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An Algebra of Observables for Cross Ratios

Monday 14th March 2011 - 15:00 to 16:00
We define a Poisson Algebra called the swapping algebra using the intersection of curves in the disk. We interpret a subalgebra of the fraction swapping algebra -- called the algebra of multifractions -- as an algebra of functions on the space of cross ratios and thus as an algebra of functions on the Hitchin component as well as on the space of SL(n;R)-opers with trivial holonomy. We finally relate our Poisson structure to the Drinfel'd-Sokolov structure and to the Atiyah-Bott-Goldman symplectic structure for classical Teichmüller spaces and Hitchin components.
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons