# nLab A-hat genus

Contents

### Context

#### Index theory

index theory, KK-theory

noncommutative stable homotopy theory

partition function

genus, orientation in generalized cohomology

## Definitions

operator K-theory

K-homology

cohomology

# Contents

## 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

Given the complexfication? 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.

$d$partition function in $d$-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential formsorientation
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin \to KO$
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbersAtiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2type II superstringDirac-Ramond operatorsuperstring partition function in NS-R sectorOchanine elliptic genusSO orientation of elliptic cohomology
heterotic superstringDirac-Ramond operatorsuperstring partition functionWitten genusEisenstein seriesstring orientation of tmf
self-dual stringM5-brane charge
3w4-orientation of EO(2)-theory

## References

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

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

Last revised on December 3, 2020 at 09:29:52. See the history of this page for a list of all contributions to it.