nLab
A-hat genus

Contents

Idea

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

MSpinπ (KO) M Spin \longrightarrow \pi_\bullet(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.

Properties

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).

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

References

The A^\hat A-genus as the index of the spin complex is discussed for instance in section 3 of

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

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