Commuting Line Defects At qN = 1

We explain the physical origin of a curious property of algebras Aq$${{\mathcal {A}}}_{{\mathfrak {q}}}$$ which encode the rotation-equivariant fusion ring of half-BPS line defects in four-dimensional N=2$$\mathcal{N}=2$$ 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 logq$$\log {{\mathfrak {q}}}$$. They are known to acquire a large center, canonically isomorphic to the undeformed algebra, whenever q$${{\mathfrak {q}}}$$ 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 Aq$${{\mathcal {A}}}_{{\mathfrak {q}}}$$-modules associated to three-dimensional N=2$$\mathcal{N}=2$$ boundary conditions. Finally, we use dualities to relate this construction to a construction in the Kapustin–Witten twist of four-dimensional N=4$$\mathcal{N}=4$$ 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.