symmetric monoidal (∞,1)-category of spectra
The K-theory spectrum $KU$ (for complex K-theory) or $KO$ (for orthogonal K-theory) in the strict sense is the spectrum that represents the generalized (Eilenberg-Steenrod) cohomology theory topological K-theory. For complex topological K-theory this is periodic with period 2 (reflect Bott periodicity) of the form
More generally, to every stable (infinity,1)-category $C$ is associated a K-theory space which in good cases, such as when the category is presented by a Waldhausen category is the degree 0 piece of a corresponding algebraic K-theory spectrum. The detailed construction is known as the Waldhausen S-construction.
As a symmetric spectrum: Schwede 12, Example I.2.10
$KU$ is a 2-periodic ring spectrum. This is the original Bott periodicity.
Snaith's theorem asserts that the K-theory spectrum for periodic complex K-theory is the ∞-group ∞-ring of the circle 2-group localized away from the Bott element $\beta$:
Complex conjugation on complex vector bundles induces on the complex K-theory spectrum $KU$ an involutive automorphism. KR-theory is the corresponding $\mathbb{Z}_2$-equivariant cohomology theory.
In particular, the homotopy fixed point of KU under this automorphism is KO
and this way where in complex K-theory one has KU-modules (∞-modules), so in KR-theory one has $KO$-modules.
a proof in terms of moduli stacks is given in Mathew 13, section 3
There are close relations between K-theory and Clifford algebras. One conceptual statement relating them is this:
KO-theory is the first Weiss-derivative (in orthogonal calculus) of the K-theory of Clifford algebras. (Charles Rezk, MO comment, Sept ‘13)
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 $KSC$ |
The weak homotopy eqivalence $B U \to \Omega U$ is discussed in detail in
The incarnation of $KU$ and $KO$ as a symmetric spectrum is discussed in
and the structure of a symmetric ring spectrum on $KO$ is discussed in
The structure of a homotopy ring spectrum on $KU$ is discussed in
The E-infinity ring structure of $KU$ is discussed in
and the underlying H-infinity ring spectrum structure in
The uniqueness of the E-infinity ring structure on $KU$ is due to
Discussion of $KO$ in analogy to the construction of tmf is in
with a summary in