Comments on the Casimir energy in supersymmetric field theories

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 × S 3 , we recover the supersymmetric Casimir energy from its path integral definition. Secondly, we consider the same theories in the Hamiltonian formalism on ℝ × S 3 $$ \mathbb{R}\times {S}^3 $$ , focussing on the free limit and including a one- parameter family of background gauge fields along ℝ $$ \mathbb{R} $$ . 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