Three-sphere free energy for classical gauge groups

In this note, we calculate the S ³ free energy F of 3-d ≥ 4 supersymmetric gauge theories with U( N), O( N), and USp(2 N) gauge groups and matter hypermultiplets in the fundamental and two-index tensor representations. Supersymmetric localization reduces the computation of F to a matrix model that we solve in the large N limit using two different methods. The first method is a saddle point approximation first introduced in [1], which we extend to next-to-leading order in 1 /N. The second method generalizes the Fermi gas approach of [2] to theories with symplectic and orthogonal gauge groups, and yields an expression for F valid to all orders in 1 /N . In developing the second method, we use a non-trivial generalization of the Cauchy determinant formula.