Vertex Algebras for S-duality
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Communications in
Mathematical Physics
Vertex Algebras for S-duality Thomas Creutzig1 , Davide Gaiotto2 1 University of Alberta, Edmonton, Canada. E-mail: [email protected] 2 Perimeter Institute, Ontario, Canada
Received: 31 August 2017 / Accepted: 24 July 2020 Published online: 6 October 2020 – © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract: We define new deformable families of vertex operator algebras A[g, , σ ] associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The A[g, , σ ] vertex operator algebras are equipped with two g affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of vertex operator algebras equipped with a g affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of vertex operator algebras. The space of conformal blocks (in the derived sense, i.e. chiral homology) for A[g, , σ ] is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the A[g, , σ ] vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras. Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Geometric Langlands and branes . . . . . . . . . . . . . . . . . . . .
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T. C. is supported by the Natural Sciences and Engineering Research Council of Canada (RES0020460). The research of D.G. is supported by the Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation.
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T. Creutzig, D. Gaiotto
1.3 Vertex operator algebras . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 A brief discussion of surface defects . . . . . . . . . . . . . . . . . . A Quick Review of the Gauge Theory Setup . . . . . . . . . . . . . . . . 2.1 The basic boundary conditions and interfaces . . . . . . . . . . . . . 2.2 The weakly coupled corner VOAs . . . . . . . . . . . . . . . . . . . 2.3 The action of S-duality . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Concatenating junctions . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 General
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