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

We construct a lattice formulation of a mass-deformed two-dimensional N = (8, 8) super Yang-Mills theory with preserving two supercharges exactly. Gauge fields are represented by compact unitary link variables, and the exact supercharges on the lattice are nilpotent up to gauge transformations and SU(2) R rotations. Due to the mass deformation, the lattice model is free from the vacuum degeneracy problem, which was encountered in earlier approaches, and flat directions of scalar fields are stabilized giving discrete minima representing fuzzy S 2 . Around the trivial minimum, quantum continuum theory is obtained with no tuning, which serves a nonperturbative construction of the IIA matrix string theory. Moreover, around the minimum of k-coincident fuzzy spheres, four-dimensional N = 4 U (k) super Yang-Mills theory with two commutative and two noncommutative directions emerges. In this theory, sixteen supersymmetries are broken by the mass deformation to two. Assuming the breaking is soft, we give a scenario leading to undeformed N = 4 super Yang-Mills on R 4 without any fine tuning. As an evidence for the validity of the assumption, some computation of 1-loop radiative corrections is presented.Subject Index: 100, 110, 125, 127, 138 §1. IntroductionNonperturbative aspects of supersymmetric Yang-Mills (SYM) theories play prominent roles in physics beyond the standard model 1) as well as in superstring/M theory. 2)-5) However, to construct their nonperturbative formulations such as lattice is not a straightforward task because of the notorious difficulties of supersymmetry (SUSY) on lattice. So far, lattice formulations for SYM are constructed for one-and two-dimensional cases and N = 1 pure SYM in three and four dimensions, 6) where no requirement of fine tunings due to the ultra-violet (UV) effects can be shown at least in perturbative arguments. For one-dimensional theory (matrix quantum mechanics) more powerful "non-lattice" technique 7) is applicable. (For corresponding lattice study, see Ref. 8).) For two-dimensional N = (2, 2) SYM, nonperturbative evidence for the lattice model presented in Ref. 9) to require no fine tuning has been given by numerical simulation for the gauge group G = SU (2) in Ref. 10) and for * )

We provide a general classification of supersymmetric lattice gauge theories that can be obtained from orbifolding of theories with four and eight supercharges. We impose at least one preserved supercharge on the lattice and Lorentz invariance in the naive continuum limit. Starting with four supercharges, we obtain one two-dimensional lattice gauge theory, identical to the one already given in the literature. Starting with eight supercharges, we obtain a unique three-dimensional lattice gauge theory and infinitely many two-dimensional lattice theories. They can be classified according to seven distinct groups, five of which have two preserved supercharges while the others have only one.

We prove that the prescription for construction of supersymmetric lattice gauge theories by orbifolding and deconstruction directly leads to Catterall's geometrical discretization scheme in general. These two prescriptions always give the same lattice discretizations when applied to theories of p-form fields. We also show that the geometrical discretization scheme can be applied to more general theories. * phdamg@nbi.dk † matsuura@nbi.dk 1 For further analysis, see, e.g., refs.[8]- [12].

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