Z Extremization in Chiral-Like Chern Simons Theories

TL;DR

This paper investigates the free energy and R charge in three-dimensional N=2 Chern-Simons theories, resolving discrepancies between methods by restoring symmetry, and applies findings to models related to D3 branes probing Hirzebruch surfaces.

Contribution

It introduces a symmetry-restoring approach that aligns different computational methods for R charge in chiral-like Chern-Simons theories.

Findings

Methods agree after symmetry restoration

Matching with four-dimensional parent theories

Insights into strong coupling regime

Abstract

We study the localized free energy on S^3 of three-dimensional N=2 Chern-Simons matter theories at weak coupling. We compute the two loop R charge in three different ways, namely by the standard perturbative approach, by extremizing the localized partition function at finite N and by applying the standard saddle point approximation for large N. We show that the latter approach does not reproduce the expected result when chiral theories are considered. We circumvent these problems by restoring a reflection symmetry on the eigenvalues in the free energy. Thanks to this symmetrization we find that the three methods employed agree. In particular we match the computation for a model whose four dimensional parent is the quiver gauge theory describing D3 branes probing the Hirzebruch surface. We conclude by commenting on the application of our results and to the strong coupling regime.