Twelve-Dimensional Supersymmetric Gauge Theory as the Large N Limit

Starting with the ordinary ten-dimensional supersymmetric Yang-Mills theory for the gauge group U(N), we obtain a twelve-dimensional supersymmetric gauge theory as the large N limit. The two symplectic canonical coordinates parametrizing the unitary N X N matrices for U(N) are identified with the extra coordinates in twelve dimensions in the $N\to\infty$ limit. Applying further a strong/weak duality, we get the `decompactified' twelve-dimensional theory. The resulting twelve-dimensional theory has peculiar gauge symmetry which is compatible also with supersymmetry. We also establish a corresponding new superspace formulation with the extra coordinates. By performing a dimensional reduction from twelve dimensions directly into three dimensions, we see that the Poisson bracket terms which are needed for identification with supermembrane action arises naturally. This result indicates an universal duality mechanism that the 't Hooft limit of an arbitrary supersymmetric theory promotes the original supersymmetric theory in (D-1,1) dimensions into a theory in (D,2) dimensions with an additional pair of space-time coordinates. This also indicates interesting dualities between supermembrane theory, type IIA superstring with D0-branes, and the recently-discovered twelve-dimensional supersymmetric theories.Comment: 14 pages, latex, no figure

Phase Diagram of a 2D Vertex Model

Phase diagram of a symmetric vertex model which allows 7 vertex configurations is obtained by use of the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG). The critical indices of this model are identified as $\beta = 1/8$ and $\alpha = 0$.Comment: 2 pages, 5 figures, short not

Self-Consistent Tensor Product Variational Approximation for 3D Classical Models

We propose a numerical variational method for three-dimensional (3D) classical lattice models. We construct the variational state as a product of local tensors, and improve it by use of the corner transfer matrix renormalization group (CTMRG), which is a variant of the density matrix renormalization group (DMRG) applied to 2D classical systems. Numerical efficiency of this approximation is investigated through trial applications to the 3D Ising model and the 3D 3-state Potts model.Comment: 12 pages, 6 figure

Snapshot Observation for 2D Classical Lattice Models by Corner Transfer Matrix Renormalization Group

We report a way of obtaining a spin configuration snapshot, which is one of the representative spin configurations in canonical ensemble, in a finite area of infinite size two-dimensional (2D) classical lattice models. The corner transfer matrix renormalization group (CTMRG), a variant of the density matrix renormalization group (DMRG), is used for the numerical calculation. The matrix product structure of the variational state in CTMRG makes it possible to stochastically fix spins each by each according to the conditional probability with respect to its environment.Comment: 4 pages, 8figure

Critical Point of a Symmetric Vertex Model

We study a symmetric vertex model, that allows 10 vertex configurations, by use of the corner transfer matrix renormalization group (CTMRG), a variant of DMRG. The model has a critical point that belongs to the Ising universality class.Comment: 2 pages, 6 figures, short not