Abstract

Gromov Witten invariants Fg encodes the numbers of genus g curves in Calabi Yau threefolds and play an important role in enumerative geometry. In 1993, Bershadsky, Cecotti, Ooguri, Vafa exhibited a hidden "Feynman structure" governing all Fg’s at once, using path integral methods. The counterpart in mathematics has been missing for many years. After a decades of search, in 2018, a mathematical approach: Mixed Spin P field (MSP) moduli, is finally developed to provide the wanted “Feynman structure”, for quintic CY 3-fold. Instead of enumerating curves in the quintic 3-fold, MSP enumerate curves in a large N dimensional singular space with quintic-3-fold boundary. The “P fields” and “cosections” are used to formulate counting in the singular space via a Landau Ginzburg type construction.

In this talk, I shall focus on geometric ideas behind the MSP moduli. Some update will be provided. The results follow from a decade of joint works with Jun Li, Shuai Guo, Young Hoon Kiem, Weiping Li, Melissa C.C. Liu, Jie Zhou, and Yang Zhou.