Finite energy shifts in SU(n) supersymmetric Yang-Mills theory on T^3xR at weak coupling

TL;DR

This paper analyzes the weak coupling regime of SU(n) supersymmetric Yang-Mills theory on T^3xR, focusing on semi-classical vacua, perturbative corrections, and the role of supersymmetry in ensuring finite energy shifts.

Contribution

It introduces a semi-classical perturbative framework around specific vacua in SU(n) supersymmetric Yang-Mills theory on T^3xR, addressing Hilbert space redefinition and finite energy corrections.

Findings

Perturbative corrections to the free energy spectrum are finite due to supersymmetry.

A consistent redefinition of the Hilbert space norm is necessary for the interacting theory.

The interacting Hilbert space is unitarily inequivalent to the free one.

Abstract

We consider a semi-classical treatment, in the regime of weak gauge coupling, of supersymmetric Yang-Mills theory in a space-time of the form T^3xR with SU(n)/Z_n gauge group and a non-trivial gauge bundle. More specifically, we consider the theories obtained as power series expansions around a certain class of normalizable vacua of the classical theory, corresponding to isolated points in the moduli space of flat connections, and the perturbative corrections to the free energy eigenstates and eigenvalues in the weakly interacting theory. The perturbation theory construction of the interacting Hilbert space is complicated by the divergence of the norm of the interacting states. Consequently, the free and interacting Hilbert furnish unitarily inequivalent representation of the algebra of creation and annihilation operators of the quantum theory. We discuss a consistent redefinition of the Hilbert space norm to obtain the interacting Hilbert space and the properties of the interacting representation. In particular, we consider the lowest non-vanishing corrections to the free energy spectrum and discuss the crucial importance of supersymmetry for these corrections to be finite.