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 ofM5-branes wrapped on six manifold of product topology $M_4\times\Sigma_2$ andthe apparence of multi-loop $\theta$-functions in a supersymmetric indexcalculation. We find a total reconstruction of the g-loop heteroticcontribution in the case of a double K3 M-theory compactification. Moreover, weconsider total wrapping of M5-branes on del Pezzo surfaces $B_k$ and, bystudying the relevant amplitude, we notice the arising of $\theta$-functionsrelative to BPS strings on $T^{k-1}$, i.e. membranes on $T^k$. This happensbecause of beautiful relations between four dimensional SYM theories and CFTsin two dimensions and seems to be linked to a duality recently observed byA.Iqbal, A.Neitzke and C.Vafa in.Comment: 1+14 pages; v2: misprints corrected, clarifications and one reference adde