}e^{2\eta\lambda}\right)-3\eta\lambda\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)+$
$\displaystyle\hskip
2.84526pt2\eta^{2}\lambda^{2}\Big{)}-\log\left(a_{1}\right)(2\eta\lambda+1)-2\log^{2}\left(a_{1}\right)\Big{)}-2a_{1}a_{2}b_{1}c_{5}\kappa
e^{\eta\lambda}\Big{)}\Bigg{)},$ (154)

$\displaystyle\mathcal{H}_{3}(\eta)=$ $\displaystyle\hskip
2.84526pt2e^{2\eta\lambda}-\frac{1}{8a_{1}a_{2}^{2}\left(a_{1}e^{2\eta\lambda}-a_{2}\right)}a\Bigg{[}8a_{2}a_{1}^{3}\lambda
e^{2\eta\lambda}\Big{(}4a_{2}\kappa
e^{2\eta\lambda}\left(a_{4}\gamma+a_{6}\kappa+a_{3}\right)-2a_{2}^{2}+$
$\displaystyle\hskip 2.84526ptc_{1}^{2}\kappa
e^{2\eta\lambda}\left(2\eta\lambda-\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\right)\Big{)}+a_{1}^{2}\kappa
e^{2\eta\lambda}\Big{(}c_{1}^{4}\kappa\lambda
e^{2\eta\lambda}\Big{(}\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)$
$\displaystyle\hskip
2.84526pt-2\eta\lambda\Big{)}{}^{2}+8a_{2}a_{3}c_{1}^{2}\kappa\lambda
e^{2\eta\lambda}\left(2\eta\lambda-\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\right)+8a_{2}^{2}\Big{(}2a_{3}^{2}\kappa\lambda
e^{2\eta\lambda}-$ $\displaystyle\hskip
2.84526ptc_{1}\left(c_{1}\lambda\left(-\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)+2\eta\lambda+1\right)+2c_{5}\kappa\right)\Big{)}-32a_{2}^{3}\lambda\big{(}a_{4}\gamma+a_{6}\kappa+$
$\displaystyle\hskip
2.84526pta_{3}\big{)}\Big{)}-a_{2}a_{1}\kappa^{2}\Big{(}16a_{2}^{2}\left(a_{3}^{2}\lambda
e^{2\eta\lambda}-c_{1}c_{5}\right)+c_{1}^{4}\lambda
e^{2\eta\lambda}\Big{(}\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)^{2}-$
$\displaystyle\hskip
2.84526pt2(2\eta\lambda+1)\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)+4\eta\lambda(\eta\lambda+1)\Big{)}+8a_{2}a_{3}c_{1}^{2}\lambda
e^{2\eta\lambda}\Big{(}2\eta\lambda-$ $\displaystyle\hskip
2.84526pt\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)\Big{)}\Big{)}-8a_{2}^{3}a_{3}c_{1}^{2}\kappa^{2}\lambda+16a_{2}^{2}a_{1}^{4}\lambda
e^{4\eta\lambda}\Bigg{]},$ (155) $\displaystyle\mathcal{H}_{4}(\eta)=$
$\displaystyle\hskip 2.84526pt\frac{a}{\lambda},$ (156)

$\displaystyle\mathcal{H}_{5}(\eta)=$ $\displaystyle\hskip
2.84526pta\Bigg{[}\frac{1}{4}\kappa
e^{\eta\lambda}\Bigg{(}\frac{2b_{1}c_{1}^{2}e^{2\eta\lambda}}{a_{2}\left(a_{2}-a_{1}e^{2\eta\lambda}\right)}+\frac{b_{1}}{a_{1}a_{2}}\Big{(}c_{1}^{2}\left(-\log\left(a_{2}-a_{1}e^{2\eta\lambda}\right)+2\eta\lambda+2\right)+$
$\displaystyle\hskip
2.84526pt4a_{2}a_{3}\Big{)}-\frac{d_{1}^{2}e^{\eta\lambda}}{a_{1}a_{2}\lambda
e^{2\eta\lambda}+a_{2}^{2}\lambda}-\frac{d_{1}^{2}\tan^{-1}\left(\frac{\sqrt{a_{1}}e^{\eta\lambda}}{\sqrt{a_{2}}}\right)}{\sqrt{a_{1}}a_{2}^{3/2}\lambda}-\frac{a_{1}e_{1}^{2}e^{\eta\lambda}}{a_{1}\lambda
e^{2\eta\lambda}+a_{2}\lambda}-$ $\displaystyle\hskip
2.84526pt\frac{\sqrt{a_{1}}}{\sqrt{a_{2}}\lambda}e_{1}^{2}\tan^{-1}\left(\frac{\sqrt{a_{1}}e^{\eta\lambda}}{\sqrt{a_{2}}}\right)\Bigg{)}+\frac{\kappa^{2}e^{-\eta\lambda}}{2a_{1}^{2}\lambda}\Big{(}2a_{1}^{2}b_{4}\lambda
e^{2\eta\lambda}+2a_{2}a_{3}\lambda(2\eta\lambda+1)-$ $\displaystyle\hskip
2.84526ptb_{1}c_{1}c_{5}(2\eta\lambda+1)\Big{)}+b_{2}\gamma\kappa
e^{\eta\lambda}+b_{1}e^{\eta\lambda}\Bigg{]},$ (157)

where we denote the constant
$a=\frac{2\left(a_{1}^{2}+1\right)}{3a_{1}^{2}\lambda}-\frac{4a_{3}\kappa}{3a_{1}^{3}\lambda}$.

##  Appendix C Order $\kappa^{2}$ solutions

In this Appendix, we note down the solution of the equations (127)-(131),

$\displaystyle\omega_{3}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{1}{192\Lambda^{2}\mu^{3/2}\text{$\phi_{0}$}^{4}}\Bigg{(}64g_{1}\Lambda\mu^{3/2}m_{1}^{2}\rho\text{$\phi_{0}$}\left(\log\left(1-\frac{\rho}{\sqrt{\mu}}\right)+\log\left(\frac{\rho}{\sqrt{\mu}}+1\right)\right)-$
$\displaystyle\hskip 2.84526pt16\Lambda
l_{1}^{2}m_{1}^{2}\text{$\phi_{0}$}\Big{(}\sqrt{\mu}\log\left(\frac{\rho}{\sqrt{\mu}}+1\right)+\log\left(1-\frac{\rho}{\sqrt{\mu}}\right)\Big{(}\rho\left(-\log\left(\frac{\rho}{\sqrt{\mu}}+1\right)\right)+$
$\displaystyle\hskip
2.84526pt\sqrt{\mu}+\rho\log(4)\Big{)}+\rho\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)-2\rho\text{Li}_{2}\left(\frac{1}{2}-\frac{\rho}{2\sqrt{\mu}}\right)+\rho\Big{)}+16m_{1}\times$
$\displaystyle\hskip
2.84526pt\Big{(}3\mu\rho\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)\left(2\Lambda^{2}m_{6}\text{$\phi_{0}$}^{3}-3m_{1}^{3}\right)-\Lambda
m_{1}n_{1}^{2}\text{$\phi_{0}$}\Big{(}\sqrt{\mu}\log\left(\frac{\rho}{\sqrt{\mu}}+1\right)$
$\displaystyle\hskip
2.84526pt+\log\left(1-\frac{\rho}{\sqrt{\mu}}\right)\left(\rho\left(-\log\left(\frac{\rho}{\sqrt{\mu}}+1\right)\right)+\sqrt{\mu}+\rho\log(4)\right)+\rho\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)$
$\displaystyle\hskip
2.84526pt-2\rho\text{Li}_{2}\left(\frac{1}{2}-\frac{\rho}{2\sqrt{\mu}}\right)+\rho\Big{)}\Big{)}-96\Lambda\sqrt{\mu}q_{2}\text{$\phi_{0}$}^{2}\Big{(}\Lambda
q_{1}\text{$\phi_{0}$}^{2}\Big{(}-\sqrt{\mu}+2\left(\mu-\rho^{2}\right)\rho\times$
$\displaystyle\hskip
2.84526pt\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)\Big{)}-4\sqrt{\mu}m_{1}^{2}\rho\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)\Big{)}-3\Lambda^{2}\sqrt{\mu}q_{2}^{2}\text{$\phi_{0}$}^{4}\Big{(}\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)$
$\displaystyle\hskip 2.84526pt\times
32\left(\left(\mu-\rho^{2}\right)\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)+\sqrt{\mu}\rho\right)+45\sqrt{\mu}\rho\Big{)}+96\Lambda^{2}\sqrt{\mu}\rho
q_{1}^{2}\text{$\phi_{0}$}^{4}\Big{(}\rho-$ $\displaystyle\hskip
2.84526pt\sqrt{\mu}\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)\Big{)}+192\Lambda^{2}\mu\text{$\phi_{0}$}^{4}\left(q_{6}\left(\rho\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)-\sqrt{\mu}\right)+\rho
q_{5}\right)\Bigg{)},$ (158) $\displaystyle\Phi_{3}^{(bh)}=$
$\displaystyle\hskip
2.84526ptg_{3}\rho-\frac{1}{8\mu^{2}}\Bigg{(}2\rho^{2}\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)\left(q_{1}\left(l_{1}^{2}+n_{1}^{2}\right)-4g_{1}\mu^{3/2}q_{2}\right)+\mu\log\left(\rho-\sqrt{\mu}\right)\times$
$\displaystyle\hskip
2.84526pt\left(q_{1}\left(l_{1}^{2}+n_{1}^{2}\right)-4g_{1}\mu^{3/2}q_{2}\right)-\mu\log\left(\sqrt{\mu}+\rho\right)\left(q_{1}\left(l_{1}^{2}+n_{1}^{2}\right)-4g_{1}\mu^{3/2}q_{2}\right)-$
$\displaystyle\hskip 2.84526pt8g_{1}\mu^{3/2}\rho^{2}q_{1}-8g_{1}\mu^{2}\rho
q_{2}+2\mu
q_{2}\left(l_{1}^{2}+n_{1}^{2}\right)\log\left(1-\frac{\rho^{2}}{\mu}\right)-\mu\big{(}q_{2}\left(l_{1}^{2}+n_{1}^{2}\right)+$
$\displaystyle\hskip
2.84526pt4l_{1}l_{5}+2n_{1}n_{5}\big{)}\log\left(\mu-\rho^{2}\right)+2q_{2}\left(l_{1}^{2}+n_{1}^{2}\right)\left(\rho^{2}-\mu\right)\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)^{2}+$
$\displaystyle\hskip 2.84526pt2\sqrt{\mu}\rho
q_{1}\left(l_{1}^{2}+n_{1}^{2}\right)-\mu\left(q_{2}\left(l_{1}^{2}+n_{1}^{2}\right)-4l_{1}l_{5}-2n_{1}n_{5}\right)\log\left(\sqrt{\mu}+\rho\right)+$
$\displaystyle\hskip
2.84526pt\sqrt{\mu}\left(\rho-\sqrt{\mu}\right)\left(q_{2}\left(l_{1}^{2}+n_{1}^{2}\right)-4l_{1}l_{5}-2n_{1}n_{5}\right)\log\left(\rho-\sqrt{\mu}\right)-\sqrt{\mu}\rho\big{(}q_{2}\left(l_{1}^{2}+n_{1}^{2}\right)$
$\displaystyle\hskip
2.84526pt-4l_{1}l_{5}-2n_{1}n_{5}\big{)}\log\left(\sqrt{\mu}+\rho\right)+4\sqrt{\mu}\rho
q_{2}\left(l_{1}^{2}+n_{1}^{2}\right)\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)\Bigg{)}+g_{4},$
(159) $\displaystyle A_{t3}^{(bh)}=$ $\displaystyle\hskip
2.84526pt\frac{1}{8\Lambda\mu^{3/2}\text{$\phi_{0}$}^{2}}\Bigg{(}2\sqrt{\mu}\Big{(}m_{1}\rho\left(-4g_{1}\mu\rho+l_{1}^{2}+n_{1}^{2}+8\sqrt{\mu}\rho
q_{1}\text{$\phi_{0}$}+8\mu q_{2}\text{$\phi_{0}$}\right)+$
$\displaystyle\hskip
2.84526pt4\Lambda\mu\text{$\phi_{0}$}^{2}\left(m_{6}\rho+m_{5}\right)\Big{)}+2m_{1}\rho^{2}\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)\left(l_{1}^{2}+n_{1}^{2}+8\mu
q_{2}\text{$\phi_{0}$}\right)+$ $\displaystyle\hskip 2.84526pt\mu
m_{1}\left(\log\left(\rho-\sqrt{\mu}\right)-\log\left(\sqrt{\mu}+\rho\right)\right)\left(l_{1}^{2}+n_{1}^{2}+8\mu
q_{2}\text{$\phi_{0}$}\right)\Bigg{)},$ (160) $\displaystyle\xi_{3}^{(bh)}=$
$\displaystyle\hskip
2.84526pt\frac{l_{5}}{\sqrt{\mu}}\tanh^{-1}\left(\frac{\rho}{\sqrt{\mu}}\right)+l_{6},$
(161) $\displaystyle\chi_{3}^{(bh)}=$ $\displaystyle\hskip
2.84526ptn_{5}t+n_{6},$ (162)

where $m_{i},n_{i},l_{i},q_{i}$, $g_{j}$, $(i=5,6$, $j=3,4)$ are the
integration constants and we define $\text{Li}_{2}(x)=\text{PolyLog}(2,x)$.

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