Algebraic Geometry in String Theory Seminar

Title: Gauged Linear Sigma Models and Cohomological Field Theories

Abstract: This talk is dedicated to the memory of my friend and collaborator Bumsig Kim and based on joint work with Ciocan-Fontanine-Guere-Kim-Shoemaker.  Gauged Linear Sigma Models (GLSMs)  serve as a means of interpolating between Kahler geometry and singularity theory.  In enumerative geometry, they should specialize to both Gromov-Witten and Fan-Jarvis-Ruan-Witten theory.   In joint work with Bumsig Kim (see arXiv:2006.12182), we constructed such enumerative invariants for GLSMs.  Furthermore, we proved that these invariants form a Cohomological Field Theory.   In this lecture, I will describe GLSMs and Cohomological Field Theories, review the history of their development in enumerative geometry, and discuss the construction of these general invariants.  Briefly, the invariants are obtained by forming the analogue of a virtual fundamental class which lives in the twisted Hodge complex over a certain “moduli space of maps to the GLSM”.  This virtual fundamental class roughly comes as the Atiyah class of a “virtual matrix factorization” associated to the GLSM data.