Publication date:
9 July 2017
Source:Advances in Mathematics, Volume 314
Author(s): Thomas Creutzig, Antun Milas
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 higher rank partial and false theta functions as well as Kostant's version of these. 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. This paper substantially generalize our previous work [19] pertaining to $(1,p)$-singlet W-algebras (the ${\mathfrak{sl}}_2$ case). Results in this paper are very general and are applicable in a variety of situations.