nLab K-orientation

Idea

A K-orientation is an orientation in generalized cohomology for K-theory (typically: topological K-theory).

Definition

In topological K-theory

K-Orientation of oriented manifolds

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

In this case there is an Umkehr map

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

The universal Atiyah-Bott-Shapiro orientation

There is a universal orientation in generalized cohomology of

• complex K-theory$\,$KU over spin^c structures;

• KO over spin structure

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

MSpin$\longrightarrow$ KO

and

MSpin^c$\longrightarrow$ KU

These are (both) referred to as the Atiyah-Bott-Shapiro orientation (after Atiyah-Bott-Shapiro 64); the $E_\infty$-structure is due to (Joachim 04).

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

Of course, $KU$ is also a complex oriented cohomology theory and as such canonically comes with a homomorphism of homotopy-commutative ring spectra $MU \to KU$ (this general Prop., 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

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

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

The universal Conner-Floyd orientation

There are multiplicative natural transformations of multiplicative cohomology theories, hence homomorphisms of homotopy-commutative ring spectra

MU$\longrightarrow$ KU

and

MSU$\longrightarrow$ KO

which are compatible in that the diagram formed by the evident vertical morphisms

commutes.

This Conner-Floyd orientation is originally due to Conner-Floyd 66, Section 5.

The morphism MU$\to$KU reflects the complex orientation of KU via the universal complex orientation on MU.

In KK-theory

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

$$f! \in KK_d(B,A)$$

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

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

(BMRS 07)

Properties

As the relation between cobordisms cohomology and K-theory

The Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$ induces an equivalence

$$M Spin^c_\bullet(-)\otimes_{M Spin^c_\bullet} KU_\bullet \simeq KU_\bullet(-)$$

and $M Spin \to KO$ similarly induces

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

This is due to (Hopkins-Hovey 92), a variation of the Conner-Floyd isomorphism. See at cobordism theory determining homology theory for more.

Related concepts

• spin^c structure

• Conner-Floyd isomorphism, cobordism theory determining homology theory

• Grothendieck-Riemann-Roch theorem

partition functions in quantum field theory as indices/genera/orientations in generalized cohomology theory:

$d$	partition function in $d$-dimensional QFT	supercharge	index in cohomology theory	genus	logarithmic coefficients of Hirzebruch series
0			push-forward in ordinary cohomology: integration of differential forms			orientation
1	spinning particle	Dirac operator	KO-theory index	A-hat genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin \to KO$
	endpoint of 2d Poisson-Chern-Simons theory string	Spin^c Dirac operator twisted by prequantum line bundle	space of quantum states of boundary phase space/Poisson manifold	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
	endpoint of type II superstring	Spin^c Dirac operator twisted by Chan-Paton gauge field	D-brane charge	Todd genus	Bernoulli numbers	Atiyah-Bott-Shapiro orientation $M Spin^c \to KU$
2	type II superstring	Dirac-Ramond operator	superstring partition function in NS-R sector	Ochanine elliptic genus		SO orientation of elliptic cohomology
	heterotic superstring	Dirac-Ramond operator	superstring partition function	Witten genus	Eisenstein series	string orientation of tmf
	self-dual string		M5-brane charge
3						w4-orientation of EO(2)-theory

References

In topological K-theory

The MSpin/MSpin^c-orientation of KO/KU topological K-theory is attributed to

• Michael Atiyah, Raoul Bott, Arnold Shapiro, Clifford modules, Topology Volume 3, Supplement 1, July 1964, Pages 3-38 (doi:10.1016/0040-9383(64)90003-5, pdf)

The MSU/MU-orientation of KO/KU (Conner-Floyd orientation) is originally due to

• Pierre Conner, Edwin Floyd, Section 5 of: The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

That ABS-orientation $M Spin^c\to KU$ 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

• Michael Hopkins, Mark Hovey, Spin cobordism determines real K-theory, Mathematische Zeitschrift 210.1 (1992): 181-196. (pdf)

In KK-theory

Discussion in noncommutative topology/KK-theory is in

• Alain Connes, , Georges Skandalis, The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20,

  no. 6, 1139–1183 (1984)

• Jacek Brodzki, Varghese Mathai, Jonathan Rosenberg, Richard Szabo, Noncommutative correspondences, duality and D-branes in bivariant K-theory, Adv. Theor. Math. Phys.13:497-552,2009 (arXiv:0708.2648)