group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
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:
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
(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.
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.
…(Atiyah 66)…
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
The reduced canonical line bundle over this (the Hopf fibration) is classified by a map
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
This is in fact an Omega spectrum, by equivariant complex Bott periodicity (for instance in Dugger 03, p. 2-3).
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
The corresponding representation sphere is
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
(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 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 $KU^1$ |
type IIB superstring | super line 2-bundle | $B GL_1(KU)$ | KU-theory $KU^0$ |
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 | $\mathbb{S}$ |
KR theory was introduced in
The version of $KSC$-theory was introduced in
The dual concept of KR-homology was defined in
Discussion in the general context of real oriented cohomology theory is in
Further discussion includes
Reviews include
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
Sergei Gukov, K-Theory, Reality, and Orientifolds, Commun.Math.Phys. 210 (2000) 621-639 (arXiv:hep-th/9901042)
Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, T-duality For Orientifolds and Twisted KR-theory (arXiv:1306.1779)
Charles Doran, Stefan Mendez-Diez, Jonathan Rosenberg, String theory on elliptic curve orientifolds and KR-theory (arXiv:1402.4885)
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
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)
Disucssion of twists of KR-theory by HZR-theory in degree 3 via bundle gerbes (Jandl gerbes) includes