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Communications in Mathematical Physics

We define new deformable families of vertex operator algebras $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ 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 $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras are equipped with two $$\mathfrak {g}$$ 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 $$\mathfrak {g}$$ 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 $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ 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 $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras.

Vertex Algebras for S-duality