Quivers, Words and Fundamentals

TL;DR

This paper extends the counting of gauge invariant operators in quiver gauge theories to include fundamental matter, deriving infinite product formulas and revealing new combinatorial relations involving words and monoids.

Contribution

It introduces a refined counting method for quiver gauge theories with fundamental matter, connecting gauge theory combinatorics to the Cartier-Foata monoid and Littlewood-Richardson coefficients.

Findings

Derived infinite product formulas for theories with fundamental matter.

Connected counting problems to the Cartier-Foata monoid structure.

Expressed refined counting in terms of Littlewood-Richardson coefficients for finite ranks.

Abstract

A systematic study of holomorphic gauge invariant operators in general $\mathcal{N}=1$ quiver gauge theories, with unitary gauge groups and bifundamental matter fields, was recently presented in [1]. For large ranks a simple counting formula in terms of an infinite product was given. We extend this study to quiver gauge theories with fundamental matter fields, deriving an infinite product form for the refined counting in these cases. The infinite products are found to be obtained from substitutions in a simple building block expressed in terms of the weighted adjacency matrix of the quiver. In the case without fundamentals, it is a determinant which itself is found to have a counting interpretation in terms of words formed from partially commuting letters associated with simple closed loops in the quiver. This is a new relation between counting problems in gauge theory and the Cartier-Foata monoid. For finite ranks of the unitary gauge groups, the refined counting is given in terms of expressions involving Littlewood-Richardson coefficients.