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The converse of the Segal-Witten reciprocity law. (English) Zbl 0856.57033

Let \(G\) be a connected complex reductive group and let \(LG\) denote the loop group \(C^\infty (S^1; G)\). In this paper the authors are concerned with the central extensions \(\widetilde {LG}\) of \(LG\) by the group \(C^*\) such that the reciprocity and glueing properties are satisfied. They are interested in characterizing precisely which extensions of \(LG\) satisfy these properties. The key to solving this problem is the connection between conformal field theory and \((2 + 1)\)-dimensional topological quantum field theory discovered by Witten. The purpose of this paper is to prove the following
Theorem: Let \(G\) be a connected semisimple complex Lie group. Then any extension of \(LG\) satisfying both the reciprocity and the glueing properties must necessarily lie in the image of \(\tau\) where \(\tau\) is a natural transgression \(\tau : H^4 (BG_{\bullet \geq 1}; Z(2)_D) \to H^3 (BLG_{\bullet \geq 1}; Z(1)_D)\).
Reviewer: V.Abramov (Tartu)

MSC:

57R99 Differential topology
22E67 Loop groups and related constructions, group-theoretic treatment
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