Yixuan Li Gives an Academic Talk on McKay Correspondence for gl(m|n)
Recently, at the invitation of Associate Professor Huibin Chen of the School of Mathematical Sciences,Nanjing Normal University, Yixuan Li delivered an academic talk entitled “McKay Correspondence forgl(m|n).” The talk focused on McKay correspondence, ADE singularities, mirror symmetry, and the Lie superalgebra gl(m|n), attracting faculty members and students from related areas.The talk was based on ongoing joint work with Mina Aganagic, Jinghang Miao, Spencer Tamagni, and Peng Zhou. Li began by reviewing the McKay correspondence for simply laced Lie algebras and the two different realizations of ADE root systems arising from ADE singularities. These singularities are algebraic Poisson varieties. Starting from such a singularity, one may consider its semi-universal symplectic deformation and study the monodromy action, via Picard-Fuchs theory, on the middle-dimensional homology of smooth fibers. This action can be further categorified as a braid group action on the Fukaya category.Li then discussed the parallel picture coming from symplectic resolutions. In the ADE case, the symplectic resolution is also the minimal resolution. The exceptional divisors give rise to spherical objects, and the corresponding spherical twists generate a braid group action on the category of coherent sheaves on theresolution. In type A, these two pictures are mirror to each other, summarizing and connecting the work of Seidel, Smith, and Thomas.Building on this perspective, Li presented an analogue for the Lie superalgebra gl(m|n). The corresponding singularity is the threefold singularity xy = zᵐwⁿ. Unlike the classical ADE surface singularities, this threefold singularity admits many minimal resolutions, related to one another by local Atiyah flops. Correspondingly,gl(m|n) has many inequivalent Dynkin diagrams, organized by the Weyl groupoid. The talk explained how to categorify this groupoid action on the categories of coherent sheaves on the minimal resolutions and described their mirror counterparts.After the talk, participants exchanged ideas on geometric representation theory, Fukaya categories, spherical twists, local flops, and Weyl groupoids for Lie superalgebras. The lecture highlighted deep connections among singularity theory, symplectic geometry, mirror symmetry, and Lie theory, and offered newperspectives for future research in these areas.