2023/07/26 by Gaiotto, Davide, Moore, Gregory W., Neitzke, Andrew +1 · 1 citation
#Commutative Algebra (math.AC) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #High Energy Physics - Theory (hep-th) #Quantum Algebra (math.QA) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2307.14429
We explain the physical origin of a curious property of algebras A_\mathfrakq which encode the rotation-equivariant fusion ring of half-BPS line defects in four-dimensional 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 \mathfrakq. They are known to acquire a large center, canonically isomorphic to the undeformed algebra, whenever \mathfrakq 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 A_\mathfrakq-modules associated to three-dimensional 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 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.