W-Algebras and Universal Objects in Vertex Algebra Theory

University Of Denver, Denver CO

Investigators

Abstract

Many structures and concepts that were inspired by physics have had a profound influence on mathematics during the last half century. For example, stringy invariants of manifolds such as quantum cohomology have led to spectacular advances in enumerative geometry. On the algebraic side, a fundamental new structure called a vertex operator algebra (VOA) emerged from quantum field theory in the 1980s, and was axiomatized by Borcherds in his work on the Moonshine Conjecture. VOAs are natural generalizations of commutative rings, and in the last thirty years, they have found applications in a diverse range of subjects including finite group theory, representation theory, combinatorics, number theory, and algebraic geometry. In this project, the PI will investigate the structure and representation theory of VOAs, as well as some connections between VOAs and algebraic geometry. These projects will advance the subject and provide educational and collaborative opportunities for the PI's current and former graduate students. First, the PI's work on the coset realization of principal W-algebras of A, D, and E types resolved a 30-year-old conjecture which had been a key starting assumption in physics. It has many striking corollaries such as the unitarity of all discrete series representations of principal W-algebras, and the existence of modular tensor categories of modules for affine VOAs at admissible levels. The PI will construct coset realizations of principal W-algebras of other Lie types, as well as non-principal W-algebras and W-superalgebras. Second, the PI recently constructed a universal two-parameter VOA which interpolates between all the type A principal W-algebras, in the sense that they arise as one-parameter quotients of this structure. The PI will construct more general universal objects, which are VOAs defined over the ring of functions on some variety X. Typically, interesting VOAs are obtained by specializing the universal object along certain subvarieties of X, and unexpected isomorphisms of these VOAs correspond to intersection points on these subvarieties. Third, the PI will study two affine schemes that have been attached to a VOA by Arakawa: the associated scheme and the singular support. There is always a closed embedding of the singular support in the arc space of the associated scheme. The PI will investigate good criteria for this embedding to be an isomorphism. This has many applications in VOA theory and is connected to questions about the geometry of arc spaces that are of independent interest. 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.