Abstracts Mathematics

Twisted K-Theory and Gerbes from Hamiltonian Quantization

by Antti Harju

In this introductory part we review recent progress in the application of the methods of Hamiltonian quantization to construct twisted K-theory elements and gerbes. The important examples are the Wess-Zumino-Witten conformal field theory (WZW Model) and Hamiltonian quantization in low dimensions. The former has been studied by Mickelsson and Freed-Hopkins-Teleman in the equivariant case which provides an important connection between the representation rings of loop groups and twisted K-theory groups. There is a new line of research to apply the ideas of low dimensional quantum field theories on a product manifolds T × M to give explicit constructions of gerbes and twisted (equivariant) K-theory elements. This is the subject of the references [HM], [H3]. In the chapters 2 and 3 of the introductory part we motivate this study and introduce the relevant prerequisites. Chapter 4 is an introduction to the authors previous research project on the Fredholm index problems for the quantum groups of Drinfeld-Jimbo type, [H1], [H2]. Keywords: Twisted K-Theory, Gerbes, Dirac Operators, Index Theory, Lie Groupoids, Hamiltonian Quantization, Quantum Groups, Noncommutative Geometry. Työssä sovelletaan kvanttikenttäteoriaa geometristen ja topologisten ongelmien tutkimiseen. Pääpaino on selventää 3-kohomologia ryhmien rakennetta. Kvanttikenttäteoria luo koneiston, jolla näitä ryhmiä vastaavat geometriset oliot, ns. kerput voidaan rakentaa. Kerppuja sovelletaan K-teoriassa ja Fredholm-indeksiteoriassa.