Chiral algebra, Wilson lines, and mixed Hodge structure of Coulomb branch

TL;DR

This work links the representation theory of the non-unitary $\mathcal{N}=4$ SU($N$) chiral algebra $\mathbb{V}[\mathcal{T}_{SU(N)}]$ to the mixed Hodge structure of the Coulomb branch in 4D $\mathcal{N}=2$ theories. By computing the Wilson line Schur index and enforcing modular invariance, the authors identify an irreducible-character basis labeled by partitions of $N$ and establish a map to the chiral algebras $\mathbb{V}[\mathcal{T}_{p,N}]$, including the appearance of logarithmic modules. They show that the pure part $PH_c$ of the Coulomb-branch mixed Hodge polynomial encodes the same representation-theoretic data, yielding a concrete 4D mirror-symmetry connection between Higgs-branch representation theory and Coulomb-branch geometry. The results provide a practical route to access chiral algebra characters via exact line-operator indices and suggest broad extensions to other gauge groups and Argyres-Douglas theories, enriching the 4D-3D mirror-symmetry landscape and non-rational chiral algebra studies.

Abstract

We find an intriguing relation between the chiral algebra and the mixed Hodge structure of the Coulomb branch of four dimensional $\mathcal{N} = 2$ superconformal field theories. We identify the space of irreducible characters of the $\mathcal{N} = 4$ $SU(N)$ chiral algebra $\mathbb{V}[\mathcal{T}_{SU(N)}]$ by analytically computing the Wilson line Schur index, and imposing modular invariance. We further establish a map from the $\mathbb{V}[\mathcal{T}_{SU(N)}]$ characters to the characters of the $\mathcal{T}_{p, N}$ chiral algebra. We extract the pure part of the mixed Hodge polynomial $PH_c$ of the Coulomb branch compactified on a circle, and prove that $PH_c$ encodes the representation theory of $\mathbb{V}[\mathcal{T}_{SU(N)}]$. We expect this to be a new entry of the 4D mirror symmetry framework.