The $\operatorname{spin}^c$ index for manifolds

$$\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$$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.

• If if you think of $y$ as defined by a complex vector bundle, then $\mathrm{ind}(y)$ is the index of the $spin^c$ Dirac operator corresponding to the $spin^c$ structure $c$ and twisted by the vector bundle $y$ . This idea goes back to work of Atiyah and Hirzebruch Jul 29 '20 at 18:01
• @Liviu Nicolaescu Thank you! Jul 30 '20 at 8:58

Theorem 26.1.1. Let $$d$$ be an element of $$H^2(X, \mathbb{Z})$$ whose reduction mod 2 is the Whitney class $$w_2(X)$$, and $$\eta$$ a continuous $$GL(q, \mathbb{C})$$-bundle over $$X$$. Then
$$\hat{A}(X, \tfrac{1}{2}d, \eta) = \chi^m\left[e^{\frac{1}{2}d}\cdot\operatorname{ch}\eta\cdot\sum_{j=0}^{\infty}\hat{A}_j(p_1, \dots, p_j)\right]$$