# Contents

## Idea

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

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

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

$c \circ \nabla \;\colon\; \Gamma(\mathcal{S}^+(X)) \to \Gamma(\mathcal{S}^-(X))$

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 $\hat A$-genus.

## References

The $\hat A$-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)