Brylinski, J.-L.; McLaughlin, D. A. The converse of the Segal-Witten reciprocity law. (English) Zbl 0856.57033 Int. Math. Res. Not. 1996, No. 8, 371-380 (1996). 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 followingTheorem: 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) Cited in 4 Documents MSC: 57R99 Differential topology 22E67 Loop groups and related constructions, group-theoretic treatment Keywords:loop group; central extensions; conformal field theory; topological quantum field theory; transgression PDFBibTeX XMLCite \textit{J. L. Brylinski} and \textit{D. A. McLaughlin}, Int. Math. Res. Not. 1996, No. 8, 371--380 (1996; Zbl 0856.57033) Full Text: DOI