3d N = 4 OPE coefficients from Fermi gas

The partition function of a 3d N = 4 gauge theory with rank N can be computed using supersymmetric localization in terms of a matrix model, which often can be formulated as an ideal Fermi gas with a non-trivial one-particle Hamiltonian. We show how OPE coefficients of protected operators correspond in this formalism to averages of n-body operators in the Fermi gas, which can be computed to all orders in 1/N using the WKB expansion. We use this formalism to compute OPE coefficients in the U(N)_k× U(N)_−k ABJM theory as well as the U(N) theory with one adjoint and N_f fundamental hypermultiplets, both of which have weakly coupled M-theory duals in the large N and finite k or N_f regimes. For ABJM we reproduce known results, while for the N_f theory we compute the all orders in 1/N dependence at finite N_f for the coefficient c_T of the stress tensor two-point function.