Three-sphere free energy for classical gauge groups

Abstract

In this note, we calculate the $S^3$ free energy $F$ of 3-d $\mathcal{N}\ge 4$ supersymmetric gauge theories with $U(N)$, $O(N)$, and $USp(2N)$ 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 arXiv:1011.5487, which we extend to next-to-leading order in $1/N$. The second method generalizes the Fermi gas approach of arXiv:1110.4066 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.

47 pages, 1 figure; v2: refs added, minor improvements; v3: corrected count of eigenvalues in the Fermi gas approach

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• Holographic c-theorems in arbitrary dimensions
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