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.

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.

## Contents

* 1 Definition
* 2 Relation to K-theory
* 3 Calculations
* 3.1 Example: the three-sphere
* 4 See also
* 5 Notes
* 6 References
* 7 External links

## Definition

To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah–Jänich theorem, stating that

[math]\displaystyle{ Fred(\mathcal H), }[/math]

the Fredholm operators on Hilbert space [math]\displaystyle{ \mathcal H }[/math], is a classifying space for ordinary, untwisted K-theory. This means that the K-theory of the space [math]\displaystyle{ M }[/math] consists of the homotopy classes of maps

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

from [math]\displaystyle{ M }[/math] to [math]\displaystyle{ Fred(\mathcal H). }[/math]

A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of [math]\displaystyle{ Fred(\mathcal H) }[/math] over [math]\displaystyle{ M }[/math], that is, the Cartesian product of [math]\displaystyle{ M }[/math] and [math]\displaystyle{ Fred(\mathcal H) }[/math]. Then the K-theory of [math]\displaystyle{ M }[/math] consists of the homotopy classes of sections of this bundle.

We can make this yet more complicated by introducing a trivial

[math]\displaystyle{ PU(\mathcal H) }[/math]

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

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

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

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

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

[math]\displaystyle{ K(\mathbf Z,3) }[/math].

The generalization is then straightforward. Rosenberg has defined

[math]\displaystyle{ K_H(M) }[/math],

the twisted K-theory of [math]\displaystyle{ M }[/math] with twist given by the 3-class [math]\displaystyle{ H }[/math], to be the space of homotopy classes of sections of the trivial [math]\displaystyle{ Fred(\mathcal H) }[/math] bundle over [math]\displaystyle{ M }[/math] that are covariant with respect to a [math]\displaystyle{ PU(\mathcal H) }[/math] bundle [math]\displaystyle{ P_H }[/math] fibered over [math]\displaystyle{ M }[/math] with 3-class [math]\displaystyle{ H }[/math], that is

[math]\displaystyle{ K_H(M)=[P_H\rightarrow Fred(\mathcal H)]_{PU(\mathcal H)}. }[/math]

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

## Relation to K-theory

When [math]\displaystyle{ H }[/math] is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when [math]\displaystyle{ H }[/math] 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 [math]\displaystyle{ M }[/math] with all possible twists is a ring. In particular, the product of an element of K-theory with twist [math]\displaystyle{ H }[/math] with an element of K-theory with twist [math]\displaystyle{ H' }[/math] is an element of K-theory twisted by [math]\displaystyle{ H+H' }[/math]. 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 [math]\displaystyle{ K_0 }[/math] or the twisted [math]\displaystyle{ K^0 }[/math], and then one takes the cohomology with respect to a series of differential operators. The first operator, [math]\displaystyle{ d_3 }[/math], for example, is the sum of the three-class [math]\displaystyle{ H }[/math], which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square,[3] so

> [math]\displaystyle{ d_3^{p,q} = Sq^3 + H }[/math]

No elementary form for the next operator, [math]\displaystyle{ d_5 }[/math], has been found, although several conjectured forms exist. Higher operators do not contribute to the [math]\displaystyle{ K }[/math]-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 [math]\displaystyle{ M }[/math].[4]

After taking the cohomology with respect to the full series of differentials one obtains twisted [math]\displaystyle{ K }[/math]-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, [math]\displaystyle{ S^3 }[/math], has trivial cohomology except for [math]\displaystyle{ H^0(S^3) }[/math] and [math]\displaystyle{ H^3(S^3) }[/math] 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 [math]\displaystyle{ d_5 = H }[/math]. The later differentials increase the degree of a cohomology class by more than three and so are again trivial; thus the twisted [math]\displaystyle{ K }[/math]-theory is just the cohomology of the operator [math]\displaystyle{ d_3 }[/math] which acts on a class by cupping it with the 3-class [math]\displaystyle{ H }[/math].

Imagine that [math]\displaystyle{ H }[/math] is the trivial class, zero. Then [math]\displaystyle{ d_3 }[/math] is also trivial. Thus its entire domain is its kernel, and nothing is in its image. Thus [math]\displaystyle{ K^0_H(S^3) }[/math] is the kernel of [math]\displaystyle{ d_3 }[/math] in the even cohomology, which is the full even cohomology, which consists of the integers. Similarly [math]\displaystyle{ K^1_H(S^3) }[/math] consists of the odd cohomology quotiented by the image of [math]\displaystyle{ d_3 }[/math], in other words quotiented by the trivial group. This leaves the original odd cohomology, which is again the integers. In conclusion, [math]\displaystyle{ K^0 }[/math] and [math]\displaystyle{ K^1 }[/math] of the three-sphere with trivial twist are both isomorphic to the integers. As expected, this agrees with the untwisted [math]\displaystyle{ K }[/math]-theory.

Now consider the case in which [math]\displaystyle{ H }[/math] is nontrivial. [math]\displaystyle{ H }[/math] is defined to be an element of the third integral cohomology, which is isomorphic to the integers. Thus [math]\displaystyle{ H }[/math] corresponds to a number, which we will call [math]\displaystyle{ n }[/math]. [math]\displaystyle{ d_3 }[/math] now takes an element [math]\displaystyle{ m }[/math] of [math]\displaystyle{ H^0 }[/math] and yields the element [math]\displaystyle{ nm }[/math] of [math]\displaystyle{ H^3 }[/math]. As [math]\displaystyle{ n }[/math] is not equal to zero by assumption, the only element of the kernel of [math]\displaystyle{ d_3 }[/math] is the zero element, and so [math]\displaystyle{ K_{H=n}^0(S^3)=0 }[/math]. The image of [math]\displaystyle{ d_3 }[/math] consists of all elements of the integers that are multiples of [math]\displaystyle{ n }[/math]. Therefore, the odd cohomology, [math]\displaystyle{ \mathbb{Z} }[/math], quotiented by the image of [math]\displaystyle{ d_3 }[/math], [math]\displaystyle{ n\mathbb{Z} }[/math], is the cyclic group of order [math]\displaystyle{ n }[/math], [math]\displaystyle{ \mathbb{Z}/n }[/math]. In conclusion

> [math]\displaystyle{ K^1_{H=n}(S^3) = \mathbb{Z}/n }[/math]

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

There is an extension of this calculation to the group manifold of SU(3).[5] In this case the Steenrod square term in [math]\displaystyle{ d_3 }[/math], the operator [math]\displaystyle{ d_5 }[/math], and the extension problem are nontrivial.

## See also

* K-theory
* Wess–Zumino–Witten model
* Bundle gerbe

## Notes

1. ↑ Donavan, Peter; Karoubi, Max (1970). "Graded Brauer groups and $K$-theory with local coefficients" (in en). Publications Mathématiques de l'IHÉS 38: 5–25. http://www.numdam.org/item/PMIHES_1970__38__5_0/.
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.

## 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)

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