The geometry of M5-branes and TQFTs

The calculation of the partition function for N M5-branes is addressed for the case in which the worldvolume wraps a manifold T 2 × M4, where M4 is simply connected and Kaehler. This is done in a compactification of M-theory which induces the Vafa-Witten theory on M4 in the limit of vanishing torus volume. The results follow from the equivalence of the BPS spectrum counting in the complementary limit of vanishing M4 volumes and from a classification of the the moduli space of quantum vacua of the supersymmetric twisted theory in terms of associated spectral covers. This reduces the problem of the moduli counting to algebraic equations.