Quivers as Calculators: Counting, Correlators and Riemann Surfaces

TL;DR

This paper develops a combinatorial and topological framework for counting and analyzing chiral operators in free supersymmetric quiver gauge theories, revealing new formulas and bases that connect gauge theory to symmetric groups and Riemann surfaces.

Contribution

It introduces a split-node quiver modification, an infinite product counting formula, and a quiver character basis, advancing the understanding of operator counting in free quiver theories.

Findings

Finite N operator counting via Young diagrams and Littlewood-Richardson multiplicities.

A simple infinite product formula for large N gauge invariant operators.

Construction of an orthogonal basis using quiver characters and topological field theories.

Abstract

The spectrum of chiral operators in supersymmetric quiver gauge theories is typically much larger in the free limit, where the superpotential terms vanish. We find that the finite N counting of operators in any free quiver theory, with a product of unitary gauge groups, can be described by associating Young diagrams and Littlewood-Richardson multiplicities to a simple modification of the quiver, which we call the split-node quiver. The large N limit leads to a surprisingly simple infinite product formula for counting gauge invariant operators, valid for any quiver with bifundamental fields. An orthogonal basis for the operators, in the finite N CFT inner product, is given in terms of quiver characters. These are constructed by inserting permutations in the split-node quivers and intepreting the resulting diagrams in terms of symmetric group matrix elements and branching coefficients. The fusion coefficients in the chiral ring - valid both in the UV and in the IR - are computed at finite N. The derivation follows simple diagrammatic moves on the quiver. The large N counting and correlators are expressed in terms of topological field theories on Riemann surfaces obtained by thickening the quiver. The TFTs are based on symmetric groups and defect observables associated with subgroups play an important role. We outline the application of the free field results to the construction of BPS operators in the case of non-zero super-potential.