In the paper *Obstruction theory on 8-manifolds* (https://arxiv.org/pdf/0710.0734.pdf), the authors discussed the "$spin^c$-index" for a $spin^c$ manifold $M$ (display (3.1) of the paper):
$$y\in K^0(M)\mapsto ind(y)=(e^{c/2}\hat{A}(\tau M)ch(y))[M]\in\mathbb{Z}, $$
where $c$ is the $spin^c$ class, $\tau M$ is the tangent bundle of $M$, and $\hat{A}$ is the Hirzebruch signature:
$$\hat{A}(\tau M)=1-p_1(\tau M)+\cdots. $$
I was wondering if there is a more coherent context in which the invariant $ind(y)$ is discussed. Thank you!