The M5-brane on K3 and del Pezzo's and multi-loop string amplitudes

We study the BPS spectrum of Little String Theory for bound states of M5-branes wrapped on six manifold of product topology M4 × Σ2 and the apparence of multi-loop θ-functions in a supersymmetric index calculation. We find a total reconstruction of the g-loop heterotic contribution in the case of a double K3 M-theory compactification. Moreover, we consider total wrapping of M5-branes on del Pezzo surfaces Bk and, by studying the relevant amplitude, we notice the arising of θ-functions relative to BPS strings on Tk−1, i.e. membranes on Tk. This happens because of beautiful relations between four dimensional SYM theories and CFTs in two dimensions and seems to be linked to a duality recently observed by A. Iqbal, A. Neitzke and C. Vafa in [1].