Exact Witten Index in D=2 supersymmetric Yang-Mills quantum mechanics

TL;DR

This paper introduces a recursive method for calculating matrix elements in D=2 supersymmetric Yang-Mills quantum mechanics, enabling exact spectral and index computations with high precision, and provides the first analytical derivation of the restricted Witten index.

Contribution

A novel recursive technique for exact matrix element calculation in supersymmetric quantum mechanics, applicable to arbitrary occupation numbers and enabling precise spectral analysis.

Findings

Exact matrix elements for D=2 SU(2) system obtained

High-precision spectrum and Witten index computed numerically

First analytical derivation of the restricted Witten index

Abstract

A new, recursive method of calculating matrix elements of polynomial hamiltonians is proposed. It is particularly suitable for the recent algebraic studies of the supersymmetric Yang-Mills quantum mechanics in any dimensions. For the D=2 system with the SU(2) gauge group, considered here, the technique gives exact, closed expressions for arbitrary matrix elements of the hamiltonian and of the supersymmetric charge, in the occupation number representation. Subsequent numerical diagonalization provides the spectrum and restricted Witten index of the system with very high precision (taking into account up to $10^5$ quanta). Independently, the exact value of the restricted Witten index is derived analytically for the first time.