SUSY Localization for Coulomb Branch Operators in Omega-Deformed 3d $$\mathcal {N}$$ = 4 Gauge Theories

Abstract

We perform SUSY localization for Coulomb branch operators of 3d $$\mathcal {N}=4$$ gauge theories in $$\mathbb {R}^3$$ with $$\Omega $$ -deformation. Our study provides a path integral foundation to the so-called abelianization procedure that has been used to study the Coulomb branch. For the dressed monopole operators whose expectation values do not involve non-perturbative corrections, our computations reproduce the results of abelianization. For the expectation values of other operators and the correlation functions of multiple operators in U(N) gauge theories, we compute the non-perturbative corrections due to monopole bubbling using matrix models obtained by string theory (brane) construction. We relate the results of localization to algebraic structures discussed in the mathematical literature, and also uncover a similar relation for line operators in 4d $$\mathcal {N}=2$$ gauge theories. For U(N) (quiver) gauge theories in 3d we demonstrate a direct correspondence between wall-crossing in matrix models and the ordering of operators.