nLab A-hat genus

Idea

In terms of an operator index

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.

In terms of the universal $Spin$-orientation of $KO$

More abstractly, there is the universal orientation in generalized cohomology of KO over spin structure, known as the Atiyah-Bott-Shapiro orientation, which is a homomorphism of E-∞ rings of the form

$$M Spin \longrightarrow KO$$

from the universal spin structure Thom spectrum. The $\hat A$-genus

$$\Omega_\bullet^{SO}\longrightarrow \pi_\bullet(KO)\otimes \mathbb{Q}$$

is the corresponding homomorphism in homotopy groups.

Properties

Characteristic series

The characteristic series of the $\hat A$-genus is

$$\begin{aligned} K_{\hat A}(e) & = \frac{z}{e^{z/2} - e^{-z/2}} \\ &= \exp\left( - \sum_{k \geq 2} \frac{B_k}{k} \frac{z^k}{k!} \right) \end{aligned} \,,$$

where $B_k$ is the $k$th Bernoulli number (Ando-Hopkins-Rezk 10, prop. 10.2).

Relation to the Todd genus

On an almost complex manifold $M_{\mathrm{U}}$, the Todd class coincides with the A-hat class up to the exponential of half the first Chern class:

$$Td(M_{\mathrm{U}}) \;=\; \big(e^{c_1/2} \hat A\big)(M_{\mathrm{U}}) \,.$$

(e.g. Freed 87 (1.1.14)).

In particular, on manifolds $M_{S\mathrm{U}}$ with SU-structure, where $c_1 = 0$, the Todd class is actually equal to the A-hat class:

$$Td(M_{S\mathrm{U}}) \;=\; \hat A(M_{S\mathrm{U}}) \,.$$

Given the complexification of a real vector bundle $\mathcal{X}$ to a complex vector bundle $\mathcal{E} \otimes \mathbb{C}$, the $\hat A$-class of $\mathcal{E}$ is the square root of the Todd class of $\mathcal{E} \otimes \mathbb{C}$ (e.g. de Lima 03, Prop. 7.2.3).

As a Rozansky-Witten invariant

Proposition

(Rozansky-Witten Wilson loop of unknot is square root of A-hat genus)

For $\mathcal{M}^{4n}$ a hyperkähler manifold (or just a holomorphic symplectic manifold) the Rozansky-Witten invariant Wilson loop observable associated with the unknot in the 3-sphere is the square root $\sqrt{{\widehat A}(\mathcal{M}^{4n})}$ of the A-hat genus of $\mathcal{M}^{4n}$.

This is Roberts-Willerton 10, Lemma 8.6, using the Wheels theorem and the Hitchin-Sawon theorem.

Related entries

• Todd genus

• index of an elliptic complex

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

Review:

• Daniel Freed, Sections 1.1, 1.2 of: Geometry of Dirac operators, 1987 (pdf, pdf)

The $\hat A$-genus as the index of the spin complex is discussed for instance in:

• Peter Gilkey, Section 3 of: The Atiyah-Singer Index Theorem – Chapter 5 (pdf)

• Levi Lopes de Lima, The Index Formula for Dirac operators: an Introduction, 2003 (pdf)

The relation of the characteristic series to the Bernoulli numbers is made explicit for instance in prop. 10.2 of

• Matthew Ando, Mike Hopkins, Charles Rezk, Multiplicative orientations of KO-theory and the spectrum of topological modular forms, 2010 (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)