Counting BPS Operators in the Chiral Ring of N=2 Supersymmetric Gauge Theories or N=2 Braine Surgery

TL;DR

The paper tackles counting BPS operators in the chiral ring of ${\cal N}=2$ quiver gauge theories on D3 branes probing ALE singularities, where the moduli space splits into Higgs, Coulomb, and mixed branches. It introduces a surgery framework that combines branch-generating functions $H$, $C$, and their intersection $L$ into the mixed-branch function $G(\nu; t_1,t_2,t_3; \Gamma)=\dfrac{H C}{L}$, providing explicit constructions and examples. The authors derive explicit forms for the Higgs and Coulomb partition functions, analyze their intersections, and validate the approach with low-$N$ cases and the $A_n$ series, yielding compact, computable expressions. This approach clarifies how to count chiral ring operators across multiple branches and has potential implications for AdS/CFT checks, black hole microstate counting, and broader classes of quiver theories. The results offer a practical toolkit for exact operator counting in a broad class of ${\cal N}=2$ gauge theories with rich moduli spaces.

Abstract

This note is presenting the generating functions which count the BPS operators in the chiral ring of a N=2 quiver gauge theory that lives on N D3 branes probing an ALE singularity. The difficulty in this computation arises from the fact that this quiver gauge theory has a moduli space of vacua that splits into many branches -- the Higgs, the Coulomb and mixed branches. As a result there can be operators which explore those different branches and the counting gets complicated by having to deal with such operators while avoiding over or under counting. The solution to this problem turns out to be very elegant and is presented in this note. Some surprises with "surgery" of generating functions arises.