Asato Tsuchiya

Dynamical aspects of large-N reduced models

Abstract We study the large- N reduced model of D -dimensional Yang-Mills theory with special attention to dynamical aspects related to the eigenvalues of the N × N matrices, which correspond to the space-time coordinates in the IIB matrix model. We first put an upper bound on the extent of space-time by perturbative arguments. We perform a Monte Carlo simulation and show that the upper bound is actually saturated. The relation of our result to the SSB of the U(1) D symmetry in the Eguchi-Kawai model is clarified. We define a quantity which represents the uncertainty of the space-time coordinates and show that it is of the same order as the extent of space-time, which means that a classical space-time picture is maximally broken. We develop a 1/ D expansion, which enables us to calculate correlation functions of the model analytically. The absence of an SSB of the Lorentz invariance is shown by the Monte Carlo simulation as well as by the 1/ D expansion.

IIB matrix model

We review our proposal for a constructive definition of superstring, type IIB matrix model. The IIB matrix model is a manifestly covariant model for space-time and matter which possesses N=2 supersymmetry in ten dimensions. We refine our arguments to reproduce string perturbation theory based on the loop equations. We emphasize that the space-time is dynamically determined from the eigenvalue distributions of the matrices. We also explain how matter, gauge fields and gravitation appear as fluctuations around dynamically determined space-time.

N = 4 super Yang-Mills theory from the plane wave matrix model

We propose a nonperturbative definition of N=4 super Yang-Mills (SYM). We realize N=4 SYM on RxS{sup 3} as the theory around a vacuum of the plane wave matrix model. Our regularization preserves 16 supersymmetries and the gauge symmetry. We perform the 1-loop calculation to give evidences that the superconformal symmetry is restored in the continuum limit.

String field theory from IIB matrix model

Abstract We derive Schwinger-Dyson equations for the Wilson loops of IIB matrix model. Superstring coordinates are introduced through the construction of the loop space. We show that the continuum limit of the loop equation reproduces the light-cone superstring field theory of type IIB superstring in the large-N limit. We find that the interacting string theory can be obtained in the double scaling limit as it is expected.

Deconfinement phase transition in N=4 super Yang-Mills theory on R x S3 from supersymmetric matrix quantum mechanics.

We test the recent claim that supersymmetric matrix quantum mechanics with mass deformation preserving maximal supersymmetry can be used to study N=4 super Yang-Mills theory on R x S{3} in the planar limit. When the mass parameter is large, we can integrate out all of the massive fluctuations around a particular classical solution, which corresponds to R x S{3}. The resulting effective theory for the gauge field moduli at finite temperature reproduces the deconfinement phase transition in N=4 super Yang-Mills theory on R x S{3} at weak coupling. This transition was speculated to be a continuation of the conjectured phase transition at strong coupling, which corresponds to the Hawking-Page transition based on the gauge-gravity duality. At strong coupling, one can study the target theory by the new simulation method for supersymmetric matrix quantum mechanics.

Complex matrix model and fermion phase space for bubbling AdS geometries

We study a relation between droplet configurations in the bubbling AdS geometries and a complex matrix model that describes the dynamics of a class of chiral primary operators in dual = 4 super Yang Mills (SYM). We show rigorously that a singlet holomorphic sector of the complex matrix model is equivalent to a holomorphic part of two-dimensional free fermions, and establish an exact correspondence between the singlet holomorphic sector of the complex matrix model and one-dimensional free fermions. Based on this correspondence, we find a relation of the singlet holomorphic operators of the complex matrix model to the Wigner phase space distribution. By using this relation and the AdS/CFT duality, we give a further evidence that the droplets in the bubbling AdS geometries are identified with those in the phase space of the one-dimensional fermions. We also show that the above correspondence actually maps the operators of = 4 SYM corresponding to the (dual) giant gravitons to the droplet configurations proposed in the literature.

Embedding of theories with SU(2|4) symmetry into the plane wave matrix model

We study theories with SU(2|4) symmetry, which include the plane wave matrix model, 2+1 SYM on R ? S2 and = 4 SYM on R ? S3/Zk. All these theories possess many vacua. From Lin-Maldacenas method which gives the gravity dual of each vacuum, it is predicted that the theory around each vacuum of 2+1 SYM on R ? S2 and = 4 SYM on R ? S3/Zk is embedded in the plane wave matrix model. We show this directly on the gauge theory side. We clearly reveal relationships among the spherical harmonics on S3, the monopole harmonics and the harmonics on fuzzy spheres. We extend the compactification (the T-duality) in matrix models a la Taylor to that on spheres.

= 4 SYM on R × S3 and theories with 16 supercharges

We study N=4 SYM on R x S^3 and theories with 16 supercharges arising as its consistent truncations. These theories include the plane wave matrix model, N=4 SYM on R x S^2 and N=4 SYM on R x S^3/Z_k, and their gravity duals were studied by Lin and Maldacena. We make a harmonic expansion of the original N=4 SYM on R x S^3 and obtain each of the truncated theories by keeping a part of the Kaluza-Klein modes. This enables us to analyze all the theories in a unified way. We explicitly construct some nontrivial vacua of N=4 SYM on R x S^2. We perform 1-loop analysis of the original and truncated theories. In particular, we examine states regarded as the integrable SO(6) spin chain and a time-dependent BPS solution, which is considered to correspond to the AdS giant graviton in the original theory.

Testing a novel large-N reduction for = 4 super Yang-Mills theory on R × S3

Recently a novel large-N reduction has been proposed as a maximally supersymmetric regularization of = 4 super Yang-Mills theory on R × S3 in the planar limit. This proposal, if it works, will enable us to study the theory non-perturbatively on a computer, and hence to test the AdS/CFT correspondence analogously to the recent works on the D0-brane system. We provide a nontrivial check of this proposal by performing explicit calculations in the large-N reduced model, which is nothing but the so-called plane wave matrix model, around a particular stable vacuum corresponding to R × S3. At finite temperature and at weak coupling, we reproduce precisely the deconfinement phase transition in the = 4 super Yang-Mills theory on R × S3. This phase transition is considered to continue to the strongly coupled regime, where it corresponds to the Hawking-Page transition on the AdS side. We also perform calculations around other stable vacua, and reproduce the phase transition in super Yang-Mills theory on the corresponding curved space-times such as R × S3/Zq and R × S2.

Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory

Recently we have shown by Monte Carlo simulation that expanding (3+1)-dimensional universe appears dynamically from a Lorentzian matrix model for type IIB superstring theory in (9+1)-dimensions. The mechanism for the spontaneous breaking of rotational symmetry relies crucially on the noncommutative nature of the space. Here we study the classical equations of motion as a complementary approach. In particular, we find a unique class of SO(3) symmetric solutions, which exhibits the time-dependence compatible with the expanding universe. The space-space noncommutativity is exactly zero, whereas the space-time noncommutativity becomes significant only towards the end of the expansion. We interpret the Monte Carlo results and the classical solution as describing the behavior of the model at earlier time and at later time, respectively.