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3d $\mathcal{N}=4$ OPE Coefficients from Fermi Gas

The partition function of a 3d $\mathcal{N}=4$ 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 $U(N)_k\times U(N)_{-k}$ 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.