Nonabelian Duality and Solvable Large N Lattice Systems

TL;DR

This paper develops a nonabelian duality transformation for lattice vector-field models, introducing a solvable family of large N matrix systems with potential applications to higher-dimensional gauge theories and string representations.

Contribution

It introduces a novel nonabelian duality framework and constructs a new class of solvable large N lattice models with potential for higher-dimensional gauge theory insights.

Findings

Constructed a solvable family of SU(N) D-matrix systems.

Mapped these models onto eigenvalue-models with D symmetry.

Reconstructed a D-dimensional gauge theory with free energy related to the matrix systems.

Abstract

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' \otimes_{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\leq{k}\leq{(D-1)} of the manifest [U(N)]^{\oplus 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 \cite{Dub1} with the largest k=D symmetry. Extending \cite{Dub1}, 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\leq{k}\leq{(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.