Deformations of the Almheiri-Polchinski model

We study deformations of the Almheiri-Polchinski (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.


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 1 fermions with a random, all-to-all quartic interaction. This model is now called the Sachdev-Ye-Kitaev (SYK) model. 1 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 JHEP03(2017)173 hole in Einstein gravity is 2π/β [22,23], where β 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 AdS 2 geometries. Recently, a systematic way to perform integrable deformations, which is called the Yang-Baxter deformation [31][32][33][34], 2 has been intensively studied. 3 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 [69][70][71]. Hence it is important to get much deeper understanding of the simplest case like AdS 2 . 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 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. This paper is organized as follows. In section 2 we study the most general Yang-Baxter deformation of AdS 2 . 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

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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 AdS 2 In this section, we consider the most general Yang-Baxter deformation of the AdS 2 metric. First of all, we briefly describe a coset construction of the Poincaré AdS 2 Then we study the most general Yang-Baxter deformation of Poincaré AdS 2 . As a result, we obtain a three-parameter family of deformed AdS 2 spaces.

Coset construction of AdS 2
Let us recall a coset construction of the Poincaré AdS 2 metric (For the detail of the coset construction, for example, see [72]).
The starting point is that the AdS 2 geometry is represented by a coset By using the coordinates t and z, a coset representative g is parametrized as where H and D are the time translation and dilatation generators, respectively. By involving the special conformal generator C, the sl(2) algebra in the conformal basis is spanned as These generators can be represented by the so(1, 2) ones T I (I = 0, 1, 2) like where T I 's satisfy the commutation relations: In the following, we will work with T I 's in the fundamental representation, where σ i (i = 1, 2, 3) are the standard Pauli matrices.

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Note here that the coset (2.1) is symmetric as one can readily understand from (2.5). When vector spaces h and m are spanned as the Z 2 -grading structure is expressed as When representing the sl(2) algebra by a direct product (as vector spaces), the projection operator P : sl(2) → m can be defined as Now the Poincaré AdS 2 metric can be computed by performing coset construction. The left invariant one-form J = g −1 dg is expanded as Here e 0 and e 1 are the zweibeins, and ω 01 is the spin connection. With the parametrization (2.2), the zweibeins are given by 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 This is nothing but the AdS 2 metric in the Poincaré coordinates.
Hereafter, it is often convenient to use the the light-cone coordinates defined as Then the metric is rewritten as The exponential factor will play an important role in later discussion.

The general Yang-Baxter deformation
Let us next consider Yang-Baxter deformations of the AdS 2 metric (2.9). In the usual discussion, Yang-Baxter deformations [31][32][33][34] 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: Here η is a constant parameter which measures the deformation. Then R g is defined as a chain of operation like where g is the group element in (2.2). The key ingredient is a linear operator R : sl(2) → sl (2), and satisfy the (modified) classical Yang-Baxter equation [(m)CYBE]: (2)) . (2.14) Here c is a real constant parameter. The case with c = 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 where Ω IJ and M IJ are defined as 4 (2.16) Putting the ansatz (2.15) into the (m)CYBE (2.14) leads to an algebraic relation, After evaluating the expression (2.12) with the general ansatz (2.15), one can obtain the following metric: Here α, β and γ are defined as linear combinations of m p (p = 1, 2, 3) as follows:

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When η = 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 where we have introduced a new quantity, 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 η → 0. The expression (2.20) indicates that the deformed geometry contains both AdS and dS in general.

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.

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 Φ. The coordinates are parametrized as The classical action S is given by Here G is the Newton constant in 1+1 dimensions and U(Φ) is the dilaton potential.
In the following, we will work with the metric in the conformal gauge, where the light-cone coordinates are defined in (2.10). Then the equations of motion are given by

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The energy-momentum tenser for the matter field f is normalized as This expression (3.4) is valid for the general form of Ω(Φ).

The AP model
The AP model corresponds to a special case of 1+1 D dilaton gravity specified by the following condition: 5 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 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 AdS 2 [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 = µ/2, where µ is a real positive constant. Then by performing a coordinate transformation, the solution is rewritten into the following form: The new coordinates T and Z cover a smaller region which is in the inside of the entire Poincaré AdS 2 , as depicted in figure 1. The largest red triangle describes the Poincaré patch of AdS 2 as usual. The coordinate system (3.9) covers the inside of smaller triangle bounded by green lines. The right vertex corresponds to the black hole horizon which is specified as the point that T is finite but Z is infinity.

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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, Then the background (3.9) can be rewritten as 6 In this metric, the black hole horizon is located at ρ = √ µ, and the Hawking temperature T H can be evaluated in the standard manner as (3.12) 6 The 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.

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Thus one can see that the background (3.9) describes a black hole whose horizon is located at Z = ∞. The Bekenstein-Hawking entropy can also be computed as Here the area A is taken as A = 1 because the horizon is just a point, and the effective Newton constant G eff can be read off from the classical action as (3.14) 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 Then by using the thermodynamic relation the entropy is obtained as where S T 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.

Deforming the AP model
In this section, we consider deforming the AP model so that the deformed AdS 2 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.

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

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Here we have introduced new quantities: a coordinate vector X I and a parameter vector P I defined as The metric of the embedding space M 2,1 is taken as η IJ = diag(−1, +1, −1). The inner products are defined as These three products X · X, P · P and X · P are transformed as scalars under the SL(2, R) transformation. 7 For example, X · P is transformed as X · P =X ·P , whereX andP are new coordinate and parameter vectors, respectively. Using the SL(2, R) transformation, we can choose the vectorP freely as long as it satisfies the relationP ·P = P · P = −ω. Note that only the warped factor of the metric changes like because the rigid AdS 2 part is invariant under the SL(2, 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 Φ is determined as when the dilaton potential is deformed as 8 .

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In the undeformed limit η → 0, the dilaton (4.6) is reduced to 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 α, β and γ correspond to a, b and c in (3.7), respectively. Similarly, as η → 0, the upper branch of the potential (4.7) reduces to 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, , and the vacuum solution is given by where X · P = α + β t + γ (−t 2 + z 2 ) z .

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 ofω, we use a new parameter µ defined as 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 AdS 5 [43,44], while the split type has gotten little attention. For the recent progress on the split type, see [76,77]. It may be interesting to seek some connection between black hole geometries and solutions of split type.

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By performing the same coordinate transformation as in the undeformed case like the deformed black hole solution is obtained as (4.10) In this coordinate, the Ricci scalar (2.20) is rewritten as In the following, we impose that so as to ensure the existence of the undeformed limit. 9 Note here that this background has a naked singularity at Z = Z 0 , where This is a peculiar feature of the Yang-Baxter deformed geometry based on the modified CYBE like the η-deformation of AdS 5 [46]. 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 figure 2). In the undeformed limit η → 0, Z 0 is sent to zero and the singularity disappears because the undeformed spacetime is just AdS 2 . 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 (4.7). By performing the following coordinate transformation, the metric takes a Schwarzschild-like form 10 where the scalar function F (r) is defined as (4.16) 9 Otherwise, it is not posible to take the undeformed limit η → 0 because η 2 > 1/µ. 10 The factor 4 is included so as to reproduce the result of [28]. In this coordinate system, the dilaton takes the simplest form,

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The locations of the boundary and black hole horizon are boundary : r = ∞ , BH horizon : Bekenstein-Hawking entropy. Let us compute the Bekenstein-Hawking entropy of the deformed black hole by utilizing the coordinate system (4.15). The Hawking temperature T H is given by the standard formula: 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 eff in (3.14), the Bekenstein-Hawking entropy

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S BH can be computed as (4.20) In the undeformed limit η → 0, the entropy is reduced to and thus the result of AP model has been reproduced.

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 [69][70][71]. More precisely, by introducing a UV cut-off , the boundary is taken just before the singularity (Z = Z 0 + ).
In the conformal gauge, the total action including the Gibbons-Hawking term can be rewritten as K is the extrinsic curvature and γ is the extrinsic metric. By using the explicit expression of the deformed black hole solution in (4.10), the on-shell bulk action can be evaluated on the boundary, Recall that the regulator is introduced such that Z − Z 0 = , the on-shell action can be expanded as To cancel the divergence that occurs as the bulk action approaches the boundary, it is appropriate to add the following counter-term: 11 The 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 [78]. We are grateful to Ioannis Papadimitriou for this point.

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Here the scalar function F is already given in (4.16) and hence Note that the inside of the root of (5.4) is positive due to the condition (4.12). The extrinsic metric γ tt on the boundary is obtained as In the undeformed limit η → 0, this counter-term reduces to because Φ 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,Φ + S ct becomes finite on the boundary by using the expanded form of the counter-term (5.4): Around the boundary, the warped factor of the metric in (4.10) can be expanded as Hence, by normalizing the boundary metric aŝ γ tt = η γ tt , the boundary stress tensor can be defined as After all, T tt has been evaluated as To compute the associated entropy, T tt should be identified with energy E like

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Here S T 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 (4.20), 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 [69][70][71] 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).

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 AdS 2 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 AdS 2 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 [79] by following [72,80,81].
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 η-deformed AdS 5 [69-71] but it has not been supported by concrete evidence before this paper. It is JHEP03(2017)173 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 AdS 2 geometries and 1D quantum mechanical system like the SYK model or its cousins.