Clash of discrete symmetries for the supersymmetric kink on a circle

TL;DR

This paper investigates the supersymmetric kink on a finite circle, revealing that boundary conditions cannot preserve all discrete symmetries simultaneously, leading to the necessity of averaging over boundary choices to maintain symmetry.

Contribution

It demonstrates that no single locally invisible boundary condition preserves all three discrete symmetries, and proposes averaging over boundary conditions to restore symmetry in quantum energy calculations.

Findings

Single boundary conditions violate some discrete symmetries.

Averaging over boundary conditions restores symmetry.

Parity and time reversal are violated under twisted boundary conditions.

Abstract

We consider the N=1 supersymmetric kink on a circle, i.e., on a finite interval with boundary or transition conditions which are locally invisible. For Majorana fermions, the single-particle Dirac Hamiltonian as a differential operator obeys simultaneously the three discrete symmetries of charge conjugation, parity, and time reversal. However, no single locally invisible transition condition can satisfy all three. When calculating sums over zero-point energies by mode number regularization, this gives a new rationale for a previous suggestion that one has to average over different choices of boundary conditions, such that for the combined set all three symmetries are obeyed. In particular it is shown that for twisted periodic or twisted antiperiodic boundary conditions separately both parity and time reversal are violated in the kink sector, as manifested by a delocalized momentum that cancels only in the average.