Identities from Vertex Operator Algebras on the Moduli of Curves

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In algebraic geometry, one aims to understand varieties, modeled on solutions of polynomial equations. By varying the coefficients of the polynomials, one obtains families of related objects. Often the family itself can be described as a variety, and by studying it, one learns about the objects it parameterizes. Moduli spaces of curves are successful examples of this approach, giving insight into families of curves and their degenerations, and serving as a prototype for other moduli spaces. Amenable to analogies and recursive arguments, they have achieved the status of special varieties on which the theory of algebraic geometry has been tested and explored. Most importantly, moduli spaces of curves have benefited from their connections to many different areas of mathematics and mathematical physics. The grant will also provide support for mathematics enrichment activities for high-school students in Philadelphia.More specifically, vertex operator algebras (VOAs) and their representations define sheaves of coinvariants (and dual sheaves of conformal blocks) on moduli spaces of curves. Studied in special cases since the 1980's, these were recently extended to the moduli space of stable pointed curves by the PI and her coauthors, who proved that when defined by VOAs satisfying finite and semi-simplicity assumptions, sheaves of coinvariants have factorization and sewing properties, and give rise to vector bundles. This project has three major objectives: (1) to generalize these results to the context where VOAs are finite but not semi-simple, as is expected from results in log-conformal field theory; (2) to study important properties and features of such sheaves in general, including their ranks and higher Chern classes; and (3) to find geometric interpretations of vector spaces of VOA conformal blocks.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.