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!

$\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$In the paper Čadek, Crabb, and Vanžura - Obstruction theory on 8-manifolds, 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.