Twisted K-theory

In mathematics, twisted K-theory (also called K-theory with local coefficients^[1]) 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

Fred({\mathcal {H}}),

the Fredholm operators on Hilbert space ${\mathcal {H}}$ , 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

[M\rightarrow Fred({\mathcal {H}})]

from $M$ to $Fred({\mathcal {H}}).$

A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of $Fred({\mathcal {H}})$ over $M$ , that is, the Cartesian product of $M$ and $Fred({\mathcal {H}})$ . 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

PU({\mathcal {H}})

bundle $P$ over $M$ , where $PU({\mathcal {H}})$ is the group of projective unitary operators on the Hilbert space ${\mathcal {H}}$ . Then the group of maps

[P\rightarrow Fred({\mathcal {H}})]_{PU({\mathcal {H}})}

from $P$ to $Fred({\mathcal {H}})$ which are equivariant under an action of $PU({\mathcal {H}})$ is equivalent to the original groups of maps

[M\rightarrow Fred({\mathcal {H}})].

This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that $PU({\mathcal {H}})$ bundles on $M$ are classified by elements $H$ of the third integral cohomology group of $M$ . This is a consequence of the fact that $PU({\mathcal {H}})$ topologically is a representative Eilenberg–MacLane space

K(\mathbf {Z} ,3)

The generalization is then straightforward. Rosenberg has defined

K_{H}(M)

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 $Fred({\mathcal {H}})$ bundle over $M$ that are covariant with respect to a $PU({\mathcal {H}})$ bundle $P_{H}$ fibered over $M$ with 3-class $H$ , that is

K_{H}(M)=[P_{H}\rightarrow Fred({\mathcal {H}})]_{PU({\mathcal {H}})}.

Equivalently, it is the space of homotopy classes of sections of the $Fred({\mathcal {H}})$ bundles associated to a $PU({\mathcal {H}})$ bundle with class $H$ .

Relation to K-theory

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.

Calculations

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 $K_{0}$ or the twisted $K^{0}$ , and then one takes the cohomology with respect to a series of differential operators. The first operator, $d_{3}$ , 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,^[3] so

$d_{3}^{p,q}=Sq^{3}+H$

No elementary form for the next operator, $d_{5}$ , 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 $M$ .^[4]

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, $S^{3}$ , has trivial cohomology except for $H^{0}(S^{3})$ and $H^{3}(S^{3})$ 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 $d_{5}=H$ . 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 $d_{3}$ which acts on a class by cupping it with the 3-class $H$ .

Imagine that $H$ is the trivial class, zero. Then $d_{3}$ is also trivial. Thus its entire domain is its kernel, and nothing is in its image. Thus $K_{H}^{0}(S^{3})$ is the kernel of $d_{3}$ in the even cohomology, which is the full even cohomology, which consists of the integers. Similarly $K_{H}^{1}(S^{3})$ consists of the odd cohomology quotiented by the image of $d_{3}$ , in other words quotiented by the trivial group. This leaves the original odd cohomology, which is again the integers. In conclusion, $K^{0}$ and $K^{1}$ 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 $n$ . $d_{3}$ now takes an element $m$ of $H^{0}$ and yields the element $nm$ of $H^{3}$ . As $n$ is not equal to zero by assumption, the only element of the kernel of $d_{3}$ is the zero element, and so $K_{H=n}^{0}(S^{3})=0$ . The image of $d_{3}$ consists of all elements of the integers that are multiples of $n$ . Therefore, the odd cohomology, $\mathbb {Z}$ , quotiented by the image of $d_{3}$ , $n\mathbb {Z}$ , is the cyclic group of order $n$ , $\mathbb {Z} /n$ . In conclusion

$K_{H=n}^{1}(S^{3})=\mathbb {Z} /n$

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).^[5] In this case the Steenrod square term in $d_{3}$ , the operator $d_{5}$ , and the extension problem are nontrivial.

Notes

^ Donavan, Peter; Karoubi, Max (1970). "Graded Brauer groups and $K$-theory with local coefficients". Publications Mathématiques de l'IHÉS. 38: 5–25.
^ A guide to such calculations in the case of twisted K-theory can be found in E8 Gauge Theory, and a Derivation of K-Theory from M-Theory by Emanuel Diaconescu, Gregory Moore and Edward Witten (DMW).
^ (DMW) also provide a crash course in Steenrod squares for physicists.
^ In Twisted K-theory and cohomology.
^ In D-Brane Instantons and K-Theory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg.

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