Event Details

Speaker: Sebastiano Carpi (Rome Tor Vergata) Title: Vertex operator algebras, Teichmüller modular forms and the monster

Abstract: Classical modular functions and modular forms are meromorphic functions on the upper half plane H satisfying certain functional equations related to the action of the modular group SL(2,Z) on H. They play an important role in number theory and they are deeply related to the geometry of the moduli space of genus one compact complex curves. Vertex operator algebras (VOAs) can be seen as an axiomatization of chiral conformal (quantum) field theories in two space-time dimensions (chiral CFT). An important example is given by the Frenkel-Lepowsky-Meurman moonshine VOA. It is a holomorphic VOA i.e. a VOA with trivial representation theory. Its genus one partition function can be directly related to the Klein modular function j and its automorphism group is the Fisher-Griess monster group. This gives an explanation of McKay's observation relating the Klein modular function j and the representation theory of the monster group and led Borcherds to prove the Conway-Norton moonshine conjecture. More generally the genus one partition function of a holomorphic VOA with central charge c gives rise to a modular form of weight c/2. This modular form is an important invariant of the VOA but in general it is not sufficient to determine it, as shown by examples coming from isospectral self-dual positive definite even lattices. Almost fifty years ago Friedan and Shenker conjectured, on physical grounds, that the collection of all genus g partition functions of a two-dimensional CFT completely determines the theory. Correspondingly, they argued that CFTs can be completely described in terms of the geometry of the moduli spaces of genus g compact complex curves.

Teichmüller modular forms are higher genus generalizations of classical modular forms. In this talk I will review some recent results of an ongoing joint work with Giulio Codogni. If V is a holomorphic VOA of central charge c we show that the genus g partition function of V gives rise in a natural way to a Teichmüller modular form of weight c/2. This gives strong constraints on the partition functions of holomorphic VOAs. Moreover, we clarify the relation between unitary VOAs having the same genus g partition function for all g and give various examples in which the collection of all the genus g partition functions determines the VOA. Finally, we relate the important open problem of the uniqueness of the moonshine VOA with a weak form of the Harrison-Morrison slope conjecture about the geometry of the module spaces of compact Riemann surfaces.

The talk is a joint event of the pure maths seminar and the AQFTUK network. More information about the meeting on 22 January can be found here:

https://sites.google.com/view/aqftuk/home/meeting-january-2025

The meeting is open to everyone. While the other talks will be aimed at experts, this one is meant to be accessible to the whole department.