We present a study of $\mathcal{N}=4$ supersymmetric QED in three dimensions, on a three-sphere, with $2N$ massive hypermultiplets and a Fayet-Iliopoulos parameter. We identify the exact partition function of the theory with a conical (Mehler) function. This implies a number of analytical formulas, including a recurrence relation and a second-order differential equation. In the large $N$ limit, the theory undergoes a second-order phase transition on a critical line in the parameter space. We will discuss the critical behavior and compute the two-point correlation function of a gauge invariant mass operator.

(Joint work with Jorge G. Russo, arXiv:1610.08527)