Once more on the Witten index of 3d supersymmetric YM-CS theory

TL;DR

This paper clarifies the calculation of the Witten index in 3d supersymmetric Yang-Mills-Chern-Simons theory, resolving discrepancies between large and small volume approaches by addressing singularities in the moduli space.

Contribution

It demonstrates that the Hamiltonian Born-Oppenheimer method accurately reproduces Witten's original large-volume result by properly handling singularities.

Findings

The small-volume calculation has uncertainties due to singularities.

Proper treatment of singularities aligns small-volume results with Witten's original calculation.

The Hamiltonian Born-Oppenheimer method effectively resolves the controversy.

Abstract

The problem of counting the vacuum states in the supersymmetric 3d Yang-Mills-Chern-Simons theory is reconsidered. We resolve the controversy between its original calculation by Witten at large volumes and the calculation based on the evaluation of the effective Lagrangian in the small volume limit. We show that the latter calculation suffers from uncertainties associated with the singularities in the moduli space of classical vacua where the Born-Oppenheimer approximation breaks down. We also show that these singularities can be accurately treated in the Hamiltonian Born-Oppenheimer method, where one has to match carefully the effective wave functions on the Abelian valley and the wave functions of reduced non-Abelian QM theory near the singularities. This gives the same result as original Witten's calculation.