Commuting Line Defects At $q^N=1$

TL;DR

This paper explains the origin of special properties of algebras related to line defects in supersymmetric quantum field theories when the deformation parameter is a root of unity, connecting algebraic structures to physical dualities.

Contribution

It provides a physical explanation for the large center of these algebras at roots of unity and generalizes the construction to boundary conditions, linking to dualities and known algebraic structures.

Findings

Large center of algebras at roots of unity explained physically

Generalization to boundary condition modules in 3D theories

Connections established with skein algebras, cluster varieties, and quantum groups

Abstract

We explain the physical origin of a curious property of algebras $\mathcal{A}_\mathfrak{q}$ which encode the rotation-equivariant fusion ring of half-BPS line defects in four-dimensional $\mathcal{N}=2$ supersymmetric quantum field theories. These algebras are a quantization of the algebras of holomorphic functions on the three-dimensional Coulomb branch of the SQFTs, with deformation parameter $\log \mathfrak{q}$. They are known to acquire a large center, canonically isomorphic to the undeformed algebra, whenever $\mathfrak{q}$ is a root of unity. We give a physical explanation of this fact. We also generalize the construction to characterize the action of this center in the $\mathcal{A}_\mathfrak{q}$-modules associated to three-dimensional $\mathcal{N}=2$ boundary conditions. Finally, we use dualities to relate this construction to a construction in the Kapustin-Witten twist of four-dimensional $\mathcal{N}=4$ gauge theory. These considerations give simple physical explanations of certain properties of quantized skein algebras and cluster varieties, and quantum groups, when the deformation parameter is a root of unity.