It is special among the various twists that K-theory admits for two reasons.
de l'IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg in Continuous-Trace Algebras from the Bundle Theoretic Point of View.
In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory.
In the broader context of K-theory, in each subject it has numerous isomorphic formulations and, in many cases, isomorphisms relating definitions in various subjects have been proven.
It also has numerous deformations, for example, in abstract algebra K-theory may be twisted by any integral cohomology class.
To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah–Jänich theorem, stating that the Fredholm operators on Hilbert space
consists of the homotopy classes of sections of this bundle.
We can make this yet more complicated by introducing a trivial bundle
is the group of projective unitary operators on the Hilbert space
is equivalent to the original groups of maps This more complicated construction of ordinary K-theory is naturally generalized to the twisted case.
topologically is a representative Eilenberg–MacLane space The generalization is then straightforward.
, to be the space of homotopy classes of sections of the trivial
, that is Equivalently, it is the space of homotopy classes of sections of the
This element can be constructed directly from the above definition by using adjoints of Fredholm operators and construct a specific 2 x 2 matrix out of them (see the reference 1, where a more natural and general Z/2-graded version is also presented).
Physicist typically want to calculate twisted K-theory using the Atiyah–Hirzebruch spectral sequence.
[2] The idea is that one begins with all of the even or all of the odd integral cohomology, depending on whether one wishes to calculate the twisted
, and then one takes the cohomology with respect to a series of differential operators.
, which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square,[3] so
-theory of a 10-manifold, which is the dimension of interest in critical superstring theory.
Over the rationals Michael Atiyah and Graeme Segal have shown that all of the differentials reduce to Massey products of
[4] After taking the cohomology with respect to the full series of differentials one obtains twisted
-theory as a set, but to obtain the full group structure one in general needs to solve an extension problem.
Because the three-sphere is of dimension three, which is less than five, the third Steenrod square is trivial on its cohomology and so the first nontrivial differential is just
The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted
consists of the odd cohomology quotiented by the image of
This leaves the original odd cohomology, which is again the integers.
of the three-sphere with trivial twist are both isomorphic to the integers.
is defined to be an element of the third integral cohomology, which is isomorphic to the integers.
In string theory this result reproduces the classification of D-branes on the 3-sphere with
-flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric