Higher genus categorical Gromov-Witten invariants

这个项目涉及代数几何的研究，多项式方程解的几何研究。这一主题在过去几年中取得了重大进展。其中最引人注目的是现代不变量的发明，它可以计算出空间中具有某些性质的曲线的数量，这些曲线预计与我们当前宇宙的时空有关。一个重大的突破来自于这样的理解，即这些不变量的计算可以用与所谓镜像空间几何相关的某些方程的解来理解。虽然这些不变量的最初描述是几何性质的，但Kontsevich和Costello的工作表明，代数方法将导致更大的灵活性和更有效的计算。这个项目的目标是扩展这项工作，并以这样一种方式定义不变量，即同调镜像对称猜想的现有结果将自动暗示现有的曲线计数的数值预测是正确的。 凯文·科斯特洛在2005年引入了Gromov-Witten不变量的范畴推广，定义了所有的属。尽管相当大的兴趣，很少有人知道这些不变量。第一次计算，对于椭圆曲线的通用族，是由PI在2017年与涂俊武共同完成的。在这个项目中，PI扩展了目前对这些不变量的理解。PI将完成最初由K提出的想法。Costello，然后利用所得的理论计算了高维簇的正亏格的B-模型Gromov-Witten不变量，包括五次三重簇. 这将验证镜像对称预言在高等亏格中的有效性。主要的方法是用称为矩阵分解范畴的代数结构来代替几何空间。这本身是独立的利益：这些类别的不变量应该与Fan-Jarvis-Ruan-Witten不变量密切相关，但这种关系并不明确。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.