Higher genus categorical Gromov-Witten invariants

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This project concerns research in algebraic geometry, the geometric study of solutions of polynomial equations. This subject has seen major developments in the last few years. One of the most striking is the invention of modern invariants counting the number of curves with certain properties in spaces that are expected to be related to our current universe's space-time. A major breakthrough came from understanding that the computation of some of these invariants can be understood in terms of solutions of certain equations related to the geometry of so-called mirror spaces. While the initial description of these invariants was geometric in nature, work of Kontsevich and Costello suggests that an algebraic approach would lead to increased flexibility and more efficient computation. The goal of this project is to expand that work and define the invariants in such a way that existing results on the Homological Mirror Symmetry conjecture will automatically imply that existing numerical predictions on curve-counts are correct. Kevin Costello introduced in 2005 a categorical generalization of Gromov-Witten invariants, defined for all genera. Despite considerable interest, very little is known about these invariants. The first calculation of them, for the universal family of elliptic curves, was achieved by the PI in 2017, in joint work with Junwu Tu. In this project the PI expand the current understanding of these invariants. The PI will complete ideas originally proposed by K. Costello, and then use the resulting theory to compute B-model Gromov-Witten invariants of positive genus for higher dimensional varieties, including the quintic threefold. This will verify the validity of the mirror symmetry predictions in higher genus. The main approach will be to replace the geometric spaces with algebraic structures called categories of matrix factorizations. This is of independent interest in itself: the invariants of these categories should be closely related to the Fan-Jarvis-Ruan-Witten invariants, but such a relationship is not explicitly known.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.