Supersymmetric index of the M-theory 5-brane and little string theory

We propose a six-dimensional framework to calculate the supersymmetric index of M-theory 5-branes wrapped on a six-manifold with product topology M4×T2, where M4 is a holomorphic 4-cycle in a Calabi–Yau three-fold. This is obtained by zero-modes counting of the self-dual tensor contribution plus “little” string states and correctly reproduces the known results which can be obtained by shrinking or blowing the T2 volume parameter. We also extract the geometric moduli space of the multi M5-brane system and infer the generic structure of the supersymmetric index for more general geometries.