We study $\mathcal{N}=4$ supersymmetric QED in three dimensions, on a 3-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, associated with an integrable system. In the large $N$ limit, the theory undergoes a second-order phase transition on a critical line in the parameter space. We discuss the critical behavior and compute the two-point correlation function of a gauge invariant mass operator, which is shown to diverge as one approaches criticality from the subcritical phase. Finally, we comment on the asymptotic $1/N$ expansion and on mirror symmetry.

• Received 1 November 2016

DOI:https://doi.org/10.1103/PhysRevD.95.031901