Comments on the Casimir energy in supersymmetric field theories

TL;DR

This paper investigates the supersymmetric Casimir energy in 4D gauge theories, comparing localized partition function results with Hamiltonian formalism calculations, revealing an interpolation between supersymmetric and ordinary Casimir energies.

Contribution

It provides a detailed comparison of supersymmetric Casimir energy calculations using localization and Hamiltonian methods, including a novel interpolation analysis.

Findings

Revealed the relation between supersymmetric and ordinary Casimir energies.

Computed vacuum expectation values using zeta function regularization.

Demonstrated interpolation between different Casimir energy definitions.

Abstract

We study the Casimir energy of four-dimensional supersymmetric gauge theories in the context of the rigid limit of new minimal supergravity. Firstly, revisiting the computation of the localized partition function on $S^1\times S^3$, we recover the supersymmetric Casimir energy. Secondly, we consider the same theories in the Hamiltonian formalism on $\mathbb{R}\times S^3$, focussing on the free limit and including a one-parameter family of background gauge fields along $\mathbb{R}$. We compute the vacuum expectation value of the canonical Hamiltonian using zeta function regularization, and show that this interpolates between the supersymmetric Casimir energy and the ordinary Casimir energy of a supersymmetric free field theory.