Logarithmic Geometry and the Gauged Linear Sigma Model

Boston College, Chestnut Hill MA

Investigators

Abstract

Algebraic varieties are a class of geometric objects obtained by gluing together sets of solutions of polynomial equations. In string theory, a branch of theoretical physics, algebraic varieties are used to describe fine pieces of our universe. In this theory, everything is made of tiny strings which travel through spacetime, and trace out algebraic curves in some algebraic varieties. Gromov-Witten invariants, originating from physics, are virtual counts of algebraic curves in algebraic varieties satisfying prescribed incidence constraints. They are used in physic to describe the structures of our universe. They also provide new approaches and insights to classical problems from algebraic geometry. Despite their importance, these invariants are very difficult to compute. The primary goal of this project is to develop a new method to calculate Gromov-Witten invariants by investigating the boundary of the gauged linear sigma model from physics using tools of logarithmic structures from algebraic geometry. This project provides research training opportunities for graduate students. In more detail, this project focuses on studying the geometry of the gauged linear sigma model (GLSM) using stable log maps of Abramovich-Chen-Gross-Siebert. The GLSM proposed by Witten in the 1990s can be viewed as a deep generalization of the hyper-plane property of Gromov-Witten invariants in all genus. However, the moduli stacks in GLSM which carry the perfect obstruction theory for defining GLSM invariants are in general non-proper. This presents a major difficulty in calculating GLSM invariants. Recently, log compactifications of hybrid-type GLSM were constructed by Chen, Janda, and Ruan using stable log maps. These compactifications provide proper moduli stacks carrying a reduced perfect obstruction theory whose associated virtual cycles recover the GLSM virtual cycles. This project is an integrated study aiming at a new computational method for calculating Gromov-Witten invariants by investigating the structures of the virtual cycles of these log compactifications. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.