nLab KR cohomology theory

Context

Stable Homotopy theory

stable homotopy theory

Contents

Idea

What is called KR-theory (Atiyah 66) is variant of topological K-theory on spaces equipped with a $\mathbb{Z}_2$-action (by homeomorphism, hence equipped with one involutive homeomorphism – a “real space”).

In terms of cocycle models, classes of KR-theory are represented by complex vector bundles over $X$ on which the involution on their base space lifts to an anti-linear involution of the total space. Over manifolds with trivial involution these are precisely the complexification of real vector bundles and hence over such spaces $KR$-theory reduces to KO-theory. Conversely, over two copies $X \cup X$ lof $X$ equipped with the involution that interchanges the two, $KR$-theory reduces to KU-theory. Finally over $X \times S^1$ with the involution the antipodal identification on the second (circle) factor , $KR$-theory reduces to the self-conjugate KSC-theory (Anderson 64). So in general $KR$-theory interpolates between all these cases. For instance on $X \times S^1$ with the reflection-involution on the circle (the real space denoted $S^{1,1}$, the non-trivial $\mathbb{Z}_2$-representation sphere) it behaves like $KO$-theory at the two involution fixed points (the two O-planes) and like $KU$ in their complement (a model that makes this very explicit is given in DMR 13, section 4), schematically:

$KR(S^{1,1}) = ( KO --- KU --- KO )$

More abstractly, complex conjugation of complex vector bundles induces on the complex K-theory spectrum KU an involutive automorphism. $KR$ is the corresponding $\mathbb{Z}_2$-equivariant spectrum, and $KR$-theory the corresponding $\mathbb{Z}_2$-equivariant cohomology theory. In particular, the homotopy fixed point of KU under this automorphism is KO

$KO \simeq (KU)^{\mathbb{Z}/2}$

(e.g.Karoubi 01, Dugger 03, corollary 7.6, Hill-Hopkins-Ravenel, section 7.3) and this way where in complex K-theory one has KU-modules (∞-modules), so in KR-theory one has $KO$-modules.

KR is an example of a real-oriented cohomology theory, together with for instance MR-theory and BPR-theory.

Remark on terminology

An involution on a space by a homeomorphism (or diffeomorphism) as they appear in KR theory may be thought of as a “non-linear real structure”, and therefore spaces equipped with such involutions are called “real spaces”. Following this, $KR$-theory is usually pronounced “real K-theory”. But beware that this terminology easily conflicts with or is confused with KO-theory. For disambiguation the latter might better be called “orthogonal K-theory”. But on abstract grounds maybe $KR$-theory would best be just called $\mathbb{Z}_2$-equivariant complex K-theory.

Definition

…(Atiyah 66)…

As a genuine $\mathbb{Z}_2$-Spectrum

The following gives $KR$ as a genuine G-spectrum for $G = \mathbb{Z}_2$.

Using that every orthogonal representation of $\mathbb{Z}_2$ is contained in an $\mathbb{C}^n$ with its complex conjugation action, one can restrict attention to these. Write

$\mathbb{C}P^1 = S^{2,1} = S^{\mathbb{C}} \,.$

The reduced canonical line bundle over this (the Hopf fibration) is classified by a map

$S^{2,1}= \mathbb{C}P^1 \to \mathbb{Z}\times BU$

to the classifying space for topological K-theory. The homotopy-associative multiplication on this space then yields the structure map of a $\mathbb{Z}_2$-spectrum

$S^{2,1} \wedge (\mathbb{Z} \times BU)\to \mathbb{Z}\times BU \,.$

This is in fact an Omega spectrum, by equivariant complex Bott periodicity (for instance in Dugger 03, p. 2-3).

Properties

As any genuine equivariant cohomology theory $KR$-theory is naturally graded over the representation ring $RO(\mathbb{Z}_2)$. Write $\mathbb{R}$ for the trivial 1-dimensional representation and $\mathbb{R}_-$ for that given by the sign involution. Then the general orthogonalrepresentation decomposes as a direct sum

$V = \mathbb{R}^+\oplus \mathbb{R}_-^q \,.$

The corresponding representation sphere is

$S^V = (some\; convention) \,.$

As induced from the derived moduli stack of tori

The relation between $KU$, $KO$ and $KR$ naturally arises in chromatic homotopy theory as follows.

Inside the moduli stack of formal group laws sits the moduli stack of one dimensional tori $\mathcal{M}_{\mathbb{G}_m}$ (Lawson-Naumann 12, def. A.1, A.3). This is equivalent to the quotient stack of the point by the group of order 2

$\mathcal{M}_{\mathbb{G}_m}\simeq \mathbf{B}\mathbb{Z}_2$

(Lawson-Naumann 12, prop. A.4). Here the $\mathbb{Z}_2$-action is the inversion involution on abelian groups.

Using the Goerss-Hopkins-Miller theorem this stack carries an E-∞ ring-valued structure sheaf $\mathcal{O}^{top}$ (Lawson-Naumann 12, theorem A.5); and by the above equivalence this is a single E-∞ ring equipped with a $\mathbb{Z}_2$-∞-action. This is KU with its involution induced by complex conjugation, hence essentially is $KR$.

Accordingly, the global sections of $\mathcal{O}^{top}$ over $\mathcal{M}_{\mathbb{G}_m}$ are the $\mathbb{Z}_2$-homotopy fixed points of this action, hence is $KO$. This is further amplified in (Mathew 13, section 3) and (Mathew, section 2).

As suggested there and by the main (Lawson-Naumann 12, theorem 1.2) this realizes (at least localized at $p = 2$) the inclusion $KO \to KU$ as the restriction of an analogous inclusion of tmf built as the global sections of the similarly derived moduli stack of elliptic curves.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

cohomology theories of string theory fields on orientifolds

string theoryB-field$B$-field moduliRR-field
bosonic stringline 2-bundleordinary cohomology $H\mathbb{Z}^3$
type II superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KR-theory $KR^\bullet$
type IIA superstringsuper line 2-bundle$B GL_1(KU)$KU-theory $KU^1$
type IIB superstringsuper line 2-bundle$B GL_1(KU)$KU-theory $KU^0$
type I superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KO-theory $KO$
type $\tilde I$ superstringsuper line 2-bundle$Pic(KU)//\mathbb{Z}_2$KSC-theory $KSC$

Substructure of the moduli stack of curves and the (equivariant) cohomology theory associated with it via the Goerss-Hopkins-Miller-Lurie theorem:

covering moduli spaceof level-n structures (modular curve)
$\ast = Spec(\mathbb{Z})$$\to$$Spec(\mathbb{Z}[ [q] ])$$\to$$\mathcal{M}_{\overline{ell}}[n]$
structure group$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
$\mathcal{M}_{1dTori}$$\hookrightarrow$$\mathcal{M}_{Tate}$$\hookrightarrow$$\mathcal{M}_{\overline{ell}}$$\hookrightarrow$$\mathcal{M}_{cub}$$\to$$\mathcal{M}_{FG}$
moduli stackof 1d toriof Tate curvesof elliptic curvesof cubic curvesof formal groups
$\mathcal{O}^{top}_{\Sigma}$KU$KU[ [q] ]$elliptic spectrumcomplex oriented cohomology theory
$\Gamma(-, \mathcal{O}^{top}) =$(KO $\hookrightarrow$ KU) = KR-theoryTate K-theory ($KO[ [q] ] \hookrightarrow KU[ [q] ]$)(Tmf $\to$ Tmf(n)) (modular equivariant elliptic cohomology)tmf$\mathbb{S}$

References

KR theory was introduced in

• Michael Atiyah, K-theory and reality, The Quarterly Journal of Mathematics. Oxford. Second Series 17 (1) (1966),: 367–386, doi:10.1093/qmath/17.1.367, ISSN 0033-5606, MR 0206940 (pdf)

The version of $KSC$-theory was introduced in

• D. W. Anderson, The real K-theory of classifying spaces Proc. Nat. Acad. Sci. U. S. A., 51(4):634–636, 1964.

The dual concept of KR-homology was defined in

• Gennady Kasparov, The operator K-functor and extensions of $C^\ast$-algebras, Izv. Akad. Nauk. SSSR Ser. Mat. 44, 571-636 (1980).

Discussion in the general context of real oriented cohomology theory is in

• Igor Kriz, Real-oriented homotopy theory and an analogue of the Adams}Novikov spectral sequence, Topology 40 (2001) 317-399 (pdf)

Further discussion includes

• Johan Dupont, Symplectic bundles and $KR$-theory (1967) (pdf)

Reviews include

• Wikipedia, KR-theory

• Paolo Masulli, Equivariant homotopy: $KR$-theory, Master thesis (2011) (pdf)

Remarks on homotopy-theoretic KR in the context of algebraic K-theory are in

Explicit groupoid/stack models for equivariant and twisted KR-theory theory are discussed in

• El-kaïoum M. Moutuou, Twistings of KR for Real groupoids (arXiv:1110.6836)

• Daniel Freed, Lectures on twisted K-theory and orientifolds, lectures at ESI Vienna, 2012 (pdf)

This is with motivation from orientifolds, see the references given there for more. A long list of computations of twisted $KR$-classes on tori with applications to T-duality on orientifolds is in

Discussion of $KO$ as the $\mathbb{Z}_2$-homotopy fixed points of $KU$ (or $KR$) is in

Discussion of $KU$ with its $\mathbb{Z}_2$-action as the E-∞ ring-valued structure sheaf of the moduli stack of tori is due to

which is reviewed and amplified further in

• Akhil Mathew, section 3 of The homology of $tmf$ (arXiv:1305.6100)

• Akhil Mathew, section 2 of The homotopy groups of $TMF$, talk notes (pdf)

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Revised on May 21, 2014 02:09:02 by Urs Schreiber (92.68.97.89)