ADE double scaled little string theories, mock modular forms and Umbral Moonshine

We consider double scaled little string theory on K3. These theories are labelled by a positive integer k ≥ 2 and an ADE root lattice with Coxeter number k. We count BPS fundamental string states in the holographic dual of this theory using the super-conformal field theory K3×SL2ℝkU1×SU2kU1/ℤk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ K3\times \left(\frac{\mathrm{SL}{\left(2,\mathbb{R}\right)}_k}{\mathrm{U}(1)}\times \frac{\mathrm{SU}{(2)}_k}{\mathrm{U}(1)}\right)/{\mathbb{Z}}_k $$\end{document}. We show that the BPS fundamental string states that are counted by the second helicity supertrace of this theory give rise to weight two mixed mock modular forms. We compute the helicity supertraces using two separate techniques: a path integral analysis that leads to a modular invariant but non-holomorphic answer, and a Hamiltonian analysis of the contribution from discrete states which leads to a holomorphic but not modular invariant answer. From a mathematical point of view the Hamiltonian analysis leads to a mixed mock modular form while the path integral gives the completion of this mixed mock modular form. We also compare these weight two mixed mock modular forms to those that appear in instances of Umbral Moonshine labelled by Niemeier root lattices X that are powers of ADE root lattices and find that they are equal up to a constant factor that we determine. In the course of the analysis we encounter an interesting generalization of Appell-Lerch sums and generalizations of the Riemann relations of Jacobi theta functions that they obey.