Geometry of Moduli Spaces and Metaplectic Representations

Conformal field theories, rooted in theoretical physics, offer a powerful framework for geometers to study algebraic curves. While traditionally focused on curves, recent advancements in the geometric manifestations of infinite-dimensional algebras suggest the emergence of a rich theory applicable to higher-dimensional varieties as well. Motivated by promising preliminary findings, the PI will lead an exploration of conformal field theories extending beyond curves, probing fundamental questions concerning the geometry of higher-dimensional varieties. Similarly, the PI will investigate invariants for knots and 3-manifolds. This project will fuel the integration of ideas from several fields of mathematics, such as representation theory, algebraic geometry, and quantum topology. It will also feature experiential learning initiatives tailored for middle school students, alongside research training opportunities designed for undergraduate and graduate students. Furthermore, the project aims to foster interdisciplinary collaborations and enhance mathematical literacy within the general public through a series of public lectures and events. More specifically, the PI will build upon recent work on coinvariants of vertex algebras to explore geometric realizations of metaplectic modules on abelian varieties and their moduli spaces. These investigations will offer a novel perspective on the theory of theta functions and vector bundles equipped with a projectively flat connection on families of abelian varieties. Furthermore, the PI will investigate the factorization properties of spaces of coinvariants on decomposable abelian varieties, followed by an assessment of the persistence of these properties at boundary points across various compactifications. Finally, the PI will explore various refinements of the theory of homological blocks for knots and 3-manifolds. 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.