Projective unitary positive-energy representations of \(Diff(S^ 1)\)
Let \({\mathfrak d}\) be the Lie algebra of complex vector fields on \(S^ 1\) whose coefficients are the Fourier polynomials. Its central extension - the Virasoro algebra \(\hat{\mathfrak d}\)- admits a triangular decomposition \(\hat{\mathfrak d}={\mathfrak n}^+\oplus {\mathfrak h}\oplus {\mathfrak n}^-\), which enables us to construct highest weight modules L(h,c) for \(\hat{\mathfrak d}\) depending on two complex parameters h and c determined by the action of the subalgebra \({\mathfrak h}\subset \hat {\mathfrak d}\). \textit{V. G. Kac} conjectured [Lect. Notes Math. 933, 117-126 (1982; Zbl 0493.17011)] that if the contravariant \((=\) Hermitian Shapavolov) form on L(h,c) is positive definite, then the module L(h,c) ``integrates'' to a continuous projective representation of the group \({\mathcal D}\) of orientation preserving diffeomorphisms of \(S^ 1\). The authors prove the Kac conjecture. Let us remark that the arguments in the proof are rather delicate.
scientific article; zbMATH DE number 4036098
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