$\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$In the paper [Čadek, Crabb, and Vanžura - *Obstruction theory on 8-manifolds*](https://arxiv.org/abs/0710.0734), 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:
$$\hat{A}(\tau M)=1-p_1(\tau M)+\dotsb. $$
I was wondering if there is a more coherent context in which the invariant $\ind(y)$ is discussed.