Chiral de Rham complex. II

This paper is a sequel to math.AG/9803041. It consists of three parts. In the first part we give certain construction of vertex algebras which includes in particular the ones appearing in op. cit. In the second part we show how the cohomology ring $H^{\ast }(X)$ of a smooth complex variety $X$ could be restored from the correlation functions of the vertex algebra $R\mathrm{\Gamma }(X;{\mathrm{\Omega }}_X^{ch})$. In the third part, we prove first a useful general statement that the sheaf of loop algebras over the tangent sheaf $\text{Cal}{T}_X$ acts naturally on ${\mathrm{\Omega }}_X^{ch}$ for every smooth $X$ (see \S 1). The Z-graded vertex algebra $H^{\ast }(X;{\mathrm{\Omega }}_X^{ch})$ seems to be a quite interesting object (especially for compact $X$). In \S 2, we compute $H^0(C{P}^N;{\mathrm{\Omega }}_{C{P}^N}^{ch})$ as a module over $\widehat{sl}(N+1)$.