  We study $\mathcal{N}=4$ 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, 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.
