\mathcal{G}_{\xi}=&\hspace{1mm}-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),\\
\mathcal{G}_{\Phi}=&\hspace{1mm}2K-2\lambda,
\end{align}

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, 127] corresponding to the action (<ref>)

\begin{align}\label{rstressdef}
T^{ab}=\frac{2}{\sqrt{-h}}\frac{\delta I_{\text{boundary}}}{\delta h_{ab}}=\frac{\mathcal{G}^{ab}}{8\pi G_2},
\end{align}

where $\mathcal{G}^{ab}$ is given in (<ref>).

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

Under the diffeomorphism,

\begin{align}\label{ctdef}
x^{\mu}\rightarrow x^{\mu}+\epsilon^{\mu} (x),
\end{align}

the background fields  (<ref>)-(<ref>) transform as

\begin{align}
\delta_{\epsilon}A_{\mu}=&\hspace{1mm}\epsilon^{\nu}\nabla_{\nu}A_{\mu}+A_{\nu}\nabla_{\mu}\epsilon^{\nu},\label{diffeotranscoordi}\\
\delta_{\epsilon}g_{\mu\nu}=&\hspace{1mm}\nabla_{\mu}\epsilon_{\nu}+\nabla_{\nu}\epsilon_{\mu},\label{diffeotranscoordm}\\
\delta_{\epsilon}\mathcal{S}=&\hspace{1mm}\epsilon^{\mu}\nabla_{\mu}\mathcal{S},\label{diffeotranscoordf}
\end{align}

where $\mathcal{S}$ collectively denotes the scalar fields $\Phi,\hspace{1mm}\xi$ and $\chi$.

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

\begin{align}\label{parametersdiffeo}
\epsilon_t=e^{2\eta\lambda}f(t)+\frac{1}{2\lambda}\Bigg(\frac{2 \left(a_1^2+1\right)}{3 a_1^2 \lambda }-\frac{4 a_3 \kappa
}{3 a_1^3 \lambda }\Bigg)\partial^2_tf(t) \hspace{1mm},\hspace{2mm}\epsilon_{\eta}=\Bigg(\frac{2 \left(a_1^2+1\right)}{3 a_1^2 \lambda }-\frac{4 a_3 \kappa
}{3 a_1^3 \lambda }\Bigg)\partial_tf(t),
\end{align}

where $f(t)$ is some function[In the Fefferman-Graham gauge [127, 191], the variation of the space-time metric (under diffeomorphism (<ref>)) 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 $\mathcal{A}$ (<ref>) and $\mathcal{B}$ (<ref>).] of time [48].

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 (<ref>). On the other hand, under the diffeomorphism (<ref>), $A_{\eta}$ transforms as

\begin{align}\label{sigmagaugeeqn}
\delta_{\epsilon}A_{\eta}=A_t\partial_{\eta}\Bigg(\frac{\epsilon_t}{h_{tt}}\Bigg)\neq 0,
\end{align}

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

\begin{align}\label{sigmaintermedstep}
\Sigma= \hspace{1mm}-\int d\eta A_t\partial_{\eta}\Bigg(\frac{\epsilon_t}{h_{tt}}\Bigg),
\end{align}

where we have used the variation (<ref>).

Now, one can perform the above integration (<ref>) using the background fields (<ref>)-(<ref>) and the diffeomorphism parameter (<ref>), which yields

\begin{align}\label{sigmavalue}
\Sigma=&\hspace{1mm}\frac{e^{-2 \eta  \lambda } }{12 a_1^5 a_2 \lambda ^3}\Bigg(f''(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 +\nonumber\\
&\hspace{1mm}\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)-4 a_2 a_3\right) \left(2 \log
\left(a_1\right)+2 \eta  \lambda +1\right)+\nonumber\\
&\hspace{1mm}c_1 \kappa  \lambda
\left(c_1^2 \log \left(a_1\right)-8 a_2 a_3\right) \left(2 \log
\left(a_1\right)+2 \eta  \lambda +1\right)\big)-3 a_1 a_2 c_1^3
\kappa  \lambda ^3 f(t) \big(2 \log \left(a_1\right)+\nonumber\\
&\hspace{1mm}2 \eta
\lambda +1\big)\Bigg).
\end{align}

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 (<ref>).

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

\begin{align}
(\delta_{\epsilon}+\delta_{\Sigma})T_{tt}=&\hspace{1mm}\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)\nonumber\\
&+\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].\label{variationstresstensor}
\end{align}

The variations of the background fields $h_{tt}, \hspace{1mm} A_t$ and $ \Phi$ can be obtained using (<ref>)-(<ref>) and (<ref>), which yields the following

\begin{align}
(\delta_{\epsilon}+\delta_{\Sigma})A_t=&\hspace{1mm}\mathcal{H}_1(\eta)\partial_t f(t)+\mathcal{H}_2(\eta)\partial_t^3 f(t)\label{fv1},\\
\delta_{\epsilon} h_{tt} =&\hspace{1mm} \mathcal{H}_3(\eta) \partial_tf(t) +\mathcal{H}_4(\eta)\partial_t^3f(t),\\
\delta_{\epsilon} \Phi =&\hspace{1mm}\mathcal{H}_5(\eta)\partial_t f(t)\label{fv2},
\end{align}

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

Using these variations (<ref>)-(<ref>), the transformation of the boundary stress tensor (<ref>) can be expressed in a more elegant way

\begin{align}
(\delta_{\epsilon}+\delta_{\Sigma})\Tilde{T}_{tt}=2\Tilde{T}_{tt}\partial_tf(t)+f(t)\partial_t\Tilde{T}_{tt}- c_M\partial^3_tf(t).\label{stvariation}
\end{align}

Here, we define the re-scaled stress tensor as

\begin{align}
\Tilde{T}_{tt}= \frac{T_{tt}}{b_3(1+a_2^2b_3^2)},
\end{align}

and identify the coefficient “$c_M$” (coefficient of $\partial_t^3f(t)$) as being the central charge [48, 127] of the boundary theory,

\begin{align}\label{cfinalexp}
c_M=\frac{1}{144 \sqrt{3} \pi  G_2}\Big(\kappa-12 \gamma \kappa +2 \kappa^2\Big),
\end{align}

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

It should be noted that the above expression of the central charge (<ref>) is a perturbative result up to quadratic order in the ModMax coupling ($\gamma$) and the $U(1)$ gauge coupling ($\kappa$). Furthermore, $c_M$ must be positive for the unitary theory [124], implying that the gauge couplings must lie within the range $\frac{1}{12} \leq \gamma < 1$ and $\frac{1}{2}(12\gamma - 1) < \kappa < 1$. Clearly, in the limit $\gamma\rightarrow 0$, the central charge (<ref>) reduces to $\sim\frac{1}{G_2}$ which is consistent with the existing result in the literature [48].

§ 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 (<ref>). 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 [131, 132, 193] associated with these 2D black holes. Finally, we also comment on the possibilities for extremal black hole solutions in two dimensions.

§.§ Black hole solutions

We estimate the 2D black hole solutions of (<ref>) 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 (<ref>). However, one can perform an elegant calculation using the light cone gauge. In this gauge, the space-time metric can be expressed as

\begin{align}\label{bhlcgauge}
\Phi=\hspace{1mm}&\Phi(z),\hspace{1mm}\chi=\chi(t),\hspace{1mm}\xi=\xi(z).
\end{align}

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

\begin{align}
& \mathcal{A}^{(bh)}=\mathcal{A}_0^{(bh)}+\kappa\mathcal{A}_1^{(bh)}+\gamma\kappa\mathcal{A}_2^{(bh)}+\kappa^2\mathcal{A}_3^{(bh)}...,\label{fieldexpansionblackhole1}\\
& \mathcal{B}^{(bh)}=\mathcal{B}_1^{(bh)}+\gamma\mathcal{B}_2^{(bh)}+\kappa \mathcal{B}_3^{(bh)}...,\hspace{2mm}|\kappa|<<1,\hspace{1.5mm}|\gamma|<<1,\label{fieldexpansionblackhole2}
\end{align}

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.

§.§ Zeroth order solution

In order to calculate black hole solutions at zeroth order, we switch off the $U(1)$ gauge couplings ($\kappa\rightarrow0$, $\gamma\rightarrow0$) in the equations of motion (<ref>)-(<ref>), which yields the following set of equations

\begin{align}
\Phi_0''-\omega_0'\Phi_0'+\Lambda e^{2\omega_0}\Phi_0=&\hspace{1mm}0,\label{bhzeroeomi}\\
\omega_0'\Phi_0'+\Lambda e^{2\omega_0}\Phi_0=&\hspace{1mm}0,\\
\omega_0''+e^{2\omega_0}\Lambda=&\hspace{1mm}0,\label{bhzeroeomf}
\end{align}

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

On solving the equations (<ref>)-(<ref>), one finds

\begin{align}\label{bhzerosollc}
e^{2\omega_{0}^{(bh)}}=-\frac{4\mu}{\Lambda \sinh^2{(2\sqrt{\mu}}z)}\hspace{1mm},\hspace{2mm}\Phi_0^{(bh)}=\phi_0,
\end{align}

where $\phi_0$ is a constant.

It should be noted that we treat the dilaton ($\Phi$) as constant while taking the limits $\kappa\rightarrow0$ and $\gamma\rightarrow0$. However, it possesses a non-trivial profile in the presence of $U(1)$ gauge fields (see Section (<ref>) and (<ref>)).

§.§ Order $\kappa$ solution

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

\begin{align}
\Phi_1''-2\big(\omega_0'\Phi_1'+\omega_1'\Phi_0'\big)+2\big(\dot{\chi}_1^2+\xi_1'^2\big)=&\hspace{1mm}0,\label{bhoneeomi}\\
\omega_1''+2\Lambda\omega_1e^{2\omega_0}-e^{-2\omega_0}A_{t1}'^2=&\hspace{1mm}0,\\
\partial_{z}\Big(\Phi_0e^{-2\omega_0}A_{t1}'\Big)=&\hspace{1mm}0,\\
\xi_1''=&\hspace{1mm}0,\\
\ddot{\chi}_1=&\hspace{1mm}0,\label{bhoneeomf}
\end{align}

where . and $'$ denote the derivatives with respect to $t$ and $z$ respectively.

In order to solve the above differential equations (<ref>)-(<ref>), we adopt the following change in coordinates

\begin{align}\label{bhctrhoz}
\rho=\sqrt{\mu}\coth{(2\sqrt{\mu}z)}.
\end{align}

Using the zeroth order solutions (<ref>) together with (<ref>), one finds

\begin{align}
\omega_1^{(bh)}=&\hspace{1mm}\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,\label{bhsolonei}\\
\Phi_1^{(bh)}=&\hspace{1mm}-\frac{(n_1^2+l_1^2)\rho}{4\mu^{\frac{3}{2}}}\tanh^{-1}{\Bigg(\frac{\rho}{\sqrt{\mu}}\Bigg)}+g_1\rho,\label{phi1bh}\\
\xi_1^{(bh)}=&\hspace{1mm}\frac{l_1}{2\sqrt{\mu}}\coth^{-1}
\chi_1^{(bh)}=&\hspace{1mm}n_1t+n_2\label{bhsolonef},
\end{align}

where $m_i,n_i,l_i,q_i$ and $g_1$, $(i=1,2)$ are the integration constants.

§.§ Order $\gamma\kappa$  solution

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

\begin{align}
\Phi_2''-2\big(\omega_0'\Phi_2'+\omega_2'\Phi_0'\big)+4\big(\dot{\chi}_1\dot{\chi}_2+\xi_1'\xi_2'\big)-\frac{2s_0}{\sqrt{s_0^2+p_0^2}}\big(\dot{\chi}_1^2+\xi_1'^2\big)=&\hspace{1mm}0,\label{bhtwoeomi}\\
\omega_2''+2\Lambda\omega_2e^{2\omega_0}-2e^{-2\omega_0}A_{t1}'A_{t2}'-e^{2\omega_0}\sqrt{s_0^2+p_0^2}+f_1=&\hspace{1mm}0,\\
\partial_z\Bigg[\xi_2'-\frac{1}{\sqrt{s_0^2+p_0^2}}\Big(s_0\xi_1'-p_0\dot{\chi}_1\Big)\Bigg]=&\hspace{1mm}0,\\
\partial_z\Bigg[e^{-2\omega_0}\Phi_0\Bigg(A_{t2}'-\frac{A_{t1}'s_0}{\sqrt{s_0^2+p_0^2}}\Bigg)\Bigg]=&\hspace{1mm}0,\\
\ddot{\chi}_2=&\hspace{1mm}0,\label{bhtwoeomf}
\end{align}

where we define the above quantities as

\begin{align}
f_1=&\hspace{1mm}\frac{1}{\Phi_0\sqrt{s_0^2+p_0^2}}\Big(s_0\big(-\dot{\chi}_1^2+\xi_1'^2\big) -2p_0\dot{\chi}_1\xi_1'\Big)\hspace{1mm},\hspace{2mm} p_0=-2\Phi_0^{-1}e^{-2\omega_0}\dot{\chi}_1\xi_1',\nonumber\\ s_0=&\hspace{1mm}-e^{-4\omega_0}A_{t1}'^2+\Phi_0^{-1}e^{-2\omega_0}\big(-\dot{\chi}_1^2+\xi_1'^2\big).
\end{align}

Clearly, the above differential equations (<ref>)-(<ref>) 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 (<ref>), the location of the horizon $(\rho_H)$ can be determined by noting the spacetime metric (<ref>)

\begin{align}\label{expbhmetric}
\end{align}

which yields $\rho=\rho_H=\sqrt{\mu}$.

Finally, the near horizon solutions of the equations of motion (<ref>)-(<ref>) could be listed as

\begin{align}
\Phi_2^{(bh)}=&\hspace{1mm}\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_1^8l_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
&\hspace{1mm}-3n_1^4l_1^4\Bigg)-15 \sqrt{\mu
\rho \Bigg)+2 n_1 n_3 \rho  \left(2 \rho -15 \sqrt{\mu }\right)+\rho  l_1 \Big(l_1
\left(\rho -12 \sqrt{\mu }\right)+\nonumber\\
&3 l_3 \left(\rho -9 \sqrt{\mu }\right)\Big)+192 \mu
^{5/2} g_2\Bigg)+\frac{\rho  \left(n_1^2+2 n_3 n_1+l_1 \left(l_1+2
l_3\right)\right) \log \left(\rho -\sqrt{\mu }\right)}{8 \mu ^{3/2}}\label{phi2bh},\\
\xi_2^{(bh)}=&\hspace{1mm}\frac{1}{16 \mu ^{3/2}}\Bigg(-\frac{256 m_1^4 n_1^2 l_1 \mu ^2 \rho  \left(\rho -2 \sqrt{\mu }\right)}{\Lambda ^2 \text{$\phi_0 $}^2(n_1^2+l_1^2)^3}-\frac{32 \mu  \rho  \left(2 \sqrt{\mu }
\left(n_1^2-2 l_1^2\right)+l_1^2 \rho \right)}{n_1^2l_1}+\nonumber\\
&\hspace{1mm}(l_{3}+l_1)\Bigg(-\frac{1}{2} \rho\left(\rho -6
\sqrt{\mu }\right)-4 \mu \log \left(\rho -\sqrt{\mu }\right)\Bigg)\Bigg)+l_{4},\label{xi2bh}\\
A_{t2}^{(bh)}=&\hspace{1mm}\frac{32 \sqrt{\mu } m_1^3 n_1^2  l_1^2  \rho\left(\rho -2 \sqrt{\mu }\right)}{\Lambda ^2 \text{$\phi_0$}^2(n_1^2+l_1^2)^3
}-m_3 \rho +m_4,\\
\omega_2^{(bh)}=&\hspace{1mm}\frac{m_1 }{2
\text{$\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),\\
\chi_2^{(bh)}=&\hspace{1mm}n_3t+n_4,\label{chi2bh}
\end{align}

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 [220] of the first $(I_{n}(\Tilde{\rho}))$ and the second kind $(K_{n}(\Tilde{\rho}))$.

§.§ Order $\kappa^2$  solution

The contribution due to the Maxwell (<ref>) term alone at quadratic ($\kappa^2$) could be estimated by solving the equations (<ref>)-(<ref>) at order $\kappa^2$

\begin{align}
\omega_3''+2\Lambda e^{2\omega_0}\big(\omega_1^2+\omega_3\big)-2e^{-2\omega_0}\big(-\omega_1 A_{t1}'^2+A_{t1}'A_{t3}'\big)&=0,\\
\Phi_3''-2\big(\omega_3'\Phi_0'+\omega_0'\Phi_3'+\omega_1'\Phi_1'\big)+4(\dot{\chi}_1\dot{\chi}_3+\xi_1'\xi_3')&=0,\\
\xi_3''&=0,\\
\ddot{\chi}_3&=0.\label{bhjhepfeq}
\end{align}

The solutions of the above equations (<ref>)-(<ref>) are quite complicated, therefore we mention them in the Appendix <ref>. Like before, one can further simplify these solutions (<ref>)-(<ref>) and (<ref>)-(<ref>) by making use of the residual gauge freedom in the light cone gauge (<ref>). In particular, the re-scaling of the time coordinate, $t\rightarrow n_1t$  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 (<ref>), (<ref>), (<ref>) and (<ref>)  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 (<ref>) and (<ref>)). However, for a particular choice of constants

\begin{align}\label{n3l3}
n_3=&\hspace{1mm}\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),\\
l_3=&\hspace{1mm}\frac{1}{\gamma l_1}\Big(-l_1^2(1+\gamma)+\kappa n_5\sqrt{1-l_1^2}\Big),
\end{align}

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.

§.§ 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 [131, 132, 193] 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 [72] for the 2D black holes which receives quadratic order corrections due to $U(1)$ gauge and ModMax couplings

\begin{align}\label{hawktdef}
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^2p\Big)\Bigg],
\end{align}

where we set the constants $q_4=q_{2}=q$ and $p$ is defined as

\begin{align}\label{jpcostbh}
\end{align}

The Wald entropy [131, 132, 193] is defined as

\begin{align}\label{walddefblackhole}
S_W=-2\pi\frac{\delta \mathcal{L}}{\delta R_{\mu\nu\alpha\beta}}\epsilon_{\mu\nu}\epsilon_{\alpha\beta},
\end{align}

where $R_{\mu\nu\alpha\beta}$ is the Riemann curvature tensor, $\mathcal{L}$ is the Lagrangian density[Here we used the convention, $I=\int d^2x\sqrt{-g}\mathcal{L}$. ] 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 (<ref>), the Wald entropy (<ref>) for 2D black holes turns out to be[Here, the entities $\phi_1$, $\phi_2$ and $\phi_3$ are respectively the values of  $\Phi_1^{(bh)}$ (<ref>), $\Phi_2^{(bh)}$ (<ref>) and $\Phi_3^{(bh)}$ (<ref>) at the horizon $\rho=\rho_H=\sqrt{\mu}$.]

\begin{align}\label{waldentexp01}
\end{align}

where we denote the above entities as

\begin{align}
\phi_1=&\hspace{1mm}\sqrt{\mu } g_1-\frac{1}{192 \mu }\Big(12 \log (4 \mu )+2 l_1^2-13\Big),\label{phih}\\
\phi_2=&\hspace{1mm}\frac{64 \mu  }{3 l_1^2}\frac{\left(1-2 l_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}\label{ph2h},\\
\phi_3=&\hspace{1mm}\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 )\nonumber\\
&+13-24 \log (2))-48 q_1\Bigg),\label{jph3h}
\end{align}

and $\phi_0$ is the usual constant dilaton solution in the limit $\kappa\rightarrow0$ and $\gamma\rightarrow0$ (<ref>).

§.§ 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 (<ref>)

\begin{align}\label{extremalconstraint}
\kappa q+ \gamma\kappa q+\kappa^2p=1,
\end{align}

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

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

\begin{align}
\end{align}

where the entities $p$, $\phi_1$, $\phi_2$ and $\phi_3$ are respectively given in (<ref>), (<ref>), (<ref>) and (<ref>).

§ CONCLUDING REMARKS

To summarise, in this chapter, 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.

This completes the second part of the thesis, whose primary goal was to study the transformation properties of the 1D boundary stress-energy tensor under the combined action of both diffeomorphism and $U(1)$ gauge transformation and hence compute the central charge associated with the boundary theory at strong coupling.

CHAPTER:  CONCLUSION AND OUTLOOK

We conclude the thesis first by summarising the key observations in Chapters 2-5 and then highlighting some of the future extensions down the line. As mentioned before, this thesis contains two parts: (i) in the first part, we investigated the phase transitions in 2D gravitational theories, and (ii) in the second part, we examined the transformation properties of the 1D boundary stress-energy tensor under the combined action of both diffeomorphism and $U(1)$ gauge transformation which eventually leads to the central charge associated with the 1D boundary theory at strong coupling.

Below, we summarize the key outcome of each of the individual chapters.

In Chapter 2, we constructed a 2D theory of gravity starting from a 5D gravity coupled with $U(1)$ and $SU(2)$ Yang-mills fields following a dimensional reduction. We explored the thermal properties of the vacuum and the black hole solutions for the 2D gravity model using a canonical ensemble. In particular, we computed the free energy and thermal entropy of the system and found that the thermal radiation beyond a certain critical temperature would collapse to form a globally stable black hole, indicating the analog of the Hawking-Page (HP) transition in 2D gravity.

In Chapter 3, we investigated the phase stability of $U(1)$ gauge charged Euclidean wormhole solutions in two dimensions. To be specific, we obtained the free energy (density) and the total charge of the system and explored their dependencies on the temperature at a fixed chemical potential. To our surprise, the free energy and the total charge of the system suffer from a discontinuity at a critical temperature ($T=T_0$), indicating the onset of a first-order phase transition. Furthermore, we found that beyond the critical temperature, the wormhole solutions transit into a pair of black holes at finite chemical potential via a first-order phase transition.

In Chapter 4, we build up a model of JT gravity that contains 2-derivative as well as 4-derivative interactions between the $U(1)$ gauge fields and the space-time metric. We computed the renormalized boundary stress-energy tensor, examined its transformation properties under the diffeomorphism and the $U(1)$ gauge transformation in the Fefferman-Graham gauge, and obtained the central charge associated with the 1D boundary theory. We notice that, the central charge goes inversely with the coupling constant associated with the 4-derivative interactions.

We further obtained the black hole solutions and investigated the effects of quartic interactions on their thermal properties. We found that the Wald entropy associated with the black hole diverges at the horizon due to the presence of higher derivative interactions in the theory. We interpreted the near-horizon divergences in terms of the density of states. Finally, we investigated the near-horizon modes of the theory and computed the central charge associated with the CFT$_{1}$ in the near-horizon limit. In addition, we found that the near-horizon CFT could be recast as 2D Liouville theory with generalized potential. We examined the Weyl invariance of the Liouville theory and computed the associated Weyl anomaly.

In Chapter 5, following a dimension reduction, we constructed a 2D theory of gravity starting from a 4D gravity in the presence of “ModMax” interactions. We explored the effects of these non-trivial interactions on the central charge associated with the 1D boundary theory up to quadratic order in the ModMax and $U(1)$ gauge couplings. We further computed the associated black hole solutions in two dimensions and examined the effects of ModMax interactions on their thermal properties. Interestingly, we found that the ModMax interactions play a crucial role in obtaining the finite values of the background fields near the horizon.

After briefly summarizing the key results of the thesis, below we outline some interesting projects that can be persuaded in the future.

* It would be indeed an interesting project to look for the dual 1D boundary theory pertinent to the modified models of JT gravity as discussed in Chapters 2-5 and identify the corresponding Schwarzian degrees of freedom. The Schwarzian  [13] action can be obtained following a standard procedure in which one integrates out the dilaton ($ \Phi $). In the process, one is left with the boundary term which is regarded as the low energy effective theory living on the boundary. Clearly, for the class of theories that we study, one would expect a modified Schwarzian in the presence of a chemical potential ($ \mu $).

As for example, below, we outline a simple toy model calculation for a particular case [123], in which we show schematically how does this extra term is generated due to the presence of the chemical potential ($ \mu $). To begin with, we consider an $ AdS_2 $ metric of the form

\begin{equation}
\end{equation}

where the boundary is located at, $z \sim \varepsilon \sim 0$.

Taking the first term on the R.H.S. of (<ref>) into account, one can show that the boundary action typically looks like (here we absorb the coupling constants of the bulk theory into $ \mu $)

\begin{eqnarray}
S_B \sim \int du~\Phi_r (u)  \text{Sch}\lbrace t(u), u\rbrace +\mu  \int du \frac{\Phi_r^2 (u)}{\varepsilon},
\end{eqnarray}

where we parameterize the coordinates ($t , z$) in terms of the boundary time coordinate ($u$) and $\Phi_r (u)$ is the coupling constant of the boundary field theory which is defined in terms of the boundary data of the form $\Phi|_{bdy}=\frac{\Phi_r}{\epsilon}$ [13]. It would be indeed an interesting project to explore such theories in detail an extract Green's functions etc. out of it. We leave all these issues for future investigations.

* It would be an interesting project to generalise the results of Chapter 2 in the presence of ModMax interactions (see Chapter 5) and investigate the possible deviations in the Hawking-Page transition in the presence of projected ModMax interactions. Moreover, one can further compute their imprints on various physical observables like the boundary stress-energy tensor and the central charge associated with the 1D boundary theory.

* It would be nice to construct the 2D wormhole solutions [123] and explore their thermal stability for the ModMax corrected JT gravity models [128] and/or the 2D gravity theory that contains the most generic quartic interactions [127].

* Recall that, in Section <ref> we compute the “annealed” Free energy [55] of the wormhole solution and explore the associated phase stability (see Section <ref>). However, one can further refine the computation by including the corrections due to the replica wormholes [221, 78] and compute the “quenched” Free energy [55]. It would be nice to explore the properties of “quenched” Free energy and study the corresponding phase stability of the configuration at finite chemical potential ($\mu$).

* It is natural to further generalise the results of Chapter 4 in the presence of $SU(2)$ Yang-Mills fields and look for its imprints on the holographic stress tensor as well as central charge associated with the 1D boundary theory. It is noteworthy to mention that the $SU(2)$ Yang-Mills fields are responsible for the first order phase transition in 2D gravity [122]. Therefore, it would be an interesting project to explore the phase transition in the presence of such quartic couplings.

* 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 ($S_C$)  [48, 51, 198]

\begin{align}\label{cardyformula}
S_C= 2\pi\sqrt{\frac{c_M \Delta}{6}},
\end{align}

where $\Delta$ is the  eigen value of the associated Virasoro generator $L_0$.

The authors in  [48] establish a 2D/3D dictionary which by virtue of the Cardy formula (<ref>) 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 into three dimensions and establish a suitable 2D/3D mapping in the presence  most generic quartic interactions [127] and/or 2D projected ModMax interactions [128].

* It would be an interesting project to explore the holographic renormalisation group flow and holographic c-theorem [222]-[224] in the context of 2D gravity models that contain the most generic quartic interactions [127] and/or 2D projected ModMax interactions [128].

* As mentioned in the introduction (see Section <ref>), the authors in [117] investigated the phase transition in the deformed JT gravity and the matrix integral model. It would be interesting to further generalize the matrix integral models associated with JT gravity in the presence of non-trivial gauge couplings (see Chapters 2-5) and explore the possible deviations in the Hawking-Page transition.

CHAPTER: DIMENSIONAL REDUCTION FROM $5D$ TO $2D$

In order to analyze the dimensional reduction of the $5D$ theory (<ref>) to $2D$, in the following we rewrite the ansatz
given in (<ref>):

\begin{equation}\label{anz:red2}
ds^{2}=\Phi^{\alpha}\;d\tilde{s}^{2}+\Phi^{\beta}dx_{i}^{2},\quad A_{M}^{a}dx^{M}
=A_{\mu}^{a}dx^{\mu}, \quad A_{M}dx^{M}=A_{\mu}dx^{\mu}~;~\alpha,\beta  \in \mathbb{R}.
\end{equation}

With this choice of the metric we note that

\begin{equation}\label{g:measure}
\sqrt{-g}=\sqrt{-\tilde{g}}\;\Phi^{\alpha+3\beta/2},
\end{equation}

where $\tilde{g}$ is the determinant of the two-dimensional metric.

It is then easy to check that the first two terms in the action (<ref>) can be written as

\begin{equation}\label{RL:5D}
\sqrt{-g}(\mathcal{R}-3\Lambda)=\sqrt{-\tilde{g}}\Phi^{3\beta/2}(\tilde{\mathcal{R}}
\end{equation}

In the next step, we rewrite the third term within the curly braces in (<ref>) as a total derivative term given by

\begin{equation}\label{totDv}
\Phi^{\alpha+3\beta/2}g^{ij}\mathcal{R}_{ij}\sim \nabla_{\mu}\Big[g^{\mu\nu}
\Phi^{\alpha+\beta/2}\Big],
\end{equation}

which therefore does not contribute to the bulk equations of motion (<ref>)-(<ref>). Notice that, the JT gravity
model (<ref>) can be obtained once we set the parameters $\alpha=0$ and $\beta=2/3$. In other words, these
choices of parameters are quite pertinent to the JT gravity model discussed in this paper. The remaining terms in
(<ref>) involving gauge fields can be obtained in a similar manner.

CHAPTER: EQUATIONS OF MOTION AT DIFFERENT ORDER IN THE PERTURBATION SERIES

In the following, we note down the equations of motion upto leading order in perturbative expansion. For example, substituting
(<ref>) into (<ref>) we obtain

\begin{align}
\mathcal{O}(0): \quad& 2\Phi_{(0)}''+6\Lambda\Phi_{(0)} e^{2\omega_{(0)}}=0,  \label{dil:zero}\\
\mathcal{O}(\xi): \quad& 2\left(\Phi_{(1)}^{ab}\right)''+6\Lambda e^{2\omega_{(0)}}\left(2
\omega_{(1)}^{ab}\Phi_{(0)}+\Phi_{(1)}^{ab}\right)+\frac{Q^{2}}{\Phi_{(0)}} e^{2\omega_{(0)}}=0,
\label{dil:xi}\\
\mathcal{O}(\kappa): \quad& 2\left(\Phi_{(1)}^{na}\right)''+6\Lambda e^{2\omega_{(0)}}\left(2 \omega_{(1)}^{na}\Phi_{(0)}+\Phi_{(1)}^{na}\right)+\frac{1}{g_{s}^{2}}\Phi_{(0)}e^{-2\omega_{(0)}}
\left(\left(\chi_{(0)}'\right)^{2}+\chi_{(0)}^{2}\eta_{(0)}^{2}\right)\nonumber\\
&=0.  \label{dil:kapa}
\end{align}

Similarly (<ref>) could be arranged upto leading order in the perturbative expansion as

\begin{align}
\mathcal{O}(0): \quad& 4\omega_{(0)}''+6\Lambda e^{2\omega_{(0)}}=0,  \label{omga:zero}\\
\mathcal{O}(\xi): \quad& 4\left(\omega_{(1)}^{ab}\right)''+12\Lambda\omega_{(1)}^{ab}
e^{2\omega_{(0)}}-\frac{Q^{2}}{\Phi_{(0)}^{2}}e^{2\omega_{(0)}}=0,  \label{omga:xi}\\
\mathcal{O}(\kappa): \quad& 4\left(\omega_{(1)}^{na}\right)''+12\Lambda\omega_{(1)}^{na}
+\chi_{(0)}^{2}\eta_{(0)}^{2}\right)=0.  \label{omga:kapa}
\end{align}

Substituting (<ref>) into the constraint equation (<ref>) we obtain

\begin{align}
\mathcal{O}(0): &\quad \Phi_{(0)}e^{-2\omega_{(0)}}\chi_{(0)}^{2}\eta_{(0)}=0,  \label{cnst:0}\\
\mathcal{O}(\xi): &\quad  \Phi_{(0)}\left(\chi_{(0)}\eta_{(1)}^{ab}+2\eta_{(0)}\chi_{(1)}^{ab}\right)
e^{-2\omega_{(0)}}\chi_{(0)}^{2}=0, \label{cnst:xi}\\
\mathcal{O}(\kappa): &\quad\Phi_{(0)}\left(\chi_{(0)}\eta_{(1)}^{na}+2\eta_{(0)}\chi_{(1)}^{na}\right)
e^{-2\omega_{(0)}}\chi_{(0)}^{2}=0. \label{cnst:kapa}
\end{align}

Finally, from (<ref>) and (<ref>) the zeroth order equations could be recast as

\begin{align}
\partial_{z}\left(\Phi_{(0)}e^{-2\omega_{(0)}}\chi_{(0)}\eta_{(0)}\right)
\label{zero:nab1}\\
\partial_{z}\left(\Phi_{(0)}e^{-2\omega_{(0)}}\left(\partial_{z}\chi_{(0)}\right)
\right)-\Phi_{(0)}e^{-2\omega_{(0)}}\chi_{(0)}\eta_{(0)}^{2}&=0. \label{zero:nab2}
\end{align}

CHAPTER: DERIVATIONS OF ZEROTH ORDER SOLUTIONS

In the conformal gauge [34]

\begin{equation}\label{conf:guge}
ds^{2}=-e^{2\omega(x^{+},x^{-})}dx^{+}dx^{-},\quad x^{\pm}=(t\pm z),
\end{equation}

the zeroth order equations of motion for the metric and the dilaton can be rewritten as

\begin{align}
8\partial_{+}\partial_{-}\omega_{(0)}-3\Lambda e^{2\omega_{(0)}}&=0, \label{B2}\\
\partial_{+}\left(e^{2\omega_{(0)}}\partial_{+}\Phi_{(0)}\right)&=0, \label{B3}\\
\partial_{-}\left(e^{2\omega_{(0)}}\partial_{-}\Phi_{(0)}\right)&=0. \label{B4}
\end{align}

Clearly, the solution of (<ref>) is given by

\begin{equation}\label{sol:B2}
\end{equation}

Now integrating (<ref>) and (<ref>) we may write

\begin{align}
\partial_{+}\Phi_{(0)} &= \left(-\frac{8}{3\Lambda}\right)\frac{f(x^{-})}{\left(x^{+}
\partial_{-}\Phi_{(0)} &= \left(-\frac{8}{3\Lambda}\right)\frac{g(x^{+})}{\left(x^{+}
\end{align}

Differentiating (<ref>) w.r.to $x^{-}$ and (<ref>) w.r.to $x^{+}$ and substituting them in (<ref>) we obtain

\begin{align}\label{sol:phigen}
\Phi_{(0)} =
\begin{cases}
\frac{2}{3\Lambda}\frac{\partial_{-}f(x^{-})\cdot (x^{+}-x^{-})+2f(x^{-})}
{(x^{+}-x^{-})}  \\[10pt]
\frac{2}{3\Lambda}\frac{\partial_{+}g(x^{+})\cdot (x^{+}-x^{-})-2g(x^{+})}
{(x^{+}-x^{-})} .
\end{cases}
\end{align}

After a few easy algebraic steps, the general solution for the dilaton may be written as [34]

\begin{equation}\label{sol:phiF}
\Phi_{(0)}=\left(-\frac{2}{3\Lambda}\right)\frac{a+b(x^{+}+x^{-})+cx^{+}x^{-}}
\end{equation}

where $a$, $b$, $c$ are real constants. For the vacuum we may choose $a=1$, $b=0$ and $c=0$ [34, 37].

In the next step, we find the corresponding solutions for the black hole by exploiting the $SL(2,\mathbb{R})$ invariance of the
metric (<ref>). The general solutions can then be written as

\begin{align}
\omega^{-}(x^{-})}{\left[\omega^{+}(x^{+})-\omega^{-}(x^{-})\right]^{2}},
\label{met:w}\\[10pt]
\Phi_{(0)}&=\left(-\frac{2}{3\Lambda}\right)\frac{1-\mu\;\omega^{+}(x^{+})
\omega^{-}(x^{-})}{\left[\omega^{+}(x^{+})-\omega^{-}(x^{-})\right]}, \label{dil:w}
\end{align}

where $\mu(>0)$ can be considered as the mass of the black hole, and $\omega^{\pm}(x^{\pm})$ are monotonic
functions [34].

Finally, using the conformal transformations

\begin{equation}
\omega^{\pm}(x^{\pm})=\frac{1}{\sqrt{\mu}}\tanh\sqrt{\mu}x^{\pm}
\end{equation}

(<ref>) and (<ref>) can be rewritten as

\begin{align}
\sqrt{\mu}(x^{+}-x^{-})\right)},
\label{met:H}\\[10pt]
\Phi_{(0)}&=\left(-\frac{2}{3\Lambda}\right)\sqrt{\mu}\coth\left(\sqrt{\mu}(x^{+}
-x^{-})\right). \label{dil:H}
\end{align}

CHAPTER: 3D TO 2D REDUCTION

We begin by considering $ AdS_3 $ gravity coupled to Maxwell Chern-Simons term of topological gauge theories [130, 158]

\begin{align}\label{3daction}
S_{(3)}=\int d^3x\sqrt{-g_{(3)}}\Bigg(R_{(3)}+2+a_1F_{MN}F^{MN}+a_2\epsilon^{MNP}A_M\partial_N A_P+2(\partial_M f)^2\Bigg),
\end{align}

where $R_{(3)}$ is the 3D Ricci scalar, $f$ is the scalar field and $\epsilon^{MNP}=\frac{\varepsilon^{MNP}}{\sqrt{-g_{(3)}}}$ is the Levi-Civita tensor. Here $(M,N)$ represent the 3 dimensional indices.

In order to obtain the desired JT gravity model (<ref>), we propose a reduction ansatz of the following form

\begin{align}\label{3dansatz}
A_{\mu}&\equiv A_{\mu}(x^{\nu}),\hspace{1mm}A_z\equiv \kappa(x^{\mu}),\hspace{1mm}f\equiv f(x^{\mu}),
\end{align}

where $(\mu,\nu)$ are the 2 dimensional indices and $z$ is the compact dimension.

Substituting  (<ref>) into (<ref>) and integrating over compact dimension, we find

\begin{align}\label{2dactionreduce}
S_{(2)}=&\int d^2x\sqrt{-g_{(2)}}\Bigg(\Phi R_{(2)}+2\big(1+(\partial_{\mu}f)^2\Phi\big)-\Phi^{-1}(\partial_{\mu}\Phi)^2+a_1\Phi^2F^2+\nonumber\\
&2a_1\Phi^{-1}(\partial_{\mu}\kappa)^2 +a_2\epsilon^{\mu\nu}_{(2)}F_{\mu\nu}\kappa\Bigg).
\end{align}

Next, we redefine the fields as

\begin{align}\label{redefine}
\kappa (\rho)=\frac{\Phi}{\sqrt{2a_1}},\hspace{2mm}f(\rho)=\int d\rho\sqrt{\frac{\Tilde{g}_{\rho\rho}(\Phi-1)}{\Phi}},
\end{align}

where $\rho$ is the radial direction of $ AdS_2 $.

Finally, substituting (<ref>) into (<ref>), we obtain the 2D gravity model

\begin{align}
S_{(2)}=\int d^2x\sqrt{-g_{(2)}}\Bigg(\Phi \big(R_{(2)}+2\big)+a_1\Phi^2F^2+\Tilde{a}_2\epsilon^{\mu\nu}_{(2)}F_{\mu\nu}\Phi\Bigg),
\end{align}

where we define $\Tilde{a}_2=\frac{a_2}{\sqrt{2a_1}}$.

CHAPTER: QUANTUM STRESS-ENERGY TENSOR FOR GAUGE FIELDS

In this Appendix, we compute the expectation value for the quantum stress-energy tensor [55] in the double trumpet background (<ref>). We use the point-splitting method of [159]. As mentioned previously, the following derivation is technically different from that of the scalar field ($\chi$) [55]. The reason for this stems from the fact that gauge fields in 2D are non-conformal. Therefore, one has to carry out a first principle derivation of the quantum stress-energy tensor considering the double trumpet geometry as the background.

The equation of motion for the gauge field, $A_{\mu}$ (<ref>) in the double trumpet background (<ref>) turns out to be

\begin{align}\label{cylgaugeeom}
\hat{\textbf{F}}A_{\tau}=J(\rho),
\end{align}

where $\hat{\textbf{F}}=\partial^2_{\rho}+(f(\rho)-2\tan(\rho))\partial_{\rho}$ is the corresponding differential operator.

Furthermore, here we define

$$f(\rho)=2\Phi^{-1}\partial_{\rho}\Phi\hspace{1mm},\hspace{2mm} J(\rho)=\frac{ a_2}{2a_1}\frac{\partial_{\rho}\Phi}{\Phi^{2}\cos^2\rho}.$$

In order to proceed further, we segregate out gauge field components from (<ref>)

\begin{align}\label{gaugeaction}
S_{gauge}=a_1\int_{(I)} d^2x\sqrt{-g}\Phi^{2}F^2+a_2\int_{(II)} d^2x\sqrt{-g}\Phi\varepsilon^{\mu\nu}F_{\mu\nu}.
\end{align}

Notice that, the variation of the second integral $(II)$ with respect to the metric $g_{\mu\nu}$ vanishes. Therefore, only the first integral $(I)$ contributes to the expectation value of the quantum stress-energy tensor for gauge fields (<ref>).

In order to take the variation of (<ref>), we decompose the (first) integral $(I)$ into the bulk and the boundary pieces as follows

\begin{align}\label{gaugeactionsplit}
S_{gauge}&=2a_1\int d^2xA_{\nu}\partial_{\mu}\Big[\sqrt{-g}\Phi^{2}K^{\mu\alpha\nu\beta}\partial_{\beta}A_{\alpha}\Big]+a_2\int d^2x\sqrt{-g}\Phi\varepsilon^{\mu\nu}F_{\mu\nu}+S_{boundary},\nonumber\\
\end{align}

The variation of (<ref>) with respect to the bulk metric $g^{\eta\kappa}$  yields the following expression

\begin{align}\label{gaugevariation1}
&\frac{\delta S_{gauge}}{\delta g^{\eta\kappa}}=2a_1\int d^2x\Bigg[A_{\nu}\Big[\Phi^{2}\Big\{-\frac{1}{2}\Big(\partial_{\mu}(g_{\eta\kappa}\sqrt{-g})+\sqrt{-g}g_{\lambda\sigma}\frac{\delta(\partial_{\mu}g^{\lambda\sigma})}{\delta g^{\eta \kappa}}\Big)K^{\mu\alpha\nu\beta}-\nonumber\\
&\frac{1}{2}\sqrt{-g}g_{\eta\kappa}(\partial_{\mu}K^{\mu\alpha\nu\beta})+\sqrt{-g}\Big(g^{\nu\beta}\frac{\delta(\partial_{\mu}g^{\mu\alpha})}{\delta g^{\eta\kappa}}+g^{\mu\alpha}\frac{\delta(\partial_{\mu}g^{\nu\beta})}{\delta g^{\eta\kappa}}-g^{\nu\alpha}\frac{\delta(\partial_{\mu}g^{\mu\beta})}{\delta g^{\eta\kappa}}-\nonumber\\
&g^{\mu\beta}\frac{\delta(\partial_{\mu} g^{\nu\alpha})}{\delta g^{\eta\kappa}}\Big)\Big\}\Big]\dot(\partial_{\beta}A_{\alpha})+\Phi^{2}\Big\{A_{\nu}(\partial_{\eta}\sqrt{-g})g^{\nu\beta}\partial_{\beta}A_{\kappa}+A_{\eta}(\partial_{\mu}\sqrt{-g})g^{\mu\alpha}\partial_{\kappa}A_{\alpha}-\nonumber\\
&A_{\nu}g^{\nu\alpha}\partial_{\eta}(\Phi^{2}\partial_{\kappa} A_{\alpha})-A_{\eta}g^{\mu\beta}\partial_{\mu}(\Phi^{2}\partial_{\beta}A_{\kappa})\Big\}\Bigg].
\end{align}

In the double trumpet background (<ref>), the above expression (<ref>) further simplifies as

\begin{align}\label{gaugevariation2}
\frac{\delta S_{gauge}}{\delta g^{\tau\tau}}= \frac{\delta S_{gauge}}{\delta g^{\rho\rho}}=-a_1\int d^2x\sqrt{-g}A_{\tau}\hat{\textbf{L}} A_{\tau},
\end{align}

where we ignore all the derivatives of the metric variation i.e. $ \Big|\frac{\partial_{\mu}(\delta g^{\mu\alpha})}{\delta g^{\eta\kappa}}\Big|<<1$ and  $\hat{\textbf{L}}$ is the differential operator such that $\hat{\textbf{L}}A_{\tau}=\cos^2\rho\partial_{\rho}(\Phi^{2}\partial_{\rho}A_{\tau})$.

Using (<ref>) as well as the point splitting method [55], the expectation value for the quantum stress-energy tensor[Notice that, there are two possible ways in which the differential operator $\hat{\textbf{L}}$ can act on the Green's function $G(\rho,\tau;\rho',\tau')$. However, both the possibilities give the same result. Therefore, we consider the average of both the possibilities in the definition of the expectation value of the stress tensor.] turns out to be

\begin{align}\label{stcylgauge}
<T_{\tau\tau }^{(gauge)}>_{qm}\hspace{1mm}=\hspace{1mm}<T_{\rho\rho}^{(gauge)}>_{qm}\hspace{1mm}= \frac{a_1}{2}\Big[&\lim_{x'\rightarrow x}\cos^2\rho\partial_{\rho}\Big(\Phi^{2}(\rho')\partial_{\rho'}G(\rho,\tau;\rho',\tau')\Big)+\nonumber\\
&\lim_{x\rightarrow x'}\cos^2\rho'\partial_{\rho'}\Big(\Phi^{2}(\rho)\partial_{\rho}G(\rho,\tau;\rho',\tau')\Big)\Big],
\end{align}

where $x\equiv(\rho,\tau)$,  $x'\equiv(\rho',\tau')$.

Here $G(\rho,\tau;\rho',\tau')$ is the Green's function which satisfies the following equation

\begin{align}\label{green1}
\hat{\textbf{F}}G(\rho,\tau;\rho',\tau')=-\delta(\rho-\rho')\delta(\tau-\tau'),
\end{align}

where the operator $\hat{\textbf{F}}$ is given by (<ref>).

In order to solve (<ref>), we consider the following Fourier decomposition of $G(\rho,\tau;\rho',\tau')$ and $\delta(\tau-\tau')$

\begin{align}
G(\rho,\tau;\rho',\tau')&=\sum_{m,m'\in \mathbb{Z}}\Tilde{G}(\rho,m;\rho',m')e^{2\pi i (m\tau+m'\tau')/b}\label{fd1},\\
\delta(\tau-\tau')&=\frac{1}{b}\sum_{m,m'\in \mathbb{Z}}e^{2\pi i (m\tau+m'\tau')/b}\delta_{m+m'}.\label{fd2}
\end{align}

Plugging (<ref>) and (<ref>) into (<ref>), we obtain

\begin{align}\label{green2}
\Big[\partial^2_{\rho}+(f(\rho)-2\tan(\rho))\partial_{\rho}\Big]\Tilde{G}(\rho,m;\rho',m')=-\frac{1}{b}\delta(\rho-\rho')\delta_{m+m'}.
\end{align}

Now, we solve (<ref>) using the following properties:

$\bullet$ $\Tilde{G}$ is continuous in the limit $\rho\rightarrow \rho'$ : $\Tilde{G}(\rho)=\Tilde{G}(\rho')$.

$\bullet$ The derivative of $\Tilde{G}$ is discontinuous in the limit[Here, we integrate (<ref>) with respect to $\rho$ from $\rho=\rho'-\epsilon$ to $\rho=\rho'+\epsilon$ and take the limit $\epsilon\rightarrow 0.$] $\epsilon\rightarrow 0$ namely,

\begin{align}\label{disc}
\frac{d \Tilde{G}}{d\rho}\Big|_{\rho'+\epsilon}-\frac{d \Tilde{G}}{d\rho}\Big|_{\rho'-\epsilon}=-\frac{1}{b}.
\end{align}

Notice that, $f(\rho)$ in (<ref>) is an unknown function of $\rho$. Therefore one cannot solve (<ref>) exactly for all values of $\rho$. However, we are interested in the near boundary analysis, therefore we solve (<ref>) in the near boundary limits, $\rho\rightarrow\pm\frac{\pi}{2}$.

$\bullet$ Case 1 : Near  $\rho\sim\rho_R\sim\frac{\pi}{2} $ and $-\frac{\pi}{2}<\rho'<\rho<\frac{\pi}{2}$

Setting $m'=-m$ and upon solving equation (<ref>) near $\rho\sim\frac{\pi}{2}$, we obtain

\begin{align}\label{npbgp1}
\Tilde{G}_+(\rho,\rho',m)= &\hspace{1mm} \frac{1}{4}A(\rho',m)\Bigg(\frac{2\exp(p_0(\pi-2\rho)/2)}{\pi-2\rho}-p_0\text{Ei}(p_0(\pi-2\rho)/2)\Bigg)\nonumber \\
\end{align}

where $p_0=f\Big(\frac{\pi}{2}\Big)$ and the subscript $`+$' denotes the value of Green's function near the right boundary $\rho_R\sim+\frac{\pi}{2}$.

$\bullet$ Case 2 : Near $\rho\sim\rho_L\sim-\frac{\pi}{2}$ and $-\frac{\pi}{2}<\rho<\rho'<\frac{\pi}{2}$

On setting $m'=-m$ and solving equation (<ref>) near $\rho\sim-\frac{\pi}{2}$, we obtain

\begin{align}\label{npbgp2}
\Tilde{G}_-(\rho,\rho',m)= &\hspace{1mm}  \frac{1}{4}C(\rho',m)\Bigg(-\frac{2\exp(-q_0(\pi+2\rho)/2)}{\pi+2\rho}-q_0\text{Ei}(-q_0(\pi+2\rho)/2)\Bigg)\nonumber \\
\end{align}

where $q_0=f\Big(-\frac{\pi}{2}\Big)$ and the subscript $`-$' denotes the value of Green's function near the left boundary $\rho_L\sim-\frac{\pi}{2}$.

Notice that, the differential equation (<ref>) contains the derivative of
$\Tilde{G}(\rho,m;\rho',m')$ with respect to the quantity “$\rho$”. Therefore, the entities $A(\rho',m)$, $B(\rho',m)$, $C(\rho',m)$ and $D(\rho',m)$ in (<ref>) and (<ref>) appears as an integration constants. However, one can compute the functions $A(\rho',m)$ and $C(\rho',m)$ using the properties of the Green's function those were mentioned in (<ref>) but the functions $B(\rho',m)$ and $D(\rho',m)$ still remain undetermined. Therefore, we perform all the analysis keeping the general form of the functions $B(\rho',m)$ and $D(\rho',m)$ and finally impose a suitable condition on these functions (<ref>) using the fact that the left temperature of the wormhole near $\rho_L\sim-\frac{\pi}{2}$ must be identified with the right temperature near $\rho_R\sim\frac{\pi}{2}$.

Finally, (<ref>) can be systematically expressed as

\begin{align}\label{greenfunc}
G(\rho,\tau;\rho',\tau')= \left\{
\begin{array}{lr}
\sum\limits_{m\in \mathbb{Z}} \Tilde{G}_+(\rho,\rho',m)e^{2\pi i (\tau-\tau')m/b} & -\frac{\pi}{2}<\rho'<\rho<\frac{\pi}{2} \\
\sum\limits_{m\in \mathbb{Z}} \Tilde{G}_-(\rho,\rho',m)e^{2\pi i (\tau-\tau')m/b}  & -\frac{\pi}{2}<\rho<\rho'<\frac{\pi}{2}
\end{array}
\right.,
\end{align}

where the function $A(\rho',m)$ and $C(\rho',m)$ could be expressed as

\begin{align}
&A(\rho',m)=-\frac{1}{b}(\pi-2\rho')^2\Big(4b B(\rho',m)-4b D(\rho',m)+(\pi+2\rho')\big(2+\nonumber\\
&C(\rho',m)=\frac{1}{b}\exp(\pi q_0/2+(p_0+q_0)\rho')\Big(2\exp(p_0(\pi-2\rho')/2)(\pi+2\rho')^2(\pi-2\rho'-\nonumber\\
&2b B(\rho',m)+2b D(\rho',m))-p_0(\pi^2-4\rho'^2)^2\text{Ei}(p_0(\pi-2\rho')/2)\Big)\Big(4\exp(p_0\pi/2)\pi-\nonumber\\
\end{align}

Finally, plugging (<ref>) into (<ref>), we obtain

\begin{align}
<T_{\tau\tau +}^{(gauge)}>_{qm}\hspace{1mm}=\hspace{1mm}<T_{\rho\rho +}^{(gauge)}>_{qm}\hspace{1mm}=&\hspace{1mm}\frac{1}{\mathcal{A}_4(\rho)}\Bigg[a_1\exp\big(p_0(\pi-2\rho)/2\big)\cos^2\rho\Phi^2\Big(4\mathcal{A}_1(\rho)+\nonumber\\
<T_{\tau\tau -}^{(gauge)}>_{qm}\hspace{1mm}=\hspace{1mm}<T_{\rho\rho -}^{(gauge)}>_{qm}\hspace{1mm}=&\hspace{1mm}\frac{1}{\mathcal{B}_3(\rho)}\Bigg[a_1\exp(p_0\rho)\cos^2\rho\Phi^2\Big(\mathcal{B}_1(\rho)+\nonumber\\
\end{align}

where the functions $\mathcal{A}_1(\rho),.. \mathcal{A}_4(\rho), \mathcal{B}_1(\rho),.. \mathcal{B}_3(\rho)$ are given by

\begin{align}
&\mathcal{A}_1(\rho)=\Bigg(4b B(\rho,m)-4b D(\rho,m)+(\pi+2\rho)\Big(2+\exp(q_0(\pi+2\rho)/2)q_0(\pi+2\rho)\times\nonumber\\
& \mathcal{A}_2(\rho)=\Bigg(4b B(\rho,m)-4b D(\rho,m)+(\pi+2\rho)\Big(2+\exp(q_0(\pi+2\rho)/2)q_0(\pi+2\rho)\times\nonumber\\
&(4+\pi q_0+2q_0\rho)\text{Ei}(-q_0(\pi+2\rho)/2)\Big)\Bigg),\\
&\mathcal{A}_3(\rho)= \Bigg(-p_0(\pi-2\rho)^2\text{Ei}(p_0(\pi-2\rho)/2)+\exp(p_0(\pi-2\rho)/2)\Big(4\pi+\nonumber\\
&q_0(\pi+2\rho)\text{Ei}(-q_0(\pi+2\rho)/2)+q_0(\pi+2\rho)\Big(2+\exp(q_0(\pi+2\rho)/2)(2+\pi q_0+\nonumber\\
& \mathcal{A}_4(\rho)=b(\pi-2\rho)\Bigg(p_0(\pi-2\rho)^2\text{Ei}(p_0(\pi-2\rho)/2)-\exp(p_0(\pi-2\rho)/2)\Big(4\pi+\nonumber\\
\end{align}

\begin{align}
&\mathcal{B}_2(\rho)=2\Bigg(4\exp(p_0\pi/2)\pi-\exp(p_0\rho)p_0(\pi-2\rho)^2\text{Ei}(p_0(\pi-2\rho)/2)+ q_0(\pi+2\rho)^2\times\nonumber\\
& \mathcal{B}_3(\rho)=b(\pi+2\rho)^2\Bigg(4\exp(p_0\pi/2)\pi-\exp(p_0\rho)p_0(\pi-2\rho)^2 \text{Ei}(p_0(\pi-2\rho)/2)+\nonumber\\
&\exp\big((p_0+q_0)\pi/2+q_0\rho\big)q_0(\pi+2\rho)^2  \text{Ei}(-q_0(\pi+2\rho)/2)\Bigg)^2.
\end{align}

Here the subscript $`\pm$' denotes the  expectation value of the quantum stress-energy tensor near the boundary  $\rho=\pm\frac{\pi}{2}$.

The leading order terms in the quantum stress-energy tensor could be expressed as follows

\begin{align}
&<T_{\tau\tau +}^{(gauge)}>_{qm}\Big|_{\rho\rightarrow\frac{\pi}{2}}\hspace{1mm}=\hspace{1mm}<T_{\rho\rho +}^{(gauge)}>_{qm}\Big|_{\rho\rightarrow\frac{\pi}{2}}\hspace{1mm}=\hspace{1mm}\frac{a_1}{b \pi}(\pi-2\rho)\Big(\pi+b B(\pi/2,m)-\nonumber\\
&b D(\pi/2,m)+\exp(\pi q_0)\pi^2q_0\text{Ei}(-\pi q_0)\Big)\Phi\Big(\frac{\pi}{2}\Big)^{2}\Big(1+\exp(\pi q_0)\pi q_0\text{Ei}(-\pi q_0)\Big)^{-1},\label{stcylgaugep}
\end{align}

\begin{align}
&<T_{\tau\tau -}^{(gauge)}>_{qm}\Big|_{\rho\rightarrow-\frac{\pi}{2}}\hspace{1mm}=\hspace{1mm}<T_{\rho\rho -}^{(gauge)}>_{qm}\Big|_{\rho\rightarrow-\frac{\pi}{2}}\hspace{1mm}=\hspace{1mm}\frac{a_1}{b \pi}(\pi+2\rho)\Big(b\exp(p_0\pi)\times \nonumber\\
&B(-\pi/2,m)-b \exp(p_0\pi)D(-\pi/2,m)+\pi\big(-\exp(p_0\pi)+p_0\pi\text{Ei}(\pi p_0)\big)\Big)\Phi\Big(-\frac{\pi}{2}\Big)^{2}\nonumber\\
&\times\Big(-\exp(p_0\pi)+\pi p_0\text{Ei}(\pi p_0)\Big)^{-1}.\label{stcylgaugem}
\end{align}

Before we conclude, it is noteworthy to mention that one can also evaluate the expectation value of the quantum stress-energy tensor for the scalar field $\chi$ following the method as described above. In the flat space-time limit ($\rho\rightarrow0$), these results boil down into an expression in the cylindrical coordinates [55] as given below

\begin{align}
<T_{\rho\rho}^{(\chi)}>_{qm}&\approx\sum_{m\in \mathbb{Z}}-\frac{m\pi}{b^2\tanh(2\pi^2m/b)}+\Bigg(-\frac{m\pi\coth(2\pi^2m/b)}{b^2}-\frac{\tanh(m\pi^2/b)}{4m\pi}\Bigg)\rho^2\nonumber\\
\end{align}

CHAPTER: DETAILED EXPRESSIONS OF $F_{\PM}$

The functions $F_{\pm}$, are given by

\begin{align}
&r_1\alpha \log\Big(\frac{2}{\pi}\Big)+\frac{\pi}{2}X_{\tau\tau}(b)+\frac{1}{36}\Bigg(-1-9C_1^2\pi+\frac{9a_1(8+\pi^2)}{b \pi}\Big(\pi+b B(\pi/2,m)\nonumber\\
&-b D(\pi/2,m)+\exp(\pi q_0)\pi^2q_0\text{Ei}(-\pi q_0)\Big)\Phi\Big(\frac{\pi}{2}\Big)^{2}\Big(1+\exp(\pi q_0)\pi q_0\times\nonumber\\
&\text{Ei}(-\pi q_0)\Big)^{-1}\Bigg)\Bigg),\label{fphi1}\\
&+s_3))\big)\beta+s_1\beta \log\Big(\frac{\pi}{2}\Big)-\frac{\pi}{2}X_{\tau\tau}(b)+\frac{a_1}{4b \pi}(8+\pi^2)\Big(b\exp(p_0\pi) B(-\pi/2,m)\nonumber\\
&-b \exp(p_0\pi)D(-\pi/2,m)+\pi\big(-\exp(p_0\pi)+p_0\pi\text{Ei}(\pi p_0)\big)\Big)\Phi\Big(-\frac{\pi}{2}\Big)^{2}\times\nonumber\\
&\Big(\exp(p_0\pi)-\pi p_0\text{Ei}(\pi p_0)\Big)^{-1}\Bigg),\label{fphi2}
\end{align}

where we denote $r_0=\hspace{1mm}\Phi\Big |_{\rho=\frac{\pi}{2}},\hspace{1mm}r_1=\Phi'\Big|_{\rho=\frac{\pi}{2}},\hspace{1mm}r_2=\Phi''\Big|_{\rho=\frac{\pi}{2}},\hspace{1mm}r_3=\Phi'''\Big|_{\rho=\frac{\pi}{2}},$

\begin{align}
\hspace{1mm}s_0=\Phi\Big|_{\rho=-\frac{\pi}{2}},\hspace{1mm}s_1=\Phi'\Big|_{\rho=-\frac{\pi}{2}},\hspace{1mm}s_2=\Phi''\Big|_{\rho=-\frac{\pi}{2}},\hspace{1mm}s_3=\Phi'''\Big|_{\rho=-\frac{\pi}{2}}.
\end{align}

Here $\gamma$ is the Euler's constant and $C_3,C_4$ are the integration constants.

CHAPTER: BLACK HOLE SOLUTION

In the following section, we evaluate the black hole solution by substituting (<ref>) into the equations of motion (<ref>)-(<ref>) and solve them at different orders in the coupling.

§ ZEROTH ORDER SOLUTIONS

Zeroth order solutions are obtained by setting the expansion parameters as $a_1=a_2=0$. The corresponding equations of motion (<ref>)-(<ref>) turn out to be

\begin{align}
\Phi_0''-2e^{2\omega_0}\Phi_0&=0,\label{bh01}\\
\omega_0''-e^{2\omega_0}&=0,\label{bh02}\\
\chi_0''&=0.\label{bh03}
\end{align}

On solving (<ref>)-(<ref>), we find the zeroth order solutions as

\begin{align}\label{bh0sol}
\omega_0=\frac{1}{2}\log\Bigg(\frac{4r_H}{\sinh^2(2\sqrt{r_H}z)}\Bigg),\hspace{1mm}\Phi_0=\sqrt{r_H}\coth(2\sqrt{r_H}z),\hspace{1mm}\chi_0=b_1z+b_2,
\end{align}

where $r_H,b_1$ and $b_2$ are integration constants.

§ FIRST ORDER SOLUTIONS

We now estimate the leading order contributions due to $U(1)$ gauge fields. The corresponding equations of motion (<ref>)-(<ref>) turn out to be

\begin{align}
\Phi_1''-2(\omega_1'\Phi_0'+\omega_0'\Phi_1')-2\chi_0'\chi_1'&=0,\label{bh11}\\
\omega_1''-2\omega_0''\omega_1-2\Phi_0e^{-2\omega_0}A_{\tau1}'^2&=0,\label{bh12}\\
\partial_z(\Phi_0^2e^{-2\omega_0}A_{\tau1}')&=0,\label{bh13}\\
\chi_1''-2\chi_0''\omega_1&=0.\label{bh14}
\end{align}

In order to solve (<ref>)-(<ref>), we adopt the following change in coordinates

\begin{align}
\end{align}

where $r_H$ denotes the location of the black hole horizon.

Upon solving (<ref>)-(<ref>) we find first order corrections to the background fields as

\begin{align}
\omega_1=&\hspace{1mm}-\frac{b_3^2}{r_H^2r}\Big(r_H+r^2(-2\log(r)+\log(-\sqrt{r_H}+r)+\log(\sqrt{r_H}+r))\Big)-b_5+\nonumber\\
\chi_1=&\hspace{1mm}\frac{b_6}{\sqrt{r_H}}\tanh^{-1}\Bigg(\frac{r}{\sqrt{r_H}}\Bigg)+b_7,\label{bhsol13}\\
\Phi_1=&\hspace{1mm}\frac{b_3^2}{r_H^2}\Big((-r_H+r^2)(2\log r-\log(-\sqrt{r_H}+r)-\log(\sqrt{r_H}+r))\Big)+\frac{1}{8r_H^{\frac{3}{2}}}\Bigg(8r_H r^2b_4\nonumber\\
&+b_8+\rho b_{9}\label{bhsol14},
\end{align}

where $b_3,b_4,.,b_{9}$ are the integration constants and $\mu_{bh}$ is the chemical potential for the black hole phase.

The black hole solution up to leading order in $a_1$ can be summarised as

\begin{align}
\omega_{bh}=&\hspace{1mm}\frac{1}{2}\log(-4r_H+4r^2)+a_1\Bigg(
\frac{r}{\sqrt{r_H}}\Bigg(b_4+\tanh^{-1}\Bigg(\frac{r}{\sqrt{r_H}}\Bigg)b_5\Bigg)\Bigg),\\
\Phi_{bh}=&\hspace{1mm}r+a_1\Bigg(\frac{b_3^2}{r_H^2}\Big((-r_H+r^2)(2\log r-\log(-\sqrt{r_H}+r)-\log(\sqrt{r_H}+r))\Big)+\nonumber\\
&\frac{1}{8r_H^{\frac{3}{2}}}\Bigg(8r_H r^2b_4+4r_H\Bigg(2\sqrt{r_H}r+2r^2\tanh^{-1}\Bigg(\frac{r}{\sqrt{r_H}}\Bigg)+r_H\log(-\sqrt{r_H}+r)-\nonumber\\
&\rho b_{9}\Bigg),\\
\chi_{bh}=&\hspace{1mm}\frac{b_1\coth^{-1}(r/\sqrt{r_H})}{2\sqrt{r_H}}+b_2+a_1\Bigg(\frac{b_6}{\sqrt{r_H}}\tanh^{-1}\Bigg(\frac{r}{\sqrt{r_H}}\Bigg)+b_7\Bigg),\\
\end{align}

CHAPTER: NON-ABELIAN GENERALIZATION OF JT GRAVITY

In this Appendix, we will derive the most general 2D action for Einstein-dilaton gravity coupled with U(1) gauge and SU(2) Yang-Mills fields that contains the 4-derivative interaction terms.

We start with the most general 5D action that contains the 2-derivative interaction terms
\begin{align}\label{Na 5D}
S^{(2)}_{5D}=& \int d^5x\sqrt{-g_{(5)}}\Big[\alpha_1(\lambda + R)-\frac{\alpha_2}{4}F^2+\frac{\alpha_3}{3}\epsilon^{MNOPQ}A_MF_{NO}F_{PQ}-\frac{\alpha_4}{4}\big(F^{(a)}\big)^2+\nonumber\\
\end{align}

where $\alpha_i$ $(i=1,..,5)$ are the respective coupling constants.

Next, we will add the following 4-derivative gauge invariant interaction terms to the above action (<ref>)

\begin{align}\label{Na4 5D}
S^{(4)}_{5D}=&\int d^5x\sqrt{-g_{(5)}}\Big[\beta_1 R^2+\beta_2 (R_{MN})^2 + \beta_3 (R_{MNOP})^2+\beta_4 RF^2+ \beta_5R^{MN}F_{MO}{F_{N}}^O +\nonumber\\
&\beta_6 R_{MNOP}F^{MN}F^{OP}+\beta_7 R_{MNOP}F^{MO}F^{NP}+\beta_8 (F^2)^2+ \beta_9 F^{MN}F_{NO}F^{OP}F_{PM}+\nonumber\\
&(\beta_{16}F_{OP}\bigtriangledown^RF_{RQ}+\beta_{17}F_{OR}\bigtriangledown^RF_{PQ}+\beta_{18}F_{OR}\bigtriangledown_PF_{QS}g^{RS})+\beta_{19}A_M R_{NOKL}\times\nonumber\\
&+\delta_3 R_{MNOP}F^{(a)MN}F^{(a)OP}+\delta_4 R_{MNOP}F^{(a)MO}F^{(a)NP}+\delta_5((F^{(a)})^2)^2+\nonumber\\
\end{align}

where $\beta_i$ and $\delta_j$ $(i=1,..,20$ and $j=1,..,28)$ are the respective coupling constants.

On adding (<ref>) and (<ref>), we get the required action that contains all 2-derivative as well as 4-derivative interactions. However, we can eliminate the various interaction terms in (<ref>) using a proper redefinition of fields.

Consider the following redefinition of fields

\begin{align}
g^{RS}\rightarrow g^{RS}+\delta g^{RS},\hspace{2mm}A_N\rightarrow A_N+\delta A_N,\hspace{2mm}A_N^{(a)}\rightarrow A_N^{(a)}+\delta A_N^{(a)}
\end{align}

such that the action transform as $S\rightarrow S'=S+\delta S$ where $S=S_{5D}^{(2)}+S_{5D}^{(4)}$ and $S'$ is called the transformed action.

The most general variation of fields that contains 2-derivative interactions takes the form

\begin{align}
\delta g^{RS} = & \mu_1R^{RS}+\mu_2F^{RP}{F^S}_P+\mu_3Rg^{RS}+\mu_4F^2g^{RS}+\mu_5g^{RS}+\mu_6(F^{(a)})^2g^{RS}+\nonumber\\
\delta A_N= &\lambda_1A_N+\lambda_2\bigtriangledown^MF_{MN}+\lambda_3\epsilon_{NOPQR}F^{OP}F^{QR}+\lambda_4\epsilon_{NOPQR}F^{(a)OP}F^{(a)QR},\label{naa2}\\
\delta A^{(a)}_N= &\sigma_1A^{(a)}_N+\sigma_2\bigtriangledown^MF^{(a)}_{MN}+\sigma_3\epsilon_{NOPQR}F^{(a)OP}F^{QR},\label{naa3}
\end{align}

where $\mu_i$, $\lambda_j$ and $\sigma_k$ $(i=1,..,7,j=1,..,4$ and $ k=1,..,3)$ are the respective constants.

In order to get rid of various interaction terms in (<ref>) and (<ref>), we transformed the action $S$ (where $S=S_{5D}^{(2)}+S_{5D}^{(4)}$) into $S'$ using (<ref>)-(<ref>) with the following particular choice of constants

\begin{align}
&\mu_1=-\frac{4\beta_2}{3\alpha_1},\hspace{2mm}\mu_2 = \frac{-2\alpha_2\beta_2}{3\alpha_1^2} ,\hspace{2mm}\mu_3=\frac{4}{9\alpha_1}(2\beta_1+\beta_2), \hspace{2mm}\mu_4=\frac{2\alpha_2}{27\alpha_1^2}(2\beta_2+\beta_1),\hspace{2mm}\mu_5 = -\frac{2}{3},\nonumber\\
&\mu_6 = \frac{2}{27\alpha_1}[9\delta_2+6\delta_1+\frac{\alpha_4}{\alpha_1}(2\beta_2+\beta_1)],\hspace{2mm} \mu_7=\frac{-2}{3\alpha_1}[\delta_2+\frac{\alpha_4}{\alpha_1}\beta_2],\hspace{2mm}\lambda_1=\frac{-1}{3},\nonumber\\
&\lambda_2=\frac{-4\alpha_2}{3\alpha_3\alpha_5}[2\delta_{18}-\frac{\alpha_5}{\alpha_3}\beta_{20}],\hspace{3mm}\lambda_3=\frac{4\beta_{20}}{3\alpha_3},\hspace{3mm} \lambda_4=\frac{1}{3\alpha_3}[2\delta_{18}-\frac{\alpha_5}{\alpha_3}\beta_{20}]
\end{align}

and $\sigma_k=0$ [We choose $\sigma_k=0$ to preserve the gauge invariance.] which yields

\begin{align}\label{Na4f 5D}
S'=&\int d^5x\sqrt{-g_{(5)}}\Big[\eta_1\lambda +\eta_2R-\frac{\eta_3}{4}F^2-\frac{\eta_4}{4}\big(F^{(a)}\big)^2+
\frac{\eta_5}{4}\epsilon^{MNOPQ}A_MF^{(a)}_{NO}F^{(a)}_{PQ}+\nonumber\\
\end{align}

where $\eta_i$ $(i=1,..,22)$ are the new coupling constants respectively.

We can express the new coupling constants $\eta_i$ in terms of the old coupling constants $\alpha_j$, $\beta_k$ and $\delta_l$ as follows

\begin{align}
&\eta_1 =\frac{8}{3}\alpha_1, \hspace {2mm}\eta_2=2\alpha_1-\frac{4\lambda}{9}(5\beta_1+\beta_2),\hspace{2mm}\eta_3=\frac{\lambda\alpha_2}{27\alpha_1}(4\beta_2+20\beta_1)+\frac{2\alpha_2}{3},\nonumber\\
\end{align}

\begin{align}
&\eta_{14}=\frac{\alpha_4}{108\alpha_1}\Big[6\delta_1+\frac{\alpha_4}{\alpha_1}(\beta_1-7\beta_2)\Big], \hspace{2mm}\eta_{15}=\frac{\alpha_4\alpha_1}{54\alpha_1^2}(\beta_1-7\beta_2)+\frac{\alpha_2\delta_1}{\alpha_118}-\frac{2\delta_{15}}{3},\nonumber\\
\end{align}

With all these preliminaries, equation (<ref>) is the most general 5D action of gravity coupled with U(1) gauge and SU(2) Yang-Mills fields that contains all the 4-derivative interaction terms.

Our next step is to truncate the 5D action to 2D using the following ansatz

\begin{align}
&A_Mdx^M=A_{\mu}dx^{\mu},\hspace{2mm} A_{\mu}\equiv A_{\mu}(x^{\nu}),\label{AAA}\\
&A_M^{(a)}dx^M=A_{\mu}^{(a)}dx^{\mu},\hspace{2mm} A_{\mu}^{(a)}\equiv A_{\mu}^{(a)}(x^{\nu}).\label{ANA}
\end{align}

Using the above ansatz (<ref>)-(<ref>) in (<ref>) we finally obtain

\begin{align}\label{NA 2D}
S_{(2D)} =& \int d^2x\sqrt{-g_{(2)}}\phi\Bigg[\eta_1\lambda+\eta_2R-\frac{\eta_3}{4}F^2-\frac{\eta_4}{4} \big(F^{(a)}\big )^2+\eta_6\Big[(R_{\mu\nu\alpha\beta})^2+\nonumber\\
\end{align}

where we identify  (<ref>) as the most general 2D action of gravity coupled with U(1) gauge and SU(2) Yang-Mills fields that contains all the 4-derivative interaction terms.

CHAPTER: MOST GENERAL 4-DERIVATIVE ACTION IN 5D

The purpose of this Appendix is to discuss the most general 5D action of gravity coupled with U(1) gauge fields that contains the 4-derivative interaction terms[This can be achieved by simply removing the SU(2) Yang-Mills fields in Appendix <ref>.].

In the first place, we consider the most general 2-derivative action of the following form

\begin{eqnarray}\label{A2d}
S^{(2)}=&& \int d^5x\sqrt{-g}\Bigg[12 + R-\frac{\alpha_1}{4}F^2+\frac{\alpha_2}{3}\epsilon^{MNOPQ}A_MF_{NO}F_{PQ}\Bigg],
\end{eqnarray}

where $\alpha_1$ and $\alpha_2$ are the respective coupling constants.

Next, we note down the most general 4-derivative gauge invariant interaction terms as follows

\begin{align}\label{A4d}
S^{(4)}=&\int d^5x\sqrt{-g} \Big [\beta_1 R^2+\beta_2 [R_{MN}]^2 + \beta_3 [R_{MNOP}]^2+\beta_4 RF^2+ \beta_5R^{MN}F_{MO}{F_{N}}^O +\nonumber\\
&\beta_6 R_{MNOP}F^{MN}F^{OP}+\beta_7 R_{MNOP}F^{MO}F^{NP}+\beta_8 F^4+ \beta_9 F^{MN}F_{NO}F^{OP}F_{PM}+\nonumber\\
&\beta_{19}A_M R_{NOKL}{R_{PQ}}^{KL}\big\}+\beta_{20}\epsilon_{NOPQR}\epsilon^{NIJKL}F^{OP}F^{QR}F_{IJ}F_{KL}\Big],
\end{align}

where $\beta_i$ $(i=1,..,20)$ are the respective coupling constants. On adding (<ref>) in (<ref>), we obtain the most general action for gravity (coupled to U(1) gauge fields) that contains 4-derivative interaction terms. Moreover, it is also possible to eliminate the various interaction terms in (<ref>) using a proper redefinition of fields as discussed below.

Consider the following redefinition of fields
$$g^{RS}\rightarrow g^{RS}+\delta g^{RS},\hspace{2mm}A_N\rightarrow A_N+\delta A_N$$
such that the action transform as $S\rightarrow S'=S+\delta S$.

The most general 2-derivative variation of fields are given by

\begin{eqnarray}
\delta g^{RS} &=& \mu_1R^{RS}+\mu_2F^{RP}{F^S}_P+\mu_3Rg^{RS}+\mu_4F^2g^{RS}+\mu_5g^{RS}\label{aa1}\\
\delta A_N&=&\lambda_1A_N+\lambda_2\bigtriangledown^MF_{MN}+\lambda_3\epsilon_{NOPQR}F^{OP}F^{QR}.\label{aa2}
\end{eqnarray}

where $\mu_i$ and $\lambda_j$ ($i=1,..,4$ and $j=1,..,3$) are the respective coupling constants.

In order to get rid of various interaction terms in (<ref>) and (<ref>), we transformed the action $S$ (where $S=S_{5D}^{(2)}+S_{5D}^{(4)}$) into $S'$ using (<ref>) and (<ref>) with the following particular choice of constants

\begin{eqnarray}
&&\mu_1=-\frac{4\beta_2}{3},\hspace{3mm}\mu_2=-\frac{2}{3}\alpha_1\beta_2 ,\hspace{3mm}\mu_3=\frac{4}{9}(2\beta_1+\beta_2), \hspace{3mm}\mu_4=\frac{2\alpha_1}{27}(2\beta_2+\beta_1),\nonumber\\
&&\mu_5 = -\frac{2}{3},\hspace{3mm}\lambda_1=\frac{-1}{3},\hspace{3mm}\lambda_2=\frac{4\alpha_1}{3\alpha_2^2}\beta_{20},\hspace{3mm}\lambda_3=\frac{4\beta_{20}}{3\alpha_2},
\end{eqnarray}

which yields

\begin{align}\label{FA4d}
S'=&\int d^5x\sqrt{-g}\Big [(12+R)-\frac{\eta_1}{4}F^2+\eta_2[R_{MNOP}]^2+\eta_3 F^4+\eta_4F^{SP}F_{PR}F^{RQ}F_{QS}+ \nonumber\\
&\eta_{8}F_{MN}F_{OR}\bigtriangledown_P{F_Q}^R+\eta_{9}A_MR_{NOIJ}{R_{PQ}}^{IJ}) \Big ]
\end{align}

where $\eta_i$ ($i=1,..,9$) are the new coupling constants respectively.

We can express the new coupling constants, $\eta_i$ in terms of old coupling constants, $\alpha_j$ and $\beta_k$ as follows

\begin{align}
&\eta_1=\frac{\alpha_1}{6}(4\beta_2+20\beta_1)+\frac{\alpha_1}{4},\hspace{2mm}\eta_2=\frac{\beta_3}{2},\hspace{2mm}\eta_3=\frac{\alpha_1^2}{288}(\beta_1-7\beta_2)-\frac{\beta_8}{2},\hspace{2mm}\eta_4=\frac{1}{8}(\alpha_1^2\beta_2\nonumber\\ &-4\beta_9),\eta_5=\frac{\alpha_1^2}{2\alpha_2^2}\beta_{20},\hspace{2mm}
\eta_{6}=\alpha_1\Big(\frac{\beta_{20}}{\alpha_2}\Big),\hspace{3mm}\eta_{7}=\frac{-\beta_{17}}{4},\hspace{3mm}\eta_{8}=\frac{-\beta_{18}}{4},\hspace{3mm}\eta_{9}=\frac{\beta_{19}}{4}.
\end{align}

Notice that, we have taken out a common factor $\frac{8}{3}$ in (<ref>) and impose the condition on $\beta_{1,2}$ such that $5\beta_1+\beta_2=-\frac{1}{8}$. With all these preliminaries, we end up with a most general 5D action (<ref>) of gravity coupled with U(1) gauge fields that contains all the 4-derivative interaction terms.

CHAPTER: COVARIANCE OF THE 2D ACTION (<REF>)

In this section, we will discuss the general covariance of the action (<ref>). After a careful observation,  we realize that the $\Gamma^{\alpha}_{\beta\lambda}$s appear in such a combination that the entire expression transform as a “scalar” under general coordinate transformation (GCT).

As an illustration, let's look at a particular term in (<ref>) as mentioned below:

\begin{align}
\Gamma^{\sigma}_{\alpha\mu}(\bigtriangledown_{\sigma}\phi)(\bigtriangledown^{\alpha}\phi)(\bigtriangledown^{\mu}\phi)=\Gamma^{\sigma}_{\alpha\mu}A_{\sigma}B^{\alpha}C^{\mu},\label{C1}
\end{align}

where $A_{\sigma},B^{\alpha}$ and $C^{\mu}$ are the tensors of rank 1.

A straightforward computation reveals that under GCT

\begin{align}\label{C3}
\Gamma^{\sigma}_{\alpha\mu}A_{\sigma}B^{\alpha}C^{\mu}\rightarrow\Gamma^{'\sigma}_{\alpha\mu}A'_{\sigma}B^{'\alpha}C^{'\mu}
=&\frac{\partial x^{'\sigma}}{\partial x^{\beta}}\frac{\partial x^{\rho}}{\partial x^{'\mu}}\frac{\partial x^{\tau}}{\partial x^{'\alpha}}\Gamma^{\beta}_{\tau\rho}\frac{\partial x^{\tilde{\beta}}}{\partial x^{'\sigma}}\frac{\partial x^{'\alpha}}{\partial x^{\tilde{\tau}}}\frac{\partial x^{'\mu}}{\partial x^{\tilde{\rho}}}A_{\tilde{\beta}}B^{\tilde{\tau}}C^{\tilde{\rho}}\nonumber\\
&-\frac{\partial x^{\tilde{\tau}}}{\partial x^{'\alpha}}\frac{\partial^2x^{'\sigma}}{\partial x^{\tilde{\tau}}\partial x^{\tilde{\beta}}}\frac{\partial x^{\tilde{\beta}}}{\partial x^{'\mu}}\frac{\partial x^{\beta}}{\partial x^{'\sigma}}\frac{\partial x^{'\alpha}}{\partial x^{\tau}}\frac{\partial x^{'\mu}}{\partial x^{\rho}}A_{\beta}B^{\tau}C^{\rho} \nonumber\\\nonumber
=&\Gamma^{\beta}_{\tau\rho}A_{\tilde{\beta}}B^{\tilde{\tau}}C^{\tilde{\rho}}\delta^{\tilde{\beta}}_{\beta}\delta^{\rho}_{\tilde{\rho}}\delta^{\tau}_{\tilde{\tau}}-\delta^{\tilde{\tau}}_{\tau}\delta^{\tilde{\beta}}_{\rho}\frac{\partial}{\partial x^{\tilde{\tau}}}\Big(\frac{\partial x^{'\sigma}}{\partial x_{\tilde{\beta}}}\Big)\frac{\partial x^{\beta}}{\partial x^{'\sigma}}A_{\beta}B^{\tau}C^{\rho}\nonumber\\
=&\Gamma^{\beta}_{\tau\rho}A_{\beta}B^{\tau}C^{\rho}-\frac{\partial}{\partial x^{\tau}}\Big(\frac{\partial x^{'\sigma}}{\partial x_{\tilde{\beta}}}\Big)\frac{\partial x^{\beta}}{\partial x^{'\sigma}}A_{\beta}B^{\tau}C^{\tilde{\beta}}.
\end{align}

Notice that, the last term in (<ref>) vanishes identically as shown below

\begin{align}\label{C4}
\Bigg[\frac{\partial}{\partial x^{\tau}}\Big(\frac{\partial x^{'\sigma}}{\partial x_{\tilde{\beta}}}\Big)\frac{\partial x^{\beta}}{\partial x^{'\sigma}}\Bigg]A_{\beta}B^{\tau}C^{\tilde{\beta}}=&\Bigg[\frac{\partial}{\partial x^{\tau}}\Big(\frac{\partial x^{'\sigma}}{\partial x_{\tilde{\beta}}}\frac{\partial x^{\beta}}{\partial x^{'\sigma}}\Big)-\frac{\partial x^{'\sigma}}{\partial x_{\tilde{\beta}}}\frac{\partial^2x^{\beta}}{\partial x^{\tau}\partial x^{'\sigma}}\Bigg]A_{\beta}B^{\tau}C^{\tilde{\beta}}\nonumber\\
=&\Bigg[\frac{\partial}{\partial x^{\tau}}\Big(\delta^{\beta}_{\tilde{\beta}}\Big)-\frac{\partial x^{'\sigma}}{\partial x_{\tilde{\beta}}}\frac{\partial}{\partial x^{'\sigma}}\Big(\frac{\partial x^{\beta}}{\partial x^{\tau}}\Big)\Bigg]A_{\beta}B^{\tau}C^{\tilde{\beta}}\nonumber\\
=& 0.
\end{align}

CHAPTER: SOLVING THE CONSTANTS DI

In order to cure the divergences in the Gibbons-Hawking-York term and Einstein-Hilbert action (<ref>)-(<ref>), we set the constants $d_{i}$s (<ref>) such that the coefficients of each divergent term vanish identically.

Below, we note  down the coefficients of each divergent terms as

\begin{align}
&\text{coefficient of $\frac{1}{z^\frac{10}{3}}$}\hspace{2mm}:\hspace{2mm}\frac{-2032}{45}-\frac{103172}{585}D_1-\frac{14188}{195}\sqrt{\frac{2}{3}}d_1L+\frac{11104}{45}d_2/C_1=0,\label{CONST1}\\
&\text{coefficient of $\frac{1}{z^\frac{11}{3}}$}\hspace{2mm}:\hspace{2mm}-\frac{24}{7}D_1-\frac{4}{7}\sqrt{6}Ld_1=0,\label{CONST2}\\
&\text{coefficient of $\frac{1}{z^\frac{14}{3}}$}\hspace{2mm}:\hspace{2mm}\frac{64}{21}+\frac{18}{7}D_1+\frac{4}{7}\sqrt{\frac{2}{3}}Ld_1-\frac{32}{7}d_2/C_1=0,\label{CONST3}\\
&\text{coefficient of $\xi\log(z)$}\hspace{2mm}:\hspace{2mm}d_3/C_3+\frac{11213}{3123}=0,\label{CONST4}\\
&\text{coefficient of $\kappa\log(z)$}\hspace{2mm}:\hspace{2mm}12d_3+\frac{19868C_3}{347}+\frac{d_3}{6C_3}C_9+\frac{1844}{9369}C_9=0,\label{CONST5}\\
&\text{coefficient of $\frac{1}{z^2}$}\hspace{2mm}:\hspace{2mm}d_6\kappa C_1+\frac{16112}{1041}\kappa-\frac{871}{6246}=0,\label{CONST6}\\
&\text{coefficient of $\frac{\xi}{z^3}$}\hspace{2mm}:\hspace{2mm}d_4C_1^2/C_6+\frac{871}{3123}=0,\label{CONST7}\\
&\text{coefficient of $\frac{\kappa}{z^3}$}\hspace{2mm}:\hspace{2mm}d_5C_1^2/C_6+\frac{871}{3123}=0.\label{CONST8}
\end{align}

On solving (<ref>)-(<ref>), we get the following values as solutions

\begin{align}
& d_6=-\frac{0.16}{\kappa C_1}\left(-0.871+96.6\kappa\right),\hspace{1mm}D_1=-2.5795,\hspace{1mm}C_3=0.0283C_9.
\end{align}

CHAPTER: CARDY FORMULA

Cardy formula [51] measures the degrees of freedom (and hence the entropy) of a two-dimensional conformal field theory (CFT$_{2}$)[For the  generalization to d-dimensional CFT, see [225]] using the central charge (c) and the conformal weight for the ground state ($\Delta$). In the present Section, we derive the expression for the Cardy formula. We start by computing the partition function of CFT$_{2}$ on torus (T$^{2}$) and then use this partition function to estimate the entropy.

The partition function of CFT$_{2}$ on torus (T$^{2}$) of modular parameter $\tau=x^1+ix^0$ is given by [51]

\begin{align}\label{cd1}
\end{align}

where $H$ and $P$ are the time and space translation on the cylinder respectively and the factor of $2\pi$ in the definition of $Z$ is merely a convention.

One can derive $H$ and $P$ from the stress tensor $T_{\mu\nu}$ as

\begin{align}\label{cd2}
H=\frac{1}{2\pi}\int dx^1T_{00}\hspace{2mm}\text{and}\hspace{2mm}  P=\frac{1}{2\pi}\int dx^1T_{01},
\end{align}

where $T_{00}$ and $T_{01}$ can be expressed in terms of the stress tensor on the cylinder ($T_{cyl}$) as

\begin{align}\label{cd3}
\end{align}

The stress tensor on the cylinder, $ T_{cyl}(z)$ is given by [226]-[227]

\begin{align}\label{cd4}
T_{cyl}(z)=-\sum e^{-inz}L_n+\frac{c}{24},
\end{align}

where $L_n$ are the virasoro generators that satisfy the following commutation relations

\begin{align}\label{cd5}
\end{align}

Using equations (<ref>)-(<ref>) in (<ref>), we get[Since central charge is the real number, therefore we use $\overline{c}=c$ in (<ref>).]

\begin{align}\label{cd6}
Z(\tau,\overline{\tau})=\text{Tr}\Big[e^{2\pi i\tau\big(L_0-\frac{c}{24}\big)}e^{2\pi i\overline{\tau}\big(\overline{L}_0-\frac{c}{24}\big)}\Big].
\end{align}

Next, we trace (<ref>) over the energy eigen state $|E\rangle$ that satisfy the following eigen value equation

\begin{align}
\end{align}

which yields[For simplicity, we have consider only first part of the trace.]

\begin{align}\label{cd7}
Z(\tau)=\int_0^{\infty}d\Delta\rho(\Delta)e^{2\pi i\tau\big(\Delta-\frac{c}{24}\big)},
\end{align}

where $\rho(\Delta)$ is the density of states correspond to the energy $\Delta$.

One can evaluate $\rho(\Delta)$ by taking the inverse Laplace transformation of (<ref>) as follows

\begin{align}\label{cd8}
\rho(\Delta)=\oint_c d\tau Z(\tau)e^{-2\pi i\tau\big(\Delta-\frac{c}{24}\big)}.
\end{align}

We are interested in computing the entropy of CFT$_2$ in the high temperature limit (or $\Delta>>1$). Therefore, in this limit, the integral (<ref>) is dominated by $Z(\tau\rightarrow0)$. In order to find $Z(\tau)$ in the limit $\tau\rightarrow0$, we utilize the important fact that the partition function (<ref>) is modular invariant i.e. $Z(\tau)=Z(-\frac{1}{\tau})$. This can be understood in terms of the geometry of torus as described below.

Consider a torus (T$^2$) which is parametrize by the two coordinates i.e. $\sigma_1$ and $\sigma_2$ in the range $0\leq\sigma_1\leq2\pi$ and $0\leq\sigma_2\leq2\pi$. The general metric of the torus is given by

\begin{align}\label{t1}
ds^2=|d\sigma_1+\tau d\sigma_2|^2 \hspace{2mm},
\end{align}

where $\tau\in\mathbb{C}$ is the modular parameter. It is easy to check that (<ref>) is invariant (up to conformal factor) under the following $SL(2,\mathbb{Z})$ transformation

\begin{align}
\begin{pmatrix}
\sigma_1\\
\sigma_2
\end{pmatrix}=\begin{pmatrix}
d & b\\
c & a
\end{pmatrix}\begin{pmatrix}
\sigma'_1\\
\sigma'_2
\end{pmatrix}\hspace{2mm},\hspace{2mm}\tau'=\frac{a\tau+b}{c\tau+d}\hspace{2mm},
\end{align}

where $a,b,c,d\in\mathbb{Z}$ and $ad-bc=1$.

Therefore we conclude that the modular parameter $\tau'$ is equivalent to $\tau$ and for a particular choice of constants i.e. $a=0,b=-1,c=1,d=0$, we get $\tau'=-\frac{1}{\tau}$.

Using this property in (<ref>), we obtain

\begin{align}\label{cd9}
Z(\tau)=e^{2\pi i\frac{1}{\tau}\frac{c}{24}}\overline{Z}(-\frac{1}{\tau}),\hspace{2mm}\text{where}\hspace{2mm}\overline{Z}(-\frac{1}{\tau})\equiv \text{Tr}e^{-2\pi i \frac{1}{\tau}L_0}.
\end{align}

In the limit $\tau\rightarrow 0$, the dominant contribution in $\overline{Z}(-\frac{1}{\tau})$ comes from the lowest eigen value of the operator $L_0$ which is set to be zero without any loss of generality. Therefore, (<ref>) reduces to $ Z(\tau)=e^{2\pi i\frac{1}{\tau}\frac{c}{24}}$.

Using this expression in (<ref>), we finally obtain

\begin{align}\label{cd10}
\rho(\Delta)=\oint_c d\tau e^{-2\pi i \big(\tau\big(\Delta-\frac{c}{24}\big)-\frac{1}{\tau}\frac{c}{24}\big)}.
\end{align}

One can approximate the above integral (<ref>) using the saddle point approximation as

\begin{align}\label{cd11}
\rho(\Delta)\approx e^{-2\pi i f(\tau*)},\hspace{2mm}\text{where}\hspace{2mm}f(\tau)=\tau\big(\Delta-\frac{c}{24}\big)-\frac{1}{\tau}\frac{c}{24}
\end{align}

and $\tau*$ is calculated using the following condition

\begin{align}\label{cd12}
\frac{d f(\tau)}{d \tau}\Big|_{\tau=\tau*}=0,\hspace{2mm}\text{which yields}\hspace{2mm}\tau*=i\sqrt{\frac{c}{24\big(\Delta-\frac{c}{24}\big)}}.
\end{align}

On plugging the above value of $\tau*$ (<ref>) in (<ref>), we obtain the density of states $\rho(\Delta)$ in the limit $\Delta>>1$.

The Cardy formula for the black hole is defined in terms of density of states $\rho(\Delta)$ [227] as

\begin{align}\label{CDF}
\end{align}

Using (<ref>) and (<ref>) in (<ref>), we obtain the Cardy formula for a 2D black hole in the limit $\Delta>>1$ as

\begin{align}\label{CDF1}
\end{align}

Equation (<ref>) is the entropy of a two-dimensional CFT.

It has been found in [198] that the entropy computed using the Cardy formula (<ref>) matches with the black hole entropy in the bulk. In particular, the author of [198] considered the three-dimensional theory of gravity coupled with matter fields

\begin{align}\label{ath1}
S=\frac{1}{16\pi G}\int d^3x\sqrt{-g}\Big(R+\frac{2}{l^2}\Big)+S_m,
\end{align}

where $S_m$ contains the matter field.

Next, they computed the central charge for the boundary theory corresponding to (<ref>) and estimated the boundary degrees of freedom using the Cardy formula (<ref>). Remarkably, these boundary degrees of freedom precisely matched with the Bekenstein-Hawking entropy of the black holes[In literature, these black holes are called the BTZ black holes [228]] corresponding to (<ref>). Finally, they claimed that this result held for any consistent theory of quantum gravity.

A similar analysis has been performed by authors in [48]. In particular, they considered the 2D Einstein-dilaton gravity in the presence of $U(1)$ gauge fields and computed the central charge associated with the corresponding boundary theory. They established the 2D/3D dictionary and determined the boundary degrees of freedom using the Cardy formula, showing that it precisely matched with the corresponding Bekenstein-Hawking entropy formula for 2D black holes.

Following a similar spirit, one can also construct a suitable 2D/3D dictionary for 2D black holes in the presence of quartic interactions and compute their entropy using the Cardy formula (<ref>), as discussed by the authors in [48]. Alternatively, one can also determine the eigenvalue of the dilatation operator $L_0$ by comparing the Wald entropy (<ref>) with the Cardy formula, provided that there exists a suitable 2D/3D dictionary.

Below, we compute the $\Delta$ for the quartic interactions by assuming that the boundary degrees of freedom for the ground state are equivalent to the black hole entropy, or that there exists a suitable 2D/3D dictionary that establishes a connection between them.

On comparing the Wald entropy of a 2D black hole (<ref>) with the Cardy formula  (<ref>), we obtain

\begin{align}
\Delta=\frac{3}{2}\frac{S_W^2}{c\pi^2}.
\end{align}

As our analysis reveals, $\Delta$  receives corrections both due to the presence of 2-derivative and 4-derivative interaction terms in (<ref>). We re-scale  $\Delta$ $\rightarrow$ $\tilde{\Delta}=\frac{\Delta}{w}$ ( with $w=3.53\times10^{-5}\mu \kappa^{1.5}$) which finally yields the eigen value for the UV CFT as

\begin{align}\label{s3}
\Delta=\tilde{\Delta}\big|_{UV}=&\frac{c}{24}+\xi F_1+\kappa F_2,
\end{align}

where $F_1$ and $F_2$ are the corrections due to 2-derivative and 4-derivative interactions present in (<ref>)

\begin{align}\label{s4}
F_1=&\frac{ \kappa^{0.75} }{w}\Big\{\Big(0.0044587-0.0029725\log\Big(-\frac{\delta}{\sqrt{\mu}}\Big)+0.0029725\log(\delta\sqrt{\mu})+\nonumber\\
&-0.000990832\log\Big(-\frac{\delta}{\sqrt{\mu}}\Big)-0.000495416\log(\mu)\Big)\frac{3\log(\mu)}{4}Q^2+0.01189\mu d_4\Big\},\nonumber\\
\end{align}

$\bullet$ Note : In arriving at (<ref>), we write the full solution of the gauge field (<ref>) and the dilaton (<ref>)

\phi^{bh}=\phi_{(0)}^{bh}+\xi\phi_{(1)}^{bh}+\kappa\phi_{(2)}^{bh}.

In the near horizon limit, we absorb the integration constant $d_3$ in $\phi_{(1)}^{bh}$ (<ref>) into the constant $d_{11}$ in $\phi_{(2)}^{bh}$ (<ref>) without any loss of generality. Similarly, we absorb the additive constant $d_1$ in $A_{t(0)}^{bh}$ (<ref>) into the constant $d_6$ in $A_{t(1)}^{bh}$ (<ref>). Furthermore, we write the constant $d_2$ in terms of the charge $Q$ using (<ref>).

Finally, we have used the fact that $\delta<<\sqrt{\mu}$ and retain terms up to leading order in the couplings $\xi$ and $\kappa$.

CHAPTER: PROPERTIES OF THE POTENTIAL VPSI

In this Appendix, we discuss the stability of the potential function $V(\psi)$ for the generalised Liouville theory as constructed in (<ref>). Finally, we generalise this theory (<ref>) by considering 4-derivative interactions.

In order to study the stability of the potential, we first note down the extrema of $V(\psi)$ (<ref>) by setting

\begin{align}\label{fa3}
\frac{dV(\psi)}{d\psi}\Bigg|_{\psi=\psi_{i}}=0,\hspace{2mm}(i=1,2,3)
\end{align}

which reveals

\begin{align}
\psi_1&=-\frac{1}{2}\sqrt{\frac{c_H}{3\pi}}\label{fa4},\\
\psi_2&= \frac{1}{2}\sqrt{\frac{c_H}{3\pi}}\text{Productlog}\Bigg[-\pi\sqrt{\frac{3b^2\xi}{2c_H^2}}\Bigg]\label{fa5},\\
\psi_3&= \frac{1}{2}\sqrt{\frac{c_H}{3\pi}}\text{Productlog}\Bigg[\pi\sqrt{\frac{3b^2\xi}{2c_H^2}}\Bigg]\label{fa6},
\end{align}

where we express $\Phi_H$ in terms of the central charge $c_H$ (<ref>) associated with the theory (<ref>).

Using (<ref>)-(<ref>), we find that the potential $V(\psi)$ exhibits local minima at $\psi_1$ and $\psi_3$ (if $2c_H^2>3\xi\pi^2 b^2 e$) which we identify as the possible vacuua of the theory (<ref>). However, notice that in the limit of large central charge $\psi_1\rightarrow-\infty$ which therefore corresponds to the most stable vacuua of the theory (<ref>).

Finally, we write down the most general action  (<ref>) by considering 4-derivative interactions. Under the following field redefinition

\begin{align}
\phi=q\Phi_H\psi\hspace{2mm},\hspace{2mm}g_{\mu\nu}\rightarrow e^{2\sigma}g_{\mu\nu}\hspace{2mm},\text{where}\hspace{2mm}\sigma=\frac{\psi}{q\Phi_H},
\end{align}

4-derivative interactions in (<ref>) transform as
\begin{align}
S^{(4)}=&q\Phi_H\kappa\int d^2x\sqrt{-g}\psi\Bigg[e^{-2\sigma}\Big\{\big[R_{\mu\nu\alpha\beta}-\bigtriangledown^2\sigma(g_{\mu\alpha}g_{\nu\beta}-g_{\mu\beta}g_{\nu\alpha})\big]\big[R^{\mu\nu\alpha\beta}-\bigtriangledown^2\sigma(g^{\mu\alpha}g^{\nu\beta}\nonumber\\
&g_{\rho\bar{\sigma}}g^{\beta\lambda}\bigtriangledown_{\lambda}e^{2\sigma}))g^{\alpha\rho}g^{\mu\bar{\sigma}}+\frac{2}{3}    (\Gamma^{\lambda}_{\alpha\mu}+\frac{e^{-2\sigma}}{2}(\delta^{\lambda}_{\alpha}\bigtriangledown_{\mu}e^{2\sigma}+\delta^{\lambda}_{\mu}\bigtriangledown_{\alpha}e^{2\sigma}-g_{\alpha\mu}g^{\lambda\bar{\sigma}}\bigtriangledown_{\bar{\sigma}}e^{2\sigma}))\times\nonumber\\
\end{align}

Notice that, in arriving at (<ref>), we express the Riemann tensor in 2D as
where $R$ is the Ricci scalar in 2D. Equation (<ref>) represents corrections to the Liouville theory (<ref>) due to the presence of 4-derivative interactions in (<ref>).

CHAPTER: EQUATIONS OF MOTION

In this Appendix, we note down the most general form of the equations of motion (<ref>)-(<ref>) in the Fefferman-Graham gauge (<ref>),

\begin{align}
A_{t}\hspace{1mm}:\hspace{1mm}&\frac{\kappa}{\sqrt{-h_{tt}}}\partial_{\eta}\Bigg[\frac{\Phi A_t'}{\sqrt{-h_{tt}}}\Bigg(\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^2+p^2}}\Bigg)\Bigg]=0,\label{gaugeeomiap}\\
\chi\hspace{1mm}:\hspace{1mm}&\frac{\kappa}{\sqrt{-h_{tt}}}\partial_t\Bigg(-\frac{\dot{\chi}}{\sqrt{-h_{tt}}}\cosh{\gamma}+\frac{s\dot{\chi}+p\xi'\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\nonumber\\
\xi\hspace{1mm}:\hspace{1mm}&\frac{\kappa}{\sqrt{-h_{tt}}}\partial_t\Bigg(-\frac{\dot{\xi}}{\sqrt{-h_{tt}}}\cosh{\gamma}-\frac{-s\dot{\xi}+p\chi'\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\nonumber\\
\end{align}

\begin{align}
\Phi\hspace{1mm}:\hspace{1mm}&\sqrt{-h_{tt}}(\sqrt{-h_{tt}})''+(\sqrt{-h_{tt}})^2\Lambda-\kappa A_t'^2\cosh{\gamma}+\frac{\kappa s A_t'^2}{\sqrt{s^2+p^2}}\sinh{\gamma}=0,\\
g_{\eta t}\hspace{1mm}:\hspace{1mm}&-(\dot{\Phi})'+\frac{\dot{\Phi}(\sqrt{-h_{tt}})'}{\sqrt{-h_{tt}}}-2\kappa \Bigg(\cosh{\gamma}-\frac{s\sinh{\gamma}}{\sqrt{s^2+p^2}}\Bigg)(\dot{\xi}\xi'+\dot{\chi}\chi')=0,\label{gaugeeomfap}
\end{align}

along with the functions

\begin{align}
\end{align}

where . and $'$ denote the derivatives with respect to $t$ and $\eta$ respectively.

CHAPTER: DETAILS OF THE FUNCTIONS $\MATHCAL{H}_I$'S

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

\begin{align}
\mathcal{H}_1(\eta)=&\hspace{1mm}\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)+\nonumber\\
&\hspace{1mm}\frac{1}{a_1^2}\Big(4 (\gamma +1) \eta
\lambda -4 \log \left(a_2-a_1 e^{2 \eta  \lambda }\right)\Big)-4 a (\gamma +1) \lambda
+\frac{8 a a_1 \lambda  e^{2 \eta  \lambda }}{a_1 e^{2 \eta  \lambda }-a_2}\Bigg)-\nonumber\\
}{a_1^2}\left(\kappa  \left(c_5 \left(\eta -a a_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(2 a_1 e^{2 \eta  \lambda }\Big(2 \eta ^2 \lambda ^2+
\nonumber\\
&\hspace{1mm}\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],
\end{align}

\begin{align}
\mathcal{H}_2(\eta)=&\hspace{1mm}-\frac{1}{8 a_1^3
a_2 b_1 \lambda ^2}\Bigg(a e^{-3 \eta  \lambda } \Big(c_1 \Big(-2 a_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)\nonumber\\
&\hspace{1mm}-\gamma +4
\eta  \lambda +1\Big)+4 a_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\nonumber\\
+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)+\nonumber\\
&\hspace{1mm}2 \eta ^2 \lambda ^2\Big)-\log \left(a_1\right) (2 \eta  \lambda +1)-2
\log ^2\left(a_1\right)\Big)-2 a_1 a_2 b_1 c_5 \kappa  e^{\eta  \lambda }\Big)\Bigg),
\end{align}

\begin{align}
\mathcal{H}_3(\eta)=&\hspace{1mm}2 e^{2 \eta  \lambda }-\frac{1}{8 a_1 a_2^2 \left(a_1 e^{2 \eta  \lambda }-a_2\right)}a \Bigg[8 a_2 a_1^3 \lambda  e^{2 \eta  \lambda } \Big(4 a_2 \kappa  e^{2 \eta
\lambda } \left(a_4 \gamma +a_6 \kappa +a_3\right)-2 a_2^2+\nonumber\\
&\hspace{1mm}c_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)\nonumber\\
&\hspace{1mm}-2 \eta  \lambda
\Big){}^2+8 a_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)+8 a_2^2 \Big(2 a_3^2 \kappa  \lambda  e^{2 \eta  \lambda }-\nonumber\\
&\hspace{1mm}c_1 \left(c_1 \lambda
\left(-\log \left(a_2-a_1 e^{2 \eta  \lambda }\right)+2 \eta  \lambda +1\right)+2 c_5 \kappa \right)\Big)-32
a_2^3 \lambda  \big(a_4 \gamma +a_6 \kappa +\nonumber\\
&\hspace{1mm} a_3\big)\Big)-a_2 a_1 \kappa ^2 \Big(16 a_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-\nonumber\\
&\hspace{1mm}2 (2 \eta  \lambda +1) \log \left(a_2-a_1 e^{2 \eta  \lambda }\right)+4 \eta  \lambda
(\eta  \lambda +1)\Big)+8 a_2 a_3 c_1^2 \lambda  e^{2 \eta  \lambda } \Big(2 \eta  \lambda -\nonumber\\
&\hspace{1mm}\log \left(a_2-a_1
e^{2 \eta  \lambda }\right)\Big)\Big)-8 a_2^3 a_3 c_1^2 \kappa ^2 \lambda +16 a_2^2 a_1^4 \lambda  e^{4 \eta
\lambda }\Bigg],\\
\mathcal{H}_4(\eta)=&\hspace{1mm}\frac{a}{\lambda},
\end{align}

\begin{align}
\mathcal{H}_5(\eta)=&\hspace{1mm} a \Bigg[\frac{1}{4} \kappa  e^{\eta  \lambda } \Bigg(\frac{2 b_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)+\nonumber\\
&\hspace{1mm}4 a_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
\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 } }{2 a_1^2 \lambda }\Big(2 a_1^2 b_4 \lambda  e^{2 \eta  \lambda }+2 a_2 a_3
\lambda  (2 \eta  \lambda +1)-\nonumber\\
&\hspace{1mm}b_1 c_1 c_5 (2 \eta  \lambda +1)\Big)+b_2 \gamma  \kappa
e^{\eta  \lambda }+b_1 e^{\eta  \lambda }\Bigg],
\end{align}

where we denote the constant $a= \frac{2 \left(a_1^2+1\right)}{3 a_1^2 \lambda }-\frac{4 a_3 \kappa
}{3 a_1^3 \lambda }$.

CHAPTER: ORDER $\KAPPA^2$ SOLUTIONS

In this Appendix, we note down the solution of the equations (<ref>)-(<ref>),

\begin{align}
\omega_3^{(bh)}=&\hspace{1mm}\frac{1}{192 \Lambda ^2 \mu ^{3/2} \text{$\phi_0$}^4}\Bigg(64 g_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)-\nonumber\\
&\hspace{1mm}16 \Lambda
l_1^2 m_1^2 \text{$\phi_0 $} \Big(\sqrt{\mu } \log
\left(\frac{\rho }{\sqrt{\mu }}+1\right)+\log
\left(1-\frac{\rho }{\sqrt{\mu }}\right) \Big(\rho
\left(-\log \left(\frac{\rho }{\sqrt{\mu
&\hspace{1mm}\sqrt{\mu }+\rho  \log (4)\Big)+\rho
\tanh ^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right)-2 \rho
\text{Li}_2\left(\frac{1}{2}-\frac{\rho }{2 \sqrt{\mu
}}\right)+\rho \Big)+16 m_1 \times\nonumber\\
&\hspace{1mm}\Big(3 \mu  \rho  \tanh
^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right) \left(2
\Lambda ^2 m_6 \text{$\phi_0$}^3-3 m_1^3\right)-\Lambda
m_1 n_1^2 \text{$\phi_0$} \Big(\sqrt{\mu } \log
\left(\frac{\rho }{\sqrt{\mu }}+1\right)\nonumber\\
\left(1-\frac{\rho }{\sqrt{\mu }}\right) \left(\rho
\left(-\log \left(\frac{\rho }{\sqrt{\mu
}}+1\right)\right)+\sqrt{\mu }+\rho  \log (4)\right)+\nonumber\\
\tanh ^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right)-2 \rho
\text{Li}_2\left(\frac{1}{2}-\frac{\rho }{2 \sqrt{\mu
}}\right)+\rho \Big)\Big)-96 \Lambda  \sqrt{\mu } q_2
\text{$\phi_0$}^2 \Big(\Lambda  q_1 \text{$\phi_0$}^2\times
\nonumber\\
&\hspace{1mm} \Big(-\sqrt{\mu }+2 \left(\mu -\rho ^2\right)
\rho\tanh
^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right)\Big)-4 \sqrt{\mu } m_1^2 \rho  \tanh
^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right)\Big)\nonumber\\
\Lambda ^2 \sqrt{\mu } q_2^2 \text{$\phi_0$}^4 \Big(
\tanh ^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right)32\left(\left(\mu -\rho ^2\right) \tanh
^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right)+\sqrt{\mu }
\rho \right)\nonumber\\
&\hspace{1mm}+45 \sqrt{\mu } \rho \Big)+96 \Lambda ^2
\sqrt{\mu } \rho  q_1^2 \text{$\phi_0$}^4 \Big(\rho
-\sqrt{\mu } \tanh ^{-1}\left(\frac{\rho }{\sqrt{\mu
}}\right)\Big)+192 \Lambda ^2 \mu  \text{$\phi_0$}^4\times\nonumber\\
&\Big(q_6 \Big(\rho  \tanh ^{-1}\left(\frac{\rho
}{\sqrt{\mu }}\right)-\sqrt{\mu }\Big)+\rho
\Phi_3^{(bh)}=&\hspace{1mm}g_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)-4 g_1 \mu
^{3/2} q_2\right)+\mu  \log \left(\rho -\sqrt{\mu }\right)\times\nonumber\\
&\hspace{1mm}\left(q_1 \left(l_1^2+n_1^2\right)-4 g_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)-4 g_1 \mu ^{3/2}
&\hspace{1mm}8 g_1 \mu ^{3/2} \rho ^2 q_1-8 g_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)+\nonumber\\
&\hspace{1mm}4 l_1 l_5+2 n_1 n_5\big) \log
\left(\mu -\rho ^2\right)+2 q_2 \left(l_1^2+n_1^2\right)
\left(\rho ^2-\mu \right) \tanh ^{-1}\left(\frac{\rho
}{\sqrt{\mu }}\right)^2+\nonumber\\
&\hspace{1mm}2 \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)-4 l_1 l_5-2 n_1 n_5\right) \log
\left(\sqrt{\mu }+\rho \right)+\nonumber\\
&\hspace{1mm}\sqrt{\mu } \left(\rho
-\sqrt{\mu }\right) \left(q_2 \left(l_1^2+n_1^2\right)-4
l_1 l_5-2 n_1 n_5\right) \log \left(\rho -\sqrt{\mu
}\right)-\sqrt{\mu } \rho  \big(q_2
\left(l_1^2+n_1^2\right)\nonumber\\
&\hspace{1mm}-4 l_1 l_5-2 n_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,\label{jhepeopmkappa2i1}
\end{align}

\begin{align}
A_{t3}^{(bh)}=&\hspace{1mm}\frac{1}{8 \Lambda  \mu ^{3/2}
\text{$\phi_0$}^2}\Bigg(2 \sqrt{\mu } \Big(m_1 \rho  \left(-4 g_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)+\nonumber\\
&\hspace{1mm}4 \Lambda  \mu  \text{$\phi
_0$}^2 \left(m_6 \rho +m_5\right)\Big)+2 m_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)+\nonumber\\
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),\\
\xi_3^{(bh)}=&\hspace{1mm}\frac{l_5}{\sqrt{\mu}}\tanh ^{-1}\left(\frac{\rho }{\sqrt{\mu }}\right)+l_6,\label{jhepeopmkappa2i2}\\
\chi_3^{(bh)}=&\hspace{1mm}n_5t+n_6,\label{jhepeopmkappa2f}
\end{align}

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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