Etingof, Pavel I.; Frenkel, Igor B. Central extensions of current groups in two dimensions. (English) Zbl 0822.22014 Commun. Math. Phys. 165, No. 3, 429-444 (1994). From the paper: This paper is devoted to a generalization of the theory of loop groups to the two-dimensional case. To every complex Riemann surface, we assign a central extension of the group of smooth maps from this surface to a simple complex Lie group \(G\) by the Jacobian of this surface. This extension is topologically non-trivial, as in the loop group case. When the surface is the torus (elliptic curve), classification of coadjoint orbits is related to linear difference equations in one variable, and to classification of conjugacy classes in the loop group. We study integral orbits, for which the Kirillov-Kostant form is a curvature form for some principal torus bundle. The number of such orbits for a given level is finite, as in the loop group case; conjecturally, they correspond to analogues of integrable modules occurring in conformal field theory. Reviewer: A.N.Pressley (London) Cited in 1 ReviewCited in 21 Documents MSC: 22E67 Loop groups and related constructions, group-theoretic treatment 81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics 58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 57S05 Topological properties of groups of homeomorphisms or diffeomorphisms Keywords:loop groups; complex Riemann surfaces; group of smooth maps; simple complex Lie groups; integral orbits; Kirillov-Kostant form PDFBibTeX XMLCite \textit{P. I. Etingof} and \textit{I. B. Frenkel}, Commun. Math. 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