Abstract
We develop an approach to the study of Coulomb branch operators in 3D mathcal{N} = 4 gauge theories and the associated quantization structure of their Coulomb branches. This structure is encoded in a one-dimensional TQFT subsector of the full 3D theory, which we describe by combining several techniques and ideas. The answer takes the form of an associative and noncommutative star product algebra on the Coulomb branch. For “good” and “ugly” theories (according to the Gaiotto-Witten classification), we also exhibit a trace map on this algebra, which allows for the computation of correlation functions and, in particular, guarantees that the star product satisfies a truncation condition. This work extends previous work on abelian theories to the non-abelian case by quantifying the monopole bubbling that describes screening of GNO boundary conditions. In our approach, monopole bubbling is determined from the algebraic consistency of the OPE. This also yields a physical proof of the Bullimore-Dimofte-Gaiotto abelianization description of the Coulomb branch.
Highlights
Gauge theories in three dimensions contain special local defect operators called monopole operators, which are defined by requiring certain singular behavior of the gauge field close to the insertion point [1]
We develop an approach to the study of Coulomb branch operators in 3D N = 4 gauge theories and the associated quantization structure of their Coulomb branches
We focus on a class of 3D gauge theories with N = 4 supersymmetry constructed by coupling a vector multiplet with gauge group G to a matter hypermultiplet that transforms in some representation of G.1
Summary
Gauge theories in three dimensions contain special local defect operators called monopole operators, which are defined by requiring certain singular behavior of the gauge field close to the insertion point [1]. The goal of this paper is to present the first direct computations of operator product expansion (OPE) coefficients and correlation functions of monopole operators in 3D non-abelian gauge theories. The mathematical physics motivation for studying the 1D TQFT is that it provides a “quantization” of the ring of holomorphic functions defined on the Coulomb branch MC The 3D theories that we study have two distinguished branches of the moduli space of vacua: the Higgs branch and the Coulomb branch These are each parametrized, redundantly, by VEVs of gauge-invariant chiral operators whose chiral ring relations determine the branches as generically singular complex algebraic varieties. Further examples, and comments on connections between our approach and existing ones can be found in the appendices
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