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In mathematics, twisted K-theory (also called K-theory with local coefficients) 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. ==The definition== To motivate Rosenberg's geometric formulation of twisted K-theory, start from the Atiyah-Jänich theorem, stating that : the Fredholm operators on Hilbert space , 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 : from ''M'' to A slightly more complicated way of saying the same thing is as follows. Consider the trivial bundle of over M, that is, the Cartesian product of M and . 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 : bundle over M, where is the group of projective unitary operators on the Hilbert space . Then the group of maps : from to which are equivariant under an action of is equivalent to the original groups of maps : This more complicated construction of ordinary K-theory is naturally generalized to the twisted case. To see this, note that bundles on M are classified by elements H of the third integral cohomology group of M. This is a consequence of the fact that topologically is a representative Eilenberg-MacLane space : 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 bundle over M that are covariant with respect to a bundle fibered over M with 3-class H, that is : Equivalently, it is the space of homotopy classes of sections of the bundles associated to a bundle with class H. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Twisted K-theory」の詳細全文を読む スポンサード リンク
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