Abstract
We formulate a conjectural relation between the category of line defects in topologically twisted 3d mathcal{N} = 4 supersymmetric quantum field theories and categories of modules for Vertex Operator Algebras of boundary local operators for the theories. We test the conjecture in several examples and provide some partial proofs for standard classes of gauge theories.
Highlights
There are various intertwined relations between supersymmetric gauge theories and Vertex Operator Algebras [1,2,3,4,5,6,7,8,9,10,11]
Holomorphicity guarantees that the local operators at that location will have meromorphic OPE’s and form a vertex algebra
Up to a singular gauge transformation, this is precisely how a gauge theory monopole operator of charge ±1 looks like! the VOA dictionary is compatible with the standard mirror symmetry dictionary
Summary
There are various intertwined relations between supersymmetric gauge theories and Vertex Operator Algebras [1,2,3,4,5,6,7,8,9,10,11]. The objective of this paper is to study the relation between the bulk topological data and the properties of modules for the corresponding boundary VOAs. In particular, we would like to compare the algebra of functions on the Higgs or Coulomb branches with the algebra of derived endomorphisms of the vacuum module for the boundary VOAs. The analysis of the most basic examples will immediately show us the importance of the “derived” part of this statement. The VOAs themselves can be effective computational tools to study the bulk TFTs. One final observation is that the algebras of endomorphisms of line defects in twisted 3d N = 4 gauge theories admit interesting “quantum deformations” associated to Ω deformations of the theory [21]. We expect these quantum deformations to arise from VOA constructions, perhaps working equivariantly for loop rotations
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