# Contents

## Idea

For $X$ a smooth manifold of even dimension and with spin structure, write $𝒮\left(X\right)$ for the spin bundle and

$𝒮\left(X\right)\simeq {𝒮}^{+}\left(X\right)\oplus {𝒮}^{-}\left(X\right)$\mathcal{S}(X) \simeq \mathcal{S}^+(X) \oplus \mathcal{S}^-(X)

for its decomposition into chiral spinor? bundles. For $\left(X,g\right)$ the Riemannian manifold structure and $\nabla$ the corresponding Levi-Civita spin connection consider the map

$c\circ \nabla \phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\Gamma \left({𝒮}^{+}\left(X\right)\right)\to \Gamma \left({𝒮}^{-}\left(X\right)\right)$c \circ \nabla \;\colon\; \Gamma(\mathcal{S}^+(X)) \to \Gamma(\mathcal{S}^-(X))

given by composing the action of the covariant derivative on sections with the symbol map. This is an elliptic operator. The index? of this operator is called the $\stackrel{^}{A}$-genus.

## References

The $\stackrel{^}{A}$-genus as the index of the spin complex is discussed for instance in section 3 of

A construction via a 1-dimensional Chern-Simons theory is in

Revised on January 24, 2013 19:30:10 by Urs Schreiber (82.113.99.233)