A geometrical representation of the quantum information metric in the gauge/gravity correspondence

We study a geometrical representation of the quantum information metric in the gauge/gravity correspondence. We consider the quantum information metric that measures the distance between the ground states of two theories on the field theory side, one of which is obtained by perturbing the other. We show that the information metric is represented by a back reaction to the volume of a codimension-2 surface on the gravity side if the unperturbed field theory possesses the Poincare symmetry.

In [5], we found a universal formula that represents the quantum information metric in terms of a back reaction to a geometrical quantity in the bulk. We considered a CFT and a theory obtained by perturbing the CFT by a primary operator, and calculated the quantum information metric that measures the distance between the ground states of the two theories. We showed that the quantum information metric is represented by a back reaction to the volume of a codimension-2 surface. This is universal in the sense that it holds for perturbations by scalar, vector and tensor operators.
In this letter, we push forward with the above project. We show that the above formula holds also for a field theory, which is not necessarily a CFT and has a gravity dual whose background geometry is not necessarily AdS. We introduce a covariant calculation on the gravity side, which allows us to derive a condition that must be satisfied by the dual geometry in order that the quantum information metric can be represented by the back reaction to the volume of the codimension-2 surface. We find that the condition implies that the original field theory possesses the Poincare invariance.

On-shell action and Einstein equation
In this section, we consider a back reaction caused by a scalar field in a background geometry in the bulk. The background geometry is supposed to be dual to a field theory on the boundary which is perturbed by an operator corresponding to the scalar field in the bulk.
We represent the on-shell action for the scalar field in terms of the back reaction to the background geometry. As seen in the next section, the information metric that measures the distance between the ground states of the original and perturbed field theories corresponds to the on-shell action for the scalar field. Thus, we eventually obtain a formula that represents the information metric in terms of the back reaction to the background geometry.
The gravity action consists of the Einstein-Hilbert action 1 and the Gibbons-Hawking term: where the bulk cosmological constant Λ is given by The matter action is given by where Ψ is a scalar field that gives the background metricĝ, and Φ is a perturbation that gives a back reaction corresponding tog µν in (2.2). In what follows, we keep the contribution up to the second order in Φ.
The Einstein equations are given by We substitute (2.2) into (2.6) and expand (2.6) in terms ofg µν . Then, the zeroth and first orders readR respectively, wherê Because of (2.3), there exist d Killing vectors corresponding to translations in the (d + 1)dimensional background spacetime. We denote one of those by ξ µ , which satisfieŝ We can further assume that the invariance under the translation corresponding to ξ µ is preserved by the perturbation: We contract ξ µ with the Einstein equations (2.8) as where ξ 2 =ĝ µν ξ µ ξ ν and expandR µν to the first order ing µν as (2.14) By using (2.7), (2.11), (2.12) and (2.14), we obtain from (2.13) By using the equation of motion for Φ, we see that the RHS of (2.15) is a total derivative term, −4πG N∇µ Φ∇ µ Φ . The third and fourth lines in the LHS are calculated as Then, in order for the LHS to be total derivative terms, the above expression must vanish.
This leads us to impose a condition We assume that the boundary where the dual field theory lives is specified by z = ǫ. Then, integrating both sides of (2.18) over the bulk yields where (2.1) is used, and the on-shell action for Φ in the RHS is given by Hereafter, we identify the direction of ξ i with that of x 0 such that ξ z = 0, ξ 0 = 1 and ξ a = 0. We make an ADM-like decomposition of the metric g ij in d dimensions as where a, b = 1, . . . , d − 1. Expanding N, N a and γ ab around the background as N =N +Ñ , N a =N a +Ñ a and γ ab =γ ab +γ ab , leads tô (2.25) The boundary specified by z = ǫ where the dual field theory lives is a codimension-2 hyperplane perpendicular to ξ i . The volume of hyperplane is given up to the first order iñ We consider the z derivative of δV : where the prime represents the z-derivative. Here the first term and the second term in the RHS of (2.28) represent the canonical scaling contribution and a nontrivial scaling contribution, respectively. Then , (2.25) is rewritten as (2.30) Here we require the second term in the RHS of (2.30) to vanish. Then,γ ab is determined aŝ with C ab being a constant tensor. Furthermore, (2.24) implies thatĝ ij is expressed aŝ with D ij being a constant tensor.
D ij is diagonalizable so thatN a can be set to zero and we redefine g zz → 1 and we reproduce the result in [5].
(2.32) indicates that the background spacetime has d-dimensional Poincare invariance.
This implies that the original dual field theory also has it.
The background spacetimeĝ µν (z) satisfying (2.32) and the background matter field Ψ(z) are determined by the equations of motion where we define 1 As an example of solutions, we consider the GPPZ flow [15] which is dual to 4-dimensional N = 1 super Yang-Mills theory, where the scalar field has a non-zero VEV. In this case, the potential U 1 (Ψ) is given by and the solution to (2.37) and (2.38) is given by (2.40)

Information metric for a dual operator to bulk scalar field
In this section, we introduce the quantum information metric and show that the one for the original and perturbed theories on the field theory side is represented by the geometrical quantity δv in (2.33).
We consider a field theory defined by a Lagrangian density L 0 on d-dimensional Euclidean spacetime whose coordinates are x i (i = 0, . . . , d − 1), where x 0 ≡ τ is viewed as the Euclidean time. We also consider another field theory with a Lagrangian density L obtained by perturbing the theory as where O(x) is a scalar operator and φ (0) ( x) is a source independent of the time τ .
We denote the ground states of the theories L 0 and L by |Ω 0 and |Ω , respectively.
Then, the inner product Ω|Ω 0 is given by a path integration where Z 0 and Z is the partition functions of the theories L 0 and L, respectively. We assume that O(x) 0 = 0 and the time reversal symmetry: 3) The information metric G that measures the distance between the ground states of the two theories is obtained by expanding Ω|Ω 0 up to the second order in φ (0) : where T is the volume of the time direction. Here we assume that we make an appropriate regularization for the two point function of O to suppress a divergence occurring at τ = τ ′ = 0. In the case in which L 0 is a CFT with O a primary operator, a regularization is given in section 3 of [5].
We apply the above result to the case in the previous section: the theory L 0 possesses a gravity dual corresponding to the background geometry with the d-dimensional Poincare invariance, and the operator O corresponds to a scalar field Φ, which coincides with φ (0) on the boundary. The τ -independence of φ (0) is consistent with (2.12). We consider a situation where the classical approximation on the gravity side is valid. By using the GKP-Witten we can show that Thus, by using (2.33), we obtain a formula, This formula was obtained in [5] in the case where a CFT is perturbed by a primary operator O on the field theory side and the background geometry on the gravity side is given by the AdS.

Conclusion
In this letter, we considered a field theory that has a gravity dual, and perturbed it by an operator which corresponds to a scalar field in the bulk. We performed a covariant calculation to find the condition that must be satisfied by the bulk geometry in order that the on-shell action for the scalar field is represented by a back reaction to the volume of a codimension-2 surface. The condition implies the Poincare invariance of the original field theory. We saw that the quantum information metric that measures the distance between the ground states of the original and perturbed theories is represented by the on-shell action. While we considered only a perturbation by a scalar field, we should obtain the same results for perturbations by vector and tensor fields. Thus, we conclude that the universal formula in [5] that represents the quantum information metric in terms of the back reaction to the volume of the codimension-2 surface is extended to the case of a general gauge/gravity correspondence if the above condition is satisfied. It is interesting to elucidate what is represented by the extra terms in the RHS of (2.30) which we have if we do not impose the condition of the Poincare invariance. We hope that our result leads us to gain deeper understanding of the relationship between quantum information and quantum geometry.