Integrability of one-parameter groups generated by the Virasoro operators (Q1370398)

The Virasoro algebra is a central extension of the Lie algebra \(\text{Vect}_{\mathbb{C}}(S^1)\) of the \(\mathbb{C}\)-valued polynomial vector fields on the unit circle. It has a basis \(\{d_p= ie^{ip\theta}(d/d\theta): p\in\mathbb{Z}\}\oplus c\) with commutation relations given by \[ [dp,dq]= (p-q)d_{p+q}+ \frac{p^3-p}{12} \delta_{p+q,0}c \quad\text{and}\quad [d_p,c]=0. \] Denote by \(\pi\) the standard representation of Vir on the Fock space \(V\); Vir denotes here the Virasoro algebra. The operator \(L_p= \pi(d_p)\in \text{End}(V)\) is called the Virasoro operator. The purpose of this paper is to prove that the one-parameter groups \(e^{zL_p}\) generated by the Virasoro operators \(L_p\) satisfy the commutation relation \[ \frac{\partial^2}{\partial z\partial w}\Biggl|_{z=w=0} e^{zL_p} e^{wL_q} e^{-zL_p} e^{-wL_q}= [L_p,L_q] \] and the integrability condition \[ e^{-zL_p} \pi(X)e^{zL_p}= \exp[(-z\text{ ad }L_p)\pi(X) \] for \(z,w\in\mathbb{C}\), \(p,q\in\mathbb{Z}\) and \(X\in\text{Vir}\).