Abstract
We use supersymmetric localization to calculate correlation functions of half-BPS local operators in 3d mathcal{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 mathcal{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 mathcal{N}=2 subalgebra of the mathcal{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 S3 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.
Highlights
Correlation functions of local operators are fundamental observables in quantum field theory
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
We 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
Summary
Correlation functions of local operators are fundamental observables in quantum field theory. We focus on 3d quantum field theories with N = 4 supersymmetry defined by general Lagrangians constructed from vectormultiplets coupled to hypermultiplets.2 In these theories, we provide new formulas for calculating correlation functions of certain operators. At the SCFT fixed point, the 2- and 3-point correlators of non-monopole Coulomb branch operators, as well as npoint functions of their twisted analogs, can be calculated by inserting gauge-invariant polynomials in σ into the matrix model. In terms of the fields of the 3d SCFT, the twisted Coulomb branch operators represented by gauge-invariant polynomials in σ correspond to position-dependent linear combinations of polynomials in the vectormultiplet scalars. We discuss the supersymmetry algebras preserved by these actions
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