Holographic renormalization group flows in two-dimensional gravity and AdS black holes

We look into the AdS black holes from two-dimensional gravity perspective. In this work, we extend the previous results of holographic renormalization group flows to dimensions two. By introducing a superpotential, we derive the flow equations in two-dimensional dilaton gravity. We also find a quantity which monotonically decreases along flows and give some comments on holographic c-theorem. As examples, we show that recently studied supersymmetric AdS black hole solutions generically dimensionally reduce to two-dimensional dilaton gravity, and obtain the flow equations for black hole solutions.


Introduction
Recently, via the AdS/CFT correspondence, [1], following the grounding works of [2,3,4], there has been remarkable development in microscopic counting of microstates of supersymmetric AdS black holes via twisted or superconformal indices of dual field theories.One of the main observation was the discovery of entropy functional which extremizes to give the Bekenstein-Hawking entropy of the black holes.In the dual field theory, this extremization procedure is presumably understood as the I-extremization, where the Witten index of 1d superconformal quantum mechanics is extremized to give the entropy.Here, the 1d quantum mechanics is dual to the AdS 2 horizon geometry of the black hole solutions.
On the other hand, even though supergravity in two dimensions has been studied long enough, [15,16,17,18,19,20], our understanding is elementary and two-dimensional supergravity models are rare compare to the other dimensional theories.
2 Review of gravity in dimensions higher than two For domain wall backgrounds in dimensions higher than two, the second order equations of motion reduce to first order flow equations, [21,22,23].The flow equations arises naturally, as they are the Hamilton-Jacobi equations of dynamical system of gravity-scalar theories, [24].In supergravity, the first order equations reproduce the BPS equations obtained from the supersymmetry variations, [21].Solutions of the equations are known to be non-perturbatively stable, [26].In this section, we review section 9 of [27].We consider the gravity coupled to a scalar field, in Euclidean spacetime.The equations of motion are Therefore, at critical points, the scalar potential satisfies

Flow equations
We consider the domain wall background, The equations of motion reduce to where the primes denote the derivative with respect to r.However, due to the Bianchi identity, the first equation is obtained from the last two equations.Hence, there are only two independent equations of motion, Miraculously, by employing a superpotential, W , the second order equations reduces to first order flow equations, where the superpotential generates the scalar potential, This result trivially extends to theories with multiple scalar fields.A large class of solutions in supergravity falls into the flat domain wall backgrounds we considered, like, holographic RG flows, and easily extended to wrapped branes, and black hole solutions.For more complicated backgrounds, like Janus solutions, [26,34,35,36], which are AdS domain walls, and also sphere domain walls, the basic structure of the flow equations stays the same, but gets more involved.

Holographic c-theorem
One of the immediate applications of holographic RG flows is the c-theorem, [25,21,22].See section 10 of [27] for a review.For the domain wall background, from the difference of the first and the second equations in (2.5), we find that If we define a function, due to (2.9), derivative of the function will be always nonnegative, (2.11) Therefore, this function naturally introduces holographic c-function, which monotonically decreases along holographic RG flows, from ultraviolet to infrared.Moreover, from the flow equations, (2.7), we have (2.12) See, for example, [9,10], for the a-function of 4d SCFTs.

Two-dimensional dilaton gravity
We consider two-dimensional dilaton gravity coupled to a scalar field, where Φ is the dilaton and φ is the scalar field.In two dimensions, we cannot go to Einstein frame by performing conformal transformations, and they just transform to other string frames.
Choosing value of the constant parameter, α, is equivalent of conformal transformations.When we study examples from higher dimensions latter in (4.17), it will be clear why we have chosen the factor to be (2α + 1).The equations of motion are On the left hand side of the Einstein equation in (3.2), in addition to the Einstein tensor, there are derivative terms of the dilaton.In two dimensions, the Einstein tensor vanishes identically, Therefore, at critical points, the scalar potential satisfies

Flow equations
We consider the domain wall background, The equations of motion reduce to where the primes denote the derivative with respect to r.
From the sum of the first and the second equations in (3.6), we obtain a relation between derivatives of functions without the scalar potential, Starting from this relation, we look for the first order flow equations by trial and error.By introducing a superpotential, W , we obtain the flow equations, Therefore, at critical points, the superpotential satisfies The superpotential produces the scalar potential by This result trivially extends to theories with multiple scalar fields.To our knowledge, it is the first derivation of the first order flow equations in two-dimensional dilaton gravity.

Comment on the holographic c-theorem
Unlike the gravity in dimensions higher than two, in two-dimensional dilaton gravity, from the equations of motion, (3.6), we do not find analogous properties of A , Φ , or (∂W/∂Φ) , e.g., we have from the difference of the first and the second equations in (3.6).It would be interesting to search for a quantity which could be analogously used as a holographic c-function, if there is such a quantity.The relation of the presumable holographic c-function and the entropy functional introduced in [4,37], is also intriguing.

Entropy of higher-dimensional black holes
We make observation that for higher-dimensional supersymmetric AdS d+1 black holes which dimensionally reduce to two-dimensional dilaton gravity on their AdS 2 horizons, the Bekenstein-Hawking entropy is given by the value of the dilaton, Φ, at AdS 2 fixed point by1 where vol horizon is volume of the horizon geometry of unit radius.We will test the observation for some supersymmetric AdS black holes in the next section.
4 Two-dimensional dilaton gravity from gauged supergravity 4.1 Supersymmetric AdS 4 black holes We review the supersymmetric AdS 4 black hole solutions of [2] and [38,39].We employ the conventions of appendix A in [4].Their microstates are microscopically counted by topologically twisted index of 3d SCFTs, [3,4].The action of gauged N = 2 supergravity in four dimensions is2 where the scalar potential is If we define the superpotential by then, the scalar potential is obtained from We set the gauge coupling constant to be We introduced a parametrization of three real scalar fields, where The field strength of four U (1) gauge fields are For the supersymmetric AdS 4 black hole solutions of [2], we consider the background of where Σ denotes the Riemann surfaces of curvatures, k = ±1.The field strength of the gauge fields are where the magnetic charges, a a , are constant and V ol Σ is the unit volume form.The first order BPS equations are obtained by solving the supersymmetry variations of the fermionic fields, and there is a twist condition on the magnetic charges, One can solve the BPS equations and obtain the supersymmetric black hole solutions which are interpolating between the AdS 4 boundary and the AdS 2 horizon.
There is an AdS 2 × Σ solution of the horizon.We introduce another parametrization of the scalar fields, or, equivalently, The horizon is at and where Now we dimensionally reduce the action on the background of supersymmetric AdS 4 black holes.The reduction ansatz for the metric is ds 2  4 = e 2αg ds 2 2 + e 2g ds 2 Σ , (4.17 where α is a constant parameter.The reduced action is two-dimensional dilaton gravity, where the scalar potential is The scalar potential satisfies the relations, (3.4), at the AdS 2 , critical point in (4.14) and (4.15).We consider the background, We obtain the flow equations which are equivalent to the equations of motion of two-dimensional dilaton gravity, where the superpotential is given by The superpotential satisfies the relations, (3.9), at the AdS 2 , critical point in (4.14) and (4.15).The scalar potential is obtained from By introducing the superpotential, we note that the flow equations are merely a rewriting of the BPS equations in AdS 4 , (4.10), as it should.Moreover, we could reproduce the flow equations from the general analysis of section 2 by replacing and setting α = 0 .( Finally, by employing the formula we proposed in (3.12), we reproduce the Bekenstein-Hawking entropy of the supersymmetric AdS 4 black holes obtained in [4], where vol Σ is the area of Riemann surfaces of unit radius, in this case.It provides a test that the twodimensional dilaton contains the information of the Bekenstein-Hawking entropy of higher dimensional AdS black holes.
The bosonic field content of pure F (4) gauged supergravity, [41], consists of the metric, g µν , a real scalar, φ, an SU (2) gauge field, A I µ , I = 1, 2, 3, a U (1) gauge field, A µ , and a two-form gauge potential, B µν .The field strengths are defined by The action is given by where g is the SU (2) gauge coupling constant and m is the mass parameter of the two-form field.The scalar potential is If we define the superpotential by then, it gives the scalar potential by We employ the mostly-minus signature.
For the supersymmetric AdS 6 black hole solutions, we consider the background of The gauge fields are where the magnetic charges, a 1 and a 2 , are constant, and the U (1) gauge field is A µ = 0.The two-form field is given by and the three-form field strength of the two-form gauge potential vanishes identically.The first order BPS equations are obtained by solving the supersymmetry variations of the fermionic fields, and there are twist conditions on the magnetic charges, where k = ±1 are curvatures of the Riemann surfaces and λ = ±1.The AdS 2 × Σ g 1 × Σ g 2 horizon solution is given by where g 1 and g 2 are genus of the Riemann surfaces.The full black hole solutions are interpolating between the AdS 6 boundary and the AdS 2 horizon.Now we dimensionally reduce the action on the background of supersymmetric AdS 6 black holes.The reduction ansatz for the metric is The reduced action is two-dimensional dilaton gravity, where the scalar potential is The scalar potential satisfies the relations, (3.4), at the AdS 2 , critical point, (4.37).This reduction was previously performed in [42].We consider the background, We only consider the case of We obtain the flow equations which are equivalent to the equations of motion of two-dimensional dilaton gravity, where the superpotential is given by The superpotential satisfies the relations, (3.9), at the AdS 2 , critical point, (4.37).The scalar potential is obtained from By introducing the superpotential, we note that the flow equations are merely a rewriting of the BPS equations in AdS 6 , (4.35), as it should.Finally, by employing the formula we proposed in (3.12), we reproduce the Bekenstein-Hawking entropy of the supersymmetric AdS 6 black holes obtained in [29], It provides a test that the two-dimensional dilaton contains the information of the Bekenstein-Hawking entropy of higher dimensional AdS black holes.

Supersymmetric AdS 5 black holes
We review the supersymmetric AdS 5 black hole solutions of gauged N = 4 supergravity in five dimensions in [30].See also [43].The microstates are microscopically counted by topologically twisted index of 4d N = 4 super Yang-Mills theory, [33].
The bosonic field content of SU (2) × U (1)-gauged N = 4 supergravity in five dimensions, [44], consists of the metric, g µν , a real scalar, ϕ, 4 an SU (2) gauge field, A I µ , I = 1, 2, 3, a U (1) gauge field, a µ , and two-form gauge potentials, B α µν .The field strengths are defined by The action is given by where g 1 and g 2 are the U (1) and SU (2) gauge coupling constants, respectively, and we define The scalar potential is If we define the superpotential by then, it gives the scalar potential by We employ the mostly-minus signature.
For the supersymmetric AdS 5 black hole solutions, we consider the background of where The gauge fields are where the magnetic charges, a, b, and c, are constant.The U (1) gauge field and two-form field are vanishing.The first order BPS equations are obtained by solving the supersymmetry variations of the fermionic fields, and there are twist conditions on the magnetic charges, The AdS 2 × H 3 horizon solution is given by where g = 2 √ 2. The full black hole solutions are interpolating between the AdS 5 boundary and the AdS 2 horizon.Now we dimensionally reduce the action on the background of supersymmetric AdS 5 black holes.The reduction ansatz for the metric is The reduced action is two-dimensional dilaton gravity, where the scalar potential is We obtain the flow equations which are equivalent to the equations of motion of two-dimensional dilaton gravity, where the superpotential is given by

.65)
By introducing the superpotential, we note that the flow equations are merely a rewriting of the BPS equations in AdS 5 , (4.56), as it should.Finally, by employing the formula we proposed in (3.12), we reproduce the Bekenstein-Hawking entropy of the supersymmetric AdS 5 black holes obtained in [33], It provides a test that the two-dimensional dilaton contains the information of the Bekenstein-Hawking entropy of higher dimensional AdS black holes.

Conclusions
In this paper, we derived the flow equations in two-dimensional dilaton gravity.We also observed that value of the two-dimensional dilaton at the AdS 2 fixed point encodes the Bekenstein-Hawking entropy of higher-dimensional AdS black holes with AdS 2 horizons.As we have seen in the examples, twodimensional dilaton gravity is ubiquitous from dimensional reduction of AdS black hole solutions in string and M-theory.Their dynamics and entropy could be understood from two-dimensional perspective.This opens several intriguing directions we may pursue.Following the studies of higher-dimensional holographic renormalization group flows, [21,22,23], it is natural to further investigate the flow equations in two-dimensional dilaton gravity.First of all, it would be interesting to find the holographic c-theorem, [25,21,22], which we could not find in section 3.2.Also there could be the Hamilton-Jacobi formulation origin of the holographic RG flows analogous to the higher-dimensional flows, [24].It is also interesting to study the non-perturbative stability of solutions, [26].
In relation of AdS black holes from two-dimensional perspective, further understanding of dual 1d superconformal quantum mechanics could be pursued, e.g., appendix B of [3].
Our construction provides examples of two-dimensional dilaton gravity from string and M-theory.See also [54].It is interesting to understand the physics of two-dimensional theories along the line of [55,56,57,58].