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On unitary representations of the Virasoro algebra. (English) Zbl 0748.17025
Infinite-dimensional Lie algebras and their applications, Proc. Workshop, MontrĂ©al 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 $c=1-6/(m(m+1)),\quad h=(((m+1)l-mq)^ 2-1)/(4m(m+1)).$

##### MSC:
 17B68 Virasoro and related algebras 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
##### Keywords:
unitary representations; Verma module; Virasoro algebra