Higher rank partial and false theta functions and representation theory

TL;DR

This paper explores higher rank generalizations of partial and false theta functions linked to rational lattices, deriving their modular properties and applying these to compute quantum dimensions in logarithmic W-algebra representations.

Contribution

It introduces higher rank partial and false theta functions, establishes their modular transformation properties, and applies these results to representation theory of W-algebras, extending previous work.

Findings

Derived modular transformation formulas for higher rank theta functions.

Computed regularized quantum dimensions of modules in logarithmic W-algebras.

Generalized earlier results from the sl_2 case to higher rank lattices.

Abstract

We study higher rank Jacobi partial and false theta functions (generalizations of the classical partial and false theta functions) associated to positive definite rational lattices. In particular, we focus our attention on certain Kostant's partial theta functions coming from ADE root lattices, which are then linked to representation theory of W-algebras. We derive modular transformation properties of regularized Kostant's partial and certain higher rank false theta functions. Modulo conjectures in representation theory, as an application, we compute regularized quantum dimensions of atypical and typical modules of "narrow" logarithmic W-algebras associated to rescaled root lattices. Results in this paper substantially generalize our previous work [19] pertaining to (1,p)-singlet W-algebras (the sl_2 case).