English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We consider double scaled little string theory on $K3$. These theories arelabelled by a positive integer $k \ge 2$ and an $ADE$ root lattice with Coxeternumber $k$. We count BPS fundamental string states in the holographic dual ofthis theory using the superconformal field theory $K3 \times \left(\frac{SL(2,\mathbb{R})_k}{U(1)} \times \frac{SU(2)_k}{U(1)} \right) \big/\mathbb{Z}_k$. We show that the BPS fundamental string states that are countedby the second helicity supertrace of this theory give rise to weight two mixedmock modular forms. We compute the helicity supertraces using two separatetechniques: a path integral analysis that leads to a modular invariant butnon-holomorphic answer, and a Hamiltonian analysis of the contribution fromdiscrete states which leads to a holomorphic but not modular invariant answer.From a mathematical point of view the Hamiltonian analysis leads to a mixedmock modular form while the path integral gives the completion of this mixedmock modular form. We also compare these weight two mixed mock modular forms tothose that appear in instances of Umbral Moonshine labelled by Niemeier rootlattices $X$ that are powers of $ADE$ root lattices and find that they areequal up to a constant factor that we determine. In the course of the analysiswe encounter an interesting generalization of Appell-Lerch sums andgeneralizations of the Riemann relations of Jacobi theta functions that theyobey.

ADE Double Scaled Little String Theories, Mock Modular Forms and Umbral Moonshine