Twist fields and broken supersymmetry

Abstract

A twist field on a cylindrical space–time has the defining property that translation about a spatial circle results in multiplying the field by a phase. In this paper we investigate how such multivalued twist fields fit into the framework of constructive quantum field theory. Twisted theories have an interest in their own right; the twists also serve as infrared regulators that partially preserve the underlying symmetries of the Hamiltonian. The main focus of this paper is to investigate the extent that boson–fermion twist-field systems are compatible with the Lie symmetry and with the N=2 supersymmetry that one expects in the same examples without twists. We consider free systems and nonlinear boson–fermion interactions that arise from a holomorphic, quasihomogeneous, polynomial superpotential. We choose the twisting angles to lie on a chosen line in twist parameter space (leaving one free twist parameter). Doing this, we can obtain Lie symmetry and half the number of supersymmetry generators that one expects in our examples without the twists. We also show that the Hamiltonians for scalar twist fields yield twisted, positive-temperature expectations with the “twist-positivity” property. This is important because it justifies the existence of a functional integral representation for twisted, positive-temperature trace functionals. We regularize these systems in a way that preserves symmetry to the maximal extent. We pursue elsewhere other aspects and applications of this method, including bounding the extent of supersymmetry breaking.