Twisted Ktheory
In mathematics, twisted Ktheory (also called "Ktheory with local coefficients") is a variation on Ktheory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory.
More specifically, twisted Ktheory with twist H is a particular variant of Ktheory, in which the twist is given by an integral 3dimensional cohomology class. It is special among the various twists that Ktheory 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'IHES) by Peter Donovan and Max Karoubi [1]; the second one in 1988 by Jonathan Rosenberg in ContinuousTrace Algebras from the Bundle Theoretic Point of View.
In physics, it has been conjectured to classify Dbranes, RamondRamond field strengths and in some cases even spinors in type II string theory. For more information on twisted Ktheory in string theory, see Ktheory (physics).
In the broader context of Ktheory, 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 Ktheory may be twisted by any integral cohomology class.
Contents
 The definition 1
 What is it? 2

How to calculate it 3
 Example: the threesphere 3.1
 See also 4
 References 5
 External links 6
The definition
To motivate Rosenberg's geometric formulation of twisted Ktheory, start from the AtiyahJänich theorem, stating that
 Fred(\mathcal H),
the Fredholm operators on Hilbert space \mathcal H, is a classifying space for ordinary, untwisted Ktheory. This means that the Ktheory 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 Ktheory 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 Ktheory 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 EilenbergMacLane space
 K(\mathbf Z,2).
The generalization is then straightforward. Rosenberg has defined
 K_{H}(M),
the twisted Ktheory of M with twist given by the 3class 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 3class 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.
What is it?
When H is the trivial class, twisted Ktheory is just untwisted Ktheory, 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 Ktheories of M with all possible twists is a ring. In particular, the product of an element of Ktheory with twist H with an element of Ktheory with twist H' is an element of Ktheory 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/2graded version is also presented). In particular twisted Ktheory is a module over classical Ktheory.
How to calculate it
Physicist typically want to calculate twisted Ktheory using the AtiyahHirzebruch spectral sequence.^{[1]} 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^{1}, 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 threeclass H, which in string theory corresponds to the NeveuSchwarz 3form, and the third Steenrod square.^{[2]} No elementary form for the next operator, d_{5}, has been found, although several conjectured forms exist. Higher operators do not contribute to the Ktheory of a 10manifold, 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.^{[3]}
After taking the cohomology with respect to the full series of differentials one obtains twisted Ktheory as a set, but to obtain the full group structure one in general needs to solve an extension problem.
Example: the threesphere
The threesphere, 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 threesphere 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_{3} = H. The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted Ktheory is just the cohomology of the operator d_{3} which acts on a class by cupping it with the 3class 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^{0}_{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^{1}_{H=0}(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 threesphere with trivial twist are both isomorphic to the integers. As expected, this agrees with the untwisted Ktheory.
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}_{H=n}(S^{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}, nZ, 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 Dbranes on the 3sphere with n units of Hflux, 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).^{[4]} In this case the Steenrod square term in d_{3}, the operator d_{5}, and the extension problem are nontrivial.
See also
References
 Graded Brauer groups and Ktheory with local coefficients, by Peter Donovan and Max Karoubi. Publ. Math. IHES Nr 38, p. 525 (1970).[2]
 DBrane Instantons and KTheory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg
 Twisted Ktheory and Cohomology by Michael Atiyah and Graeme Segal
 Twisted Ktheory and the Ktheory of Bundle Gerbes by Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray and Danny Stevenson.
 Twisted Ktheory, old and new
 ^ A guide to such calculations in the case of twisted Ktheory can be found in E8 Gauge Theory, and a Derivation of KTheory from MTheory by Emanuel Diaconescu, Gregory Moore and Edward Witten (DMW).
 ^ (DMW) also provide a crash course in Steenrod squares for physicists.
 ^ In Twisted Ktheory and cohomology.
 ^ In DBrane Instantons and KTheory Charges by Juan Maldacena, Gregory Moore and Nathan Seiberg.
External links
 Strings 2002, Michael Atiyah lecture, "Twisted Ktheory and physics"
 (PDF)The Verlinde algebra is twisted equivariant Ktheory
 (PDF)RiemannRoch and index formulae in twisted Ktheory