Quantum cohomology via D-modules

Details Quantum cohomology via D-modules Quantum cohomology via D-modules

2002-06-20

arxiv.org/abs/math/0206212v3

Mathematics

Differential Geometry

21 pages, AMS-TeX. Revised version with many minor corrections and clarifications. Material on flag manifolds has been removed

Scientific paper

We propose a new point of view on quantum cohomology, strongly motivated by the work of Givental and Dubrovin, but closer to differential geometry than the existing approaches. The central object is the D-module which "quantizes" a commutative algebra associated to the (uncompactified) space of rational curves. A standard loop group factorization procedure converts the D-module to Givental's D-module and the commutative algebra to the quantum cohomology algebra. We apply this only to the small quantum cohomology of full flag manifolds and semi-positive toric manifolds, but even in these cases the method is effective. In particular it gives an algorithm (requiring construction of a Groebner basis and solution of a system of o.d.e.) for the quantum product.

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