Workshop: Current Groups and Twisted K-theory

Lectures by Professor Jouko Mickelsson (University of Helsinki)

Short biography

Jouko Mickelsson is a Professor in Mathematics at the University of Helsinki, 2004-. He obtained his PhD at the University of Helsinki in 1972 and was an Associate Professor in Mathematics at the University of Jyväskylä (Finland) until 1993. He has held several prestigious visiting positions, including a Research Fellowship at the Mittag-Leffler Institute (Stockholm) during the academic year 1981/82, a Fulbright Scholarship for research work at Massachusetts Institute of Technology (MIT) 1986/87, an Alexander von Humboldt Fellowship for research in Freiburg (Germany) during 1987/88 and 1992/1993, and a guest scientist at MIT in 1992. During the period 1993-2012 he also held a dual professorship in Mathematical Physics at the Royal Institute of Technology (Stockholm).
Professor Mickelsson is a distinguished researcher with more than 90 research articles.

He is an international expert on the subject of current groups and is the author of a highly influential book "Current Algebras and Groups" which has become a standard reference in the area. He is best known for inventing step algebras in Lie algebra representation theory, the Mickelsson-Faddeev extension in gauge theory, the Mickelsson-Rajeev cocycle in current algebras and related work on the theory of gerbes and twisted K-theory.

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Abstract

We start looking at how gauge anomalies in quantum field theory give rise to abelian extensions of current algebras. In the hamiltonian formulation of the quantized Dirac field in Yang-Mills background the chiral anomaly defines a gerbe on the moduli space of gauge connections, the Dixmier-Douady class of the gerbe can be computed from families index theorem.

Using the gerbe as input one can define twisted K-theory on the moduli space. In particular, in 1+1 space-time dimensions one can use the supersymmetric Wess-Zumino-Witten model for constructing twisted K-theory classes on compact Lie groups (related to the Freed-Hopkins-Teleman correspondence between twisted K-theory and the Verlinde ring). We shall describe analogous constructions in the case of decomposable Dixmier-Douady classes.

Finally, we shall explain a recent work of F. Wagemann and C. Wockel on group cohomology, relating it to gauge group extensions and transgression in Lie algebra cohomology. More precisely, we shall extend the higher group cocycles to the transformation groupoid setting (motivated by QFT) and discuss potential obstructions in the construction due to a nonvanishing of low dimensional homology groups of the gauge group. The resolution of the obstruction is obtained by an application of the Cheeger-Simons differential characters.

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Schedule

Participant information

Venue

Conference Room 7.15, Level 7, Ingkarni Wardli (see on the map)

Registration

There will be no registration fees: all are welcome. However, if you are interested in attending, kindly send an e-mail to Pedram Hekmati by 25 March 2013, with the following information:

• Name
• Position and Affiliation
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Please also indicate in your e-mail whether you would like to attend the workshop dinner (Wednesday evening, 3 April), which will be $5 per person.

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Participants, other than the invited speakers, are asked to make their own travel and accommodation arrangements.

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