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Vertex Algebras for S-duality

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Abstract

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.

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Notes

  1. The global form of the gauge group does not affect directly the algebra of local operators but will select a specific class of modules for the VOA.

  2. This setup has obvious generalizations involving two-dimensional junctions of multiple topological interfaces.

  3. Or even webs of junctions between topological interfaces.

  4. This statement is true up to important fermion number shifts, which are the reason our construction produces super-algebras rather than algebras, and up to restricting the weights of the representations in a manner associated to the action of Langlands duality on the global form of the group.

  5. This actually makes the second map in the above list redundant, thanks to the coset description of \(W_\Psi [\mathfrak {g}]\).

  6. The non-critically-shifted level would be \(\Psi - n_G-h^\vee _G\).

  7. Up to important fermion number shifts due to the DS reduction.

  8. Notice that if \({}^L G = G\) then \(L_1[G]\) has only the vacuum module and is a good chiral CFT. For more general G it is a relative theory and one may worry why is it OK to use it as extra junction degrees of freedom. This can be explained by a careful analysis of how the global form of the group changes under S-duality, leading to subtle discrete anomalies which are cancelled by the coupling to \(L_1[G]\). We will not do so here.

  9. A very careful reader may wonder about the appearance of fermionic degrees of freedom at junctions in a bosonic theory. Some questions may also be raised about subtle interplay of fermionic and bosonic notions of mutual locality of modules in the coset. Such a reader is invited to explore related subtleties about the electric-magnetic duality group of Abelian gauge theories [Met15], such as the fact that the ST transformation we use here maps standard gauge connections to \(\hbox {Spin}_{\mathbb {C}}\) gauge connections.

  10. The fact that such conformal blocks can be defined in a manner which is algebraic in the \(G_{\mathrm {out}}\) flat connection is rather non-trivial and it is intimately related to the fact that the \(G_{\mathrm {out}}\) outer automorphism symmetry is the remnant of a G Kac–Moody algebra whose level is sent to infinity.

  11. In order to understand the level shifts, recall that the critical level for U(N) is \(-N\) and the non-shifted level of the \(U(N-1)\) sub-algebra equals the non-shifted level of the U(N) Kac–Moody algebra. Hence the non-critically shifted levels are both \(\Psi -N+1\).

  12. In order to understand the level shifts, recall that the critical level for U(N|M) is \(M-N\) and hence the non-shifted level of \(U(N|N-1)_{\Psi ^{-1}}\) is \(\Psi ^{-1}-1\). The DS reduction modifies the non-shifted level of the U(N) sub-algebra from \(\Psi ^{-1}-1\) to the desired \(\Psi ^{-1}-N+1\).

  13. Recall that the VOA of \(2N+1\) fermions is a simple current extension of \(L_1(so(2N+1))\) and here this is (as usual in CFT) meant by \(L_1[SO(2N+1)]\).

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Acknowledgements

DG thanks S. Braverman, K. Costello, P. Yoo for many instructive conversations. TC appreciates various discussions with T. Arakawa and A. Linshaw on related topics.

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Authors and Affiliations

  1. University of Alberta, Edmonton, Canada

    Thomas Creutzig

  2. Perimeter Institute, Ontario, Canada

    Davide Gaiotto

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  1. Thomas Creutzig
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Correspondence to Thomas Creutzig.

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Communicated by X. Yin

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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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Creutzig, T., Gaiotto, D. Vertex Algebras for S-duality. Commun. Math. Phys. 379, 785–845 (2020). https://doi.org/10.1007/s00220-020-03870-6

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