In 1945 Samuel Eilenberg and Norman E. Steenrod set forth the essential properties of a homology theory in terms of seven axioms; the last stipulating that the reduced homology of point is trivial. A number of years later (1957) Alexander Grothendieck introduced K-theory and expressed the Riemann-Roch theorem for nonsingular projective varieties by saying that the mapping $E\to ch\left(E\right)*\mathrm{Td}\left(X\right)$ from $K^0\left(X\right)$ to $H^{*}\left(X\right)$ is a natural transformation of covariant functors. Here $K^0\left(X\right)$ denotes the Grothendieck group of algebraic vector bundles on $X$, $H^{*}\left(X\right)$ denotes a suitable cohomology theory, $ch$ is the Chern character, and $\mathrm{Td}\left(X\right)$ is the Todd class of the tangent bundle of $X$. Michael Atiyah and Friedrich Hirzebruch developed K-theory in the context of topological spaces and showed that topological K-theory satisfies the first six axioms of Eilenberg and Steenrod. Using Bott periodicity one readily shows that the K-theory of a point is infinite cyclic in even degrees and vanishes in odd degrees.

Recently Edward Witten has argued that K-theory is relevant to the classification of Ramond-Ramond (RR) charges as well as noncommutative Yang-Mills theory or open string field theory. In order to consider D-branes with a topologically non-trivial Neveu-Schwarts 3-form field $H$, one needs to work with a twisted version of topological K-theory. If $H$ represents a torsion class, one may use the twisted K-theory developed by Peter Donavan and Max Karoubi. In this talk I shall describe two constructions of twisted K-theory one set forth by Michael Atiyah and Graeme Segal and the other by Daniel Freed, Michael J. Hopkins and Constantin Teleman. Due to personal limitations I shall give a braneless presentation.

References

1. M Atiyah and F Hirzebruch, Vector bundles and homogenuous spaces, Proc. of Symposia in Pure Maths vol 3, Differential Geometry, Amer. Math. Soc. 1961, 7-38.
2. M Atiyah and G Segal, Twisted K-theory, math.kt/0407054.
3. A Borel and J-P Serre, Le théorème de Riemann-Roch, Bull. Soc. Math. France 86 (1958) 97-136.
4. P Donavan and M Karoubi, Graded Brauer groups and K-theory with local coefficients, Publ. Math. IHES 38 (1970) 5-25.
5. S Eilenberg and N E Steenrod, Axiomatic approach to homology theory, Proc. Nat. Acad. Sci. USA 31 (1945), 117-120.
6. D Freed, M J Hopkins and C Teleman, Twisted K-theory and loop groups representations I, math.AT/0312155.
7. E Witten, Overview of K-theory Applied to Strings, hep-th/0007175.