# Anomalies and Holomorphy in Supersymmetric Chern-Simons-Matter Theories

## TL;DR

This paper explores the holomorphic factorization of partition functions in three-dimensional supersymmetric Chern-Simons-matter theories, linking gauge invariance, parity anomalies, and group representations.

## Contribution

It extends the understanding of holomorphic factorization from abelian to non-abelian gauge groups in supersymmetric Chern-Simons-matter theories.

## Key findings

- Factorization aligns with the parity anomaly for classical groups and G_2.
- Partition functions admit holomorphic factorization when the Chern-Simons level is properly quantized.
- Analytic continuation of torus knot observables is discussed in the appendix.

## Abstract

For Chern-Simons-matter theories in three dimensions, gauge invariance may require the Chern-Simons level k to be half-integral, in which case parity is violated. As noted by Pasquetti for abelian theories with N=2 supersymmetry, the partition function on the ellipsoid also admits a suitable holomorphic factorization precisely when the value of k is properly quantized. Using known formulas for the partition function, we investigate analytic aspects of this factorization for non-abelian gauge groups and general matter representations. We verify that factorization occurs in accord with the parity anomaly for the classical matrix groups and for the exceptional group G_2. In an appendix, we discuss the analytic continuation of torus knot observables in the SU(2) Chern-Simons-matter theory.

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Source: https://tomesphere.com/paper/1812.02832