Index of the Transversally Elliptic Complex from N=2 Localization in Four Dimensions

2021-12-20

Authors: Roman Mauch and Lorenzo Ruggeri

Preprint number: UUITP-64/21

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 T2-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 S1-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.