  In this note we present a formula for the equivariant index of the
cohomological complex obtained from localization of $\mathcal{N}=2$ SYM on
simply-connected compact four-manifolds with a $T^2$-action. Knowledge of said
index is essential to compute the perturbative part of the partition function
for the theory. In the topologically twisted case, 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 to be
only transversally elliptic. While the index of such a complex has been
computed for some cases where the manifold can be lifted to a Sasakian
$S^1$-fibration in five dimensions, a general four-dimensional treatment was
still lacking. We provide a formal, purely four-dimensional treatment of the
cohomological complex, showing that the Laplacian part can be globally split
off while the remaining part can be trivialized uniquely in the
group-direction. This ultimately produces a simple formula for the index
applicable for any compact simply-connected four-manifold. Finally, the index
formula is applied to examples on $S^4$, $\mathbb{CP}^2$ and $\mathbb{F}^1$.
For the latter, we use the result to compute the perturbative partition
function.
