Twisted K-theory

By Michael Francis Atiyah, G. Segal. 2004

312 citations recorded by the source corpus.

Twisted complex K-theory can be defined for a space X equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C ∗ -algebras. Up to equivalence, the twisting corresponds to an element of H 3 (X; Z). We give a systematic account of the definition and basic properties of the twisted theory, emphasizing some points where it behaves differently from ordinary K-theory. (We omit, however, its relations to classical coho- mology, which we shall treat in a sequel.) We develop an equivariant version of the theory for the action of a compact Lie group, proving that then the twistings are classified by the equivariant cohomology group H 3 G (X; Z). We also consider some basic examples of twisted K-theory classes, related to those appearing in the recent work of Freed-Hopkins-Teleman.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/8a3e6e10e7521bc4d465e4fa6349ce143106191e

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In Nuclear and High Energy Physics

1. G. Moore
2. A. Kapustin
3. T. Bridgeland
4. P. Aspinwall
5. D. Freed
6. G. Segal
7. E. Witten
8. N. Seiberg

Citations in the corpus

• The topological period–index problem over 6‐complexes (2012, unknown): “The map B PUn → K(Z, 3) of diagram (4) can be factored as B PUn → B PUn[2] ≃ K(Z/n, 2) and β : K(Z/n, 2) → K(Z, 3).”
• Dirichlet Branes and Mirror Symmetry (2009, methodology): “We shall explain the relationship with the description of [26] at the end of this subsection.”
• Twisted K-Homology and Group-Valued Moment Maps (2010, background): “The definition involves a distinguished G-DD bundle ASpin → G, originally defined in [6] in terms of a bundle of projective Hilbert spaces.”
• A vanishing theorem for characteristic classes of odd-dimensional manifold bundles (2009, background): “There is a technical problem to overcome at this point; it is discussed and solved in [7], pp.”
• Twisted equivariant K‐theory with complex coefficients (2002, background): “) For torsion, non-equivariant twistings, the relevant K-theory was first introduced in [14]; subsequent treatments ([28] and, more recently, [1, 5, 8]) remove the torsion assumption.”
• On the twisted K-homology of simple Lie groups (2004, background): “ action by some model for BU(1) (typically the space of projective unitary 8 operators on H); see, for example, Atiyah [3]. (The referee has pointed out that the recent exposition by Atiyah and Segal [5] is another source for operator-theoretic definitions of twisted K-theory.) Whatever the formal de”
• A Dixmier-Douady Theory for strongly self-absorbing C*-algebras II: the Brauer group (2014, background): “(An example from [1] for D = C shows that in general one cannot always arrange that k = 1.”
• Geometric cycles, index theory and twisted K-homology (2007, background): “The norm topology could be too restrictive for some examples, one might have to use the compact-open topology instead as discussed in [ 5 ].”
• Parametrized homotopy theory (2004, background): “The standard foundational reference is Atiyah and Segal [5], and we shall not repeat their arguments.”
• Loop groups and twisted K-theory III (2011, methodology): “This is known to physicists as the Dirac-Ramond operator [Mic04]; in the mathematical literature, it may have been first considered by Taubes [Tau89], and was more recently studied in detail by Landweber [Lan01], based on Kostant’s compact group analogue.”