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We calculate the small quantum orbifold cohomology of arbitrary weighted projective spaces. We generalize Givental’s heuristic argument, which relates small quantum cohomology to S 1-equivariant Floer cohomology of loop space, to weighted projective spaces and use this to conjecture an explicit formula for the small J-function, a generating function for certain genus-zero Gromov–Witten invariants. We prove this conjecture using a method due to Bertram. This provides the first non-trivial example of a family of orbifolds of arbitrary dimension for which the small quantum orbifold cohomology is known. In addition we obtain formulas for the small J-functions of weighted projective complete intersections satisfying a combinatorial condition; this condition naturally singles out the class of orbifolds with terminal singularities. [ABSTRACT FROM AUTHOR]

This article discusses Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups. The results of the paper were completed in the Summer 98 and reported by the authors at a number of conferences and seminars. In this version of the paper researchers decided to leave the material of Section 5 in the form close to the preliminary text written in 1998. In particular, researchers did not try to match the quantum group description of finite-difference Toda lattices given in Section 5 with the construction that has become standard since then due to the paper by writer P. Etingof.