Twisted K-theory explained

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( l H), l 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 → Fred( l H)]

from

M to Fred( l H).

A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of

Fred( l H) over M , that is, the Cartesian product of M and Fred( l 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( l H)

bundle

P over M , where PU( l H) is the group of projective unitary operators on the Hilbert space l H . Then the group of maps[P → Fred( l H)] _{PU(lH)}

from

P to Fred( l H) which are equivariant under an action of PU( l H) is equivalent to the original groups of maps[M → Fred( l H)].

This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that

PU( l 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( l H) topologically is a representative Eilenberg–MacLane spaceK( 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( l H) bundle over M that are covariant with respect to a PU( l H) bundle P _{H} fibered over M with 3-class H , that isK _{H(M)=[P} _{H →} Fred( l H)] _{PU(lH)} .

Equivalently, it is the space of homotopy classes of sections of the

Fred( l H) bundles associated to a PU( l 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 ^{p,q} _{3} = 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 ^{3 )} _{H(S} is the kernel of d _{3} in the even cohomology, which is the full even cohomology, which consists of the integers. Similarly K ^{3 )} _{H(S} 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 ^{0 (S} _{H = n} ^{3)=0} . The image of d _{3} consists of all elements of the integers that are multiples of n . Therefore, the odd cohomology, Z , quotiented by the image of d _{3} , n Z , is the cyclic group of order n , Z /n . In conclusion

K ^{1} _{H = n} (S ^{3)} = 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.

See also

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

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)

Notes and References

1. Donavan. Peter. Karoubi. Max. 1970. Graded Brauer groups and $K$-theory with local coefficients. Publications Mathématiques de l'IHÉS. en. 38. 5–25.
2. 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).
3. (DMW) also provide a crash course in Steenrod squares for physicists.
4. In Twisted K-theory and cohomology.
5. In D-Brane Instantons and K-Theory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg.