Twisted K-theory

In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory.
More specifically, twisted K-theory with twist "H" is a particular variant of K-theory, in which the twist is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation. This was provided in two steps; the first one was done in 1970 (Publ. Math. 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. For more information on twisted K-theory in string theory, see K-theory (physics).
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.
The definition.
To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah–Jänich theorem, stating that
the Fredholm operators on Hilbert space formula_2, is a classifying space for ordinary, untwisted K-theory. This means that the K-theory of the space formula_3 consists of the homotopy classes of maps
from formula_3 to formula_6
A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of formula_7 over formula_3, that is, the Cartesian product of formula_3 and formula_7. Then the K-theory of formula_3 consists of the homotopy classes of sections of this bundle.
We can make this yet more complicated by introducing a trivial
bundle formula_13 over formula_3, where formula_12 is the group of projective unitary operators on the Hilbert space formula_2. Then the group of maps
from formula_13 to formula_7 which are equivariant under an action of formula_12 is equivalent to the original groups of maps
This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that formula_12 bundles on formula_3 are classified by elements formula_24 of the third integral cohomology group of formula_3. This is a consequence of the fact that formula_12 topologically is a representative Eilenberg–MacLane space
The generalization is then straightforward. Rosenberg has defined
the twisted K-theory of formula_3 with twist given by the 3-class formula_24, to be the space of homotopy classes of sections of the trivial formula_7 bundle over formula_3 that are covariant with respect to a formula_12 bundle formula_34 fibered over formula_3 with 3-class formula_24, that is
Equivalently, it is the space of homotopy classes of sections of the formula_7 bundles associated to a formula_12 bundle with class formula_24.
What is it?
When formula_24 is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when formula_24 is nontrivial this theory is no longer a ring. It has an addition, but it is no longer closed under multiplication.
However, the direct sum of the twisted K-theories of formula_3 with all possible twists is a ring. In particular, the product of an element of K-theory with twist formula_24 with an element of K-theory with twist formula_45 is an element of K-theory twisted by formula_46. 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). In particular twisted K-theory is a module over classical K-theory.
How to calculate it.
Physicist typically want to calculate twisted K-theory using the Atiyah–Hirzebruch spectral sequence. 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 formula_47 or the twisted formula_48, and then one takes the cohomology with respect to a series of differential operators. The first operator, formula_49, for example, is the sum of the three-class formula_24, which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square, soformula_51No elementary form for the next operator, formula_52, has been found, although several conjectured forms exist. Higher operators do not contribute to the formula_53-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 formula_3.
After taking the cohomology with respect to the full series of differentials one obtains twisted formula_53-theory as a set, but to obtain the full group structure one in general needs to solve an extension problem.
Example: the three-sphere.
The three-sphere, formula_56, has trivial cohomology except for formula_57 and formula_58 which are both isomorphic to the integers. Thus the even and odd cohomologies are both isomorphic to the integers. 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 formula_59. The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted formula_53-theory is just the cohomology of the operator formula_49 which acts on a class by cupping it with the 3-class formula_24.
Imagine that formula_24 is the trivial class, zero. Then formula_49 is also trivial. Thus its entire domain is its kernel, and nothing is in its image. Thus formula_65 is the kernel of formula_49 in the even cohomology, which is the full even cohomology, which consists of the integers. Similarly formula_67 consists of the odd cohomology quotiented by the image of formula_49, in other words quotiented by the trivial group. This leaves the original odd cohomology, which is again the integers. In conclusion, formula_48 and formula_70 of the three-sphere with trivial twist are both isomorphic to the integers. As expected, this agrees with the untwisted formula_53-theory.
Now consider the case in which formula_24 is nontrivial. formula_24 is defined to be an element of the third integral cohomology, which is isomorphic to the integers. Thus formula_24 corresponds to a number, which we will call formula_75. formula_49 now takes an element formula_77 of formula_78 and yields the element formula_79 of formula_80. As formula_75 is not equal to zero by assumption, the only element of the kernel of formula_49 is the zero element, and so formula_83. The image of formula_49 consists of all elements of the integers that are multiples of formula_75. Therefore, the odd cohomology, formula_86, quotiented by the image of formula_49, formula_88, is the cyclic group of order formula_75, formula_90. In conclusionformula_91In string theory this result reproduces the classification of D-branes on the 3-sphere with formula_75 units of formula_24-flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric formula_94 WZW model at level formula_95.
There is an extension of this calculation to the group manifold of SU(3). In this case the Steenrod square term in formula_49, the operator formula_52, and the extension problem are nontrivial.