nLab KR cohomology theory

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 $X \cup X$ of $X$ 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”.

Definition

As the Grothendieck group of complex vector bundles with real structure

…(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

Bigrading

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.

Related concepts

• real-oriented cohomology theory

  • MR-theory

  • BPR-theory

• real algebraic K-theory

• orientifold

• KO-dimension

chromatic homotopy theory

chromatic level	complex oriented cohomology theory	E-∞ ring/A-∞ ring	real oriented cohomology theory
0	ordinary cohomology	Eilenberg-MacLane spectrum $H \mathbb{Z}$	HZR-theory
	0th Morava K-theory	$K(0)$
1	complex K-theory	complex K-theory spectrum $KU$	KR-theory
	first Morava K-theory	$K(1)$
	first Morava E-theory	$E(1)$
2	elliptic cohomology	elliptic spectrum $Ell_E$
	second Morava K-theory	$K(2)$
	second Morava E-theory	$E(2)$
	algebraic K-theory of KU	$K(KU)$
3 …10	K3 cohomology	K3 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 cohomology	MU	MR-theory

cohomology theories of string theory fields on orientifolds

string theory	B-field	$B$-field moduli	RR-field
bosonic string	line 2-bundle	ordinary cohomology $H\mathbb{Z}^3$
type II superstring	super line 2-bundle	$Pic(KU)//\mathbb{Z}_2$	KR-theory $KR^\bullet$
type IIA superstring	super line 2-bundle	$B GL_1(KU)$	KU-theory $KU^0$
type IIB superstring	super line 2-bundle	$B GL_1(KU)$	KU-theory $KU^1$
type I superstring	super line 2-bundle	$Pic(KU)//\mathbb{Z}_2$	KO-theory $KO$
type $\tilde I$ superstring	super 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					by of level-n structures (modular curve)
	$\ast = Spec(\mathbb{Z})$	$\to$	$Spec(\mathbb{Z}[ [q] ])$	$\to$	$\mathcal{M}_{\overline{ell}}[n]$
structure group of covering	$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{\mathbb{Z}/2\mathbb{Z}}$		$\downarrow^{SL_2(\mathbb{Z}/n\mathbb{Z})}$ (modular group)
moduli stack	$\mathcal{M}_{1dTori}$	$\hookrightarrow$	$\mathcal{M}_{Tate}$	$\hookrightarrow$	$\mathcal{M}_{\overline{ell}}$ (M_ell)	$\hookrightarrow$	$\mathcal{M}_{cub}$	$\to$	$\mathcal{M}_{fg}$ (M_fg)
of	1d tori		Tate curves		elliptic curves		cubic curves		1d commutative formal groups
value $\mathcal{O}^{top}_{\Sigma}$ of structure sheaf over curve $\Sigma$	KU		$KU[ [q] ]$		elliptic spectrum				complex oriented cohomology theory
spectrum $\Gamma(-, \mathcal{O}^{top})$ of global sections of structure sheaf	(KO $\hookrightarrow$ KU) = KR-theory		Tate 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, pdf, ISSN:0033-5606]

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.

Early discussion in the broader context of real-oriented cohomology theory and Real cobordism cohomology:

• Shôrô Araki, $\tau$-Cohomology Theories, Japanese Journal of Mathematics 4 2 (1978) [doi:10.4099/math1924.4.363]

The dual concept of KR-homology:

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

Discussion for algebraic varieties:

• Max Karoubi, Charles Weibel, Algebraic and Real K-theory of Algebraic varieties, Topology 42 (2003) 715-742 [arXiv:math/0509412, doi:10.1016/S0040-9383(02)00069-1]

Discussion in equivariant homotopy theory:

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

On equivariant K-theory and KR-theory for finite equivariance groups:

• Robert Bruner, John Greenlees, Connective Real K-Theory of Finite Groups, Mathematical Surveys and Monographs 169 AMS 2010 (ISBN:978-0-8218-5189-0)

Computations over compact Lie groups in the context of twisted ad-equivariant K-theory:

• Chi-Kwong Fok, The Real K-theory of compact Lie groups, SIGMA 10 (2014), 022, (arXiv:1308.3871, doi:10.3842/SIGMA.2014.022)

• Chi-Kwong Fok, Equivariant twisted Real K-theory of compact Lie groups, Journal of Geometry and Physics 124 (2018) 325-349 (arXiv:1503.00957, doi:10.1016/j.geomphys.2017.11.013)

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

• Po Hu, 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

Relation to topological Hermitian K-theory via the hyperbolic functor:

• Michael C. Crabb, §9 of: $\mathbb{Z}/2$-Homotopy Theory, Lond. Math. Soc. Lecture Notes 44, Cambridge University Press (1980) [ark:/13960/t3jx7bg4w, pdf]

• Po Hu, Igor Kriz, Appendix of: Topological Hermitian cobordism, Journal of Homotopy and Related Structures 11 (2016) 173–197 [doi:10.1007/s40062-014-0100-9]

• Max Karoubi, Charles Weibel, Thm. 3.5 in: Twisted $K$-theory, Real $A$-bundles and Grothendieck–Witt groups, Journal of Pure and Applied Algebra 221 7 (2017) 1629-1640 [doi:10.1016/j.jpaa.2016.12.020]

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

• Vigleik Angeltveit, Andrew Blumberg, Teena Gerhardt, Michael Hill, Algebraic K-theory and equivariant homotopy theory (2012) [pdf]

Discussion of equivariant and twisted versions of KR-theory

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

• El-kaïoum M. Moutuou, Graded Brauer groups of a groupoid with involution, J. Funct. Anal. 266 (2014), no.5 (arXiv:1202.2057)

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

• Daniel Freed, Gregory Moore, Section 7 of: Twisted equivariant matter, Ann. Henri Poincaré (2013) 14: 1927 (arXiv:1208.5055, doi:10.1007%2Fs00023-013-0236-x)

• Kiyonori Gomi, Freed-Moore K-theory (arXiv:1705.09134, spire:1601772)

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/O-planes is in

• Sergei Gukov, K-Theory, Reality, and Orientifolds, Commun.Math.Phys. 210 (2000) 621-639 (arXiv:hep-th/9901042)

On D-brane charge and T-duality in orientifolds formulated in KR-theory (cf. also KSC-theory):

• Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg: T-duality For Orientifolds and Twisted KR-theory, Lett. Math. Phys. 104 11 (2014) 1333-1364 [arXiv:1306.1779, doi:10.1007/s11005-014-0715-0]

  (but see HMSV 19, p.5 footnote 1)

• Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg: String theory on elliptic curve orientifolds and KR-theory, Commun. Math. Phys. 335 (2015) 955–1001 [arXiv:1402.4885, doi:10.1007/s00220-014-2200-0]

• Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, String theory on elliptic curve orientifolds and KR-theory (arXiv:1402.4885)

A general proposal for differential equivariant KR-theory of orientifolds and O-plane charge

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)

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

• Max Karoubi, A descent theorem in topological K-theory, K-theory 24 (2001), no. 2, 109–114 (arXiv:math/0509396)

• Daniel Dugger, An Atiyah-Hirzebruch spectral sequence for $KR$-theory, Ktheory

  35 (2005), 213–256. (arXiv:0304099)

• Michael Hill, Michael Hopkins, Douglas Ravenel, section 7.3 of The Arf-Kervaire problem in algebraic topology: Sketch of the proof (pdf)

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

• Tyler Lawson, Niko Naumann, appendix of Strictly commutative realizations of diagrams over the Steenrod algebra and topological modular forms at the prime 2 (arXiv:1203.1696)

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)

Discussion of twists of KR-theory by HZR-theory in degree 3 via bundle gerbes (Jandl gerbes) suitable for classifying D-brane charge on orientifolds:

• Pedram Hekmati, Michael Murray, Richard Szabo, Raymond Vozzo, Real bundle gerbes, orientifolds and twisted KR-homology, Adv. Theor. Math. Phys. 23 (2019) 2093-2159, doi:10.4310/ATMP.2019.v23.n8.a5 (arXiv:1608.06466)

• Pedram Hekmati, Michael Murray, Richard Szabo, Raymond Vozzo, Sign choices for orientifolds, Commun. Math. Phys. 378, 1843–1873 (2020) (arXiv:1905.06041)