Orbifold quantum cohomology of the symmetric product of A_r

Details Orbifold quantum cohomology of the symmetric product of A_r Orbifold quantum cohomology of the symmetric product of A_r

2009-10-05

arxiv.org/abs/0910.0629v1

Mathematics

Algebraic Geometry

39 pages, 4 figures, also available at http://www.its.caltech.edu/~keng/

Scientific paper

Let A_r be the minimal resolution of the cyclic quotient singularity C^2/Z_{r+1}. We study the equivariant quantum cohomology ring of the n-fold symmetric product stack [Sym^n(A_r)] of A_r. We calculate the operators of quantum multiplication by divisor classes. Under the assumption of the nonderogatory conjecture, these operators completely determine the ring structure, which provides an affirmative answer to the Crepant Resolution Conjecture on [Sym^n(A_r)] and Hilb^n(A_r). More strikingly, this allows us to complete a tetrahedron of equivalences relating the Gromov-Witten theories of [Sym^n(A_r)]/Hilb^n(A_r) and the relative Gromov-Witten/Donaldson-Thomas theories of A_r x P^1.

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