Deformations of the Almheiri-Polchinski model

TL;DR

This paper explores deformations of the Almheiri-Polchinski model using Yang-Baxter techniques, resulting in modified black hole solutions with altered spacetime structures and entropy, challenging holographic interpretations.

Contribution

It introduces a novel deformation of the AP model via Yang-Baxter methods, leading to new solutions with hyperbolic dilaton potentials and modified black hole properties.

Findings

Deformed AdS$_2$ metric solutions obtained.

Black hole solutions with altered entropy.

Significant changes in boundary spacetime structure.

Abstract

We study deformations of the Almheiri-Polchinski (AP) model by employing the Yang-Baxter deformation technique. The general deformed AdS$_2$ metric becomes a solution of a deformed AP model. In particular, the dilaton potential is deformed from a simple quadratic form to a hyperbolic function-type potential similarly to integrable deformations. A specific solution is a deformed black hole solution. Because the deformation makes the spacetime structure around the boundary change drastically and a new naked singularity appears, the holographic interpretation is far from trivial. The Hawking temperature is the same as the undeformed case but the Bekenstein-Hawking entropy is modified due to the deformation. This entropy can also be reproduced by evaluating the renormalized stress tensor with an appropriate counter term on the regularized screen close to the singularity.

Equations

$$\text{mbox}{AdS}_2=SL(2)/U(1).$$

$$g=\exp \left[tH\right]\exp \left[(\log z)D\right],$$

$$[D,H]=H,[C,H]=2D,[D,C]=-C.$$

$$H\equiv T_0+T_2,C\equiv T_0-T_2,D\equiv T_1,$$

$$[T_0,T_1]=-T_2,[T_1,T_2]=T_0,[T_2,T_0]=-T_1.$$

$$T_0=\frac{i}{2}{\sigma }_1,T_1=\frac{1}{2}{\sigma }_2,T_2=\frac{1}{2}{\sigma }_3,$$

$$\mathfrak{h}=\text{mbox}{span}_{\mathbb{R}}⟨T_2⟩,\mathfrak{m}=\text{mbox}{span}_{\mathbb{R}}⟨T_0,T_1⟩,$$

$$[\mathfrak{h},\mathfrak{h}]\subset \mathfrak{h},[\mathfrak{m},\mathfrak{h}]\subset \mathfrak{m},[\mathfrak{m},\mathfrak{m}]\subset \mathfrak{h}.$$

$$\mathfrak{sl}(2)=\mathfrak{h}\oplus \mathfrak{m},$$

$$P(X)\equiv \frac{\mathrm{Tr}(X{T}_0)}{\mathrm{Tr}(T_0{T}_0)}{T}_0+\frac{\mathrm{Tr}(X{T}_1)}{\mathrm{Tr}(T_1{T}_1)}{T}_1,X\in \mathfrak{sl}(2).$$

$$J=e^0{T}_0+e^1{T}_1+\frac{1}{2}{\omega }^{01}{T}_2.$$

$$e^0=\frac{\mathrm{d}t}{z},e^1=\frac{\mathrm{d}z}{z}.$$

$$\mathrm{d}{s}^2$$

$$x^{\pm }\equiv t\pm z.$$

$$\mathrm{d}{s}^2=-{\mathrm{e}}^{2\omega (x^{+},x^{-})}\mathrm{d}{x}^{+}\mathrm{d}{x}^{-}=-\frac{4\mathrm{d}{x}^{+}\mathrm{d}{x}^{-}}{(x^{+}-x^{-}{)}^2}.$$

$$\mathrm{d}{s}^2=2\text{mbox}Tr\left[J\frac{1}{1-2\eta R_g\circ P}P(J)\right].$$

$$R_g(X)\equiv g^{-1}\circ R(gX{g}^{-1})\circ g,$$

$$[R(X),R(Y)]-R([R(X),Y]+[X,R(Y)])=c\cdot [X,Y](X,Y\in \mathfrak{sl}(2)).$$

$$R(T_I)$$

$${\tilde{\Omega }}_{IJ}$$

$$-m_1^2-m_2^2+m_3^2=4c.$$

$$\mathrm{d}{s}^2=\frac{-\mathrm{d}{t}^2+\mathrm{d}{z}^2}{z^2-{\eta }^2{\left(\alpha +\beta t+\gamma (-t^2+z^2)\right)}^2}.$$

$$\alpha \equiv \frac{1}{2}(m_1+m_3),\beta \equiv -m_2,\gamma \equiv \frac{1}{2}(m_1-m_3).$$

$$R=-2\left(1-\tilde{\omega }{\eta }^2\right)\frac{z^2+{\eta }^2{\left(\alpha +\beta t+\gamma (-t^2+z^2)\right)}^2}{z^2-{\eta }^2{\left(\alpha +\beta t+\gamma (-t^2+z^2)\right)}^2},$$

$$\tilde{\omega }$$

$$S=S_{g,\Phi }+S_{\mathrm{matter}},$$

$$S_{g,\Phi }=\frac{1}{16\pi G}\int d^{2}x\sqrt{-g}\left({\Phi }^{2}R-U(\Phi )\right),$$

$$S_{\mathrm{matter}}=\frac{1}{32\pi G}\int d^{2}x\sqrt{-g}\Omega (\Phi )(\nabla f{)}^2.$$

$$\mathrm{d}{s}^2$$

$${\partial }_{+}(\Omega {\partial }_{-}f)+{\partial }_{-}(\Omega {\partial }_{+}f)$$

$$4{\partial }_{+}{\partial }_{-}{\Phi }^2-{\mathrm{e}}^{2\omega }U(\Phi )$$

$$2{\partial }_{+}({\mathrm{e}}^{-2\omega }{\partial }_{-}{\mathrm{e}}^{2\omega })-\frac{1}{2}{e}^{2\omega }{\partial }_{{\Phi }^2}U(\Phi )$$

Full text

1. 1 Introduction

2. 2 Yang-Baxter deformations of AdS2

3. 2.1 Coset construction of AdS2

4. 2.2 The general Yang-Baxter deformation

5. 3 A brief review of the AP model

6. 3.1 1+1 D dilaton gravity system

7. 3.2 The AP model

8. 4 Deforming the AP model

9. 4.1 The deformed AP model

10. 4.2 A deformed black hole solution

11. 5 The boundary computation of entropy

12. 6 Conclusion and discussion

1 Introduction

In the recent study of string theory, one of the most important issues is to understand a holographic principle [1, 2] at the full quantum level (For a review see [3]). The AdS/CFT correspondence [4, 5, 6] is a realization of the holography. This is, however, a conjectured relation and there is no rigorous proof so far. The integrable structure behind the correspondence at the planar level has played an important role in checking conjectured relations in non-BPS regions (For a comprehensive review, see [7]). But the proof is still far from the completion and furthermore it does not seem likely that the integrability would work in the presence of a black hole.

Towards the complete understanding of holography, it is significant to try to construct a simple toy model of quantum holography. In fact, Kitaev proposed such a model [8], which is a variant of the Sachdev-Ye (SY) model [9]. More concretely, this model is a one-dimensional system composed of $N\gg 1$ fermions with a random, all-to-all quartic interaction. This model is now called the Sachdev-Ye-Kitaev (SYK) model111For the recent progress on the SYK model, see [10, 11, 13, 14, 15, 16, 17, 18]. . It should be remarked that the Lyapunov exponent computed from an out-of-time-order four-point function [19, 20] in the SYK model saturates the bound presented in [21]. This is the onset to open a window to a toy model of holography because the Lyapunov exponent of black hole in Einstein gravity is $2\pi/\beta$ [22, 23], where $\beta$ is the inverse of the Hawking temperature (For a black hole S-matrix approach, see [24].).

A promising candidate of the gravity dual for the SYK model is a 1+1 D dilaton gravity. This system was originally introduced by Jackiw [25] and Teitelboim [26] (For a nice reviw of the 1+1 D dilaton gravity system, see [27]). From a renewed interest, the dilaton gravity with a certain dilaton potential was intensively studied in the recent work [28], and this model is called the Almheiri-Polchinski (AP) model. A black hole solution exists as a vacuum solution of the AP model. They studied its various properties like the RG flow structure at zero temperature, the Bekenstein-Hawking entropy, the renormalized boundary stress tensor, and the contribution of conformal matter fields to the entropy. For the recent progress on the AP model, see [29, 30].

In this paper, we are concerned with deformations of the AP model. Why is it so interesting to study the deformations? There are some observation and motivation based on the recent progress. The first is an observation that the SY model is constructed by performing a disordered quench to an isotropic quantum Heisenberg magnet [9]. The Heisenberg model itself is integrable. Hence, supposing that the conjectured duality is true, it is natural to expect that integrable deformations of it lead to the associated deformations of the AP model. The second is a motive to understand the holographic duals of deformed AdS2 geometries. Recently, a systematic way to perform integrable deformations, which is called the Yang-Baxter deformation [31, 32, 33]222For the (affine) symmetry algebras, see [34] and [35])., has been intensively studied333For Yang-Baxter deformations of type IIB superstring on AdS${}_{5}\times$S5 , see [36, 38, 39, 40, 41, 42, 43, 37, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. However, the holographic duals of the deformed geometries have been poorly understood. In particular, even the location of the holographic screen has not been clarified, though there is a proposal [60]. Hence it is important to get much deeper understanding of the simplest case like AdS2 . Furthermore, the Yang-Baxter deformation is not applicable to black hole geometries in general, because those cannot be described usually as a coset, homogeneous space. However, it is not the case for a 1+1 D black hole presented in this paper.

Based on the observation and motivation described above, we will study deformations of the AP model by employing the Yang-Baxter deformation technique. The general deformed AdS2 metric becomes a solution of a deformed AP model. In particular, the dilaton potential is deformed from a simple quadratic form to a hyperbolic function-type potential similarly to integrable deformations. A specific solution is a deformed black hole solution. Because the deformation makes the spacetime structure around the boundary change drastically and a new naked singularity appears, the holographic interpretation is far from trivial. The Hawking temperature is the same as the undeformed case but the Bekenstein-Hawking entropy is modified due to the deformation. This entropy can also be reproduced by evaluating the renormalized stress tensor with an appropriate counter term on the regularized screen close to the singularity.

This paper is organized as follows. In section 2 we study the most general Yang-Baxter deformation of AdS2 . Section 3 introduces the classical action of 1+1 D dilaton gravity and the AP model as a special case. In section 4 we study deformations of the AP model. The most general deformed metric constructed in section 2 becomes a solution with a deformed dilaton potential. This deformed system allows a black hole solution as a specific solution like in the AP model. The Bekenstein-Hawking entropy is also computed. In section 5, we revisit the black hole entropy from the viewpoint of the renormalized boundary stress tensor. Putting a regularized screen close to a singularity, we evaluate the renormalized boundary stress tensor with an appropriate counter term. The resulting entropy nicely agrees with the Bekenstein-Hawking entropy computed in section 4. Section 6 is devoted to conclusion and discussion.

2 Yang-Baxter deformations of AdS2

In this section, we consider the most general Yang-Baxter deformation of the AdS2 metric. First of all, we briefly describe a coset construction of the Poincaré AdS2 Then we study the most general Yang-Baxter deformation of Poincaré AdS2 . As a result, we obtain a three-parameter family of deformed AdS2 spaces.

2.1 Coset construction of AdS2

Let us recall a coset construction of the Poincaré AdS2 metric (For the detail of the coset construction, for example, see [61]).

The starting point is that the AdS2 geometry is represented by a coset

[TABLE]

By using the coordinates $t$ and $z$ , a coset representative $g$ is parametrized as

[TABLE]

where $H$ and $D$ are the time translation and dilatation generators, respectively. By involving the special conformal generator $C$ , the $\mathfrak{sl}(2)$ algebra in the conformal basis is spanned as

[TABLE]

These generators can be represented by the $\mathfrak{so}(1,2)$ ones $T_{I}~{}~{}(I=0,1,2)$ like

[TABLE]

where $T_{I}$’s satisfy the commutation relations:

[TABLE]

In the following, we will work with $T_{I}$’s in the fundamental representation,

[TABLE]

where $\sigma_{i}~{}(i=1,2,3)$ are the standard Pauli matrices.

Note here that the coset (2.1) is symmetric as one can readily understand from (2.5) . When vector spaces $\mathfrak{h}$ and $\mathfrak{m}$ are spanned as

[TABLE]

the $\mathbb{Z}_{2}$-grading structure is expressed as

[TABLE]

When representing the $\mathfrak{sl}(2)$ algebra by a direct product (as vector spaces) ,

[TABLE]

the projection operator $P:~{}\mathfrak{sl}(2)\,\to\,\mathfrak{m}$ can be defined as

[TABLE]

Now the Poincaré AdS2 metric can be computed by performing coset construction. The left invariant one-form $J=g^{-1}{\rm d}g$ is expanded as

[TABLE]

Here $e^{0}$ and $e^{1}$ are the zweibeins, and $\omega^{01}$ is the spin connection. With the parametrization (2.2) , the zweibeins are given by

[TABLE]

By using the projection operator $P$ in (2.8) and the explicit expressions of the zweibeins $e^{0}$ and $e^{1}$ , the resulting metric is obtained as

[TABLE]

This is nothing but the AdS2 metric in the Poincaré coordinates.

Hereafter, it is often convenient to use the the light-cone coordinates defined as

[TABLE]

Then the metric is rewritten as

[TABLE]

The exponential factor will play an important role in later discussion.

2.2 The general Yang-Baxter deformation

Let us next consider Yang-Baxter deformations of the AdS2 metric (2.9) . In the usual discussion, Yang-Baxter deformations [31, 32, 33] are performed for 2D non-linear sigma models. Then the anti-symmetric two-form is also involved as well as the metric. Here we will concentrate on the metric part only.

The prescription of the deformation is very simple. It is just to insert a factor as follows:

[TABLE]

Here $\eta$ is a constant parameter which measures the deformation. Then $R_{g}$ is defined as a chain of operation like

[TABLE]

where $g$ is the group element in (2.2) . The key ingredient is a linear operator $R:~{}\mathfrak{sl}(2)\to\mathfrak{sl}(2)$ , and satisfy the (modified) classical Yang-Baxter equation [(m)CYBE]:

[TABLE]

Here $c$ is a real constant parameter. The case with $c\neq 0$ is the mCYBE and the case with $c=0$ is the homogeneous CYBE.

We consider the most general deformations with the following $R$-operator

[TABLE]

where $\widetilde{\Omega}_{IJ}$ and $M^{IJ}$ are defined as444A similar study was done for the Nappi-Witten model [62] in [63]. Note that the present definition of $M^{IJ}$ is slightly different from the one in [63].

[TABLE]

Putting the ansatz (2.15) into the (m)CYBE (2.14) leads to an algebraic relation,

[TABLE]

After evaluating the expression (2.12) with the general ansatz (2.15) , one can obtain the following metric:

[TABLE]

Here $\alpha$ , $\beta$ and $\gamma$ are defined as linear combinations of $m_{p}~{}~{}(p=1,2,3)$ as follows:

[TABLE]

When $\eta=0$ , the undeformed metric (2.9) is reproduced. Note here that the four constant parameters $m_{p}~{}(p=1,2,3)$ and $c$ appear in our discussion. Then a constraint (2.17) , which comes from the (m)CYBE, is imposed. Hence, three of them are independent each other.

The Ricci scalar curvature of the metric (2.18) is

[TABLE]

where we have introduced a new quantity,

[TABLE]

At the last equality, the (m)CYBE (2.14) has been utilized. The scalar curvature (2.20) changes (even its sign) depending on the values of parameters and coordinates, while it becomes a constant $-2$ in the undeformed limit $\eta\to 0$ . The expression (2.20) indicates that the deformed geometry contains both AdS and dS in general.

3 A brief review of the AP model

In this section, we shall introduce the classical action of 1+1 D dilaton gravity system. Then we briefly describe the AP model and its properties related to our later discussion.

3.1 1+1 D dilaton gravity system

The dilaton gravity system in 1+1 dimensions is composed of the metric $g_{ab}~{}(a,b=0,1)$ and the dilaton $\Phi$ . The coordinates are parametrized as $x^{a}=(x^{0},x^{1})=(t,z)$ .

The classical action $S$ is given by

[TABLE]

Here $G$ is the Newton constant in 1+1 dimensions and $U(\Phi)$ is the dilaton potential.

In the following, we will work with the metric in the conformal gauge,

[TABLE]

where the light-cone coordinates are defined in (2.10) .

Then the equations of motion are given by

[TABLE]

The energy-momentum tenser for the matter field $f$ is normalized as

[TABLE]

This expression (3.4) is valid for the general form of $\Omega(\Phi)$ .

3.2 The AP model

The AP model corresponds to a special case of 1+1 D dilaton gravity specified by the following condition:

[TABLE]

This model exhibits nice properties. Among them, we are concerned with the vacuum solution of this model. For our later convenience, we shall give a brief review of the work [28] by focusing upon the vacuum solution in the following.

The general vacuum solution is given by

[TABLE]

and depends on three real constants $a$ , $b$ and $c$ . This three-parameter family contains interesting solutions as specific examples. For example, the case with $a=1/2$ , $b=0$ and $c=0$ corresponds to a renormalization group flow solution from a conformal Lifshitz spacetime to AdS2 [28], with an appropriate lift-up to higher dimensions.

Another intriguing example is a black hole solution specified with $a=1/2$ , $b=0$ and $c=\mu/2$ , where $\mu$ is a real positive constant. Then by performing a coordinate transformation,

[TABLE]

the solution is rewritten into the following form:

[TABLE]

The new coordinates $T$ and $Z$ cover a smaller region which is in the inside of the entire Poincaré AdS2 , as depicted in Fig. 1.

The background (3.9) indeed describes a black hole geometry, but it may not be so manifest. To figure out the black hole geometry, it is nice to move to the Schwarzschild coordinates by performing a further coordinate transformation,

[TABLE]

Then the background (3.9) can be rewritten as555The factor 4 is included so that the Bekenstein-Hawking entropy should match with the holographic computation. This normalization guarantees the matching of the bulk and boundary times (or temperatures). We are grateful to Ahmed Almheiri for this point.

[TABLE]

In this metric, the black hole horizon is located at $\rho=\sqrt{\mu}$ , and the Hawking temperature $T_{\rm H}$ can be evaluated in the standard manner as

[TABLE]

Thus one can see that the background (3.9) describes a black hole whose horizon is located at $Z=\infty$ .

The Bekenstein-Hawking entropy can also be computed as

[TABLE]

Here the area $A$ is taken as $A=1$ because the horizon is just a point, and the effective Newton constant $G_{\rm eff}$ can be read off from the classical action as

[TABLE]

On the other hand, the holographic entropy can be computed by using the renormalized boundary stress tensor. For the detailed computation like the regularization and the counter term, see [28]. As a result, the renormalized boundary stress tensor is evaluated as

[TABLE]

Then by using the thermodynamic relation

[TABLE]

the entropy is obtained as

[TABLE]

where $S_{T_{\rm H}=0}$ is an integration constant. Thus the holographic entropy agrees with the Bekenstein-Hawking entropy, up to the temperature-independent constant.

The main goal of this paper is to realize this correspondence of the entropies for a deformed black hole solution introduced in the next section.

4 Deforming the AP model

In this section, we consider deforming the AP model so that the deformed AdS2 metric (2.18) is supported as a solution. For simplicity, the matter fields are turned off hereafter. Along this line, as well as the dilaton itself, the dilaton potential also has to be deformed from a simple quadratic one (3.5) to a hyperbolic function, similarly to integrable deformations.

4.1 The deformed AP model

The deformed metric

Before discussing the dilaton and the dilaton potential, it is helpful to rewrite the deformed metric (2.18) as

[TABLE]

Here we have introduced new quantities: a coordinate vector $X^{I}$ and a parameter vector $P_{I}$ defined as

[TABLE]

The metric of the embedding space $\mathbb{M}^{2,1}$ is taken as $\eta_{IJ}=\text{diag}(-1,+1,-1)$ . The inner products are defined as

[TABLE]

These three products $X\cdot X$, $P\cdot P$ and $X\cdot P$ are transformed as scalars under the $SL(2,\mathbb{R})$ transformation666 This $SL(2,\mathbb{R})$ transformation is the usual one generated by three transformations,

1. time translation, 2) dilatation and 3) special conformal transformation.. For example, $X\cdot P$ is transformed as $X\cdot P=\tilde{X}\cdot\tilde{P}$, where $\tilde{X}$ and $\tilde{P}$ are new coordinate and parameter vectors, respectively.

Using the $SL(2,\mathbb{R})$ transformation, we can choose the vector $\tilde{P}$ freely as long as it satisfies the relation $\tilde{P}\cdot\tilde{P}=P\cdot P=-\tilde{\omega}$ . Note that only the warped factor of the metric changes like

[TABLE]

because the rigid AdS2 part is invariant under the $SL(2,\mathbb{R})$ transformation.

The dilaton sector

Given the deformed metric (2.18) [or equivalently (4.1)] , by solving the equations of motion (3.3) without the matter fields, the dilaton $\Phi$ is determined as

[TABLE]

when the dilaton potential is deformed as

[TABLE]

Here $c_{1}$ and $c_{2}$ are arbitrary constants

In the undeformed limit $\eta\to 0$ , the dilaton (4.6) is reduced to

[TABLE]

and thus the dilaton (3.7) in the AP model has been reproduced when $c_{1}=1$ and $c_{2}=1$ . Remarkably, the three parameters $\alpha$ , $\beta$ and $\gamma$ correspond to $a$ , $b$ and $c$ in (3.7) , respectively. Similarly, as $\eta\to 0$ , the upper branch of the potential (4.9) reduces to

[TABLE]

while the lower branch vanishes. Thus the dilaton potential of the AP model is reproduced when $c_{2}=1$ . In total, the case with $c_{1}=c_{2}=1$ is associated with the AP model and hence we will work with $c_{1}=c_{2}=1$ hereafter.

The vacuum solution in the deformed AP model

In summary, the deformed AP model is specified by the deformed dilaton potential,

[TABLE]

and the vacuum solution is given by

[TABLE]

where

[TABLE]

4.2 A deformed black hole solution

In this subsection, we study a deformed black hole solution contained as a special case of the general vacuum solution (obtained in the previous subsection). This solution can be regarded as a deformation of the black hole solution presented in [28].

In the following, instead of $\tilde{\omega}$ , we use a new parameter $\mu$ defined as

[TABLE]

so as to make our notation the same as that of the AP model. Here it may be worth noting that the black hole temperature is related to the modification of the CYBE. The zero temperature case corresponds to the homogeneous CYBE and the temperature is measured by negative values of $c$ . Solutions of the mCYBE with negative (positive) $c$ are called the split (non-split) type. The well-known example of the non-split type is the $q$-deformation of AdS5 [36], while the split type has gotten little attention. For the recent progress on the split type, see [64]. It may be interesting to seek some connection between black hole geometries and solutions of split type.

By performing the same coordinate transformation as in the undeformed case like

[TABLE]

the deformed black hole solution is obtained as

[TABLE]

In this coordinate, the Ricci scalar (2.20) is rewritten as

[TABLE]

In the following, we impose that

[TABLE]

so as to ensure the existence of the undeformed limit777 Otherwise, it is not posible to take the undeformed limit $\eta\rightarrow 0$ because $\eta^{2}>1/\mu$ .. Note here that this background has a naked singularity at $Z=Z_{0}$ , where

[TABLE]

This is a peculiar feature of the Yang-Baxter deformed geometry based on the modified CYBE like the $\eta$-deformation of AdS5 [38]. From (2.20), in the region with $Z>Z_{0}$ the Ricci scalar takes negative values, while for $0<Z<Z_{0}$, it has positive values (See Fig. 2) . In the undeformed limit $\eta\rightarrow 0$ , $Z_{0}$ is sent to zero and the singularity disappears because the undeformed spacetime is just AdS2 . In the following discussion, we focus upon the negative-curvature region ($Z>Z_{0}$) . Therefore, we are concerned with only the upper branch of the potential (4.9) .

By performing the following coordinate transformation,

[TABLE]

the metric takes a Schwarzschild-like form888The factor 4 is included so as to reproduce the result of [28].

[TABLE]

where the scalar function $F(r)$ is defined as

[TABLE]

In this coordinate system, the dilaton takes the simplest form,

[TABLE]

The locations of the boundary and black hole horizon are

[TABLE]

Bekenstein-Hawking entropy

Let us compute the Bekenstein-Hawking entropy of the deformed black hole by utilizing the coordinate system (4.20) .

The Hawking temperature $T_{\rm H}$ is given by the standard formula:

[TABLE]

This is the same result as the undeformed case. By assuming that the horizon area A is $1$ and using the effecting Newton constant $G_{\rm eff}$ in (3.14) , the Bekenstein-Hawking entropy $S_{\rm BH}$ can be computed as

[TABLE]

In the undeformed limit $\eta\to 0$ , the entropy is reduced to

[TABLE]

and thus the result of AP model has been reproduced.

5 The boundary computation of entropy

In this section, we compute the entropy of the deformed black hole by evaluating the renormalized boundary stress tensor. Now that the boundary structure is drastically changed, the first thing is to determine the location of the holographic screen. In the following, we take the screen on the singularity by following the proposal of [60]. More precisely, by introducing a UV cut-off $\epsilon$ , the boundary is taken just before the singularity ($Z=Z_{0}+\epsilon$) .

In the conformal gauge, the total action including the Gibbons-Hawking term can be rewritten as

[TABLE]

$K$ is the extrinsic curvature and $\gamma$ is the extrinsic metric. By using the explicit expression of the deformed black hole solution in (4.15), the on-shell bulk action can be evaluated on the boundary,

[TABLE]

Recall that the regulator $\epsilon$ is introduced such that $Z-Z_{0}=\epsilon$ , the on-shell action can be expanded as

[TABLE]

To cancel the divergence that occurs as the bulk action approaches the boundary, it is appropriate to add the following counter term:999The dual-theory interpretation of it is not so clear because it cotains an infinite number of polynomials and also depends on the temperature explicitly. Another counter term may be allowed and it would be nice to seek for it by following the procedure in [65]. We are grateful to Ioannis Papadimitriou for this point.

[TABLE]

Here the scalar function $F$ is already given in (4.21) and hence

[TABLE]

Note that the inside of the root of (5.4) is positive due to the condition (4.17) . The extrinsic metric $\gamma_{tt}$ on the boundary is obtained as

[TABLE]

In the undeformed limit $\eta\to 0$, this counter term reduces to

[TABLE]

because $\Phi^{2}-1>0$. This is nothing but the counter term utilized in the AP model [28].

It is straightforward to check that the sum $S=S_{g,\Phi}+S_{\rm ct}$ becomes finite on the boundary by using the expanded form of the counter term (5.4):

[TABLE]

Around the boundary, the warped factor of the metric in (4.15) can be expanded as

[TABLE]

Hence, by normalizing the boundary metric as

[TABLE]

the boundary stress tensor can be defined as

[TABLE]

After all, $\langle\hat{T}_{tt}\rangle$ has been evaluated as

[TABLE]

To compute the associated entropy, $\langle\hat{T}_{tt}\rangle$ should be identified with energy $E$ like

[TABLE]

where we have used the expression of the Hawking temperature (4.24) . Then by solving the thermodynamic relation (3.16) again, the entropy is obtained as

[TABLE]

Here $S_{T_{\rm H}=0}$ has appeared as an integration constant that measures the entropy at zero temperature. Thus the resulting entropy precisely agrees with the Bekenstein-Hawking entropy (4.25) , up to the temperature-independent constant.

Finally, it should be remarked that this agreement is quite non-trivial because the deformation changes the UV region of the geometry drastically. Hence the location of the holographic screen and the choice of the counter term are far from trivial. Although the holographic screen was supposed to be the singularity, inversely speaking, this agreement of the entropies here supports that the proposal in [60] would work well. As for the geometrical meaning of the counter term (5.4), we have no definite idea. It is significant to figure out a systematic prescription to produce the counter term (5.4) .

6 Conclusion and discussion

In this paper, we have discussed deformations of the AP model by following the Yang-Baxter deformation technique. To support the deformed AdS2 metric, the dilaton itself is deformed and the dilaton potential is also modified from the polynomial to the hyperbolic function-type potential, similarly to integrable deformations. We have obtained the general vacuum solution for the deformed potential.

A particularly interesting example is a deformed black hole solution. The deformation makes the spacetime structure around the boundary change drastically and a new naked singularity appears. The Hawking temperature is the same as in the undeformed case, but the Bekenstein-Hawking entropy is modified due to the deformation. This entropy has also been reproduced by evaluating the renormalized stress tensor with an appropriate counter term on the regularized screen close to the singularity.

There are some open problems. A possible generalization is to include matter fields, though it has not succeeded yet. The matter contribution would not be so simple in comparison to the AP model. It is also interesting to consider lifting up our results to higher dimensional setups. Possibly, the most intriguing issue is to clarify the dual quantum mechanics for the deformed black hole presented here. A candidate would be a deformed SYK model which would be constructed by performing a disordered quench for a $q$-deformed Heisenberg magnet. When an infinitesimal deformation of the deformed AdS2 geometry is considered, one would encounter a deformed Schwarzian derivative, though it seems difficult to determine what it is because there is no $SL(2)$ invariance on the boundary in comparison to the standard setup studied in [12, 29, 30]. It is also interesting to study Yang-Baxter deformations of the Callan-Giddings-Harvey-Strominger (CGHS) model [66] by following [61, 67].

There are some future directions associated with Yang-Baxter deformations as well. Now that we know the classical $r$-matrix which leads to the black hole geometry, it would be interesting to consider a Yang-Baxter deformation of higher-dimensional AdS with this $r$-matrix. In the study of Yang-Baxter deformations, it has been a long standing problem to determine where the holographic screen is, while there was a proposal for the $\eta$-deformed AdS5 [60] but it has not been supported by concrete evidence before this paper. It is significant to find out more supports to clarify the holographic interpretation for general Yang-Baxter deformations.

We hope that the deformed AP model would provide a new arena to study the correspondence between nearly AdS2 geometries and 1D quantum mechanical system like the SYK model or its cousins.

Acknowledgments

We are very grateful to Ahmed Almheiri, Heng-Yu Chen, Koji Hashimoto, Keiju Murata, Ioannis Papadimitriou, Jun-ichi Sakamoto and Yuki Yokokura for useful comments and discussions. The work of H.K. is supported by the Japan Society for the Promotion of Science (JSPS). The work of K.Y. is supported by the Supporting Program for Interaction-based Initiative Team Studies (SPIRITS) from Kyoto University and by a JSPS Grant-in-Aid for Scientific Research (C) No. 15K05051. This work is also supported in part by the JSPS Japan-Russia Research Cooperative Program and the JSPS Japan-Hungary Research Cooperative Program.