nLab
K-orientation

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Cohomology

cohomology

Special and general types

Special notions

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Extra structure

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Contents

Idea

A K-orientation is an orientation in generalized cohomology for K-theory.

Definition

In topological K-theory

K-Orientation of oriented manifolds

For ordinary topological K-theory a smooth function between smooth manifolds f:XYf \colon X \to Y is K-oriented if TXf *(TY)T X \oplus f^\ast(T Y) has a spin^c structure.

In this case there is an Umkehr map

f !:K (X)K +dim(Y)dim(X)(Y). f_! \colon K^\bullet(X) \to K^{\bullet + dim(Y) - dim(X)}(Y) \,.

The universal Atiyah-Bott-Shapiro orientation

There is a universal orientation in generalized cohomology of

hence E-infinity ring homomorphisms out of the Thom spectrum MSpin

MSpinKO M Spin \longrightarrow KO

and

MSpin cKU. M Spin^c \longrightarrow KU \,.

These are (both) referred to as the Atiyah-Bott-Shapiro orientation (after Atiyah-Bott-Shapiro), the E E_\infty-structure is due to (Joachim 04).

The genus induced by MSpinKOM Spin \to KO is the A-hat genus, that induced by MSpin cKUM Spin^c \to KU is the Todd genus.

Of course KUKU is also a complex oriented cohomology theory and as such canonically comes with an E E_\infty-map MUKUMU \to KU (see at spin^c structure – from almost complex structure).

On this level there is also the real orientation of real K-theory given by a map

MK M \mathbb{R} \longrightarrow K \mathbb{R}

from MR-theory (Kriz 01, (3.20)).

In KK-theory

In the context of KK-theory, a morphism f:ABf \colon A \to B of C*-algebra, a K-orientation of this morphism is a map

f!KK d(B,A) f! \in KK_d(B,A)

such that (…). The corresponding fiber integration/Gysin map on operator K-theory is then the postcomposition operation

f !:K (B)KK (,B)f!()KK (,A)K +d(A). f_! \colon K_\bullet(B) \simeq KK_\bullet(\mathbb{C}, B) \stackrel{f! \circ (-)}{\to} KK_\bullet(\mathbb{C}, A) \simeq K_{\bullet + d}(A) \,.

(BMRS 07)

Properties

As the relation between cobordisms cohomology and K-theory

The Atiyah-Bott-Shapiro orientation MSpin cKUM Spin^c \to KU induces an equivalence

MSpin c() MSpin cKU KU () M Spin^c_\bullet(-)\otimes_{M Spin^c_\bullet} KU_\bullet \simeq KU_\bullet(-)

and MSpinKOM Spin \to KO similarly induces

MSpin () MSpin KO KO () M Spin_\bullet(-)\otimes_{M Spin_\bullet} KO_\bullet \simeq KO_\bullet(-)

This is due to (Hopkins-Hovey 92). See at cobordism theory determining homology theory for more.

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

References

In topological K-theory

The SpinSpin/Spin cSpin^c-orientation of KO/KU topological K-theory is attributed to

That the map of spectra MSpin cKUM Spin^c\to KU extendes to a homomorphism of E-infinity rings is due to

  • Michael Joachim, Higher coherences for equivariant K-theory, in Structured ring spectra, volume 315 of London Math. Soc. Lecture Note Ser., pages 87–114. Cambridge Univ. Press, Cambridge, 2004 (pdf)

Discussion for real K-theory includes

  • Igor Kriz, Real-oriented homotopy theory and an analogue of the

    Adams-Novikov spectral sequence_, Topology 40 (2001) 317-399 (pdf)

The discussion of cobordism theory determining homology theory for the K-orientation is due to

In KK-theory

Discussion in noncommutative topology/KK-theory is in

Last revised on December 22, 2016 at 05:37:52. See the history of this page for a list of all contributions to it.