Nonabelian duality and solvable large N lattice systems

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 groupsʹ ⊗nS(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≤k≤(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≤k≤(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.