Witten index in supersymmetric 3d theories revisited

TL;DR

This paper reevaluates the Witten index in 3d supersymmetric Yang-Mills Chern-Simons theories using a Born-Oppenheimer approach, revealing that including gluon loops leads to a nonzero index and unbroken supersymmetry, contrasting with previous results.

Contribution

It introduces a new calculation method for the Witten index that accounts for gluon, fermion, and ghost loops equally, challenging earlier findings on supersymmetry breaking.

Findings

Fermion loops cause a negative shift in the Chern-Simons level.

Gluon loops induce a positive shift, preventing the index from vanishing.

The approach suggests supersymmetry remains unbroken at nonzero k.

Abstract

We have performed a direct calculation of Witten index in N = 1,2,3 supersymmetric Yang-Mills Chern-Simons 3d theories. We do it in the framework of Born-Oppenheimer (BO) approach by putting the system into a small spatial box and studying the effective Hamiltonian depending on the zero field harmonics. At the tree level, our results coincide with the results of Witten, but there is a difference in the way the loop effects are implemented. In Witten's approach, one has only take into account the fermion loops, which bring about a negative shift of the (chosen positive at the tree level) Chern-Simons coupling k. As a result, Witten index vanishes and supersymmetry is broken at small k. In the effective BO Hamiltonian framework, fermion, gluon and ghost loops contribute on an equal footing. Fermion loop contribution to the effective Hamiltonian can be evaluated exactly, and their effect amounts to the negative shift k -> k - h/2 for N =1 and k -> k - h for N = 2,3 in the tree-level formulae for the index. In our approach, with rather natural assumptions on the structure of bosonic corrections, the shift k -> k + h brought about by the gluon loops also affects the index. Since the total shift of k is positive or zero, Witten index appears to be nonzero at nonzero k, and supersymmetry is not broken. We discuss possible reasons for such disagreement.