A one-dimensional theory for Higgs branch operators

TL;DR

This work constructs and localizes a protected 1d sector of twisted Higgs branch operators in 3d ${ m N}=4$ QFTs on ${ m S}^3$, recasting their correlators into a 1d Gaussian theory coupled to a matrix model. By localizing with a carefully chosen supercharge in the ${ m su}(2|1)_{ m l} imes{ m su}(2|1)_{ m r}$ algebra, the vectormultiplet sector reduces to a simple matrix model while the hypermultiplet sector becomes a 1d Gaussian theory on a circle; together they reproduce all 2- and 3-point Higgs-branch correlators and, via operator mixing, higher-point data, in a way that generalizes to non-conformal theories with real masses and FI terms. The formalism yields a deformation quantization of the Higgs branch chiral ring, with the star product and OPE coefficients computable from 1d data; several explicit SCFT examples (SQED, necklace quivers, and U(2) with adjoint/fundamental matter) validate mirror symmetry predictions and reveal precise star-product constants. The framework also provides partial results for the Coulomb branch, and it opens avenues to monopole operators and other dimensions. Overall, the paper delivers a concrete, broadly applicable localization toolkit linking 3d ${ m N}=4$ dynamics to a tractable 1d/topological quantum-mechanical description with direct computational access to protected operator data.

Abstract

We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d ${\cal N} = 4$ superconformal field theories whose Lagrangian descriptions consist of vectormultiplets coupled to hypermultiplets. The operators we primarily study are certain twisted linear combinations of Higgs branch operators that can be inserted anywhere along a given line. These operators are constructed from the hypermultiplet scalars. They form a one-dimensional non-commutative operator algebra with topological correlation functions. The 2- and 3-point functions of Higgs branch operators in the full 3d ${\cal N}=4$ theory can be simply inferred from the 1d topological algebra. After conformally mapping the 3d superconformal field theory from flat space to a round three-sphere, we preform supersymmetric localization using a supercharge that does not belong to any 3d ${\cal N} = 2$ subalgebra of the ${\cal N}=4$ algebra. The result is a simple model that can be used to calculate correlation functions in the 1d topological algebra mentioned above. This model is a 1d Gaussian theory coupled to a matrix model, and it can be viewed as a gauge-fixed version of a topological gauged quantum mechanics. Our results generalize to non-conformal theories on $S^3$ that contain real mass and Fayet-Iliopolous parameters. We also provide partial results in the 1d topological algebra associated with the Coulomb branch, where we calculate correlation functions of local operators built from the vectormultiplet scalars.