Crossing the Walls in Enumerative Geometry

This project concerns the field of algebraic geometry, a branch of mathematics studying the geometric structure of solutions of polynomial equations. Many of the questions under study in this project are motivated by string theory, a branch of theoretical physics connected with the structure of elementary particles. This project aims to significantly enhance the intensive and fruitful interaction between cutting edge research in enumerative algebraic geometry and theoretical physics. The research aims to extend mathematical developments that verify and generalize conjectures originating from physics, and the work is expected to significantly impact development of the physical theory as well. Through conferences, a summer school, seminars, and research involvement, this project provides unique opportunities for a new generation of mathematicians to obtain the interdisciplinary knowledge and skills needed to work in this exciting research area.

The aim of the project is to study enumerative invariants in the broad sense and their dependence on various stability conditions, as well as dualities relating different enumerative invariants. The investigators plan to further develop the theory of Gauged Linear Sigma Models (GLSM) and will study the epsilon-wall-crossing conjecture and zeta-wall-crossing conjecture at all genera; Gromov-Witten (GW) and quasimap invariants are related by a sequence of epsilon-wall-crossing, whereas the Calabi-Yau/Landau-Ginzburg correspondence (relating GW invariants and FJRW invariants) and Pfaffian/Grassmannian correspondence are examples of zeta-wall-crossing. The investigators are developing the theory of Mixed-Spin-P (MSP) fields, to interpolate GW theory of quintic threefolds and FJRW theory of Fermat quintic polynomials, and to study algebraic structures of higher genus GW and FJRW invariants. The new theories of GLSM and MSP fields will provide new tools to attack the central and longstanding problem of computing higher genus GW invariants of compact Calabi-Yau threefolds. The investigators have been investigating K-theoretic Donaldson-Thomas invariants of threefolds, as well as GW and quasimap invariants of Nakajima quiver varieties. Because some of conjectures motivated by theoretical physics can only be properly formulated in terms of K-theoretic enumerative invariants, they plan to study dualities relating K-theoretic enumerative invariants of different geometries, and to lift results on traditional enumerative invariants to the K-theoretic setting.