Twisted K theory

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

from M to F r e d ( H ) .

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

We can make this yet more complicated by introducing a trivial

bundle P over M, where P U ( H ) is the group of projective unitary operators on the Hilbert space H . Then the group of maps

from P to F r e d ( H ) which are equivariant under an action of P U ( H ) 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 P U ( H ) bundles on M are classified by elements H of the third integral cohomology group of M. This is a consequence of the fact that P U ( H ) topologically is a representative Eilenberg-MacLane space

The generalization is then straightforward. Rosenberg has defined

the twisted K-theory of M with twist given by the 3-class H, to be the space of homotopy classes of sections of the trivial F r e d ( H ) bundle over M that are covariant with respect to a P U ( H ) bundle P H fibered over M with 3-class H, that is

Equivalently, it is the space of homotopy classes of sections of the F r e d ( H ) bundles associated to a P U ( H ) bundle with class H.

What is it?

When H is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when H 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 M with all possible twists is a ring. In particular, the product of an element of K-theory with twist H with an element of K-theory with twist H' is an element of K-theory twisted by H+H'. 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 K⁰ or the twisted K¹, and then one takes the cohomology with respect to a series of differential operators. The first operator, d₃, for example, is the sum of the three-class H, which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square. No elementary form for the next operator, d₅, has been found, although several conjectured forms exist. Higher operators do not contribute to the K-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 H.

After taking the cohomology with respect to the full series of differentials one obtains twisted K-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, S³, has trivial cohomology except for H⁰(S³) and H³(S³) 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 d₃ = H. The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted K-theory is just the cohomology of the operator d₃ which acts on a class by cupping it with the 3-class H.

Imagine that H is the trivial class, zero. Then d₃ is also trivial. Thus its entire domain is its kernel, and nothing is in its image. Thus K⁰_H=0(S³) is the kernel of d₃ in the even cohomology, which is the full even cohomology, which consists of the integers. Similarly K¹_H=0(S³) consists of the odd cohomology quotiented by the image of d₃, in other words quotiented by the trivial group. This leaves the original odd cohomology, which is again the integers. In conclusion, K⁰ and K¹ of the three-sphere with trivial twist are both isomorphic to the integers. As expected, this agrees with the untwisted K-theory.

Now consider the case in which H is nontrivial. H is defined to be an element of the third integral cohomology, which is isomorphic to the integers. Thus H corresponds to a number, which we will call n. d₃ now takes an element m of H⁰ and yields the element nm of H³. As n is not equal to zero by assumption, the only element of the kernel of d₃ is the zero element, and so K⁰_H=n(S³)=0. The image of d₃ consists of all elements of the integers that are multiples of n. Therefore, the odd cohomology, Z, quotiented by the image of d₃, nZ, is the cyclic group of order n, Z_n. In conclusion

In string theory this result reproduces the classification of D-branes on the 3-sphere with n units of H-flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric SU(2) WZW model at level n - 2.

There is an extension of this calculation to the group manifold of SU(3). In this case the Steenrod square term in d₃, the operator d₅, and the extension problem are nontrivial.