Factorization and degeneration of chiral homology

University Of Texas At Austin, Austin TX

Investigators

Abstract

A classical problem, with many applications in the sciences, engineering and the arts, is to determine the symmetries of an object. The branch of mathematics that studies such questions is called representation theory. In this field, symmetries are studied in part by packaging them together as abstract structures with appropriate algebraic properties, such as groups or Lie algebras. Vertex Operator Algebras (VOAs) are a generalization of Lie algebras. VOAs are tightly connected to theoretical physics in what is known as Conformal Field Theory, and, also, to the geometry of surfaces, like spheres or donut-like objects. An important way to study VOAs and their relationship to geometry is via what is known as Chiral Homology. This can be seen as a recipe that takes a VOA and a surface as ingredients and produces a collection of spaces that encode information about the symmetries of the VOA and the complexity of the surface they depend on. However, a variety of fundamental questions about the spaces produced through this recipe are still unresolved. In this project the PI will answer some of these questions. In particular, the PI will describe how Chiral Homology behaves when the surface it depends on is appropriately deformed, and provide a geometric realization of Chiral Homology. The project will also provide research training opportunities for students. In more technical terms, spaces of conformal blocks associated with projective curves--the algebraic analogue of surfaces--and Lie algebras have been a central object of study in algebraic geometry. In fact, these spaces can be identified with generalized theta functions on the moduli space of principal bundles, and they also define vector bundles on moduli spaces of stable curves. One can consider natural generalizations of these spaces: replacing Lie algebras with VOAs; considering the derived notion of conformal blocks, called Chiral Homology; and allowing the projective curve to admit worse than nodal singularities. The PI and her coauthors have shown that conformal blocks from regular VOAs satisfy factorization and sewing. These properties explicitly control the behavior of conformal blocks under nodal degeneration of the curve they depend on and have been the main tools to explicitly compute the dimensions of these spaces through the Verlinde formula. The In this project, the Pi will show that Chiral Homology from regular VOAs satisfies factorization and sewing. Furthermore, the PI will provide a geometric realization of Chiral Homology and extend this notion to curves with worse singularities. 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.