BPS Partition Functions for Quiver Gauge Theories: Counting Fermionic Operators

TL;DR

This work provides a systematic framework to construct complete 1/2 BPS partition functions for general $ abla=1$ quiver gauge theories by extending scalar chiral-ring counting to include Weyl spinor superfields $W_{}$ through a superfield formalism and a fermionic Plethystic exponential. Starting from the known $N=1$ scalar generating functions and using a generalized PE, the authors obtain finite-$N$ partition functions that simultaneously account for bosonic and fermionic statistics across mesonic and baryonic sectors. The conifold example serves as a nontrivial testbed, where the formalism yields explicit generating functions for mesonic and baryonic sectors, both in $U(N)$ and $SU(N)$ cases, and demonstrates consistency with direct field-theory operator counting. The results offer a tool to probe the full 1/2 BPS spectrum and have potential implications for holographic entropy studies and the thermodynamics of strongly coupled CFTs, including deformations and marginal perturbations.

Abstract

We discuss a general procedure to obtain 1/2 BPS partition functions for generic N=1 quiver gauge theories. These functions count the gauge invariant operators (bosonic and fermionic), charged under all the global symmetries (mesonic and baryonic), in the chiral ring of a given quiver gauge theory. In particular we discuss the inclusion of the spinor degrees of freedom in the partition functions.