Giant Gravitons, Fermionic Forms and Vertex Algebras

TL;DR

This paper explores the mathematical structure of giant graviton expansions in 3D superconformal theories, revealing connections to vertex algebras, quiver varieties, and parafermionic W-algebras.

Contribution

It uncovers a novel relation between giant graviton coefficients, quiver variety Hilbert series, and vertex algebra representations, providing explicit formulas involving affine fermionic forms.

Findings

Coefficients match characters of parafermionic W-algebras

Explicit formulas derived for specific toric hyper-Kähler cones

Connections established between physical expansions and algebraic structures

Abstract

We investigate the mathematical and physical content of the giant graviton expansion of three-dimensional $\mathcal{N}=4$ superconformal field theories in a simplifying limit. We uncover an interesting relation between the coefficients in this expansion, the Hilbert series of certain quiver varieties and the representation theory of vertex algebras. In particular, for the worldvolume theory of $N$ M2-branes at the tip of a toric hyper-K\"{a}hler four-fold cone: $X_{4}=\mathbb{C}^2 /{\mathbb{Z}_L} \times \mathbb{C}^2/{\mathbb{Z}_K}$, we derive an explicit expression for the coefficients in terms of affine fermionic forms and show that they coincide with characters of a direct sum of parafermionic W-algebras.