Deformation of Poisson structures and contact Lie (super)algebras of string theories (Q803260)

Let \(M=({\mathbb{R}}^{2d,n},\omega)\) be a symplectic supermanifold of dimension 2d,m with the even form (locally) \(\omega =\sum dp_ i\wedge dq_ i+\sum \epsilon (d\theta_ j)^ 2\), \(\epsilon_ j=\pm 1\). Let \(P=\omega^{-1}\) the bivector that defines the Poisson bracket in the superspace \({\mathcal P}\) of functions on M. Let \(tr_{\Delta}: {\mathcal P}\otimes {\mathcal P}\to {\mathcal P}\) be the operator induced by the diagonal embedding \(\Delta\) : \(M\to M\times M\), i.e. \(tr_{\Delta}(a\otimes b)=ab\). Define \(\{f,g\}_ n=tr_{\Delta}(P^ n(f\otimes g))\) and set \([f,g]^{(\hslash)}=\sum_{n\geq 1}\frac{\hslash^ n}{(2n+1)!}\{f,g\}_ n.\) It is announced that \([f,g]^{(\hslash)}\) is a nontrivial deformation of the Poisson bracket on \({\mathcal P}\) and for \(d=2k+1\) a subalgebra of \({\mathcal P}\) is distinguished which is stable under the deformation. This subalgebra \({\mathfrak h}\) is a semidirect product of the Lie superalgebra of contact vector fields on the supersphere \(S=S^{4d+1,m}\) and the direct sum of several modules of generalized densities, i.e. of the form \(F_{\lambda}=F^{\lambda}_{\alpha}\) for \(\lambda\in {\mathbb{R}}\), where \(\alpha\) is a contact form such that \(d\alpha =\omega\), F is the space of functions on S. The subalgebra \({\mathfrak h}\) is said to have a nontrivial central extension.