A-hat genus

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**.

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

$M Spin \longrightarrow \pi_\bullet(KO)$

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.

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).

**partition functions in quantum field theory as indices/genera 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 | ||||

1 | spinning particle | Dirac operator | KO-theory index | A-hat genus | Bernoulli numbers |

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 | |

endpoint of type II superstring | Spin^c Dirac operator twisted by Chan-Paton gauge field | D-brane charge | Todd genus | Bernoulli numbers | |

2 | superstring | Dirac-Ramond operator | superstring partition function | elliptic genus/Witten genus | Eisenstein series |

self-dual string | M5-brane charge |

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)

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)

Revised on March 26, 2014 08:23:55
by Urs Schreiber
(89.204.153.86)