Holographic Renormalization Group Structure in Higher-Derivative Gravity

Classical higher-derivative gravity is investigated in the context of the holographic renormalization group (RG). We parametrize the Euclidean time such that one step of time evolution in (d+1)-dimensional bulk gravity can be directly interpreted as that of block spin transformation of the d-dimensional boundary field theory. This parametrization simplifies the analysis of the holographic RG structure in gravity systems, and conformal fixed points are always described by AdS geometry. We find that higher-derivative gravity generically induces extra degrees of freedom which acquire huge mass around stable fixed points and thus are coupled to highly irrelevant operators at the boundary. In the particular case of pure R^2-gravity, we show that some region of the coefficients of curvature-squared terms allows us to have two fixed points (one is multicritical) which are connected by a kink solution. We further extend our analysis to Minkowski time to investigate a model of expanding universe described by the action with curvature-squared terms and positive cosmological constant, and show that, in any dimensionality but four, one can have a classical solution which describes time evolution from a de Sitter geometry to another de Sitter geometry, along which the Hubble parameter changes drastically.Comment: 26 pages, 6 figures, typos correcte

Lattice Formulation for 2d N=(2,2), (4,4) Super Yang-Mills Theories without Admissibility Conditions

We present a lattice formulation for two-dimensional N=(2,2) and (4,4) supersymmetric Yang-Mills theories that resolves vacuum degeneracy for gauge fields without imposing admissibility conditions. Cases of U(N) and SU(N) gauge groups are considered, gauge fields are expressed by unitary link variables, and one or two supercharges are preserved on the two-dimensional square lattice. There does not appear fermion doubler, and no fine-tuning is required to obtain the desired continuum theories in a perturbative argument. This formulation is expected to serve as a more convenient basis for numerical simulations. The same approach will also be useful to other two-dimensional supersymmetric lattice gauge theories with unitary link variables constructed so far -- for example, N=(8,8) supersymmetric Yang-Mills theory and N=(2,2) supersymmetric QCD.Comment: 19 pages, no figure, (v2) reference added, minor corrections, version to be published in JHE

Holographic Renormalization Group

The holographic renormalization group (RG) is reviewed in a self-contained manner. The holographic RG is based on the idea that the radial coordinate of a space-time with asymptotically AdS geometry can be identified with the RG flow parameter of the boundary field theory. After briefly discussing basic aspects of the AdS/CFT correspondence, we explain how the notion of the holographic RG comes out in the AdS/CFT correspondence. We formulate the holographic RG based on the Hamilton-Jacobi equations for bulk systems of gravity and scalar fields, as was introduced by de Boer, Verlinde and Verlinde. We then show that the equations can be solved with a derivative expansion by carefully extracting local counterterms from the generating functional of the boundary field theory. The calculational methods to obtain the Weyl anomaly and scaling dimensions are presented and applied to the RG flow from the N=4 SYM to an N=1 superconformal fixed point discovered by Leigh and Strassler. We further discuss a relation between the holographic RG and the noncritical string theory, and show that the structure of the holographic RG should persist beyond the supergravity approximation as a consequence of the renormalizability of the nonlinear sigma model action of noncritical strings. As a check, we investigate the holographic RG structure of higher-derivative gravity systems, and show that such systems can also be analyzed based on the Hamilton-Jacobi equations, and that the behaviour of bulk fields are determined solely by their boundary values. We also point out that higher-derivative gravity systems give rise to new multicritical points in the parameter space of the boundary field theories.Comment: 95 pages, 6 figures. Typos are corrected. References and a discussion about continuum limit are adde

A Note on the Weyl Anomaly in the Holographic Renormalization Group

We give a prescription for calculating the holographic Weyl anomaly in arbitrary dimension within the framework based on the Hamilton-Jacobi equation proposed by de Boer, Verlinde and Verlinde. A few sample calculations are made and shown to reproduce the results that are obtained to this time with a different method. We further discuss continuum limits, and argue that the holographic renormalization group may describe the renormalized trajectory in the parameter space. We also clarify the relationship of the present formalism to the analysis carried out by Henningson and Skenderis.Comment: LaTeX, 24 pages, 2 figures, typos correcte