False Theta Functions and the Verlinde formula

TL;DR

This paper explores new analytic properties of false theta functions, establishing their modular-like transformations and applying these results to derive a Verlinde-type formula for fusion rules in certain vertex algebra modules.

Contribution

It introduces novel transformation properties of false theta functions and applies them to compute fusion rules in W-algebra representation theory.

Findings

Derived modular-like transformation properties of false theta functions

Identified regularized false theta functions with characters of W-algebra modules

Proved a Verlinde-type formula for fusion rules of W(2,2p-1) modules

Abstract

We discover new analytic properties of classical partial and false theta functions and their potential applications to representation theory of W-algebras and vertex algebras in general. More precisely, motivated by clues from conformal field theory, first, we are able to determine modular-like transformation properties of regularized partial and false theta functions. Then, after suitable identification of regularized partial/false theta functions with the characters of atypical modules for the singlet vertex algebra W(2,2p-1), we formulate and prove a Verlinde-type formula for the fusion rules of irreducible W(2,2p-1)-modules.