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On unitary representations of the Virasoro algebra. (English) Zbl 0748.17025
Infinite-dimensional Lie algebras and their applications, Proc. Workshop, Montréal Can. 1986, 141-159 (1988).
[For the entire collection see Zbl 0724.00013.]
The goal of this paper is to prove the theorem by Friedan-Qiu-Shenker (FQS-theorem). Let \(V^{h,c}\) be the Verma module over the Virasoro algebra (\(h,c\in \mathbb{R}\)). There is a unique sesquilinear form \((u,v)=(u,v)^{h,c}\) with the properties: (1) \((v_ \varphi,v_ \varphi)=1\), where \(v_ \varphi\) is a highest weight vector. (2) \((u,v)=(\overline{v,u})\). (3) \((\rho(L_ m)u,v)=(u,\rho(L_{-m})v)\), \(m\in\mathbb{Z}\), \(\rho\) is a representation, corresponding to \(V^{h,c}\), \(\{L_ m\}\) are generators of the Virasoro algebra. If this form is nonnegative then the representation \(\rho\) of the Virasoro algebra on the quotient of \(V\) by the space of null vectors is unitary, in the sense that \(\rho(L_ m)^*=\rho(L_{-m})\).
Theorem. (F.Q.S.) The form \((.,.)^{h,c}\) is non-negative only if either \(c\geq 1\), \(h\geq 0\) and there exist \(m\geq 2\) and two integers \(l\), \(q\), \(1\leq l\leq m\), \(1\leq q\leq l\), such that \[ c=1-6/(m(m+1)),\quad h=(((m+1)l-mq)^ 2-1)/(4m(m+1)). \]

17B68 Virasoro and related algebras
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)