On Regularised Quantum Dimensions of the Singlet Vertex Operator Algebra and False Theta Functions

TL;DR

This paper investigates the regularised quantum dimensions of singlet vertex operator algebras, revealing how these dimensions relate to false theta functions, modular transformations, and fusion rings, with implications for quantum group theory.

Contribution

It introduces a novel regularisation method for characters of singlet vertex operator algebras using a complex parameter, linking quantum dimensions to modular forms and fusion categories.

Findings

Quantum dimensions depend on the sign of the real part of the regularisation parameter.

The character space maps to the fusion ring as a ring isomorphism for positive regularisation.

The fusion ring of a rational VOA emerges as a semi-simplification of the original category.

Abstract

We study a family of non-C2-cofinite vertex operator algebras, called the singlet vertex operator algebras, and connect several important concepts in the theory of vertex operator algebras, quantum modular forms, and modular tensor categories. More precisely, starting from explicit formulae for characters of modules over the singlet vertex operator algebra, which can be expressed in terms of false theta functions and their derivatives, we first deform these characters by using a complex parameter {\epsilon}. We then apply modular trans- formation properties of regularised partial theta functions to study asymptotic behaviour of regularised characters of irreducible modules and compute their regularised quantum dimensions. We also give a purely geometric description of the regularisation parameter as a uniformisation parameter of the fusion variety coming from atypical blocks. It turns out that the quantum dimensions behave very differently depending on the sign of the real part of {\epsilon}. The map from the space of characters equipped with the Verlinde product to the space of regularised quantum dimensions turns out to be a genuine ring isomorphism for positive real part of {\epsilon} while for sufficiently negative real part of {\epsilon} its surjective image gives the fusion ring of a rational vertex operator algebra. The category of modules of this rational vertex operator algebra should be viewed as obtained through the process of a semi-simplification procedure widely used in the theory of quantum groups. Interestingly, the modular tensor category structure constants of this vertex operator algebra, can be also detected from vector valued quantum modular forms formed by distinguished atypical characters.