Counting Chiral Operators in Quiver Gauge Theories

TL;DR

This paper develops methods to count BPS gauge invariant operators in quiver gauge theories on D-branes over toric Calabi-Yau singularities, linking geometric data with field theory and providing explicit formulas for several examples.

Contribution

It introduces a detailed approach to compute generating functions for BPS operators, incorporating baryonic charges and multiplicities, and extends the Plethystic Exponential technique to multiple D-branes.

Findings

Derived explicit formulas for counting operators in specific geometries.

Established a relation between baryonic charges and Calabi-Yau moduli.

Demonstrated the decomposition of generating functions into geometric sectors.

Abstract

We discuss in detail the problem of counting BPS gauge invariant operators in the chiral ring of quiver gauge theories living on D-branes probing generic toric CY singularities. The computation of generating functions that include counting of baryonic operators is based on a relation between the baryonic charges in field theory and the Kaehler moduli of the CY singularities. A study of the interplay between gauge theory and geometry shows that given geometrical sectors appear more than once in the field theory, leading to a notion of "multiplicities". We explain in detail how to decompose the generating function for one D-brane into different sectors and how to compute their relevant multiplicities by introducing geometric and anomalous baryonic charges. The Plethystic Exponential remains a major tool for passing from one D-brane to arbitrary number of D-branes. Explicit formulae are given for few examples, including C^3/Z_3, F_0, and dP_1.