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)x 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.