Report on “Short course on
Semi-classical approximations in conformal field theory and analytic geometry of moduli spaces”
by Dr. Claudio Meneses (DFG Principal Investigator, University of Kiel)
Dr. Claudio Meneses gave a series of 8 lectures in which he brought together conformal field theory (a major theme in mathematical physics, originating from the work of Russian researchers in the 1980s) and moduli spaces of Fuchsian differential equations (a classical subject in 19th century mathematics, dating back to Poincare).
Dr. Claudio Meneses (DFG Principal Investigator, University of Kiel)
The course began with a discussion of uniformization for noncompact Riemann surfaces, in particular the case of a two-sphere minus a finite number of points. This quickly led to modular functions, Fuchsian differential equations, and moduli spaces. The main topic was a generalization of this theory in which differential equations are replaced by connections and functions are replaced by sections of parabolic vector bundles.
Quantisation enters the picture when line bundles (based on a famous construction of Quillen) over the moduli space are studied. The geometry of these line bundles is fundamental in the study of nontrivial examples of conformal field theories such as the Liouville theory and several WZNW theories, leading to a rigorous mathematical foundation of deep physical concepts such as conformal blocks and the Verlinde formula. Similarly, semi-classical approximations of partition and correlation functions are seen to contain unexpected information of the analytic geometry of the moduli spaces in question.
This was a very well presented and comprehensive survey of a difficult subject bridging mathematics and physics, leading up to problems and results in current research.
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- Dr. Claudio Meneses (University of Kiel), Prof.Martin Guest(Waseda University)
