Superconformal partition functions and non-perturbative topological strings

A bstract We propose a non-perturbative definition for refined topological strings. This can be used to compute the partition function of superconformal theories in 5 dimensions on squashed S 5 and the superconformal index of a large number of 6 dimensional (2 , 0) and (1 , 0) theories, including that of N coincident M5 branes. The result can be expressed as an integral over the product of three combinations of topological string amplitudes. SL(3 , Z ) modular transformations acting by inverting the coupling constants of the refined topological string play a key role.