noncommutative topology, noncommutative geometry
noncommutative stable homotopy theory
genus, orientation in generalized cohomology
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
For $X$ a smooth manifold of even dimension and with spin structure, write $\mathcal{S}(X)$ for the spin bundle and
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
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
from the universal spin structure Thom spectrum. The $\hat A$-genus
is the corresponding homomorphism in homotopy groups.
The characteristic series of the $\hat A$-genus is
where $B_k$ is the $k$th Bernoulli number (Ando-Hopkins-Rezk 10, prop. 10.2).
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:
(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:
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).
(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.
partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:
Review:
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 February 22, 2021 at 12:39:09. See the history of this page for a list of all contributions to it.