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Abstract: (arXiv)
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 log⁡q\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.
Note:
  • 35 pages, 7 figures, 1 Mathematica notebook attached as ancillary files
  • dimension: 4
  • dimension: 3
  • quantization: algebra
  • defect
  • deformation
  • holomorphic
  • duality
  • Coulomb
  • gauge field theory
  • boundary condition
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