# Discrete index transforms with Bessel and modified Bessel functions

Semyon YAKUBOVICH

###### Abstract

Discrete analogs of the index transforms, involving Bessel and the modified
Bessel functions are introduced and investigated. The corresponding inversion
theorems for suitable classes of functions and sequences are established.

Keywords: Bessel functions, Modified Bessel functions, Fourier series, index
transforms

AMS subject classification: 45A05, 44A15, 42A16, 33C10

##  1 Introduction and preliminary results

Our goal in this paper is to investigate the mapping properties and prove
inversion formulas for the following six transformations between suitable
sequences $\\{a_{n}\\}_{n\geq 1}$ and functions $f$ in terms of the series and
integrals, respectively, which are associated with Bessel and modified Bessel
functions $J_{\mu}(z),I_{\mu}(z),K_{\mu}(z)$ (cf. [1], Ch. 10), namely,

$f(x)=e^{-x/2}\sum_{n=1}^{\infty}a_{n}{\rm Re}\left[I_{in}\left({x\over
2}\right)\right],\quad x>0,$ $None$

$a_{n}=\int_{0}^{\infty}{\rm Re}\left[I_{in}\left({x\over
2}\right)\right]f(x)e^{-x/2}dx,\quad n\in\mathbb{N}_{0},$ $None$

$f(x)=\sum_{n=0}^{\infty}{a_{n}\over\cosh(\pi n/2)}\ {\rm
Re}\left[J_{in}\left(2\sqrt{2x}\right)\right]K_{in}\left(2\sqrt{2x}\right),\quad
x>0,$ $None$

$a_{n}={1\over\cosh(\pi n/2)}\int_{0}^{\infty}{\rm
Re}\left[J_{in}\left(2\sqrt{2x}\right)\right]K_{in}\left(2\sqrt{2x}\right)f(x)dx,\quad
n\in\mathbb{N}_{0},$ $None$

$f(x)=\sum_{n=1}^{\infty}{a_{n}\over\sinh(\pi n/2)}\ {\rm
Im}\left[J_{in}\left(2\sqrt{2x}\right)\right]K_{in}\left(2\sqrt{2x}\right),\quad
x>0,$ $None$

$a_{n}={1\over\sinh(\pi n/2)}\int_{0}^{\infty}{\rm
Im}\left[J_{in}\left(2\sqrt{2x}\right)\right]K_{in}\left(2\sqrt{2x}\right)f(x)dx,\quad
n\in\mathbb{N}.$ $None$

Here $i$ is the imaginary unit and ${\rm Re},\ {\rm Im}$ denotes the real and
imaginary parts of a complex-valued function. We call transformations
(1.1)-(1.6) the discrete index transforms (cf. [3]). For instance, continuous
analogs of transformations (1.1), (1.2) were considered by the author in [4].
Bessel function $J_{\nu}(z)\ z,\nu\in\mathbb{C}$ of the first kind is a
solution of the Bessel differential equation

$z^{2}{d^{2}u\over dz^{2}}+z{du\over dz}+(z^{2}-\nu^{2})u=0.$ $None$

This function has the following asymptotic behavior at infinity and near the
origin

$J_{\nu}(z)=\sqrt{2\over\pi z}\cos\left(z-{\pi\over
4}(2\nu+1)\right)[1+O(1/z)],\ z\to\infty,\ |\arg z|<\pi,$ $None$
$J_{\nu}(z)=O(z^{\nu}),\ z\to 0,$ $None$

The modified Bessel functions of the first and second kind $I_{\nu}(z),\
K_{\nu}(z),\ z,\nu\in\mathbb{C}$, in turn, are solutions of the modified
Bessel differential equation

$z^{2}{d^{2}u\over dz^{2}}+z{du\over dz}-(z^{2}+\nu^{2})u=0,$ $None$

having the corresponding asymptotic behavior

$I_{\nu}(z)=O\left(|z|^{{\rm Re}\nu}\right),\ z\to 0,$ $None$
$I_{\nu}(z)=O\left({e^{z}\over\sqrt{2\pi z}}\right),\ z\to\infty,\ -{\pi\over
2}<\arg z<{3\pi\over 2},$ $None$ $K_{\nu}(z)=O\left(|z|^{-|{\rm
Re}\nu|}\right),\ z\to 0,\ \nu\neq 0,\ K_{0}(z)=O\left(\log(|z|)\right),\ z\to
0,$ $None$ $K_{\nu}(z)=O\left(\sqrt{\pi\over 2z}\ e^{-z}\right),\ z\to\infty,\
|\arg z|<{3\pi\over 2}.$ $None$

These functions are related by the equality

$K_{\nu}(z)={\pi\over 2\sin(\pi\nu)}\left[I_{-\nu}(z)-I_{\nu}(z)\right].$
$None$

The modified Bessel function of the first kind satisfies the inequality (cf.
[4], p. 138)

$\left|I_{i\tau}(x)\right|\leq
I_{0}(x)\left({\sinh(\pi\tau)\over\pi\tau}\right)^{1/2},\quad x>0,\
\tau\in\mathbb{R}.$ $None$

Meanwhile, the modified Bessel function of the second kind obeys the estimate
[3], p. 219

$\left|K_{i\tau}(x)\right|\leq A{x^{-1/4}\over\sqrt{\sinh(\pi\tau)}},\quad x,\
\tau>0,$ $None$

where $A>0$ is an absolute constant. The Mellin-Barnes integral representation
for the kernel in (1.1), (1.2) is established in [4]. We have

${\sqrt{\pi}\over\cosh(\pi\tau)}e^{-x/2}{\rm Re}\left[I_{i\tau}\left({x\over
2}\right)\right]={1\over 2\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{\Gamma\left(1/2-s\right)\Gamma\left(s+i\tau\right)\Gamma\left(s-i\tau\right)}{\Gamma(s)\Gamma(1-s)}x^{-s}ds,$
$None$

where $x>0,\tau\in\mathbb{R},\ 0<\gamma<1/2$ and $\Gamma(z)$ is Euler’s gamma
function (cf. [1], Ch. 5). In the meantime, applying the Stieltjes transform
to both sides of (1.18) and changing the order of integration by Fubini’s
theorem via the estimate

$\int_{0}^{\infty}{t^{-\gamma}dt\over
x+t}\int_{\gamma-i\infty}^{\gamma+i\infty}\left|\frac{\Gamma\left(1/2-s\right)\Gamma\left(s+i\tau\right)\Gamma\left(s-i\tau\right)}{\Gamma(s)\Gamma(1-s)}ds\right|<\infty,$

we find, employing Entries 8.4.2.5 and 8.4.23.5 in [2], Vol. III, the
following integral representation of the modified Bessel function of the
second kind (1.15)

$K_{i\tau}\left({x\over 2}\right)=e^{-x/2}\int_{0}^{\infty}{e^{-t/2}\over
x+t}{\rm Re}\left[I_{i\tau}\left({t\over 2}\right)\right]dt.$ $None$

But since (cf. Entry 2.16.6.1 in [2], Vol. II)

$\int_{0}^{\infty}e^{-x\cosh(u)}K_{in}(x)dx={\pi\sin(nu)\over\sinh(u)\sinh(\pi
n)},\quad n\in\mathbb{N},\ u\in\mathbb{R},$ $None$

we derive from (1.19)

${\pi\sin(nu)\over\sinh(u)\sinh(\pi
n)}=\int_{0}^{\infty}e^{-x(1+\cosh(u))}\int_{0}^{\infty}{e^{-t/2}\over
2x+t}{\rm Re}\left[I_{in}\left({t\over 2}\right)\right]dtdx.$ $None$

The integral with respect to $x$ is calculated in [2], Vol. I, Entry 2.3.4.3
in terms of the upper incomplete gamma function (cf. [1], Ch. 8), and we have

$\int_{0}^{\infty}{e^{-x(1+\cosh(u))}\over 2x+t}dx={1\over 2}\
e^{t\cosh^{2}(u/2)}\Gamma\left(0,t\cosh^{2}\left({u\over 2}\right)\right).$
$None$

Thus we obtain from (1.22) the value of the integral

$\int_{0}^{\infty}e^{t(\cosh^{2}(u/2)-1/2)}\Gamma\left(0,t\cosh^{2}\left({u\over
2}\right)\right){\rm Re}\left[I_{in}\left({t\over
2}\right)\right]dt={2\pi\sin(nu)\over\sinh(u)\sinh(\pi n)}.$ $None$

We note that the interchange of the order of integration in (1.22) is
guaranteed by Fubini’s theorem, owing to the estimate

$\int_{0}^{\infty}e^{-x(1+\cosh(u))}\int_{0}^{\infty}{e^{-t/2}\over
2x+t}\left|I_{in}\left({t\over 2}\right)\right|dtdx$

$\leq{1\over
2\sqrt{2}}\int_{0}^{\infty}e^{-x(1+\cosh(u))}{dx\over\sqrt{x}}\int_{0}^{1}{e^{-t/2}\over\sqrt{t}}\left|I_{in}\left({t\over
2}\right)\right|dt$

$+\int_{0}^{\infty}e^{-x(1+\cosh(u))}dx\int_{1}^{\infty}{e^{-t/2}\over
t}\left|I_{in}\left({t\over 2}\right)\right|dt<\infty,$ $None$

and asymptotic formulas (1.11), (1.12) for the modified Bessel function of the
first kind. The Mellin-Barnes integrals for kernels in (1.3)-(1.6) are given
in [2], Vol. III, Entry 8.4.23.11

${1\over\cosh(\pi\tau/2)}\ {\rm
Re}\left[J_{i\tau}\left(2\sqrt{2x}\right)\right]K_{i\tau}\left(2\sqrt{2x}\right)$

$={1\over
16\pi\sqrt{\pi}i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{\Gamma\left((1+s)/2\right)\Gamma\left((s+i\tau)/2\right)\Gamma\left((s-i\tau)/2\right)}{\Gamma(1-s/2)}x^{-s}ds,$
$None$ ${1\over\sinh(\pi\tau/2)}\ {\rm
Im}\left[J_{i\tau}\left(2\sqrt{2x}\right)\right]K_{i\tau}\left(2\sqrt{2x}\right)$

$=-{1\over
16\pi\sqrt{\pi}i}\int_{\gamma-i\infty}^{\gamma+i\infty}\frac{\Gamma\left(s/2\right)\Gamma\left((s+i\tau)/2\right)\Gamma\left((s-i\tau)/2\right)}{\Gamma((1-s)/2)}x^{-s}ds,$
$None$

where $x,\gamma>0,\ \tau\in\mathbb{R}$. Hence, we employ Entries 8.4.5.1 and
8.4.23.1 in [2], Vol. III to find from (1.25)

${1\over\cosh(\pi\tau/2)}\ \int_{0}^{\infty}\sin(xt){\rm
Re}\left[J_{i\tau}\left(2\sqrt{2t}\right)\right]K_{i\tau}\left(2\sqrt{2t}\right)dt$

$={1\over 32\pi
i}\int_{\gamma-i\infty}^{\gamma+i\infty}\Gamma\left({s+i\tau\over
2}\right)\Gamma\left({s-i\tau\over 2}\right)x^{-s}ds={1\over 4}K_{i\tau}(x).$
$None$

The interchange of the order of integration can be motivated via Fubini’s
theorem and Stirling’s asymptotic formula for the gamma function [1], Ch. 5,
splitting the integral by $t$ over $(0,1)$ and $(1,\infty)$ and choosing
$\gamma\in(0,1)$ and $\gamma>1$, correspondingly. Then with (1.20), Entry
2.5.41.1 in [2], Vol. I and asymptotic formulas (1.8), (1.9), (1.13), (1.14)
we derive the value of the integral $(n\in\mathbb{N},\ u\in\mathbb{R})$

$\int_{0}^{\infty}{t\over t^{2}+\cosh^{2}(u)}{\rm
Re}\left[J_{in}\left(2\sqrt{2t}\right)\right]K_{in}\left(2\sqrt{2t}\right)dt={\pi\sin(nu)\over
8\sinh(u)\sinh(\pi n/2)}.$ $None$

In the same manner we establish the value of the integral

$\int_{0}^{\infty}{1\over t^{2}+\cosh^{2}(u)}{\rm
Im}\left[J_{in}\left(2\sqrt{2t}\right)\right]K_{in}\left(2\sqrt{2t}\right)dt={\pi\sin(nu)\over
4\sinh(2u)\cosh(\pi n/2)}.$ $None$

In the sequel we will provide existence conditions for discrete
transformations (1.1)-(1.6) and establish their inversion formulas for
suitable sequences and functions. To do this, we will employ, in particular,
classical Fourier series for Lipschitz functions.

##  2 Inversion theorems

We begin with

Theorem 1. Let a sequence $\\{a_{n}\\}_{n\in\mathbb{N}}$ satisfy the condition

$\sum_{n=1}^{\infty}|a_{n}|{e^{\pi n/2}\over\sqrt{n}}<\infty.$ $None$

Then the discrete transformation $(1.1)$ can be inverted by the formula

$a_{n}={1\over\pi^{2}}\sinh(\pi n)\int_{0}^{\infty}\Phi_{n}(x)f(x)dx,\
n\in\mathbb{N}_{0},$ $None$

where the kernel $\Phi_{n}(x)$ is defined by

$\Phi_{n}(x)=e^{-x/2}\int_{0}^{\pi}e^{x\cosh^{2}(u/2)}\Gamma\left(0,x\cosh^{2}\left({u\over
2}\right)\right)\sinh(u)\sin(nu)du,\ x>0,\ n\in\mathbb{N}_{0},$ $None$

and integral $(2.2)$ converges absolutely.

###### Proof.

Indeed, substituting (1.1) and (2.3) on the right-hand side of (2.2), we
appeal to the estimate, using inequality (1.16), integral (1.22) and condition
(2.1), to deduce

$\int_{0}^{\infty}\left|\Phi_{n}(x)f(x)\right|dx\leq\int_{0}^{\infty}\int_{0}^{\pi}e^{x(\cosh^{2}(u/2)-1)}\Gamma\left(0,x\cosh^{2}\left({u\over
2}\right)\right)\sinh(u)$

$\times\sum_{m=1}^{\infty}|a_{m}|\left|{\rm Re}\left[I_{im}\left({x\over
2}\right)\right]\right|dxdu\leq
2\int_{0}^{\pi}\sinh(u)\int_{0}^{\infty}e^{-x/2}I_{0}\left({x\over
2}\right)\int_{0}^{\infty}{e^{-y(1+\cosh(u))}\over 2y+x}dydxdu$

$\times\sum_{m=1}^{\infty}|a_{m}|{e^{\pi
m/2}\over\sqrt{m}}\leq\sum_{m=1}^{\infty}|a_{m}|{e^{\pi
m/2}\over\sqrt{m}}\bigg{[}{1\over\sqrt{2}}\int_{0}^{\pi}\sinh(u)\int_{0}^{1}e^{-x/2}I_{0}\left({x\over
2}\right){dx\over\sqrt{x}}\bigg{.}$

$\bigg{.}\times\int_{0}^{\infty}e^{-y(1+\cosh(u))}{dydu\over\sqrt{y}}+2\int_{0}^{\pi}{\sinh(u)\over
1+\cosh(u)}du\int_{1}^{\infty}e^{-x/2}I_{0}\left({x\over 2}\right){dx\over
x}\bigg{]}<\infty.$

Consequently, changing the order of integration and summation, we recall
(1.23) to obtain

${1\over\pi^{2}}\sinh(\pi
n)\int_{0}^{\infty}\Phi_{n}(x)f(x)dx={2\over\pi}\sinh(\pi
n)\sum_{m=1}^{\infty}{a_{m}\over\sinh(\pi
m)}\int_{0}^{\pi}\sin(nu)\sin(mu)du=a_{n}.$

Theorem 1 is proved.

∎

The discrete transformation (1.2) can be inverted by the following theorem.

Theorem 2. Let $f$ be a complex-valued function on $\mathbb{R}_{+}$ which is
represented by the integral

$f(x)=\int_{-\pi}^{\pi}e^{x\left(\cosh^{2}(u/2)-1/2\right)}\Gamma\left(0,x\cosh^{2}\left({u\over
2}\right)\right)\varphi(u)du,\quad x>0,$ $None$

where $\varphi(u)=\psi(u)\sinh(u)$ and $\psi$ is a $2\pi$-periodic function,
satisfying the Lipschitz condition on $[-\pi,\pi]$, i.e.

$\left|\psi(u)-\psi(v)\right|\leq C|u-v|,\quad\forall\ u,v\in[-\pi,\pi],$
$None$

where $C>0$ is an absolute constant. Then for all $x>0$ the following
inversion formula for transformation $(1.2)$ holds

$f(x)={1\over\pi^{2}}\sum_{n=1}^{\infty}\sinh(\pi n)\Phi_{n}(x)a_{n},$ $None$

where $\Phi_{n}$ is defined by $(2.3)$.

###### Proof.

Plugging the right-hand side of the representation (2.4) in (1.2), we change
the order of integration, employ (1.23) and the definition of $\varphi$ to
obtain

$a_{n}={2\pi\over\sinh(\pi n)}\int_{-\pi}^{\pi}\psi(u)\sin(nu)du.$ $None$

This interchange is permitted due to estimates (1.24). Then we substitute
$a_{n}$ by (2.7) and $\Phi_{n}$ by (2.3) into the partial sum of the series
(2.6) $S_{N}(x)$, and it becomes

$S_{N}(x)={1\over\pi}\sum_{n=1}^{N}\int_{-\pi}^{\pi}e^{x\left(\cosh^{2}(t/2)-1/2\right)}\Gamma\left(0,x\cosh^{2}\left({t\over
2}\right)\right)\sinh(t)\sin(nt)dt$

$\times\int_{-\pi}^{\pi}\psi(u)\sin(nu)du.$ $None$

Hence we calculate the sum in (2.8), using the known identity

$\sum_{n=0}^{N}\sin(nt)\sin(nu)={1\over
4}\left[{\sin\left((2N+1)(u-t)/2\right)\over\sin((u-t)/2)}-{\sin\left((2N+1)(u+t)/2\right)\over\sin((u+t)/2)}\right],$
$None$

and equality (2.8) becomes

$S_{N}(x)={1\over
4\pi}\int_{-\pi}^{\pi}e^{x\left(\cosh^{2}(t/2)-1/2\right)}\Gamma\left(0,x\cosh^{2}\left({t\over
2}\right)\right)\sinh(t)$

$\times\int_{-\pi}^{\pi}\left[\psi(u)+\psi(-u)\right]{\sin\left((2N+1)(u-t)/2\right)\over\sin((u-t)/2)}dudt.$
$None$

Since $\psi$ is $2\pi$-periodic, we treat the latter integral with respect to
$u$ as follows

$\int_{-\pi}^{\pi}\left[\psi(u)+\psi(-u)\right]\
{\sin\left((2N+1)(u-t)/2\right)\over\sin((u-t)/2)}du$

$=\int_{t-\pi}^{t+\pi}\left[\psi(u)+\psi(-u)\right]\
{\sin\left((2N+1)(u-t)/2\right)\over\sin((u-t)/2)}du$

$=\int_{-\pi}^{\pi}\left[\psi(u+t)+\psi(-u-t)\right]\
{\sin\left((2N+1)u/2\right)\over\sin(u/2)}du.$

Moreover,

${1\over 2\pi}\int_{-\pi}^{\pi}\left[\psi(u+t)+\psi(-u-t)\right]\
{\sin\left((2N+1)u/2\right)\over\sin(u/2)}du-\psi(t)-\psi(-t)$

$={1\over
2\pi}\int_{-\pi}^{\pi}\left[\psi(u+t)+\psi(-u-t)-\psi(t)-\psi(-t)\right]\
{\sin\left((2N+1)u/2\right)\over\sin(u/2)}du.$

When $u+t>\pi$ or $u+t<-\pi$ then we interpret the value $\psi(u+t)$ by
formulas

$\psi(u+t)-\psi(t)=\psi(u+t-2\pi)-\psi(t-2\pi),$

$\psi(u+t)-\psi(t)=\psi(u+t+2\pi)-\psi(t+2\pi),$

respectively. Then due to the Lipschitz condition (2.5) we have the uniform
estimate for any $t\in[-\pi,\pi]$

${\left|\psi(u+t)+\psi(-u-t)-\psi(t)-\psi(-t)\right|\over|\sin(u/2)|}\leq
2C\left|{u\over\sin(u/2)}\right|.$

Therefore, owing to the Riemann-Lebesgue lemma

$\lim_{N\to\infty}{1\over
2\pi}\int_{-\pi}^{\pi}\left[\psi(u+t)+\psi(-u-t)-\psi(t)-\psi(-t)\right]\
{\sin\left((2N+1)u/2\right)\over\sin(u/2)}du=0$ $None$

for all $t\in[-\pi,\pi].$ Moreover, since (see (1.22))

$\int_{-\pi}^{\pi}e^{x\left(\cosh^{2}(t/2)-1/2\right)}\Gamma\left(0,x\cosh^{2}\left({t\over
2}\right)\right)\left|\sinh(t)\right|$

$\times\int_{-\pi}^{\pi}\left|\left[\psi(u+t)+\psi(-u-t)-\psi(t)-\psi(-t)\right]\
{\sin\left((2N+1)u/2\right)\over\sin(u/2)}\right|dudt$

$\leq{Be^{-x/2}\over\sqrt{x}}\int_{-\pi}^{\pi}{\left|\sinh(t)\right|dt\over(1+\cosh(t))^{1/2}}\int_{-\pi}^{\pi}\left|{u\over\sin(u/2)}\right|du<\infty,$

where $B>0$ is a constant. Therefore via the dominated convergence theorem it
is possible to pass to the limit when $N\to\infty$ under the integral sign,
and we derive from (2.11)

$\lim_{N\to\infty}\int_{-\pi}^{\pi}e^{x\left(\cosh^{2}(t/2)-1/2\right)}\Gamma\left(0,x\cosh^{2}\left({t\over
2}\right)\right)\sinh(t)$

$\int_{-\pi}^{\pi}\left[\psi(u+t)+\psi(-u-t)-\psi(t)-\psi(-t)\right]{\sin\left((2N+1)u/2\right)\over\sin(u/2)}dudt=0.$

Hence, combining with (2.10), we obtain by virtue of the definition of
$\varphi$ and $f$

$\lim_{N\to\infty}S_{N}(x)={1\over
2}\int_{-\pi}^{\pi}e^{x\left(\cosh^{2}(t/2)-1/2\right)}\Gamma\left(0,x\cosh^{2}\left({t\over
2}\right)\right)\left[\varphi(t)+\varphi(-t)\right]dt=f(x),$

where the integral (2.4) converges since $\varphi\in C[0,\pi]$. Thus we
established (2.6), completing the proof of Theorem 2.

∎

For the discrete Re-transform (1.3) with the product of Bessel functions we
have the following result.

Theorem 3. Let a sequence $a=\\{a_{n}\\}_{n\in\mathbb{N}}\in l_{1},$ i.e.
satisfy the condition

$||a||_{1}=\sum_{n=1}^{\infty}|a_{n}|<\infty.$ $None$

Then the discrete transformation $(1.3)$ has the inversion formula

$a_{n}={8\over\pi^{2}}\sinh(\pi n)\int_{0}^{\infty}\Psi_{n}(x)f(x)dx,\
n\in\mathbb{N}_{0},$ $None$

where the kernel $\Psi_{n}(x)$ is defined by

$\Psi_{n}(x)=x\int_{0}^{\pi}{\sinh(u)\sin(nu)\over x^{2}+\cosh^{2}(u)}du,\
x>0,\ n\in\mathbb{N}_{0},$ $None$

and integral $(2.13)$ converges absolutely.

###### Proof.

In fact, doing in the same manner as in the proof of Theorem 1, we substitute
(1.3) and (2.14) on the right-hand side of (2.13) and change the order of
integration and summation. Then we obtain, using integral (1.28), the
equalities

${8\over\pi^{2}}\sinh(\pi
n)\int_{0}^{\infty}\Psi_{n}(x)f(x)dx={2\over\pi}\sinh(\pi
n)\sum_{m=1}^{\infty}{a_{m}\over\sinh(\pi
m)}\int_{0}^{\pi}\sin(nu)\sin(mu)du=a_{n},$

proving (2.13). To justify the interchange of the order of integration and
summation we recall Fubini’s theorem by virtue of the estimate

$\int_{0}^{\infty}\left|\Psi_{n}(x)f(x)\right|dx\leq\int_{0}^{\infty}xI_{0}\left(2\sqrt{2x}\right)K_{0}\left(2\sqrt{2x}\right)$

$\times\int_{0}^{\pi}{\sinh(u)\over
x^{2}+\cosh^{2}(u)}dudx\bigg{[}|a_{0}|+\sum_{m=1}^{\infty}{|a_{m}|\sqrt{\sinh(\pi
m)}\over\sqrt{\pi m}\cosh(\pi m/2)}\bigg{]}$

$\leq||a||_{1}\int_{0}^{\infty}xI_{0}\left(2\sqrt{2x}\right)K_{0}\left(2\sqrt{2x}\right)\int_{0}^{\pi}{\sinh(u)\over
x^{2}+\cosh^{2}(u)}dudx$

$\leq||a||_{1}\left(\cosh(\pi)-1\right)\bigg{[}{1\over\cosh(\pi)}\int_{0}^{1}xI_{0}\left(2\sqrt{2x}\right)K_{0}\left(2\sqrt{2x}\right)dx\bigg{.}$

$\bigg{.}+\int_{1}^{\infty}I_{0}\left(2\sqrt{2x}\right)K_{0}\left(2\sqrt{2x}\right){dx\over
x}\bigg{]}<\infty,$ $None$

which is based, in turn, on definitions of Bessel functions, inequalities
(1.16), $|K_{i\tau}(x)|\leq K_{0}(x),\ x>0,\tau\in\mathbb{R}$ and asymptotic
formulas (1.11)-(1.14). Theorem 3 is proved.

∎

For the companion (1.4) the inversion formula in terms of the series is given
by the following theorem.

Theorem 4. For a class of functions $f$ being represented by the integral

$f(x)=x\int_{-\pi}^{\pi}{\varphi(u)\over x^{2}+\cosh^{2}(u)}du,\quad x>0,$
$None$

where $\varphi(u)=\psi(u)\sinh(u)$ and $\psi$ is a $2\pi$-periodic function,
satisfying the Lipschitz condition $(2.5)$ on $[-\pi,\pi]$, the following
inversion formula holds

$f(x)={8\over\pi^{2}}\sum_{n=1}^{\infty}\sinh(\pi n)\Psi_{n}(x)a_{n},$ $None$

where $a_{n}$ is given reciprocally by $(1.4)$ and $\Psi_{n}$ is defined by
$(2.14)$.

###### Proof.

Doing in the same manner, we substitute the right-hand side of (2.16) in
(1.4), we change the order of integration, employ (1.28) and the definition of
$\varphi$ to obtain

$a_{n}={2\pi\over\sinh(\pi n)}\int_{-\pi}^{\pi}\psi(u)\sin(nu)du.$ $None$

This interchange is allowed by Fubini’s theorem due to the absolute
convergence of the iterated integral (cf. (2.15)). Then the partial sum of the
series (2.17) has the form

$S_{N}(x)={x\over\pi}\sum_{n=1}^{N}\int_{-\pi}^{\pi}{\sinh(t)\sin(nt)\over
x^{2}+\cosh^{2}(t)}dt\int_{-\pi}^{\pi}\psi(u)\sin(nu)du.$ $None$

Hence, recalling identity (2.9), the latter equality (2.19) turns to be

$S_{N}(x)={x\over 4\pi}\int_{-\pi}^{\pi}{\sinh(t)\over x^{2}+\cosh^{2}(t)}$

$\times\int_{-\pi}^{\pi}\left[\psi(u)+\psi(-u)\right]{\sin\left((2N+1)(u-t)/2\right)\over\sin((u-t)/2)}dudt.$

Now the same scheme as in the proof of Theorem 3 drives us at the equality

$\lim_{N\to\infty}S_{N}(x)={x\over
2}\int_{-\pi}^{\pi}{\varphi(t)+\varphi(-t)\over x^{2}+\cosh^{2}(t)}dt=f(x),$

proving the inversion formula (2.17).

∎

The same scheme can be applied to invert discrete $Im$-transformations (1.5),
(1.6). We will state the corresponding theorems, leaving their proofs via
(1.29) to interested readers.

Theorem 5. Let a sequence $a=\\{a_{n}\\}_{n\in\mathbb{N}}\in l_{1}.$ Then the
discrete transformation $(1.5)$ has the inversion formula

$a_{n}={4\over\pi^{2}}\sinh(\pi n)\int_{0}^{\infty}\Omega_{n}(x)f(x)dx,\
n\in\mathbb{N}_{0},$ $None$

where the kernel $\Omega_{n}(x)$ is defined by

$\Omega_{n}(x)=\int_{0}^{\pi}{\sinh(2u)\sin(nu)\over x^{2}+\cosh^{2}(u)}du,\
x>0,\ n\in\mathbb{N}_{0},$ $None$

and integral $(2.20)$ converges absolutely.

Theorem 6. For a class of functions $f$ such that

$f(x)=\int_{-\pi}^{\pi}{\varphi(u)\over x^{2}+\cosh^{2}(u)}du,\quad x>0$

with $\varphi(u)=\psi(u)\sinh(2u)$ and a $2\pi$-periodic function $\psi$,
satisfying the Lipschitz condition $(2.5)$ on $[-\pi,\pi]$, the following
inversion formula holds

$f(x)={4\over\pi^{2}}\sum_{n=1}^{\infty}\sinh(\pi n)\Omega_{n}(x)a_{n},$

where $a_{n}$ is given by $(1.6)$ and $\Omega_{n}$ is defined by $(2.21)$.

Acknowledgments

The work was partially supported by CMUP, which is financed by national funds
through FCT (Portugal) under the project with reference UIDB/00144/2020.

References

  1. 1.

NIST Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release
1.0.17 of 2017-12-22. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I.
Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller and B. V. Saunders, eds.

  2. 2.

A.P. Prudnikov, Yu.A. Brychkov and O.I. Marichev, Integrals and Series. Vol.
I: Elementary Functions, Vol. II: Special Functions, Gordon and Breach, New
York and London, 1986, Vol. III : More special functions, Gordon and Breach,
New York and London, 1990.

  3. 3.

S. Yakubovich, Index Transforms, World Scientific Publishing Company,
Singapore, New Jersey, London and Hong Kong, 1996.

  4. 4.

S. Yakubovich, New index transforms of the Lebedev-Skalskaya type. Integral
Transforms and Special Functions, 27 (2016), N 2, 137-152.

S.Yakubovich
Department of Mathematics,
Faculty of Sciences,
University of Porto,
Campo Alegre st., 687
4169-007 Porto
Portugal
E-Mail<EMAIL_ADDRESS>


BRX TH-6697

CALT-TH 2022-001

A short pre-history of Quantum Gravity

S. Deser, FMRS
Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA 91125;
Physics Department, Brandeis University, Waltham, MA 02454
<EMAIL_ADDRESS>
###### Abstract

I describe the early, from the nineteen sixties, history of attempts at
quantizing General Relativity.

The period 1915-1920 marks the appearance, and rapid acceptance, of Einstein’s
classical GR, one of the most revolutionary concepts in all of Physics, by one
of the giants in quantum theory as well. Despite the war then raging in
Europe, it was rapidly accepted by the elite of the time, though not as much
by others until the verification of the “three tests” in 1919, of which the
most amazing was Einstein’s calculation of the observed value of Mercury’s
perihelion precession, a non-linear effect. Then the field lay quiescent for a
while because quantum theory and its successor, quantum mechanics, were so
fertile at microscopic scales as to absorb all research for a long time.

It was unsurprising then, that about 15 years elapsed before any attempts at
quantization were made, in the mid-thirties. These rather modest beginnings,
motivated by early quantum field theory (QFT), consisted in quantizing the
free massless spin 2 field, namely the linearized limit of GR in a weak field
expansion about flat space. The two works are by Bronstein [1] (also Trotsky’s
real name) in the USSR and by Fierz and Pauli [2] in Switzerland. Pauli had a
head-start: as a wunderkind aged 18, he wrote the first text on GR! Sadly,
Bronstein who was also a wunderkind, was an early victim of Stalin’s purges.
Their quantization was a straightforward extension of that of electrodynamics,
the other abelian gauge theory. All authors understood that this was far from
a true QG, but at least it was its first glimmer, and indeed a long time —
about 25 years — also passed until the next meaningful steps were taken.
Remember that the idea of quantizing spacetime is conceptually vastly removed
from that of ordinary matter systems, namely fermions and photons. This brings
us to about 1960, when the reformulation of GR as a (rather unusual) field
theory was successfully undertaken by Arnowitt, myself and Misner (ADM) [3]
and similarly, if less completely, by Dirac [4]. Unlike theirs, most other GR
work of the time involved its original classical geometrical form. I should
also mention deWitt’s lifetime devotion to this problem [5]. There was a
separate interlude in the mid-fifties when several people (Klein, Landau,
Pauli and I) independently suggested that QG might be a universal regulator
for the infinities then ravaging matter loop calculations, but nothing came of
it. My foolhardy attempt was the only published one [6], at the first
conference devoted to GR at Chapel Hill, in 1957, where it was rapidly shot
down by Feynman. [There seems also to have been a small meeting at the Bohr
Institute in Summer 1957, at which I was apparently present, but of which I
have no recollections whatever!]

Quite separately from this line is the lesson Heisenberg drew from Fermi’s
weak interaction model which, unlike quantum electrodynamics, had a
dimensional coupling constant. He noted that any theory of this type would be
beset by infinities of rising virulence with each perturbative order. This
rapidly understood, if tacitly, insight applied to GR with Newton’s constant,
and cast a complete pall on QG. What Heisenberg told us was that perturbative
(and there is no other way) QG was guaranteed to lose all predictive power as
soon as one left its classical, tree, level. On the other hand, GR’s
quantization is mandated because Einstein’s equations have matter as the
source of gravity, so consistency requires it (I mercifully do not cite some
feeble attempts to circumvent this by making the matter source be some sort of
expectation value of its quantum nature). Even absent matter, GR must be
quantized to avoid the ultraviolet catastrophe [7]. So the motivation to
continue any QG program was greatly diminished. Still, ADM [8] and soon after,
Schwinger [9] separately completed the formal quantization program. That is
about the end of the prehistory, since there was little incentive to proceed
further on this purely formal program, though one might add the decade later
verifications of Heisenberg’s predictions by explicit calculations of one-loop
plus matter QG corrections, then still later, the two-loop ones of pure QG
[10,11] in the seventies. A minor spinoff is the derivation of Classical GR
using QM methods [12].

In summary, the prehistory of QG was a relatively short one — consisting of
formal quantization but without follow-through calculations of much interest,
then explicit verification of non-renormalizability both of pure and matter-
coupled GQ — the mark of a field in search of motivation. How times have
changed since then: observational cosmology has reignited the field, while
Supergravity and String theory have become the chief theoretical QG arena,
with mixed results so far, although a maximal version of SUGRA has survived
finitely to at least seven loops, and of course superstrings are finite in the
ten dimensions they require. Even less successful have been some variants such
as L(oop)QG, asymptotic safety and higher curvature models. Fortunately, I
cannot discuss these within my early QG framework!

## Acknowledgements

I thank Prof. Alexander Blum for asking the many detailed questions that led
to this essay, and useful correspondence. This work was supported by the
U.S.Department of Energy, Office of Science, Office of High Energy Physics
under award number de-sc0011632.

## References

  * [1]
  * [2] M.P. Bronstein arXiv:1110.5941; Gen. Rel. Grav. 44, 263-265 (2012): Introduction to Bronstein’s ”Quantum theory of weak gravitational fields” S. Deser, A. Starobinsky. A scientific introduction and short biography to accompany the translation of Matvei P. Bronstein, ”Quantum theory of weak gravitational fields”, Phys. Zeitschr. der Sowjetunion 9, 140 157 (1936).
  * [3] M.Fierz and W.Pauli Proc Roy Soc A 173(1939)211.
  * [4] R.A. Arnowitt, S.Deser and C.W.Misner (ADM), “The Dynamics of General Relativity”(1962), reprinted as arXiv 0405109.
  * [5] P.A.M. Dirac, “General Relativity”, Princeton 1996.
  * [6] B.S. de Witt, Phys. Rev.171,1834 (1968) and references therein.
  * [7] S. Deser, Rev Mod Phys 29 (!957)417.
  * [8] S. Deser, unpublished.
  * [9] ADM, Nuov Cim 191(1961)668.
  * [10] J Schwinger, Phys Rev 130,1253; 132,1317(1965)
  * [11] “Nonrenormalizabilty of the Quantized Einstein-Maxwell System” (with P. van Nieuwenhuizen), Phys. Rev. Lett. 32, 245 (1974).; “Nonrenormalizability of the Quantized Dirac-Einstein System” (with P. van Nieuwenhuizen), Phys. Rev. D10, 401 (1974); “One-Loop Divergences of Quantized Einstein-Maxwell Fields” (with P. van Nieuwenhiuzen), Phys. Rev. D10, 411 (1974); “Nonrenormalizability of Einstein-Yang-Mills Interactions at the One-Loop Level” (with H.- S. Tsao and P. van Nieuwenhuizen), Phys. Lett. 50B, 491 (1974).;“Uniqueness and Nonrenormalizability of Quantum Gravitation” (with D. Boulware and P. van Nieuwenhuizen), Proc. of the 7th Internl. Conf. on Gravitation, Tel Aviv (1974); “One- Loop Divergences of the Einstein-Yang-Mills System” (with P. van Nieuwenhuizen and H.- S. Tsao), Phys. Rev. D10, 3337 (1974). “Nonrenormalizability of Quantized Fermion Gravitation Interactions” (with P. van Nieuwenhuizen), Nuovo Cimento Lett. 2, 218 (1974); “Quantum Gravity: Trees, Loops and Renormalization,” in Quantum Gravity, Oxford University Press (1975)’ “Quantum Gravitation: Problems and Prospects,” Proceedings of the 1974 “Texas” Astrophysics Conference, N.Y. Academy of Sciences (1975). “Quantum Gravity,” Proc. XVII Internl. Conf. on High Energy Phys., London (1974); “Scale Dependence and the Renormalization Problem of Quantum Gravity” (with M.T. Grisaru, P. van Nieuwenhuizen, and C.C. Wu), Phys. Lett. 58B, 355 (1975).
  * [12] M.H.Goroff and A.Sagnotti., Nucl Phys B266(1986)709; A.E.M. van de Van ibid 378 (1992)309.
  * [13] D. Boulware and S.Deser, Ann. Phys. 89, 193 (1975).


Holographic entanglement entropy and modular Hamiltonian in warped CFT in the
framework of GMMG model

M. R. Setare 111E-mail<EMAIL_ADDRESS>, M. Koohgard 222E-mail:
<EMAIL_ADDRESS>
Department of Science,
Campus of Bijar, University of Kurdistan, Bijar, Iran

We study some aspects of a class of non-AdS holography where the 3d bulk
gravity is given by Generalized Minimal Massive Gravity (GMMG). We consider
the spacelike warped $AdS_{3}$ ($WAdS_{3}$) black hole solution of this model
where the 2d dual boundary theory is the warped conformal field theory (WFCT).
The charge algebra of the isometries in the bulk and the charge algebra of the
vacuum symmetries at the boundary are compatible and this is an evidence for
the duality conjecture. Further evidence for this duality is the equality of
entanglement entropy and modular Hamiltonian on both sides of the duality. So
we consider the modular Hamiltonian for the single interval at the boundary in
associated to the modular flow generators of the vacuum. We calculate the
gravitational charge in associated to the asymptotic Killing vectors that
preserve the metric boundary conditions. Assuming the first law of the
entanglement entropy to be true, we introduce the matching conditions between
the variables in two side of the duality and we find equality of the modular
Hamiltonian variations and the gravitational charge variations in two sides of
the duality. According to the results of the present paper we can say with
more sure that the dual theory of the warped $AdS3$ black hole solution of
GMMG is a Warped CFT.

Keywords: Non-AdS holography, GMMG, Modular Hamiltonian, Modular flow
generator, Generalized Rindler prescription

##  1 Introduction

One way to generalize the $AdS/CFT$ duality is to use the warped Anti-se
Sitter (WAdS) spacetime in the gravity side of the theory. This is in the
context of the so-called non-AdS holography. $WAdS_{3}$ are deformations of
$AdS_{3}$ [1] by very interesting applications [2, 3, 4]. One of the features
of the warped AdS spacetime is that they exist in the near horizon of extremal
Kerr black holes [2] and they realize black holes geometries [5]. The
$SL(2,R)\times U(1)$ group is the isometry group of these spaces.

The spacelike warped $AdS_{3}$ black hole is a solution of the generalized
minimal massive gravity (GMMG) that has been introduced in the paper [6]. GMMG
is a modification of the minimal massive gravity (MMG) [7] that is an
interesting theory in the context of $AdS/CFT$ correspondence where introduces
a bulk theory with positive-energy propagating modes [6]. GMMG is a ghost
free, pure gravity theory and it can avoid the so called bulk-boundary clash
in a certain region of its parameter space. MMG and GMMG are examples of
”third-way” consistent theories, i.e. their field equations do not come from
an action, but the Bianchi identity is satisfied on-shell.

$WAdS_{3}/CFT_{2}$ duality has been investigated for the first time in [8] .
The authors in [8] investigated the duality based on the entropy calculations
utilizing the Cardy formula (for more details, one can refer to [9, 10] ).
Investigation of the asymptotic behavior and the boundary conditions of the
$WAdS_{3}$ has been done in [11]. The authors in [12], utilizing the
Lewkowycz-Maldacena procedure, find the holographic entanglement entropy in
the non-(asymptotic) AdS spacetimes and they get the result explicitly for the
$WAdS_{3}/CFT_{2}$ case.

The boundary conditions of the $WAdS_{3}/WCFT_{2}$ duality as another case of
the non-AdS holographies in 3D spacetimes has been investigated in [13]. The
bulk geometry in the $WAdS_{3}/WCFT_{2}$ duality can be considered by the
asymptotically warped $AdS_{3}$ spacetime that is dual to the warped conformal
field theories (WCFTs) where the field theoretical futures of the duality has
been investigated in [10, 14]. The WCFTs possess a global $SL(2,R)\times U(1)$
symmetry that leaves invariant the vacuum. They are 2d translation-invariant
QFTs with a chiral scaling symmetry that as a local symmetry have a Virasoro
algebra plus a $U(1)$ Kac-Moody algbera. The $SL(2,R)\times U(1)$ symmetry is
a subset of the global symmetries in CFTs given by $SL(2,R)\times SL(2,R)$.
The local symmetries in CFTs are given by two copies of the Virasoro algebra
[15, 10]. WCFTs possess an infinite dimensional group and the modular
covariance that allow us to compute the entropy using a Cardy-like formula
[16]. The entropy of a CFT using the Cardy formula is as follows [10]

$S_{CFT}=2\pi\sqrt{\frac{c_{R}}{6}L_{0}}+2\pi\sqrt{\frac{c_{L}}{6}\bar{L}_{0}}$
(1.1)

where $L_{0}$ and $\bar{L}_{0}$ are the charges associated to the $SL(2,R)$
zero modes. $c_{R}$ and $c_{R}$ are the central charges of the algebra.

A universal result for the asymptotic entropy of a WCFT is as follows [10]

$S_{WCFT}=-\frac{4\pi
iP_{0}P_{0}^{vac}}{k}+4\pi\sqrt{-(L_{0}^{vac}-\frac{(P_{0}^{vac})^{2}}{k})(L_{0}-\frac{P_{0}^{2}}{k})}$
(1.2)

where $L_{0}$ and $P_{0}$ are the charges associated to $SL(2,R)$ and $U(1)$
symmetries, respectively. $k$ is the $U(1)$\- level parameter. ”vac” labels
the vacuum expectation values of the charges. These formula (1.1) and (1.2)
are analogous to each other but they are not equal.

The entanglement entropy and the modular Hamiltonian have important roles in
establishment of $WAdS/WCFT$ duality and in verifying the first law of
entanglement entropy. WCFTs have a non-relativistic nature. This property of
the theory will be evident in the entanglement entropy of an interval in the
vacuum, where we review the result of [17] in the next section. This is based
on the Rindler method that relies on the suitable coordinate maps [17, 18]. A
Rindler transformation is a warped conformal transformation that is satisfy
some conditions. This transformation is invariant under a thermal
identification. This is a symmetry of the vacuum and the vacuum is mapped to a
state in the Rindler spacetime.

In parallel to the field theory calculations we have done in this paper, we
investigate the holographic entanglement entropy and the gravitational charge.
The holographic entanglement entropy of the $WAdS_{3}$ under the Dirichlet-
Neumann boundary conditions has been computed in [19], where the authors have
used the Rindler method to find the result. In order to be able to reproduce
the results of [17] in the WCFT, the authors of [19] have considered an
additional parameter to the Rindler transformations. They have found the
result for the holographic entanglement entropy in dualities beyond the
AdS/CFT. It can be seen the entanglement entropy in this case is different
with the result of the Ryu-Takayanagi prescription [21, 22] which is based on
a relation between the spacetime geometry and the entanglement entropy of the
field theory side of the duality.

The general holographic picture of the $WAdS_{3}/WCFT_{2}$ duality has been
analyzed in [20] by introducing the concept of the _swing surfaces_ that can
be a modification to the Ryu-Takayanagi surfaces [21, 22] in the beyond
AdS/CFT dualities. The swing surfaces can include two null geodesics that
extend beyond the entangling interval at the boundary and a spacelike geodesic
connecting the null geodesics.

Equality of the gravitational charge related to the isometries of the warped
$AdS_{3}$ spacetime and the modular Hamiltonian of WCFT is an evidence to the
holography conjecture. To compute the gravitational charge, we consider the
metric as a spacelike warped $AdS_{3}$ black hole [23, 24]. The metric
satisfies the asymptotically warped $AdS_{3}$ spacetime fall-off conditions
[24, 26]. Finding some dictionaries between the variables in two sides of the
holography is an important step to establish the first law of entanglement
entropy. This step is done by reading off the similar forms of the Fourier
modes of the charges in two sides of the duality. So entanglement entropy can
be considered as a test for $WAdS_{3}/WCFT$ conjecture, where we will study
this test in the framework of GMMG model. In another term we investigate the
holographic entanglement entropy and its first law in the context of
$WAdS_{3}/WCFT$ correspondence for asymptotically warped $AdS_{3}$ solution of
GMMG model.

This paper organized as follows. In section 2, we review some properties of
the WCFTs by defining the warped conformal transformations, Virasoro-Kac-Moody
charge algebra and the modified form of the algebra. In the preceding
subsection, we compute the modular flow generator $\zeta$ associated to a
single interval in WCFTs on the plane. The modular flow can be defined as a
linear combination of the vacuum symmetry generators. To find the universal
form of the entanglement entropy, we review the calculation in [17]. We have
permission to use its result in our calculations to implement the holography.
In section 3, we show the first law of the entanglement entropy in the field
theory side of the duality. In section 4, we introduce the warped $AdS_{3}$
black hole as a GMMG solution and as an asymptotically warped $AdS_{3}$
spacetime. The asymptotic Killing vectors that preserving the boundary
conditions will be introduced and the gravitational charge associated to this
symmetry is computed. In section 5, we find the first law of the entanglement
entropy by introducing some holographic matching equalities between the
variables on two sides of the $WAdS_{3}/CFT_{2}$ correspondence. In the last
section, we provide a conclusion for the paper and we give a summary of our
work on this class of non-AdS holography.

##  2 Warped CFT

We consider the warped conformal field theories (WCFTs) in two dimensions at
the boundary of the gauge gravity duality. To this end, it is needed to have a
good discussion on the general properties of these theories. To find an
evidence for the holographic principles, we should have the entanglement
entropy and the modular Hamiltonian, that we use the result of some earlier
works. To provide everything we need, we give a review of the related works in
[10, 17, 27] in this section.

###  2.1 Some properties

We consider a two dimensional theory defined on a plane with coordinates
$(z,w)$. The spacetime symmetries are generated by the following
transformations

$z^{\prime}=f(z^{\prime}),~{}~{}~{}w^{\prime}=w+g(z^{\prime}),$ (2.1)

where $f(z)$ and $g(z)$ are two arbitrary functions. The theories which are
invariant under the transformations (2.1) are known as Warped Conformal Field
Theories (WCFTs). The warped conformal transformation (2.1) is generated by
two operators $T(z)$ and $P(z)$. The former is generator of infinitesimal
coordinate transformation in _z_ and the latter is generator of _z_ -dependent
infinitesimal coordinate translations in _w_. $T(z)$ and $P(z)$ transform as
follows

$\displaystyle T^{\prime}(z^{\prime})$ $\displaystyle=$
$\displaystyle(\frac{\partial z}{\partial
z^{\prime}})^{2}\big{[}T(z)-\frac{c}{12}\\{z^{\prime},z\\}\big{]}+\frac{\partial
z}{\partial z^{\prime}}\frac{\partial w}{\partial
z^{\prime}}P(z)-\frac{k}{4}(\frac{\partial w}{\partial z^{\prime}})^{2},$
(2.2) $\displaystyle P^{\prime}(z^{\prime})$ $\displaystyle=$
$\displaystyle(\frac{\partial z}{\partial
z^{\prime}})[P(z)+\frac{k}{2}\frac{\partial w}{\partial z^{\prime}}],$ (2.3)

where $\\{.,.\\}$ is the Schwarzian derivative as follows

$\\{z^{\prime},z\\}=(\frac{\partial z^{\prime}}{\partial
z})^{-1}\frac{\partial^{3}z^{\prime}}{\partial
z^{3}}-\frac{3}{2}(\frac{\partial z^{\prime}}{\partial
z})^{-2}(\frac{\partial^{2}z^{\prime}}{\partial z^{2}})^{2}.$ (2.4)

$c$ is the central charge and $k$ is the level of the Virasoro-Kac-Moody
algebra [14]. We can define the charges on the plane as follows

$\displaystyle L_{n}$ $\displaystyle=$ $\displaystyle-\frac{i}{2\pi}\int
dzz^{n+1}T(z),$ $\displaystyle P_{n}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int dzz^{n}P(z).$ (2.5)

These charges satisfy a Virasoro-Kac-Moody algebra with central charge $c$ and
$U(1)$ level $k$ as follows

$\displaystyle\left[L_{n},L_{m}\right]$ $\displaystyle=$
$\displaystyle(n-m)L_{n+m}+\frac{c}{12}n(n^{2}-1)\delta_{n+m},$
$\displaystyle\left[P_{n},P_{m}\right]$ $\displaystyle=$
$\displaystyle\frac{k}{2}n\delta_{n+m},$
$\displaystyle\left[L_{n},P_{m}\right]$ $\displaystyle=$ $\displaystyle-
mP_{n+m},$ (2.6)

where $\delta_{n}$ is the Kronecker delta. The finite transformations (2.2)
and (2.3) preserve the charge algebra (2.1) unchanged but this is not true for
the vacuum expectation values of the zero-mode charges. These transformations
create changes on the vacuum expectation values that are called the spectral
flow transformations. We are interested to find a mapping from the plane to
the cylinder in coordinate $(z^{\prime},w^{\prime})$. The changes in the
coordinates are as follows [17]

$z=e^{-iz^{\prime}},~{}~{}~{}~{}w=w^{\prime}+2\alpha z^{\prime}.$ (2.7)

The Schwarzian derivative is obtained as follows

$\\{z^{\prime},z\\}=\frac{1}{2z^{2}}.$ (2.8)

Using eqs. (2.2), (2.3), (2.7) and the Schwarzian derivative (2.8), we find
$T(z)$ and $P(z)$ transformations as follows

$\displaystyle P^{cyl}(z^{\prime})$ $\displaystyle=$ $\displaystyle
izP(z)-k\alpha$ (2.9) $\displaystyle T^{cyl}(z^{\prime})$ $\displaystyle=$
$\displaystyle-z^{2}T(z)+\frac{c}{24}+2i\alpha zP(z)-k\alpha^{2}.$ (2.10)

The modes on the cylinder can be defined as follows

$\displaystyle L^{cyl}_{n}$ $\displaystyle=$ $\displaystyle-\frac{1}{2\pi}\int
dz^{\prime}e^{inz^{\prime}}T^{cyl}(z^{\prime}),$ $\displaystyle P^{cyl}_{n}$
$\displaystyle=$ $\displaystyle-\frac{1}{2\pi}\int
dz^{\prime}e^{inz^{\prime}}P^{cyl}(z^{\prime}).$ (2.11)

Substituting eqs. (2.9) and (2.10) in the charges on the cylinder, we find
these charges in terms of the charges on the plane as follows

$\displaystyle P^{cyl}_{n}$ $\displaystyle=$ $\displaystyle P_{n}+\alpha
k\delta_{n,0}$ (2.12) $\displaystyle L^{cyl}_{n}$ $\displaystyle=$
$\displaystyle L_{n}+2\alpha P_{n}+(\alpha^{2}k-\frac{c}{24})\delta_{n,0}.$
(2.13)

We can find the vacuum charges on the cylinder using the above equations in
terms of the central charge $c$ and the $U(1)$ level $k$, as follows

$\displaystyle P_{0}^{vac}\equiv\langle P^{cyl}_{0}\rangle$ $\displaystyle=$
$\displaystyle\alpha k$ (2.14) $\displaystyle L_{0}^{vac}\equiv\langle
L^{cyl}_{0}\rangle$ $\displaystyle=$ $\displaystyle\alpha^{2}k-\frac{c}{24}.$
(2.15)

where $\langle.\rangle$ is the notation for the expectation value on the
vacuum states.

There is a modified algebra as follows [10]

$\displaystyle\left[\tilde{L}_{n},\tilde{L}_{m}\right]$ $\displaystyle=$
$\displaystyle(n-m)\tilde{L}_{n+m}+\frac{c}{12}n(n^{2}-1)\delta_{n+m},$
$\displaystyle\left[\tilde{P}_{n},\tilde{P}_{m}\right]$ $\displaystyle=$
$\displaystyle 2n\tilde{P}_{0}\delta_{n+m},$
$\displaystyle\left[\tilde{L}_{n},\tilde{P}_{m}\right]$ $\displaystyle=$
$\displaystyle-m\tilde{P}_{n+m}+m\tilde{P}_{0}\delta_{n+m},$ (2.16)

in which the $U(1)$ level $k$ is charge-dependent. We use this form of the
algebra to analyze the duality in both sides of the gravity and the field
theory. We could return to the original algebra (2.1) by the following re-
definition of the charges [10]

$\displaystyle\tilde{P}_{n}$ $\displaystyle=$
$\displaystyle\frac{2}{k}P_{0}P_{n}-\frac{1}{k}P_{0}^{2}\delta_{n},$
$\displaystyle\tilde{L}_{n}$ $\displaystyle=$ $\displaystyle
L_{n}-\frac{2}{k}P_{0}P_{n}+\frac{1}{k}P_{0}^{2}\delta_{n}.$ (2.17)

###  2.2 Modular flow in WCFT

In this subsection we find the modular flow generator associated with a single
interval in WCFT from the generalized Rindler transformation [17, 18, 25, 27].
To this end, we use a symmetry transformation of the vacuum state to map an
interval $\mathcal{I}$ to a Rindler spacetime. The modular flow generator is
considered as a linear combination of the $SL(2,R)\times U(1)$ that leaves the
vacuum invariant, as follows

$\zeta=\Sigma a_{i}h_{i}$ (2.18)

where $a_{i}$ are some arbitrary constants. It is needed to satisfy the
invariance of the causal domain under the modular flow transformations to find
the coefficients $a_{i}$. We consider an interval $\mathcal{I}$ on the vacuum
state as follows

$\partial\mathcal{I}=\\{(z_{-},w_{-}),(z_{+},w_{+})\\};~{}~{}~{}l_{z}=z_{+}-z_{-},~{}~{}l_{w}=w_{+}-w_{-}.$
(2.19)

where $(z,w)$ are the coordinates on the plane. This is the reference plane
where the zero-mode charges vanish as

$P_{0}|0\rangle=L_{0}|0\rangle=0$ (2.20)

where $|0\rangle$ is the vacuum state. The symmetry generators $SL(2,R)\times
U(1)$ that leave the causal domain $\mathcal{I}$ invariant are as follows

$\displaystyle l_{n}$ $\displaystyle=$
$\displaystyle-z^{n+1}\partial_{z};~{}~{}~{}~{}n=-1,0,+1$
$\displaystyle\bar{l}_{0}$ $\displaystyle=$ $\displaystyle-\partial_{w},$
(2.21)

where $\bar{l}_{0}$ is $U(1)$ generator and $l_{n}$ are the $SL(2,R)$
generators that satisfy the following algebra

$\displaystyle\left[l_{-},l_{+}\right]$ $\displaystyle=$ $\displaystyle
2l_{0}$ $\displaystyle\left[l_{0},l_{\pm}\right]$ $\displaystyle=$
$\displaystyle\pm l_{\pm}.$ (2.22)

The modular flow generator as a linear combination of the vacuum symmetry
generators can be writen as follows

$\zeta=\zeta^{z}\partial_{z}+\zeta^{w}\partial_{w}=\Sigma_{j=-1}^{1}a_{j}l_{j}+\bar{a}_{0}\bar{l}_{0}.$
(2.23)

To find the coefficients $a_{i}$, it is needed the modular flow $\zeta$
vanishes at the boundaries of the causal domain. Satisfying this condition, we
find the $a_{i}$ as follows

$(a_{+},a_{0},a_{-})=a_{+}\big{(}1,-(z_{+}+z_{-}),z_{+}z_{-}\big{)}.$ (2.24)

To find the $a_{+}$, we have the conditions as follows [27]

$\displaystyle\partial_{s}z(s)$ $\displaystyle=$ $\displaystyle\zeta^{z}$
$\displaystyle z(s)$ $\displaystyle=$ $\displaystyle z(s+i)$ (2.25)

This is because it is needed that $e^{i\zeta}$ maps a point in the causal
domain back to itself. We use the first condition in (2.2) as follows

$\frac{dz}{ds}=-a_{+}(z^{2}-z_{+}z-z_{-}z+z_{+}z_{-})$ (2.26)

where we have used (2.24). By integration of this equation and introducing the
integration constants in $c_{0}$, we find the following result for $z(s)$

$z(s)=\frac{e^{a_{+}z_{+}s+c_{0}z_{-}}z_{+}-e^{a_{+}z_{-}s+c_{0}z}z_{-}}{e^{a_{+}z_{+}s+c_{0}z_{-}}-e^{a_{+}z_{-}s+c_{0}z}}$
(2.27)

Applying the 2nd condition in (2.2), we find the following relation that is
needed to satisfy this condition

$e^{a_{+}s(z_{+}-z_{-})i}=1.$ (2.28)

Using the relation (2.28), we find the form of the $a_{+}$ as follows

$a_{+}=\frac{2\pi}{z_{+}-z_{-}}.$ (2.29)

Substituting the $a_{+}$ in (2.24), we find the final form of the coefficients
as follows

$(a_{+},a_{0},a_{-})=\frac{2\pi}{z_{+}-z_{-}}\big{(}1,-(z_{+}+z_{-}),z_{+}z_{-}\big{)}.$
(2.30)

To find the $\bar{a}_{0}$, we pay attention to this point that this is the
coefficient of $\bar{l}_{0}$ that creates translation along $w$ direction as
follows

$w\to w-2\pi i\mu.$ (2.31)

To map a point in the causal domain back to itself, we shift the $w$
coordinate an amount $2\pi\mu$. Now we find the $\bar{a}_{0}$ as follows [27]

$\bar{a}_{0}=2\pi\mu.$ (2.32)

Substituting all the coefficients that we have found into (2.23), we find the
modular flow generator in the interval $\mathcal{I}$ as follows

$\displaystyle\zeta$ $\displaystyle=$ $\displaystyle
2\pi\mu\bar{l}_{0}+\frac{2\pi}{z_{+}-z_{-}}\big{(}l_{1}-(z_{+}+z_{-})l_{0}+z_{+}z_{-}l_{-1}\big{)}$
(2.33) $\displaystyle=$
$\displaystyle-2\pi\mu\partial_{w}-\frac{2\pi}{z_{+}-z_{-}}\big{(}z_{+}z_{-}-(z_{+}+z_{-})z+z^{2}\big{)}\partial_{z}$

where this is calculated in WCFT using the generalized Rindler method.

###  2.3 Entanglement entropy in WCFT

In this subsection we review the work of [17] that is needed to implement the
hologrpaphy in two sides of the bulk and the boundary. To calculate the
entanglement entropy of a single interval, the background geometry is
considered as a cylinder with coordinates $(t,x)$. The following
identification defines the spatial circle

$(t,x)\sim(t+\bar{L},x-L),$ (2.34)

where the interval domain will be defined as follows

$\mathcal{D}:(t,x)\in\left[(\frac{\bar{l}}{2},-\frac{l}{2}),(-\frac{\bar{l}}{2},\frac{l}{2})\right]$
(2.35)

The entanglement entropy in $\mathcal{D}$ using the density matrix
$\rho_{\mathcal{D}}$ is defined as follows

$S_{EE}=-Tr(\rho_{\mathcal{D}}\log\rho_{\mathcal{D}})$ (2.36)

Using a unitary transformation, the entanglement entropy is related to a
thermal entropy [17, 28, 29]. For a warped system, the only allowed
transformation is as follows

$\frac{\tan\frac{\pi x}{L}}{\tan\frac{\pi
l}{2L}}=\tanh\frac{\pi\tilde{x}}{L},~{}~{}~{}t+\frac{\bar{L}}{L}x=\tilde{t}+\frac{\bar{\kappa}}{\kappa}\tilde{x}$
(2.37)

where $\kappa$, $\bar{\kappa}$ are arbitrary scales and induces the following
identification in $(\tilde{t},\tilde{x})$ coordinates

$\mathcal{H}:(\tilde{t},\tilde{x})\sim(\tilde{t}-i\bar{\kappa},\tilde{x}+i\kappa).$
(2.38)

The thermal density matrix of the domain $\mathcal{H}$ is related to the
density matrix of the interval $\mathcal{D}$ under the unitary transformation
as follows

$\rho_{\mathcal{D}}=U\rho_{\mathcal{H}}U^{\dagger}$ (2.39)

where $\rho_{\mathcal{H}}=\exp(\bar{\kappa}P_{0}^{cyl}-\kappa L_{0}^{cyl})$.
Using this relation between $\rho_{\mathcal{D}}$ and $\rho_{\mathcal{H}}$, the
entanglement entropy can be defined as follows

$S_{EE}=-Tr(\rho_{\mathcal{D}}\log\rho_{\mathcal{D}})=S_{thermal}(\mathcal{H}).$
(2.40)

Introducing a cutoff parameter $\epsilon$, the regulated interval
$\mathcal{D}$ and its mapped domain $\mathcal{H}$ can be defined as follows
[17]

$\mathcal{D}:(t,x)\in\left[(\frac{\bar{l}}{2}-\frac{\bar{l}}{l}\epsilon,-\frac{l}{2}+\epsilon),(-\frac{\bar{l}}{2}+\frac{\bar{l}}{l}\epsilon,\frac{l}{2}-\epsilon)\right]$
(2.41)

and

$\mathcal{H}:(\tilde{t},\tilde{x})\in\left[(\frac{\bar{\kappa}}{2\pi}\gamma-\frac{l}{2}\frac{\bar{L}}{L}+\frac{\bar{l}}{2},-\frac{\kappa}{2\pi}\gamma),(-\frac{\bar{\kappa}}{2\pi}\gamma+\frac{l}{2}\frac{\bar{L}}{L}-\frac{\bar{l}}{2},\frac{\kappa}{2\pi}\gamma)\right],$
(2.42)

where

$\gamma=\log(\frac{L}{\pi\epsilon}\sin\frac{\pi l}{L})+O(\epsilon).$ (2.43)

To evaluate the thermal entropy, it is needed to find the partition function
for $\mathcal{H}$

$Z_{\bar{a}|a}(\bar{\theta}|\theta)$ (2.44)

where $(\bar{a},a)$ and $(\bar{\theta},\theta)$ are defined by the following
indetifications

$(\tilde{t},\tilde{x})\sim(\tilde{t}+2\pi\bar{a},\tilde{x}-2\pi
a)\sim(\tilde{t}+2\pi\bar{\theta},\tilde{x}-2\pi\theta)$ (2.45)

with

$2\pi
a=\frac{\kappa}{\pi}\gamma,~{}2\pi\bar{a}=\frac{\bar{\kappa}}{\pi}\gamma-\frac{\bar{L}}{L}l+\bar{l},~{}~{}2\pi\theta=-i\kappa,~{}2\pi\bar{\theta}=-i\bar{\kappa}.$
(2.46)

The thermal entropy is defined as follows

$S_{\bar{a}|a}(\bar{\theta}|\theta)=(1-\theta\partial_{\theta}-\bar{\theta}\partial_{\bar{\theta}})\log
Z_{\bar{a}|a}(\bar{\theta}|\theta).$ (2.47)

It could be seen the entropy is an observable that is independent of all
observers. Changing the coordinates as follows

$\hat{x}=\frac{\tilde{x}}{a},~{}~{}~{}\hat{t}=\tilde{t}+\frac{\bar{a}}{a}\tilde{x},$
(2.48)

the equality between the entropies is as follows [17]

$S_{\bar{a}|a}(\bar{\theta}|\theta)=\hat{S}(\bar{\theta}-\frac{\theta}{a}\bar{a}|\frac{\theta}{a}).$
(2.49)

In the coordinates $(\hat{t},\hat{x})$, the partition function can be
calculated as follows [17]

$\hat{Z}(\rho,\tau)=e^{i\pi\frac{k}{2}\frac{\rho^{2}}{\tau}}\hat{Z}(\frac{\rho}{\tau}|-\frac{1}{\tau})=e^{i\pi\frac{\kappa}{2}\frac{\rho^{2}}{\tau}}e^{2\pi
i\frac{\rho}{\tau}P_{0}^{vac}+2\pi i\frac{1}{\tau}L_{0}^{vac}}+...$ (2.50)

where

$\tau=-i\frac{\pi}{\gamma},~{}~{}\rho=-\frac{i}{2\gamma}(\frac{\bar{L}}{L}l-\bar{l})$
(2.51)

In the 2nd equality in (2.50), the vacuum dominates the sum.
$\hat{Z}(\rho,\tau)$ is the partition function with $(\bar{a},a)=(0,1)$ as a
canonical partition function.

The thermal entropy has the following relation with the $\hat{Z}$ partition
function,

$\hat{S}(\rho|\tau)=(1-\tau\partial_{\tau}-\rho\partial_{\rho})\log\hat{Z}(\rho|\tau).$
(2.52)

Using this relation, the thermal entropy can be defined as follows

$\hat{S}(\rho|\tau)=iP_{0}^{vac}l(\frac{\bar{L}}{L}-\frac{\bar{l}}{l})-4L_{0}^{vac}\log(\frac{L}{\pi\epsilon}\sin\frac{\pi
l}{L})$ (2.53)

Substituting the thermal entropy in (2.40), we find the entanglement entropy
as follows

$S_{EE}=iP_{0}^{vac}l(\frac{\bar{L}}{L}-\frac{\bar{l}}{l})-4L_{0}^{vac}\log(\frac{L}{\pi\epsilon}\sin\frac{\pi
l}{L})$ (2.54)

where $P_{0}^{vac}$ and $L_{0}^{vac}$ are the vacuum charge values on the
cylinder. To evaluate the entanglement entropy of one segment in the WCFT at
finite temperature, all we need to do is changing the map that we used before,
as follows [17]

$\frac{\tanh\frac{\pi x}{L}}{\tanh\frac{\pi
l}{2L}}=\tanh\frac{\pi\tilde{x}}{L},~{}~{}~{}t+\frac{\bar{\beta}}{\beta}x=\tilde{t}+\frac{\bar{\kappa}}{\kappa}\tilde{x}$
(2.55)

To find the entropy, we just to use the replacement $L\to i\beta$ and
$\bar{L}\to i\bar{\beta}$. With this replacement in (2.54), we find the
entanglement entropy as follows

$S_{EE}=iP_{0}^{vac}l(\frac{\bar{\beta}}{\beta}-\frac{\bar{l}}{l})-4L_{0}^{vac}\log(\frac{\beta}{\pi\epsilon}\sinh\frac{\pi
l}{\beta}).$ (2.56)

##  3 The first law of entanglement entropy in WCFT

The modular Hamiltonian has the following relation with the partition function

$Z=Tr\exp(-\mathcal{H}_{\zeta})=Tr\exp\big{[}2\pi i\rho P_{0}^{cyl}-2\pi i\tau
L_{0}^{cyl}\big{]}$ (3.1)

where we have used the following relation [17]

$Z_{\bar{a}|a}(\bar{\tau}|\tau)=Tr_{\bar{a}|a}\bigg{(}\exp\big{(}2\pi
i\bar{\tau}P_{0}^{cyl}-2\pi i\tau L_{0}^{cyl}\big{)}\bigg{)}$ (3.2)

Using the relation (3.1), we find the charge $\mathcal{H}_{\zeta}$ as follows

$\displaystyle\mathcal{H}_{\zeta}$ $\displaystyle=$ $\displaystyle 2\pi i\rho
P_{0}^{cyl}-2\pi i\tau L_{0}^{cyl}-\log Z$ (3.3) $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int d\hat{x}\big{\\{}2\pi
i\rho\hat{P}(\hat{x})-2\pi i\tau\hat{T}(\hat{x})\big{\\}}-\log Z,$

where in the 2nd line, we use the $L_{n}^{cyl}$ and $P_{n}^{cyl}$ definitions
on the cylinder (2.1). $\mathcal{H}_{\zeta}$ is the charge related to the
modular flow generator vector $\zeta$ (2.33) that is given in the former
subsection 2.3. The modular Hamiltonian is given by the following relation
[27]

$\mathcal{H}_{mod}=\mathcal{H}_{\zeta}+const.$ (3.4)

where the constant term makes the trace of the thermal density matrix
$\rho_{\mathcal{D}}$ equal to one. To find the charge $\mathcal{H}_{\zeta}$
using (3.1), we add $\log Z$ term to make the trace of the thermal density
matrix equal to one. The modular Hamiltonian in terms of the $(t,x)$
coordinates variables can be obtained from (3.3). To this end, we should
perform a series of warped conformal transformations
$(\hat{t},\hat{x})~{}\to~{}(\bar{t},\bar{x})~{}\to~{}(t,x)$ using (2.2) and
(2.3). The modular Hamiltonian $\mathcal{H}_{mod}$ can be find as follows [27]

$\displaystyle\mathcal{H}_{mod}$ $\displaystyle=$
$\displaystyle\int_{x_{-}}^{x_{+}}dx\bigg{\\{}2i\alpha
P(x)+\frac{\beta}{2\pi}\big{[}\frac{\cosh\frac{\pi(2x-l)}{\beta}-\cosh\frac{\pi
l}{\beta}}{\sinh\frac{\pi
l}{\beta}}\big{]}\big{[}T(x)-\frac{\bar{\beta}}{\beta}P(x)\big{]}\bigg{\\}}$
(3.5) $\displaystyle+$ $\displaystyle
ik\alpha\bar{l}-\big{[}\frac{c}{12}-\frac{k}{8\pi^{2}}\bar{\beta}^{2}\big{]}\big{[}\frac{\pi
l}{\beta}\cosh\frac{\pi l}{\beta}-1\big{]}$

where $\alpha$ is defined as a characteristic parameter of the change of the
coordinates (2.7) between the plane and the cylinder. Consequently, the
variation of the modular Hamiltonian can be find as follows

$\displaystyle\delta\mathcal{H}_{mod}$ $\displaystyle=$
$\displaystyle\int_{x_{-}}^{x_{+}}dx\bigg{\\{}2i\alpha\delta
P(x)+\frac{\beta}{2\pi}\big{[}\frac{\cosh\frac{\pi(2x-l)}{\beta}-\cosh\frac{\pi
l}{\beta}}{\sinh\frac{\pi l}{\beta}}\big{]}\big{[}\delta
T(x)-\frac{\bar{\beta}}{\beta}\delta P(x)\big{]}$ (3.6) $\displaystyle+$
$\displaystyle\frac{ik\alpha}{l}\delta\bar{l}\bigg{\\}}$

Using the $L_{n}^{cyl}$ and $P_{n}^{cyl}$ definitions on the cylinder (2.1),
we could find the result as follows

$\displaystyle\delta T(x)$ $\displaystyle=$ $\displaystyle-\delta
L_{0}^{cyl},$ $\displaystyle\delta P(x)$ $\displaystyle=$
$\displaystyle-\delta P_{0}^{cyl}.$ (3.7)

Substituting these relations into the modular Hamiltonian variation (3.6), we
could find the variation as follows

$\displaystyle\delta\mathcal{H}_{mod}$ $\displaystyle=$ $\displaystyle i\alpha
k\delta\bar{l}-2i\alpha l\delta
P_{0}^{cyl}-\frac{\beta^{2}}{4\pi^{2}}\big{(}2-\frac{2\pi
l}{\beta}\coth\frac{\pi l}{\beta}\big{)}\big{(}\delta
L_{0}^{cyl}-\frac{\bar{\beta}}{\beta}\delta P_{0}^{cyl}\big{)}$ (3.8)
$\displaystyle=$ $\displaystyle iP_{0}^{vac}\delta\bar{l}-2i\alpha l\delta
P_{0}^{cyl}-\frac{\beta^{2}}{4\pi^{2}}\big{(}2-\frac{2\pi
l}{\beta}\coth\frac{\pi l}{\beta}\big{)}\big{(}\delta
L_{0}^{cyl}-\frac{\bar{\beta}}{\beta}\delta P_{0}^{cyl}\big{)}$

where we have used the definition of $P_{0}^{vac}$ (2.14) in the 2nd line. The
first law of entanglement entropy is a statement on the equality of the
entanglement entropy variations with the modular Hamiltonian variations as
follows [27]

$\delta S_{EE}=\delta\langle\mathcal{H}_{mod}\rangle$ (3.9)

where we have the modular Hamiltonian variation $\delta\mathcal{H}_{mod}$ in
(3.8). To show this law is satisfied in the warped CFTs, we need the variation
of the entanglement entropy. The variation of the entanglement entropy (2.56)
can be found as follows

$\delta
S_{EE}=iP_{0}^{vac}l\delta(\frac{\bar{\beta}}{\beta})-iP_{0}^{vac}\delta\bar{l}-4L_{0}^{vac}\big{(}1-\frac{\pi
l}{\beta}\coth\frac{\pi l}{\beta}\big{)}\frac{\delta\beta}{\beta}$ (3.10)

Substituting the following relations in (3.10) [27]

$\displaystyle\frac{\delta\beta}{\beta^{3}}$ $\displaystyle=$
$\displaystyle-\frac{3}{c}\frac{1}{\pi^{2}}\delta\big{(}L_{0}^{cyl}-\frac{(P_{0}^{cyl})^{2}}{k}\big{)},$
$\displaystyle\delta\big{(}\frac{\bar{\beta}}{\beta}\big{)}$ $\displaystyle=$
$\displaystyle\frac{2}{k}\delta P_{0}^{cyl},$ (3.11)

we could find the variation of the entanglement entropy as follows

$\delta S_{EE}=iP_{0}^{vac}\delta\bar{l}-2i\alpha l\delta
P_{0}^{cyl}-\frac{\beta^{2}}{4\pi^{2}}\big{(}2-\frac{2\pi
l}{\beta}\coth\frac{\pi l}{\beta}\big{)}\big{(}\delta
L_{0}^{cyl}-\frac{\bar{\beta}}{\beta}\delta P_{0}^{cyl}\big{)}$ (3.12)

Comparing the entanglement entropy variation (3.12) and the variation of the
modular Hamiltonian (3.8), we find these results are equal to each other. This
is a proof of the first law of the entanglement entropy (3.9).

If we could find $\delta\bar{l}$ relation as follows

$\delta\bar{l}=f(l,P_{0}^{vac},P_{0}^{cyl})\delta P_{0}^{cyl},$ (3.13)

where $f(l,P_{0}^{vac},P_{0}^{cyl})$ is a function of the $l$, $P_{0}^{vac}$
and $P_{0}^{cyl}$, we can introduce the following functions $K(x)$ and $Y(x)$
into (3.12)

$K(x)=\big{(}\frac{\bar{\beta}}{\beta}\big{)}\bigg{[}iP_{0}^{vac}f(l,P_{0}^{vac},P_{0}^{cyl})\big{(}\frac{\beta}{\bar{\beta}}\big{)}-2i\alpha
l\big{(}\frac{\beta}{\bar{\beta}}\big{)}+\frac{\beta^{2}}{4\pi^{2}}\big{(}2-\frac{2\pi
l}{\beta}\coth\frac{\pi l}{\beta}\big{)}\bigg{]}$ (3.14)

and

$Y(x)=-\frac{\beta^{2}}{4\pi^{2}}\big{(}2-\frac{2\pi l}{\beta}\coth\frac{\pi
l}{\beta}\big{)},$ (3.15)

then we have the entropy variation as follows

$\delta S_{EE}=K(x)\delta P_{0}^{cyl}+Y(x)\delta L_{0}^{cyl}.$ (3.16)

We find the explicit form of the variation of $\bar{l}$ (3.13) in the section
4. Substituting the cylinder charges (2.1) into (3.16), we find the following
form of the entropy variation

$\delta S_{EE}=-\frac{1}{2\pi}\int dx\bigg{\\{}K(x)\delta P+Y(x)\delta
T\bigg{\\}}.$ (3.17)

For later use, we find the variation of the entanglement entropy using the
charges in the modified form (2.1). To this end, we find the relation between
the variation of the charges $\tilde{P}_{n}$ and $\tilde{L}_{n}$ in the
modified form and the variation of the charges $P_{n}$ and $L_{n}$ on the
plane as follows

$\displaystyle\delta\tilde{P}_{0}$ $\displaystyle=$
$\displaystyle\frac{2P_{0}}{k}\delta P_{0}$
$\displaystyle\delta\tilde{L}_{0}+\delta\tilde{P}_{0}$ $\displaystyle=$
$\displaystyle\delta L_{0},$ (3.18)

where we have used the relation (2.1) when $n=0$. The relation between the
variations of the cylinder charges and the variations of the plane charges can
be found as follows

$\displaystyle\delta P_{0}^{cyl}$ $\displaystyle=$ $\displaystyle\delta P_{0}$
$\displaystyle\delta L_{0}^{cyl}$ $\displaystyle=$ $\displaystyle\delta
L_{0}+2\alpha\delta P_{0},$ (3.19)

where we have used (2.12) and (2.13). Using (3) and (3), we could find the
following relations between the charges on the cylinder and the charges in the
modified form

$\displaystyle\delta P_{0}^{cyl}$ $\displaystyle=$
$\displaystyle\frac{k}{2(P_{0}^{cyl}-\alpha k)}\delta\tilde{P}_{0}$
$\displaystyle\delta L_{0}^{cyl}$ $\displaystyle=$
$\displaystyle\delta\tilde{L}_{0}+\frac{P_{0}^{cyl}}{P_{0}^{cyl}-\alpha
k}\delta\tilde{P}_{0}.$ (3.20)

The relation between the variation of the zero-mode charges and the variation
of the currents can be found as follows

$\displaystyle\delta\tilde{P}_{0}$ $\displaystyle=$
$\displaystyle-\delta\tilde{P},~{}~{}~{}~{}\delta P_{0}^{cyl}=-\delta P,$
$\displaystyle\delta\tilde{L}_{0}$ $\displaystyle=$
$\displaystyle-\delta\tilde{T},~{}~{}~{}~{}\delta L_{0}^{cyl}=-\delta T,$
(3.21)

where we have used (2.1) and the following relation for the modified charge.

$\displaystyle\tilde{L}_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int dz^{\prime}\tilde{T}(z^{\prime}),$
$\displaystyle\tilde{P}_{0}$ $\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\int dz^{\prime}\tilde{P}(z^{\prime}).$ (3.22)

Substituting (3) into (3), we have the following result

$\displaystyle\delta P$ $\displaystyle=$
$\displaystyle\frac{k}{2(P_{0}^{cyl}-\alpha k)}\delta\tilde{P}$
$\displaystyle\delta T$ $\displaystyle=$
$\displaystyle\delta\tilde{T}+\frac{P_{0}^{cyl}}{P_{0}^{cyl}-\alpha
k}\delta\tilde{P}.$ (3.23)

Substituting (3) into the entanglement entropy (3.17), we find the modified
form of the entropy as follows

$\delta S_{EE}=-\frac{1}{2\pi}\int
dx\big{\\{}\big{(}\frac{K(x)k+2P_{0}^{cyl}Y(x)}{2(P_{0}^{cyl}-\alpha
k)}\big{)}\delta\tilde{P}(x)+Y(x)\delta\tilde{T}(x)\big{\\}}.$ (3.24)

Now, we find the new form of the entanglement entropy as follows

$\delta S_{EE}=-\frac{1}{2\pi}\int
dx\big{\\{}\tilde{K}(x)\delta\tilde{P}(x)+\tilde{Y}(x)\delta\tilde{T}(x)\big{\\}}.$
(3.25)

where we have defined $\tilde{K}(x)$ and $\tilde{Y}(x)$ as follows

$\tilde{K}(x)=\frac{K(x)k+2P_{0}^{cyl}Y(x)}{2(P_{0}^{cyl}-\alpha k)}$ (3.26)

and

$\tilde{Y}(x)=Y(x).$ (3.27)

##  4 Warped AdS black holes solution of GMMG

Generalized Minimal Massive Gravity (GMMG) was introduced in [6], providing a
new example of a theory that avoids the bulk-boundary clash and therefore, as
Minimal Massive Gravity (MMG) [7], the theory possesses both, positive energy
excitations around the maximally $AdS_{3}$ vacuum as well as a positive
central charge in the dual CFT. Since these theories avoid the bulk-boundary
clash, they provide excellent areas to explore the structure of asymptotically
AdS solutions, asymptotic symmetries, their algebra and other holographically
inspired questions. The warped $AdS_{3}$ black hole is a solution of the GMMG.
The metric of this black hole is given by [23, 24]

$ds^{2}=l^{2}\bigg{(}-N(r)^{2}dt^{\prime
2}+\frac{dr^{2}}{4N(r)^{2}R(r)^{2}}+R(r)^{2}\big{(}d\phi+N^{\phi}(r)dt^{\prime}\big{)}^{2}\bigg{)}$
(4.1)

where $t^{\prime}$, $r$ and $\phi$ are time-, radial- and angular-coordinates,
respectively. We have used the $t^{\prime}$-symbol instead of the $t$-symbol
for the time coordinate, so as not to be confused with boundary calculations.
$l$ is the $AdS_{3}$ space radius. For the spacelike warped $AdS_{3}$ black
hole we have the following definitions

$\displaystyle R(r)^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}\alpha^{2}r\big{[}(1-\nu^{2})r+\nu^{2}(r_{+}+r_{-})+2\nu\sqrt{r_{+}r_{-}}\big{]}$
(4.2) $\displaystyle N(r)^{2}$ $\displaystyle=$
$\displaystyle\alpha^{2}\nu^{2}\frac{(r-r_{+})(r-r_{-})}{4R(r)^{2}},$ (4.3)
$\displaystyle N^{\phi}(r)$ $\displaystyle=$
$\displaystyle|\alpha|\frac{r+\nu\sqrt{r_{+}r_{-}}}{2R(r)^{2}},$ (4.4)

where $r_{+}$ and $r_{-}$ are the outer and inner horizon radiuses of the
black hole, respectively. The metric (4.1) has the symmetry group
$SL(2,R)\times U(1)$ as the isometry group of the spacetime. This is the
global part of the symmetry group of the Warped CFT. The appropriate boundary
conditions to introduce asymptotically spacelike warped $AdS_{3}$ spacetime,
is considered as follows [24, 26]

$\displaystyle g_{t^{\prime}t^{\prime}}$ $\displaystyle=$ $\displaystyle
l^{2},~{}~{}~{}g_{tr}=\mathcal{O}(r^{-3}),~{}~{}~{}g_{r\phi}=\mathcal{O}(r^{-2}),$
$\displaystyle g_{t^{\prime}\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{2}l^{2}|\alpha|\big{[}r+A_{t\phi}(\phi)+\frac{1}{r}B_{t\phi}(\phi)\big{]},$
$\displaystyle g_{rr}$ $\displaystyle=$
$\displaystyle\frac{l^{2}}{\alpha^{2}\nu^{2}}\big{[}\frac{1}{r^{2}}+\frac{1}{r^{3}}A_{rr}(\phi)+\frac{1}{r^{4}}B_{rr}(\phi)+\mathcal{O}(r^{-5})\big{]},$
$\displaystyle g_{\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{4}l^{2}\alpha\big{[}(1-\nu^{2})r^{2}+rA_{\phi\phi}(\phi)+B_{\phi\phi}(\phi)\big{]}+\mathcal{O}(r^{-1})$
(4.5)

These boundary conditions are in consistent with the metric (4.1) by the
$r$-dependent functions defined in (4.2), (4.3) and (4.4). The warped
$AdS_{3}$ black hole is an asymptotically spacelike warped $AdS_{3}$
spacetime. The asymptotic Killing vectors that generate the fluctuations
preservintg the boundary conditions (4) are as follows [24]

$\displaystyle\xi^{t^{\prime}}(\mathcal{K},\mathcal{Y})$ $\displaystyle=$
$\displaystyle\mathcal{K}(\phi)-\frac{2\partial^{2}_{\phi}\mathcal{Y}(\phi)}{|\alpha|^{3}\nu^{4}r}+\mathcal{O}(r^{-2}),$
$\displaystyle\xi^{r}(\mathcal{K},\mathcal{Y})$ $\displaystyle=$
$\displaystyle-r\partial_{\phi}\mathcal{Y}(\phi)+\mathcal{O}(r^{-2}),$
$\displaystyle\xi^{\phi}(\mathcal{K},\mathcal{Y})$ $\displaystyle=$
$\displaystyle\mathcal{Y}(\phi)+\frac{2\partial^{2}_{\phi}\mathcal{Y}(\phi)}{\alpha^{4}\nu^{4}r^{2}}+\mathcal{O}(r^{-3}),$
(4.6)

where $\mathcal{K}(\phi)$ and $\mathcal{Y}(\phi)$ are two arbitrary periodic
functions. The corresponding conserved charge to the asymptotic Killing
vectors (4) is given in [24] as follows

$\mathcal{Q}(\mathcal{K},\mathcal{Y})=\mathcal{P}(\mathcal{K})+\mathcal{L}(\mathcal{Y}),$
(4.7)

where

$\mathcal{P}(\mathcal{K})=-\frac{|\alpha|}{96\pi}c_{U}\int_{0}^{2\pi}\mathcal{K}(\phi)\big{[}A_{rr}(\phi)+2A_{t\phi}(\phi)\big{]}d\phi,$
(4.8)

and

$\mathcal{L}(\mathcal{Y})=\frac{\alpha^{4}\nu^{4}}{768\pi}c_{V}\int_{0}^{2\pi}\mathcal{Y}(\phi)\big{[}-3A_{rr}(\phi)^{2}+4B_{rr}(\phi)+16B_{t\phi}(\phi)\big{]}d\phi,$
(4.9)

where

$\displaystyle c_{U}$ $\displaystyle=$
$\displaystyle\frac{3l|\alpha|\nu^{2}}{G}\bigg{\\{}\sigma+\frac{\omega}{\mu}\big{(}H_{1}+l^{2}H_{2}\big{)}+\frac{1}{m^{2}}\big{(}F_{1}+l^{2}F_{2}\big{)}-\frac{|\alpha|}{2\mu
l}\bigg{\\}},$ (4.10) $\displaystyle c_{V}$ $\displaystyle=$
$\displaystyle\frac{3l}{|\alpha|\nu^{2}G}\bigg{\\{}\sigma+\frac{\omega}{\mu}\big{(}H_{1}+l^{2}H_{2}\big{)}+\frac{1}{m^{2}}\big{(}F_{1}+l^{2}F_{2}\big{)}-\frac{|\alpha|}{2\mu
l}\big{(}1-2\nu^{2}\big{)}\bigg{\\}}.$ (4.11)

$H_{1}$, $H_{2}$, $F_{1}$ and $F_{2}$ are constant parameters. $\sigma$ is a
sign and $m$ is the mass parameter of New Massive Gravity (NMG) term [30] in
the Lagrangian of GMMG model. $\omega$ is a dimensionless parameter.
Introducing $M(\phi)$ and $J(\phi)$ as follows

$\displaystyle M(\phi)$ $\displaystyle=$ $\displaystyle
A_{rr}(\phi)+2A_{t\phi}(\phi),$ $\displaystyle J(\phi)$ $\displaystyle=$
$\displaystyle-3A_{rr}(\phi)^{2}+4B_{rr}(\phi)+16B_{t\phi}(\phi),$ (4.12)

we could change the form of $\mathcal{P}(\mathcal{K})$ and
$\mathcal{L}(\mathcal{Y})$ in (4.8) and (4.9) , respectively as follows

$\mathcal{P}(\mathcal{K})=-\frac{|\alpha|}{96\pi}c_{U}\int_{0}^{2\pi}\mathcal{K}(\phi)M(\phi)d\phi,$
(4.13)

and

$\mathcal{L}(\mathcal{Y})=\frac{\alpha^{4}\nu^{4}}{768\pi}c_{V}\int_{0}^{2\pi}\mathcal{Y}(\phi)J(\phi)d\phi,$
(4.14)

In the case of the warped black hole (4.1), we have

$\displaystyle A_{rr}$ $\displaystyle=$ $\displaystyle
r_{+}+r_{-},~{}~{}~{}A_{t\phi}=\nu\sqrt{r_{+}r_{-}},$ $\displaystyle B_{rr}$
$\displaystyle=$ $\displaystyle
r_{+}^{2}+r_{-}^{2}+r_{+}r_{-},~{}~{}~{}B_{t\phi}=0.$ (4.15)

We define the Fourier modes as follows

$\displaystyle\mathcal{P}_{m}$ $\displaystyle=$
$\displaystyle\mathcal{Q}(e^{im\phi},0)=\mathcal{P}(e^{im\phi}),$
$\displaystyle\mathcal{L}_{m}$ $\displaystyle=$
$\displaystyle\mathcal{Q}(0,e^{im\phi})=\mathcal{L}(e^{im\phi}).$ (4.16)

Using this definition, the charges can be found as follows

$\mathcal{P}_{m}(\mathcal{K})=-\frac{|\alpha|}{96\pi}c_{U}\int_{0}^{2\pi}\mathcal{K}(\phi)M(\phi)e^{im\phi}d\phi,$
(4.17)

and

$\mathcal{L}_{m}(\mathcal{Y})=\frac{\alpha^{4}\nu^{4}}{768\pi}c_{V}\int_{0}^{2\pi}\mathcal{Y}(\phi)J(\phi)e^{im\phi}d\phi.$
(4.18)

The charge algebra can be written as follows [24]

$\displaystyle\left[\mathcal{L}_{n},\mathcal{L}_{m}\right]$ $\displaystyle=$
$\displaystyle(n-m)\mathcal{L}_{n+m}+\frac{c_{V}}{12}n(n^{2}-1)\delta_{n+m},$
$\displaystyle\left[\mathcal{P}_{n},\mathcal{P}_{m}\right]$ $\displaystyle=$
$\displaystyle-\frac{c_{U}}{12}\mathcal{P}_{0}\delta_{n+m},$
$\displaystyle\left[\mathcal{L}_{n},\mathcal{P}_{m}\right]$ $\displaystyle=$
$\displaystyle-m\mathcal{P}_{n+m}+\frac{m}{2}\mathcal{P}_{0}\delta_{n+m}.$
(4.19)

This charge algebra is the same as the modified algebra (2.1) that we have
used in the field theory side provided that we introduce the following state-
dependent map between the bulk $(\phi,t^{\prime})$ coordinates and the
boundary $(x,t)$ coordinates [19, 27]

$\phi=x,~{}~{}~{}~{}t^{\prime}=x+\frac{kt}{2\mathcal{P}_{0}},$ (4.20)

as a result, a mapping needs to be exist between the gravitational charges on
the bulk and the warped conformal charges on the boundary, that we find the
mapping in the next section. This correspondence between the charge algebras
on the bulk and the boundary is an evidence for the $WAdS_{3}/WCFT_{2}$
duality. Using (4.8) and (4.9) and from (4), we can find the eigenvalues of
the $\mathcal{P}_{m}$ and $\mathcal{L}_{m}$ as follows

$\displaystyle p_{m}$ $\displaystyle=$
$\displaystyle-\frac{|\alpha|c_{U}}{48}(r_{+}+r_{-}+2\nu\sqrt{r_{+}r_{-}})\delta_{m},$
(4.21) $\displaystyle l_{m}$ $\displaystyle=$
$\displaystyle\frac{\alpha^{4}\nu^{4}c_{V}}{384}(r_{+}-r_{-})^{2}\delta_{m}.$
(4.22)

##  5 The first law of entanglement entropy in $WAdS_{3}$

We can write the gravitational charge of the GMMG as follows[24]

$\displaystyle Q(\mathcal{K},\mathcal{Y})$ $\displaystyle=$
$\displaystyle-\frac{|\alpha|}{96\pi}c_{U}\int_{0}^{2\pi}\mathcal{K}(\phi)M(\phi)d\phi$
(5.1) $\displaystyle+$
$\displaystyle\frac{\alpha^{4}\nu^{4}}{768\pi}c_{V}\int_{0}^{2\pi}\mathcal{Y}(\phi)J(\phi)d\phi$

where we have used (4.7),(4.13) and (4.14) and as we have mentioned previously
$\mathcal{K}(\phi)$ and $\mathcal{Y}(\phi)$ are two arbitrary periodic
functions. The $M(\phi)$ and $J(\phi)$ function have been specified in (4).
The fourier modes of the gravitational charge can be found as follows

$\displaystyle Q_{m}(\mathcal{K},\mathcal{Y})$ $\displaystyle=$
$\displaystyle-\frac{|\alpha|}{96\pi}c_{U}\int_{0}^{2\pi}\mathcal{K}(\phi)M(\phi)e^{im\phi}d\phi$
(5.2) $\displaystyle+$
$\displaystyle\frac{\alpha^{4}\nu^{4}}{768\pi}c_{V}\int_{0}^{2\pi}\mathcal{Y}(\phi)J(\phi)e^{im\phi}d\phi$

The gravitational charge variations of (5.1) can be found as follows

$\displaystyle\delta Q(\mathcal{K},\mathcal{Y})$ $\displaystyle=$
$\displaystyle-\frac{|\alpha|}{96\pi}c_{U}\int_{0}^{2\pi}\mathcal{K}(\phi)\delta
M(\phi)d\phi$ (5.3) $\displaystyle+$
$\displaystyle\frac{\alpha^{4}\nu^{4}}{768\pi}c_{V}\int_{0}^{2\pi}\mathcal{Y}(\phi)\delta
J(\phi)d\phi$

where we have considered the variations of lengths of bulk coordinates as
follows

$\delta l_{\phi}=0,~{}~{}~{}~{}\delta l_{t^{\prime}}=0.$ (5.4)

Utilizing the relation between the bulk and the boundary coordinates in
(4.20), we could find the following dictionary between the bulk and the
boundary lengths of the coordinates

$\displaystyle l_{\phi}$ $\displaystyle=$ $\displaystyle l,$ $\displaystyle
l_{t^{\prime}}$ $\displaystyle=$ $\displaystyle
l+\frac{k}{2\mathcal{P}_{0}}\bar{l}.$ (5.5)

We could find the variations of the bulk and the boundary lengths, utilizing
(5), as follows

$\displaystyle\delta l_{\phi}$ $\displaystyle=$ $\displaystyle\delta l=0,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{k}{2\mathcal{P}_{0}}\delta\bar{l}-\frac{k\bar{l}}{2\mathcal{P}_{0}^{2}}\delta\mathcal{P}_{0}.$
(5.6)

where we have used (5.4) into the above relations. Substituting the following
relation into (5)

$\bar{l}=\frac{2\mathcal{P}_{0}}{k}(l_{t^{\prime}}-l),$ (5.7)

where we have obtained from (5), we could find the $\delta\bar{l}$ relation as
follows

$\delta\bar{l}=\frac{2}{k}(l_{t^{\prime}}-l)\delta\mathcal{P}_{0}.$ (5.8)

As one can see in the rest of this section, we have found the dictionary
between the bulk and the boundary charges in (5.15) that can be used to change
the $\delta\bar{l}$ relation as follows

$\delta\bar{l}=\frac{2}{k}(l_{t^{\prime}}-l)\delta\tilde{P}_{0},$ (5.9)

where the $\delta\tilde{P}_{0}$ and $\delta P_{0}^{cyl}$ have a relation as in
(3) that can be utilized to find the following relation for the
$\delta\bar{l}$

$\delta\bar{l}=\frac{4}{k^{2}}(l_{t^{\prime}}-l)(P_{0}^{cyl}-\alpha k)\delta
P_{0}^{cyl}.$ (5.10)

So our conjectural relation for $\delta\bar{l}$ in (3.13) is confirmed, and
the explicit form of the $f(l,P_{0}^{vac},P_{0}^{cyl})$ can be found as
follows

$\displaystyle f(l,P_{0}^{vac},P_{0}^{cyl})$ $\displaystyle=$
$\displaystyle\frac{4}{k^{2}}(l_{t^{\prime}}-l)(P_{0}^{cyl}-\alpha k)$ (5.11)
$\displaystyle=$
$\displaystyle\frac{4\alpha^{2}}{(P_{0}^{vac})^{2}}(l_{t^{\prime}}-l)(P_{0}^{cyl}-P_{0}^{vac}).$

where we have used (2.14) into the second line of the above relation.

We assume the first law of the entanglement entropy to be true in the gravity
side of the $WAdS_{3}/WCFT$ in the GMMG case (as in the Enstein gravity case
in [19, 20]). Modular Hamiltonian is the same as gravitational charge (5.3) ,
and since we have assumed the first law of entropy to be true, we can write
the relation between the holographic entanglement entropy variation $\delta
S_{HEE}$ and the gravitational charge variation as follows 333Here we should
mention that it is interesting task to one try to apply the proposal and
formula (2.29) in the paper [20] to obtain holographic entanglement entropy in
the framework of GMMG model.

$\displaystyle\delta S_{HEE}$ $\displaystyle=$ $\displaystyle\delta Q.$ (5.12)

Now we can complete the dictionaries between the fields in two sides of the
duality. The algebra (4) of the charges that generate the isometries of the
warped $AdS_{3}$ black hole in GMMG has an equal form with the charge algebra
(2.1) in WCFT. This correspondence between the bulk and the boundary symmetry
is an indication of the $WAdS_{3}/WCFT_{2}$ holography. We could find the
following dictionary between the bulk and the boundary variables,

$\displaystyle-\frac{1}{2\pi}\tilde{P}$ $\displaystyle=$
$\displaystyle-\frac{|\alpha|}{96\pi}c_{U}M(\phi)$ (5.13)
$\displaystyle\frac{1}{2\pi}\tilde{T}$ $\displaystyle=$
$\displaystyle-\frac{\alpha^{4}\nu^{4}}{768\pi}c_{V}J(\phi).$ (5.14)

Using these matching relations between the bulk and the boundary variables, we
find the following correspondence between the charges in two sides of the
duality

$\displaystyle\mathcal{P}_{m}$ $\displaystyle\to$ $\displaystyle\tilde{P}_{m}$
(5.15) $\displaystyle\mathcal{L}_{m}$ $\displaystyle\to$
$\displaystyle\tilde{L}_{m}.$ (5.16)

Substituting the dictionary variables (5.13) and (5.14) into the gravitational
charge (5.3), we find the following relation

$\delta Q=\delta\langle\mathcal{H}_{\zeta}\rangle,$ (5.17)

that it is another evidence for the duality is considered in this paper. Using
the dictionary (5.13) and (5.14), the equality between the entanglement
entropy (3.25) in WCFT and the holographic entanglement entropy (5.12) is also
established easily. From the equality, $S_{EE}=S_{HEE}$, and using (3.9) and
(5.17), we implement our claim on the establishment of the first law of the
entanglement entropy in the $WAdS_{3}/WCFT_{2}$ duality that its final form is
as follows

$\delta S_{EE}=\delta\mathcal{H}_{\zeta}=\delta Q=\delta S_{HEE}.$ (5.18)

So we have provided a holographic computation of the first law of entanglement
entropy for WAdS under the boundary condition (4). Our bulk results in this
section agree with those of WCFT results in section 3, and this is another
evidence of the conjectured WAdS/WCFT correspondence.

##  6 Conclusion

In this work, we studied one class of the non-AdS holographies. We studied the
entanglement entropy and 1st law in the $WAdS_{3}/WCFT$ correspondence in the
context of Generalized Minimal Massive Gravity. This generalizes similar
studies of $WAdS_{3}/WCFT$ in the Einstein gravity. Previously we have studied
the $Flat_{3}/BMSFT_{2}$ holography in the framework of GMMG model [31]. The
field theory at the boundary was considered a theory with $BMS_{3}$-symmetry
[32, 33, 34]. The asymptotic symmetry of the bulk side of the duality was the
same as the vacuum symmetry of the field theory side. By computing the
gravitational charge in the bulk and the modular Hamiltonian at the boundary,
we succeeded to show the first law of the entanglement entropy. In this new
work, we considered another class of the duality where the filed theory at the
boundary is a warped $CFT_{2}$ [10, 14]. In the bulk, the spacelike warped
$AdS_{3}$ black hole [24] as a solution of the GMMG was considered.

The charge algebra of the $WCFT_{2}$ is a Virasoro-Kac-Moody algebra and this
is the same as the asymptotic charge algebra of the bulk at the boundary limit
[24]. This is a permission to consider the duality as a true one. We used the
modular invariance of the WCFT to compute the modular flow generator at the
boundary’s vacuum. This was computed using the generalized Rindler
transformaton [18, 25]. The modular flow generator was computed in (2.33). To
find an evidence for the holography, we needed the modular Hamiltonian and the
entanglement entropy in the field theory side. Because of the universal form
of the entropy of the WCFT in an interval, we used the result of [17]. We
expanded this result in details to see the importance of the warped symmetry
in calculation of the entanglement entropy.

We showed the first law of the entanglement entropy (3.9) at the boundary that
states the equality of the entanglement entropy and the modular Hamiltonian
variations. This form of the first law can be implemented by some variations
in the bulk. For the bulk, we used the metric (4.1) that is an asymptotically
warped $AdS_{3}$ black hole with the $r$-dependent function (4.2), (4.3) and
(4.4). We found the gravitational charge (4.7) with (4.8) and (4.9). To
establish the first law of the entanglement entropy, introducing the
definitions (4), we found the gravitational charge variations in (5.3). This
was equal to the holographic entanglement entropy in (5.12). Finding the
relations (5.13) and (5.14) between the variables in two sides of the duality,
as a holographic dictionary, we found the completed form of the first law of
the entanglement entropy (5.18).

##  7 Acknowledgments

We thank the anonymous referee for important comments.

## References

  * [1] I. Bengtsson and P. Sandin, Anti de Sitter space, _squashed and stretched_ , _Class. Quant. Grav._ 23 (2006) 971.
  * [2] M. Guica, T. Hartman, W. Song and A. Strominger, _The Kerr/CFT correspondence_ , _Phys. Rev. D_ 80 (2009) 124008.
  * [3] D.M. Hofman and B. Rollier, _Warped conformal field theory as lower spin gravity_ , _Nucl. Phys. B_ 897 (2015) 1.
  * [4] D. Anninos, J. Samani and E. Shaghoulian, _Warped entanglement entropy_ , _JHEP_ 02 (2014) 118.
  * [5] K.A. Moussa, G. Clement and C. Leygnac, _The black holes of topologically massive gravity_ , Class. Quant. Grav. 20 (2003) L277.
  * [6] M. R. Setare, _On the Generalized Minimal Massive Gravity_ , _Nucl. Phys. B_ 898 , 259 (2015).
  * [7] E. Bergshoeff, O. Hohm, W. Merbis, A. J. Routh and P. K. Townsend, Class. Quant. Grav. 31, 145008, (2014)
  * [8] D. Anninos, W. Li, M. Padi, W. Song and A. Strominger, _Warped AdS3 black holes_ , _JHEP_ 03 (2009) 130.
  * [9] W. Song and A. Strominger, _Warped AdS3/Dipole-CFT Duality_ , _JHEP_ 05 (2012) 120.
  * [10] S. Detournay, T. Hartman and D.M. Hofman, _Warped conformal field theory_ , _Phys. Rev. D_ 86 (2012) 124018.
  * [11] G. Compere, M. Guica and M. J. Rudriguez, _Two Virasoro symmetries in stringy warped AdS3_ , _JHEP_ 12 (2014) 012.
  * [12] W. Song, Q. Weng and J. Xu, _Generalized Gravitational Entropy for Warped Anti–de Sitter Space_ , _Phys.Rev.Lett._ 117 (2016) 1, 011602.
  * [13] G. Compere and S. Detournay, _Semi-classical central charge in topologically massive gravity_ , _Class.Quant.Grav._ 26 (2009) 012001.
  * [14] D. M. Hofman and A. Strominger, _Chiral Scale and Conformal Invariance in 2D Quantum Field Theory_ ,_Phys. Rev. D_ 86 (2011) 161601.
  * [15] J. Polchinski, _Scale And Conformal Invariance In Quantum Field Theory_ , _Nucl. Phys. B_ 303, 226 (1988).
  * [16] J. L. Cardy, _Operator Content of Two-Dimensional Conformally Invariant Theories_ , _Nucl. Phys. B_ 270, 186 (1986).
  * [17] A. Castro, D.M. Hofman and N. Iqbal, _Entanglement Entropy in Warped Conformal Field Theories_ , _JHEP_ 02 (2016) 033
  * [18] H. Casini, M. Huerta and R.C. Myers, _Towards a derivation of holographic entanglement entropy_ , _JHEP_ 05 (2011) 036.
  * [19] W. Song, Q. Wen and J. Xu, _Modifications to Holographic Entanglement Entropy in Warped CFT_ , _JHEP_ 02 (2017) 067.
  * [20] L. Apolo, H. Jiang, W. Song and Y. Zhong, _Swing surfaces and holographic entanglement beyond AdS/CFT_ , _JHEP_ 12 (2020) 064.
  * [21] S. Ryu and T. Takayanagi, _Holographic derivation of entanglement entropy from AdS/CFT_ , _Phys. Rev. Lett._ 96 (2006) 181602.
  * [22] S. Ryu and T. Takayanagi, _Aspects of Holographic Entanglement Entropy_ , _JHEP_ 08 (2006) 045
  * [23] E. Tonni, _Warped black holes in 3D general massive gravity_ , _JHEP_ 1008 (2010) 070.
  * [24] M. R. Setare and H. Adami, _Asymptotically spacelike warped anti-de Sitter spacetimes in generalized minimal massive gravity_ , _Class.Quant.Grav._ 34 (2017) 12, 125008.
  * [25] H. Jiang, W. Song and Q. Wen, _Entanglement Entropy in Flat Holography_ , _JHEP_ 07 (2017) 142\.
  * [26] M. Henneaux, C. Martinez, and R. Troncoso, _Asymptotically warped anti-de Sitter spacetimes in topologically massive gravity_ _Phys. Rev. D_ 84, 124016 (2011).
  * [27] L. Apolo, H. Jiang, W. Song and Y. Zhong, _Modular Hamiltonians in flat holography and (W)AdS/WCFT_ , _JHEP_ 09 (2020) 033. .
  * [28] C. Holzhey, F. Larsen, and F. Wilczek, _Geometric and renormalized entropy in conformal field theory_ , _Nucl.Phys. B_ 424 (1994) 443-467.
  * [29] J. Hartong, _Gauging the Carroll Algebra and Ultra-Relativistic Gravity_ , _JHEP_ 08 (2015) 069\.
  * [30] E. A. Bergshoeff, O. Hohm, and P. K. Townsend, _Massive gravity in three dimensions_ _Phys. Rev. Lett._ 102 (2009) 201301.
  * [31] M. R. Setare and M. Koohgard, _Modular Hamiltonian in flat holography in the generalized minimal massive gravity (GMMG)_ , to be appeare.
  * [32] H. Bondi, M. G. J. van der Burg and A. W. K. Metzner, _Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems_ , _Proc. Roy. Soc. Lond. A_ 269 (1962) 21\.
  * [33] R. K. Sachs, _Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times_ , _Proc. Roy. Soc. Lond. A_ 270 (1962) 103.
  * [34] R. Sachs, _Asymptotic symmetries in gravitational theory, Phys. Rev._ 128 (1962) 2851.