Supersymmetry and Mass Gap in 2+1 Dimensions: A Gauge Invariant Hamiltonian Analysis

TL;DR

This paper develops a gauge-invariant Hamiltonian framework for supersymmetric Yang-Mills-Chern-Simons theories in 2+1 dimensions, analyzing mass gaps and the role of supersymmetry in their spectral properties.

Contribution

It generalizes previous non-supersymmetric approaches to include supersymmetry, providing explicit expressions for the measure, supersymmetry algebra, and mass gap analysis.

Findings

Mass gap exists in minimally supersymmetric theories.

Extended supersymmetry theories lack a mass gap.

The measure relates to the renormalization of the Chern-Simons level.

Abstract

A Hamiltonian formulation of Yang-Mills-Chern-Simons theories with $0\leq N\leq 4$ supersymmetry in terms of gauge-invariant variables is presented, generalizing earlier work on nonsupersymmetric gauge theories. Special attention is paid to the volume measure of integration (over the gauge orbit space of the fields) which occurs in the inner product for the wave functions and arguments relating it to the renormalization of the Chern-Simons level number and to mass-gaps in the spectrum of the Hamiltonians are presented. The expression for the integration measure is consistent with the absence of mass gap for theories with extended supersymmetry (in the absence of additional matter hypermultiplets and/or Chern-Simons couplings), while for the minimally supersymmetric case, there is a mass-gap, the scale of which is set by a renormalized level number, in agreement with indications from existing literature. The realization of the supersymmetry algebra and the Hamiltonian in terms of the gauge invariant variables is also presented.