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.
