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.

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 , is a classifying space for ordinary, untwisted K-theory. This means that the K-theory of the space  consists of the homotopy classes of maps

from  to 

A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of  over , that is, the Cartesian product of  and . Then the K-theory of  consists of the homotopy classes of sections of this bundle.

We can make this yet more complicated by introducing a trivial

bundle  over , where  is the group of projective unitary operators on the Hilbert space . Then the group of maps

from  to  which are equivariant under an action of  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  bundles on  are classified by elements  of the third integral cohomology group of . This is a consequence of the fact that  topologically is a representative Eilenberg–MacLane space

.

The generalization is then straightforward. Rosenberg has defined

,

the twisted K-theory of  with twist given by the 3-class , to be the space of homotopy classes of sections of the trivial  bundle over  that are covariant with respect to a  bundle  fibered over  with 3-class , that is

Equivalently, it is the space of homotopy classes of sections of the  bundles associated to a  bundle with class .

Relation to K-theory

When  is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when  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  with all possible twists is a ring. In particular, the product of an element of K-theory with twist  with an element of K-theory with twist  is an element of K-theory twisted by . 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.

Calculations

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  or the twisted , and then one takes the cohomology with respect to a series of differential operators.  The first operator, , for example, is the sum of the three-class , which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square, soNo elementary form for the next operator, , has been found, although several conjectured forms exist. Higher operators do not contribute to the -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 .

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.

Example: the three-sphere
The three-sphere, , has trivial cohomology except for  and  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 . The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted -theory is just the cohomology of the operator  which acts on a class by cupping it with the 3-class .

Imagine that  is the trivial class, zero. Then  is also trivial. Thus its entire domain is its kernel, and nothing is in its image. Thus  is the kernel of  in the even cohomology, which is the full even cohomology, which consists of the integers.  Similarly  consists of the odd cohomology quotiented by the image of , in other words quotiented by the trivial group. This leaves the original odd cohomology, which is again the integers. In conclusion,  and  of the three-sphere with trivial twist are both isomorphic to the integers. As expected, this agrees with the untwisted -theory.

Now consider the case in which  is nontrivial.  is defined to be an element of the third integral cohomology, which is isomorphic to the integers. Thus  corresponds to a number, which we will call .  now takes an element  of  and yields the element  of . As  is not equal to zero by assumption, the only element of the kernel of  is the zero element, and so .  The image of  consists of all elements of the integers that are multiples of . Therefore, the odd cohomology, , quotiented by the image of , , is the cyclic group of order , .  In conclusionIn string theory this result reproduces the classification of D-branes on the 3-sphere with  units of -flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric  WZW model at level .

There is an extension of this calculation to the group manifold of SU(3).  In this case the Steenrod square term in , the operator , and the extension problem are nontrivial.

See also
K-theory (physics)
Wess–Zumino–Witten model
Bundle gerbe

Notes

References
"Graded Brauer groups and K-theory with local coefficients", by Peter Donovan and Max Karoubi. Publ. Math. IHÉS Nr. 38, pp. 5–25 (1970).
D-Brane Instantons and K-Theory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg
Twisted K-theory and Cohomology by Michael Atiyah and Graeme Segal
Twisted K-theory and the K-theory of Bundle Gerbes by Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray and Danny Stevenson.
Twisted K-theory, old and new

External links
Strings 2002, Michael Atiyah lecture, "Twisted K-theory and physics"
The Verlinde algebra is twisted equivariant K-theory (PDF)
Riemann–Roch and index formulae in twisted K-theory (PDF)

K-theory