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.

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

${\displaystyle Fred({\mathcal {H}}),}${\displaystyle Fred({\mathcal {H}}),}

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

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

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

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

We can make this yet more complicated by introducing a trivial

${\displaystyle PU({\mathcal {H}})}${\displaystyle PU({\mathcal {H}})}

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

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

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

${\displaystyle [M\rightarrow Fred({\mathcal {H}})].}${\displaystyle [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 ${\displaystyle PU({\mathcal {H}})}${\displaystyle PU({\mathcal {H}})} bundles on ${\displaystyle M}${\displaystyle M} are classified by elements ${\displaystyle H}${\displaystyle H} of the third integral cohomology group of ${\displaystyle M}${\displaystyle M}. This is a consequence of the fact that ${\displaystyle PU({\mathcal {H}})}${\displaystyle PU({\mathcal {H}})} topologically is a representative Eilenberg–MacLane space

${\displaystyle K(\mathbf {Z} ,3)}${\displaystyle K(\mathbf {Z} ,3)}.

The generalization is then straightforward. Rosenberg has defined

${\displaystyle K_{H}(M)}${\displaystyle K_{H}(M)},

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

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

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

When ${\displaystyle H}${\displaystyle H} is the trivial class, twisted K-theory is just untwisted K-theory, which is a ring. However, when ${\displaystyle H}${\displaystyle 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 ${\displaystyle M}${\displaystyle M} with all possible twists is a ring. In particular, the product of an element of K-theory with twist ${\displaystyle H}${\displaystyle H} with an element of K-theory with twist ${\displaystyle H'}${\displaystyle H'} is an element of K-theory twisted by ${\displaystyle H+H'}${\displaystyle 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.

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 ${\displaystyle K_{0}}${\displaystyle K_{0}} or the twisted ${\displaystyle K^{0}}${\displaystyle K^{0}}, and then one takes the cohomology with respect to a series of differential operators. The first operator, ${\displaystyle d_{3}}${\displaystyle d_{3}}, for example, is the sum of the three-class ${\displaystyle H}${\displaystyle H}, which in string theory corresponds to the Neveu-Schwarz 3-form, and the third Steenrod square,^[3] so

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

No elementary form for the next operator, ${\displaystyle d_{5}}${\displaystyle d_{5}}, has been found, although several conjectured forms exist. Higher operators do not contribute to the ${\displaystyle K}${\displaystyle 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 ${\displaystyle M}${\displaystyle M}.^[4]

After taking the cohomology with respect to the full series of differentials one obtains twisted ${\displaystyle K}${\displaystyle K}-theory as a set, but to obtain the full group structure one in general needs to solve an extension problem.

The three-sphere, ${\displaystyle S^{3}}${\displaystyle S^{3}}, has trivial cohomology except for ${\displaystyle H^{0}(S^{3})}${\displaystyle H^{0}(S^{3})} and ${\displaystyle H^{3}(S^{3})}${\displaystyle 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 ${\displaystyle d_{3}=H}${\displaystyle 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 ${\displaystyle K}${\displaystyle K}-theory is just the cohomology of the operator ${\displaystyle d_{3}}${\displaystyle d_{3}} which acts on a class by cupping it with the 3-class ${\displaystyle H}${\displaystyle H}.

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

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

${\displaystyle K_{H=n}^{1}(S^{3})=\mathbb {Z} /n}${\displaystyle 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 ${\displaystyle n}${\displaystyle n} units of ${\displaystyle H}${\displaystyle H}-flux, which corresponds to the set of symmetric boundary conditions in the supersymmetric ${\displaystyle SU(2)}${\displaystyle SU(2)} WZW model at level ${\displaystyle n-2}${\displaystyle 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 ${\displaystyle d_{3}}${\displaystyle d_{3}}, the operator ${\displaystyle d_{5}}${\displaystyle d_{5}}, and the extension problem are nontrivial.

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

• "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

• 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)

• K-theory