Higher rank partial and false theta functions and representation theory

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).