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
Be?linson-Bernstein localization?
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
Computations over compact Lie groups are spelled out 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/O-planes 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)
A general proposal for differential equivariant KR-theory of orientifolds and O-plane charge
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
Discussion of combined twisted and equivariant and real K-theory
Last revised on February 4, 2019 at 04:14:26. See the history of this page for a list of all contributions to it.