Bulk Witten Indices and the Number of Normalizable Ground States in Supersymmetric Quantum Mechanics of Orthogonal, Symplectic and Exceptional Groups

TL;DR

This paper investigates the number of normalizable ground states in supersymmetric quantum mechanics with various gauge groups, using contour integrals and BRST techniques, revealing new rational bulk index values and challenging existing conjectures.

Contribution

It introduces novel rational bulk index values for different gauge groups and demonstrates limitations of previous asymptotic methods for boundary contributions.

Findings

Proposes new rational values for bulk indices.

Shows asymptotic methods fail for non-SU(N) groups.

Finds results consistent with some conjectures but contradicts others for G_2.

Abstract

This note addresses the question of the number of normalizable vacuum states in supersymmetric quantum mechanics with sixteen supercharges and arbitrary semi-simple compact gauge group, up to rank three. After evaluating certain contour integrals obtained by appropriately adapting BRST deformation techniques we propose novel rational values for the bulk indices. Our results demonstrate that an asymptotic method for obtaining the boundary contribution to the index, originally due to Green and Gutperle, fails for groups other than SU(N). We then obtain likely values for the number of ground states of these systems. In the case of orthogonal and symplectic groups our finding is consistent with recent conjectures of Kac and Smilga, but appears to contradict their result in the case of the exceptional group G_2.