23rd Geometry and Physics Seminar

Title (Pei): Hidden algebraic structures in geometry from quantum field theory

Abstract (Pei): The existence of quantum field theories in higher dimensions predicts many hidden algebraic structures in geometry and topology. In this talk, I will survey some recent developments where such algebraic structures lead to new insights into 1) the quantization of moduli spaces of Higgs bundles, 2) the categorification of quantum invariants of 3-manifolds, and 3) novel types of TQFTs in four dimensions.

Title (Dimofte): Taming Categories in QFT

Abstract: (Dimofte): The mathematical structure of a (linear) category has turned out to be exceedingly useful for organizing the information of one-dimensional extended operators (or defects) in quantum field theory --- especially QFT with some topological invariance or topological sector. I'll summarize the definition of a category, review/explain why this correspondence is particularly natural, and highlight some key examples, from boundary conditions in 2d topological QFT, to line operators in 3d Chern-Simons, to line operators in 3d and 4d twisted Yang-Mills theories (the subject of much of my work). Then I'll turn to the practical question of how one actually computes a category in QFT --- where the most accessible information lies in state spaces and operator algebras (not so much categories). A general answer is suggested by "Tannakian duality," part of representation theory in mathematics. I'll give some very old and some very new examples of Tannakian duality in QFT, the new ones leading to quantum groups & Yangians in twisted 3d & 4d gauge theories.