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Vertex Algebras for S-duality

Thomas Creutzig, Davide Gaiotto

TL;DR

The paper develops a comprehensive framework of corner vertex operator algebras $ rak A[ rak g,oldsymbol{ ext Psi}]$ arising from junctions of 4d ${ m N}=4$ gauge theory, encoding protected operator algebras that interpolate between Dirichlet boundaries and their S-duals. It constructs these VOAs via conformal extensions and DS reductions, identifies dualities as convolutions over the gauge group, and exhibits explicit kernels for simply-laced and non-simply-laced groups (notably the $U(2)$ case and classical groups), with detailed treatment of DS reductions and coset realizations. The work connects to the quantum Geometric Langlands program through spaces of conformal blocks and proposes braid-compatible module structures that realize S-duality actions on VOAs and their blocks. It further explores dual descriptions and limits (e.g., $oldsymbol{ ext Psi} ooldsymbol{ ext Psi}^{-1}$ and $oldsymbol{ ext Psi} ooldsymbol{ ext Psi}+1$), conjectures about deformable VOA families, and the role of surface defects, offering a rich mathematical and physical program linking VOAs, gauge theory, and Langlands dualities.

Abstract

We define new deformable families of vertex operator algebras $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ 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 $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ VOAs 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 VOAs equipped with a $\mathfrak{g}$ affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of VOAs. The space of conformal blocks (in the derived sense, i.e. chiral homology) for $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ 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 VOAs 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 $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ VOAs is of broader applicability and leads to many new results and conjectures about deformable families of VOAs.

Vertex Algebras for S-duality

TL;DR

The paper develops a comprehensive framework of corner vertex operator algebras arising from junctions of 4d gauge theory, encoding protected operator algebras that interpolate between Dirichlet boundaries and their S-duals. It constructs these VOAs via conformal extensions and DS reductions, identifies dualities as convolutions over the gauge group, and exhibits explicit kernels for simply-laced and non-simply-laced groups (notably the case and classical groups), with detailed treatment of DS reductions and coset realizations. The work connects to the quantum Geometric Langlands program through spaces of conformal blocks and proposes braid-compatible module structures that realize S-duality actions on VOAs and their blocks. It further explores dual descriptions and limits (e.g., and ), conjectures about deformable VOA families, and the role of surface defects, offering a rich mathematical and physical program linking VOAs, gauge theory, and Langlands dualities.

Abstract

We define new deformable families of vertex operator algebras 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 VOAs are equipped with two 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 VOAs equipped with a affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of VOAs. The space of conformal blocks (in the derived sense, i.e. chiral homology) for 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 VOAs 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 VOAs is of broader applicability and leads to many new results and conjectures about deformable families of VOAs.

Paper Structure

This paper contains 63 sections, 10 theorems, 217 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Geometric Langlands and branes
  4. Vertex operator algebras
  5. Conjectures on vertex operator algebras
  6. A brief discussion of surface defects
  7. A quick review of the gauge theory setup
  8. The basic boundary conditions and interfaces
  9. The weakly coupled corner VOAs
  10. Kac-Moody algebra at the corner
  11. $W_{G}$ algebra at the corner
  12. $W_{G,\rho}$ algebra at the corner
  13. An alternative realization of $W_{G}$ algebra at the corner
  14. The action of S-duality
  15. Concatenating junctions
  16. ...and 48 more sections

Key Result

Proposition 8.1

Consider the set-up of above with $A$ as in equation eq:extVOA. For $Y, Z$ in $C'$ then

Figures (11)

  • Figure 1: Vertex Operator Algebras at corners. Left: junctions between boundary conditions of GL-twisted SYM typically support VOAs, determined by the choice of boundary conditions and by the specific choice of junction. Center: Boundary line defects can end at the junction on vertex operators associated to modules for the junction VOA. The modules fuse and braid according to the braided tensor category of line defects. Right: There are modules associated to lines in either boundary. The modules associated to lines in one boundary braid trivially with modules associated to lines in the other boundary. They fuse to composite modules $M^{a,b}_{12}$.
  • Figure 2: The composition of junctions. Left: Two consecutive junctions supporting each a VOA. Middle: the composition of the junctions is associated to a larger VOA. Local operators at the composite junction arise from line defect segments stretched between the individual junctions. The composite VOA $\mathfrak{A}_{13}\equiv \mathfrak{A}_{12} \boxtimes_{C_2} \mathfrak{A}_{23}$ is a conformal extension of $\mathfrak{A}_{12} \times \mathfrak{A}_{23}$ by products of modules associated to these line segments. Right: Lines on outer boundaries are associated to modules for the full composite VOA, built again with the help of extra line defect segments stretched between the individual junctions.
  • Figure 3: A very general weakly coupled junction with a known corner VOA, as described in the text.
  • Figure 4: A reflection trick allows one to discuss junctions between four gauge groups, re-interpreting a boundary condition for $G_1 \times G_2$ as an interface between $G_1$ and $G_2$.
  • Figure 5: Some simple examples of corner configurations discussed in the text. Left: the semiclassical junction between $B_{1,0}$ and $B^D_{0,1}$, which supports a $G_\Psi$ Kac-Moody. Middle: the semiclassical junction between $B_{1,0}$ and the regular Nahm pole boundary condition $B_{0,1}\equiv B^{\rho_{\mathrm{reg}}}_{0,1}$, which supports a $W_G[\Psi]$ W-algebra. Right: the semiclassical junction between $B_{1,0}$ and $B_{-1,1}$ which includes auxiliary corner degrees of freedom given by a level $1$ WZW model $L_1[G]$. It supports a $W_G[1+\Psi^{-1}]$ W-algebra.
  • ...and 6 more figures

Theorems & Definitions (27)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Conjecture 1.6
  • Proposition 8.1
  • proof
  • Example 8.2
  • Example 8.3
  • ...and 17 more