Index of the Transversally Elliptic Complex in Pestunization

Abstract

We present a formula for the equivariant index of the cohomological complex obtained from localization of N = 2 SYM on simply-connected compact four-manifolds with a T 2 -action. When the theory is topologically twisted, the complex is elliptic and its index can be computed in a standard way using the Atiyah-Bott localization formula. Recently, a framework for more general types of twisting, so-called cohomological twisting, was introduced for which the complex turns out only to be transversally elliptic. While the index of such a complex was previously computed for specific manifolds and a systematic procedure for its computation was provided for cases where the manifold can be lifted to a Sasakian S 1 -fibration in five dimensions, a purely four-dimensional treatment was still lacking. In this note, we provide a formal treatment of the cohomological complex, showing that the Laplacian part can be globally split off while the remaining part can be trivialized in the group-direction. This ultimately produces a simple formula for the index applicable for any compact simply-connected four-manifold, from which one can easily compute the perturbative partition function.