Toric arc schemes and quantum cohomology of toric varieties
  We describe the quantum cohomology ring of a toric Fano variety $X$ in terms of the usual topological cohomology ring for an auxiliary infinite-dimensional scheme. This scheme is a part of an algebro-geometric model for the universal cover of the space of free loops with values in $X$. 
Sergey Arkhipov, Mikhail Kapranov