Supersymmetric Casimir energy and SL(3,Z) transformations
We provide a recipe to extract the supersymmetric Casimir energy of theories
defined on primary Hopf surfaces directly from the superconformal index. It involves an
SL(3, Z) transformation acting on the complex structure moduli of the background geometry.
In particular, the known relation between Casimir energy, index and partition
function emerges naturally from this framework, allowing rewriting of the latter as a modified
elliptic hypergeometric integral. We show this explicitly for N = 1 SQCD and N = 4
supersymmetric Yang-Mills theory for all classical gauge groups, and conjecture that it
holds more generally. We also use our method to derive an expression for the Casimir
energy of the nonlagrangian N = 2 SCFT with E6 flavour symmetry. Furthermore, we
predict an expression for Casimir energy of the N = 1 SP(2N) theory with SU(8) × U(1)
flavour symmetry that is part of a multiple duality network, and for the doubled N = 1
theory with enhanced E7 flavour symmetry.