nLab
A-hat genus

Idea

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

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

$$c\circ \nabla :\Gamma ({𝒮}^{+}(X))\to \Gamma ({𝒮}^{-}(X))$$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.

Related entries

• index of an elliptic complex

• Witten genus

References

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

• Peter Gilkey, The Atiyah-Singer Index Theorem – Chapter 5 (pdf)

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

• Owen Gwilliam, Ryan Grady, One-dimensional Chern-Simons theory and the $\hat{A}$-genus (arXiv:1110.3533)