Nonabelian duality and solvable large N lattice systems

摘要

We introduce the basics of the nonabelian duality transformation of SU(N) or U(N) vector-field models defined on a lattice. The dual degrees of freedom are certain species of the integer-valued fields complemented by the symmetric groups' x(n)S(n) variables. While the former parametrize relevant irreducible representations, the latter play the role of the Lagrange multipliers facilitating the fusion rules involved. As an application, I construct a novel solvable family of SU(N) D-matrix systems graded by the rank 1 less than or equal to k less than or equal to (D - 1) of the manifest [U(N)](+k) conjugation-symmetry. Their large N solvability is due to a hidden invariance (explicit in the dual formulation) which allows for a mapping onto the recently proposed eigenvalue-models [8] with the largest k = D symmetry. Extending [8], we reconstruct a D-dimensional gauge theory with the large N free energy given (modulo the volume factor) by the free energy of a given proposed 1 less than or equal to k less than or equal to (D - 1) D-matrix system. It is emphasized that the developed formalism provides with the basis for higher-dimensional generalizations of the Gross-Taylor stringy representation of strongly coupled 2d gauge theories. (C) 2000 Elsevier Science B.V. All rights reserved. [References: 25]