2002/06/20 by Martin A. Guest, Guest, Martin A.
Mathematics · #14N35 #53D45 #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #Differential Geometry (math.DG) #FOS: Mathematics #Quantum Algebra (math.QA) #math.DG #math.QA #msc:14N35 #msc:53D45
paper · pdf · doi:10.48550/arxiv.math/0206212
21 pages, AMS-TeX. Revised version with many minor corrections and clarifications. Material on flag manifolds has been removed and will appear, in expanded form, in a joint paper with A. Amarzaya
openalex publication_date 2002/06/20 · arxiv created 2004/10/06 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.