Integrable Structure in Quantum Cohomology

2004 Fiscal Year Final Research Report Summary

• Project/Area Number: 15540096
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Geometry
• Research Institution: Nihon University
• Principal Investigator: OTOFUJI Takashi Nihon University, College of Engineering, Lecturer, 工学部, 講師 (70339266)
• Project Period (FY): 2003 – 2004
• Keywords: Infinite dimensional flag manifolds / Quantum cohomology

Research Abstract

Our first motivation for the present research project was to understand why the integrable system, the Toda lattice, appears in the quantum cohomology of flag manifolds.
For the finite dimensional full flag manifolds, this phenomenon was found by Givental-Kim (Comm.Math Phys.1995), and an analogous one was observed for the infinite dimensional (i.e.affine) flag manifolds by the head investigator with M.A.Guest (Comm.Math.Phys.217,475-487,2001). In that work we derived the relations for divisor classes in the quantum cohomology, using only the geometry of the moduli space of rational curves in finite dimensional flag manifolds, and did not touch the infinite dimensional geometry of affine flag manifolds essentially.
If we try to investigate quantum Schubert calculus for such manifolds, we need to look into the infinite dimensional geometry. We wanted to have keys to study moduli spaces of curves in infinite dimensinal manifolds, and decided to learn tools from symplectic geometry. Hofer-Z ehnder capacity looked worth studying, because its generalization to some infinite dimensional symplectic manifolds is considered recently by Kuksin and others.
Before obtaining the foundation for quantum cohomology of aflfine flag manifolds, we thought that we should investigate its algebraic structure assuming basic properties that should be expected in comparison with the finite dimensional case. We found that the quantum product for some affine flag manifolds does not satisfy the potential property (i.e., the structure constantsfor quantum product are given by third derivatives of certain generating function), which is known to hold for finite dimensional symplectic manifolds. Besides this, we worked on D-module structures on quantum cohomology of affine flag manifolds (a work with A.L.Mare and M.A.Guest, in progress).
We observed that we can introduce an appropriate D-module structure on a small subspace of such quantum cohomology. However we have obtained neither a clue to extend it to a larger subspace, nor a geometric understanding of the trouble above.