We consider two N = 4 supersymmetric gauge theories connected by an interface and the gravity dual of this system. This interface is expressed by a fuzzy funnel solution of Nahm's equation in the gauge theory side. The gravity dual is a probe D5-brane in AdS 5 × S 5 . The potential energy between this interface and a test particle is calculated in both the gauge theory side and the gravity side by the expectation value of a Wilson loop. In the gauge theory it is evaluated by just substituting the classical solution to the Wilson loop. On the other hand it is done by the on-shell action of the fundamental string stretched between the AdS boundary and the D5-brane in the gravity. We show the gauge theory result and the gravity one agree with each other.

We analyze the symmetry realized asymptotically on the two dimensional boundary of AdS 3 geometry in topologically massive gravity, which consists of the gravitational Chern-Simons term as well as the usual Einstein-Hilbert and negative cosmological constant terms. Our analysis is based on the conventional canonical method and proceeds along the line completely parallel to the original Brown and Henneaux's. In spite of the presence of the gravitational Chern-Simons term, it is confirmed by the canonical method that the boundary theory actually has the conformal symmetry satisfying the left and right moving Virasoro algebras. The central charges of the Virasoro algebras are computed explicitly and are shown to be left-right asymmetric due to the gravitational Chern-Simons term. It is also argued that the Cardy's formula for the BTZ black hole entropy capturing all higher derivative corrections agrees with the extended version of the Wald's entropy formula. The M5-brane system is illustrated as an application of the present calculation.

We present a new exact black hole solution in three dimensional Einstein gravity coupled to a single scalar field. This is one of the extended solutions of the BTZ black hole and has in fact AdS 3 geometries both at the spatial infinity and at the event horizon. An explicit derivation of Virasoro algebras for CFT 2 at the two boundaries is shown to be possibleà la Brown and Henneaux's calculation. If we regard the scalar field as a running coupling in the dual two dimensional field theory, and its flow in the bulk as the "holographic" renormalization group flow, our black hole should interpolate the two CFT 2 living at the infinity and at the horizon. Following the Hamilton-Jacobi analysis by de Boer, Verlinde and Verlinde, we calculate the central charges c UV and c IR for the CFT 2 on the infinity and the horizon, respectively. We also confirm that the inequality c IR < c UV is satisfied, which is consistent with the Zamolodchikov's c-theorem.It has been believed that three dimensional gravity with negative cosmological constant is an important key to uncover some aspects of the quantum gravity. A vacuum solution is described by three dimensional anti-de Sitter (AdS 3 ) geometry globally [1], and a black hole solution with mass and angular momentum (BTZ black hole) is constructed from locally AdS 3 geometry with appropriate identifications of boundaries [2]. All these geometries become asymptotically AdS 3 , and two dimensional conformal field theory (CFT 2 ) is expected to exist at the boundary of the three dimensional geometry [3]. This is one of important examples of AdS/CFT correspondence [4], which enables us to evaluate physical quantities in CFT from the gravity side [5,6]. Thus many efforts have been directed to understand quantum nature of the three dimensional gravity from the viewpoint of AdS/CFT correspondence.In fact, by using the canonical formalism, Brown and Henneaux showed that general coordinate transformations which preserve the boundary behavior of the geometries form Virasoro algebras for left and right movers [3]. Furthermore they succeeded to evaluate central extensions of these algebras and found that the central charges for left and right movers take the same value. From the CFT 2 viewpoint, the globally AdS 3 corresponds to the ground state, and the BTZ black hole does to excited states. Surprisingly, the macroscopic entropy of the BTZ black hole is explained by counting the number of degenerate states in CFT 2 [7].The Brown-Henneaux's canonical approach has been applied to several interesting three dimensional theories. In ref.[8], the gravity theory with higher derivative corrections was considered. The values of central charges are scaled because of the higher derivative terms. An application to the topologically massive gravity was done in ref. [9], and left-right asymmetric central charges were derived due to the gravitational Chern-Simons term. On the other hand, the canonical approaches to the theories of Einstein gravity coupled to scalar fields were investigated in refs.[10]- ...

We consider the holographic renormalization group (RG) flow in three dimensional gravity with the gravitational Chern-Simons term coupled to some scalar fields. We apply the canonical approach to this higher derivative case and employ the Hamilton-Jacobi formalism to analyze the flow equations of two dimensional field theory. Especially we obtain flow equations of Weyl and gravitational anomalies, and derive c-functions for left and right moving modes. Both of them are monotonically non-increasing along the flow, and the difference between them is determined by the coefficient of the gravitational Chern-Simons term. This is completely consistent with the Zamolodchikov's c-theorem for parity-violating two-dimensional quantum field theories.