Categorical Invariants in Non-commutative Geometry

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Algebraic geometry, which is the geometric study of solutions of polynomial equations, has seen in the last few years major developments. Of these, one of the most striking is the invention of modern curve-counting invariants and the understanding that their computation can be understood in terms of solutions of certain differential equations related to the geometry of so-called mirror spaces. The ideas for doing this originated in physics, through the fields of string theory and mirror symmetry, but are now a major part of modern algebraic geometry. Costello introduced in 2005 a categorical generalization of curve-counting invariants, defined for all genera, called categorical enumerative invariants (CEIs). Despite considerable interest, little is known about them, primarily due to a range of difficulties that arise when trying to compute them. The first calculation of CEIs, for the universal family of elliptic curves, was achieved by the PI in 2017, in joint work with Junwu Tu. This computation led to work by Costello, Tu, and the PI which laid out a new foundation for the definition of CEIs, in a way that is explicitly computable. In addition, the PI will direct undergraduate research projects and train graduate students. A crucial ingredient in the above computations is a choice of a splitting of the non-commutative Hodge filtration. The general theory of CEIs does not specify which particular splitting to use, but in order to obtain geometrically meaningful invariants special choices must be made. In previous work the choices were guided, via mirror symmetry, by the symplectic geometry of the A-model. However, many of the choices appear ad hoc, and a general theory of splittings of the non-commutative Hodge filtration is largely missing. In this project the PI will expand our understanding of the role played by the spltting of the Hodge filtration in the definition of the categorical invariants. The main application will be to compute CEIs not only for geometric spaces, but also replace the actual 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.