Quantum cohomology of flag manifolds and Toda lattices

1993/12/13 by Alexander Givental, Bumsig Kim · 1 voice · 120 citations
Mathematics · Physics and Astronomy · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Cohomology #Equivariant cohomology #Equivariant map #Flag (linear algebra) #Geometric and Algebraic Topology #Gromov–Witten invariant #Invariant (physics) #Quantum #Quantum algebra #Quantum cohomology #hep-th #math.AG

paper · pdf · doi:10.1007/bf02101846

published in Communications in Mathematical Physics 168(3), 609-641 (Springer Science+Business Media) · 35 pages

arxiv created 1993/12/13 · arxiv published 1993/12/13 · arxiv updated 1993/12/13 · openalex publication_date 1995/04/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

Abstract

We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.

Citations

• Morse theory for Lagrangian intersections
• Pseudo holomorphic curves in symplectic manifolds
• Symplectic fixed points and holomorphic spheres
• Supersymmetry and Morse theory
• The moment map and equivariant cohomology

Cited by

• A Note on Quantization Operators on Nichols Algebra Model for Schubert Calculus on Weyl groups

Discussions

• I believe that it may be important in these. 2D sigma models and GLSMs (A-model): The chiral ring is the small quantum cohomology and the three point genus zero correlators are the GW numbers. Efficie [stackexchange, 2 points]