Witten index in N=1 and N=2 SYMCS theories with matter

TL;DR

This paper computes the Witten index for 3d supersymmetric Yang-Mills-Chern-Simons theories with matter, revealing how matter interactions and fermion loops influence the vacuum structure and index values.

Contribution

It provides explicit calculations of the Witten index in 3d N=1 and N=2 SYMCS theories with matter, including effects of Yukawa interactions and fermion loop renormalizations.

Findings

Witten index matches recent N=2 results in 3d

Extra Higgs vacua can appear due to Yukawa interactions

Fermion loops modify the index through Chern-Simons level renormalization

Abstract

We calculate the Witten index for 3d supersymmetric Yang-Mills-Chern-Simons theories with matter. For N=2 theories, our results coincide with the results of recent [1]. We compare the situation in 3d to that in 4d N = 1 theories with massive matter. In both cases, extra Higgs vacuum states may appear when the Lagrangian involves nontrivial Yukawa interactions between the matter superfields. In addition, in 3d theories, massive fermion loops affect the index via renormalization of the Chern-Simons level k.