Structure of the $\mathcal{N}=4$ chiral algebra

TL;DR

The authors analyze the structure of the chiral algebra ${\mathbb V}$ attached to 4D ${\cal N}=4$ SYM by enforcing associativity of the operator product expansions in a graded, recursive fashion. They find that the algebra is uniquely fixed by the central charge $c$, with no additional continuous parameters, and that the OPEs exhibit nontrivial truncation patterns only at specific $c$ values consistent with finite Beem-type spectra. The truncation analysis argues against a symmetric orbifold origin for generic $c$, though finite-$M$ nuances in twisted holography may permit subtler realizations; overall, the results reveal a rigid VOA structure controlled by $c$ and strongly aligned with 4D SCFT expectations for $c=-3(N^2-1)$. These findings contribute to the understanding of how 4D ${\cal N}=4$ theories encode 2D chiral algebras and suggest avenues for cross-checks with AdS$_3$/CFT$_2$ constructions and worldsheet approaches.

Abstract

The chiral algebra of 4D $\mathcal{N}=4$ SU$(N)$ super-Yang-Mills theory is an $\mathcal{N}=4$ superconformal vertex operator algebra. We analyse the structure of this algebra by studying recursively the constraints that are required by the associativity of the operator product expansion. We find that the algebra is uniquely characterized by the central charge (which can take an arbitrary value), without any additional free parameter. Furthermore, the truncation pattern of the OPE coefficients suggests that the algebra cannot arise from the symmetric orbifold.