nLab A-hat genus



Index theory



Special and general types

Special notions


Extra structure





In terms of an operator index

For XX a smooth manifold of even dimension and with spin structure, write 𝒮(X)\mathcal{S}(X) for the spin bundle and

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

for its decomposition into chiral spinor bundles. For (X,g)(X,g) the Riemannian manifold structure and \nabla the corresponding Levi-Civita spin connection consider the map

c:Γ(𝒮 +(X))Γ(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 A^\hat A-genus.

In terms of the universal SpinSpin-orientation of KOKO

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

MSpinKO M Spin \longrightarrow KO

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

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

is the corresponding homomorphism in homotopy groups.


Characteristic series

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

K A^(e) =ze z/2e z/2 =exp( k2B kkz kk!), \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 kB_k is the kkth Bernoulli number (Ando-Hopkins-Rezk 10, prop. 10.2).

Relation to the Todd genus

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

Td(M U)=(e c 1/2A^)(M U). 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 SUM_{S\mathrm{U}} with SU-structure, where c 1=0c_1 = 0, the Todd class is actually equal to the A-hat class:

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

Given the complexification of a real vector bundle 𝒳\mathcal{X} to a complex vector bundle \mathcal{E} \otimes \mathbb{C}, the A^\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


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

For 4n\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 A^( 4n)\sqrt{{\widehat A}(\mathcal{M}^{4n})} of the A-hat genus of 4n\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:

ddpartition function in dd-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 MSpinKOM 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 MSpin cKUM 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 MSpin cKUM 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



The A^\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.