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 S¹ × S³, we recover the supersymmetric Casimir energy from its path integral definition. Secondly, we consider the same theories in the Hamiltonian formalism on ℝ× (Formula presented.), focussing on the free limit and including a one- parameter family of background gauge fields along ℝ (Formula presented.). 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.