时间：2026年8月10日(星期一)14:00-16:00
地点：E14-301
主讲人：李睿轩，北京大学
报告主题：From Lattice Integrability to Conformal Symmetry: Temperley-Lieb Algebra, Annular CLE, and Virasoro Characters
报告摘要：Recent exact formulas in conformal random geometry suggest that the conformal loop ensemble (CLE) possesses a rich integrable structure. Since CLE is expected to arise as the scaling limit of critical lattice models governed by Temperley-Lieb algebras and quantum groups, it is natural to ask whether its integrability can be traced back to these discrete structures. We study this question on the cylinder and the annulus.
We consider a generalized critical loop-model partition function in which non-contractible loops are assigned an independent weight. On the discrete side, the Baxter-Kelland-Wu correspondence and quantum Schur-Weyl duality decompose this partition function into traces over Temperley-Lieb standard modules. On the continuum side, conformal welding and Liouville quantum gravity determine the generating function for non-contractible loops in annular CLE, including the non-simple regime. The two formulas have the same quantum-group coefficients and Virasoro-character expansion.
This agreement gives a character-level correspondence between the two natural local structures: the Temperley-Lieb algebra, which encodes local connectivity on the lattice, and the Virasoro algebra, which encodes infinitesimal local conformal symmetries. Assuming the convergence of critical FK interfaces on the cylinder to annular CLE, we prove that normalized Temperley-Lieb traces converge to the corresponding Virasoro Kac characters. At the free-fermion point, this convergence is proved unconditionally by an exterior-algebra reduction of the transfer matrix.
