Vertex Algebras at the Corner

TL;DR

This work introduces Y_{L,M,N}[\Psi], a broad family of VOAs realized by BRST reductions of supergroup current algebras arising from Y-junctions in GL-twisted N=4 SYM. A central idea is that S-duality generates an S_3 triality among the three integers L,M,N, with Psi transforming under PSL(2,\mathbb{Z}) actions, yielding multiple dual descriptions of the same VOA. The authors connect Y_{L,M,N}[\Psi] to W_N and W_{1+\infty} structures, show how degenerate modules D_\nu, H_\lambda, W_\mu arise and cycle under triality, and develop a rich network of abelian and non-abelian examples (including U(1), U(2) cases, parafermions, and orthosymplectic variants). A striking feature is the crystal-melting interpretation of vacuum and degenerate module characters as counting restricted 3d partitions, linking VOA data to topological vertex technology. The paper also sketches extensions to webs of interfaces, brane configurations with O3-planes, and potential connections to higher W-algebras and moduli spaces, signaling a unifying framework for VOAs from gauge-theory junctions with broad mathematical and physical implications.

Abstract

We introduce a class of Vertex Operator Algebras which arise at junctions of supersymmetric interfaces in ${\cal N}=4$ Super Yang Mills gauge theory. These vertex algebras satisfy non-trivial duality relations inherited from S-duality of the four-dimensional gauge theory. The gauge theory construction equips the vertex algebras with collections of modules labelled by supersymmetric interface line defects. We discuss in detail the simplest class of algebras $Y_{L,M,N}$, which generalizes $W_N$ algebras. We uncover tantalizing relations between $Y_{L,M,N}$, the topological vertex and the $W_{1+\infty}$ algebra.

Figures (22)

• Figure 1: The brane system engineering our Y-junction for four-dimensional ${\cal N}=4$ SYM. The three fivebranes extend along the $01456$ directions together with a ray in the $23$ plane. The stacks of D3 branes extend along the $01$ directions and fill wedges in the $23$ plane. Notice the $SO(3)_{456}\times SO(3)_{789}$ isometry of the system, which becomes the R-symmetry of a $(0,4)$ 2d super-symmetry algebra.
• Figure 2: The dualities which motivate the identification (\ref{['eq:cyclic']}) of the VOA $Y_{L,M,N}[\Psi]$, $Y_{N,L,M}[\frac{1}{1-\Psi}]$ and $Y_{M,N,L}[1-\frac{1}{\Psi}]$.
• Figure 3: The gauge theory image of a Y-junction on the $23$ plane. We denote the specific junction as $Y_{L,M,N}$. The $Y_{L,M,N}[\Psi]$ VOA will arise as a deformation of the algebra of BPS local operators at the junction.
• Figure 4: Modules $W_{\lambda},H_{\mu},D_{\sigma}$ associated to the three classes of boundary lines.
• Figure 5: Diagonal slicing of a 3d partition as described in the main text. Note that the two series of partitions are indeed interlaced $\mu_{-1}<\mu_0>\mu_1>\mu_2=\mu_3=\mu_4$.