Comments on the Casimir energy in supersymmetric field theories

TL;DR

This work analyzes the Casimir energy of 4D ${\cal N}=1$ supersymmetric gauge theories in a rigid background from new minimal supergravity, demonstrating that localization on $S^1\times S^3$ yields the supersymmetric Casimir energy $E_{\text{susy}}=\frac{4}{27}(\mathbf a+3\mathbf c)$ and that the corresponding Hamiltonian vev, computed via zeta-regularization, reproduces this result. It shows an interpolation between the SUSY BPS Hamiltonian $H_{\text{susy}}$ and the ordinary free-field Casimir energy by varying a background parameter $\mathfrak{q}$, while clarifying that, away from the special pairing point $\mathfrak{q}=0$, the vev need not be expressible as a linear combination of the anomaly coefficients $\mathbf a$ and $\mathbf c$. The paper also provides explicit quadratic Lagrangians for chiral and vector multiplets in the rigid Hopf-surface background, derives the conserved charges and their algebra, and performs canonical quantization on $\mathbb{R}\times S^3$, convincingly connecting the path-integral and Hamiltonian approaches. Collectively, these results solidify the unambiguous, coupling-independent nature of $E_{\text{susy}}$ and highlight the nuanced relationship between SUSY Casimir energy and the standard Casimir energy in free theories.

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.