Toric Arc Schemes and Quantum Cohomology of Toric Varieties. Sergey Arkhipov and Mikhail Kapranov

This paper is a part of a larger project devoted to the study of Floer cohomology in algebro-geometic context, as a natural cohomology theory defined on a certain class of ind-schemes. Among these ind-schemes are algebro-geometric models of the spaces of free loops. Let X be a complex projective variety. Heuristically, HQ(X), the quantum cohomology of X, is a version of the Floer cohomology of the universal cover of the free loop space LX = Map(S1,X). However, truly infinite-dimensional approaches to HQ are few. One of the most interesting is the unpublished result of D. Peterson: for a reductive group G the ring HQ(G/B) is identified with the usual topological cohomology ring of F, the affine flag variety of G. To be precise, if A = H2(G/B) is the coweight lattice, then H•(F) is naturally an algebra over the semigroup ring C[A+] of the dominant cone in A and