Deformations of Vertex Algebras,Quantum Cohomology of Toric Varieties,and Elliptic Genus

 We use previous work on the chiral de Rham complex and Borisov's deformation of a lattice vertex algebra to give a simple linear algebra construction of quantum cohomology of toric varieties. Somewhat unexpectedly, the same technique allows to compute the formal character of the cohomology of the chiral de Rham complex on even dimensional projective spaces. In particular, we prove that the formal character of the space of global sections equals the equivariant signature of the loop space, a well-known example of the Ochanine-Witten elliptic genus. Received: 15 July 2000 / Accepted: 17 August 2002 Published online: 8 January 2003 RID="*" ID="*" Partially supported by an NSF grant Communicated by R. H. Dijkgraaf