Quantum cohomology of flag manifolds and Toda lattices

TL;DR

This paper explores the connection between quantum cohomology of flag manifolds and Toda lattices, introducing equivariant quantum cohomology and computing its algebraic structure based on conjectures.

Contribution

It introduces equivariant quantum cohomology for flag manifolds and computes its algebra, linking it to Toda lattice invariants, based on new conjectures.

Findings

Quantum cohomology algebra of flag manifolds matches Toda lattice invariants.

Introduction of equivariant quantum cohomology for flag manifolds.

Formulation of conjectures about properties of equivariant quantum cohomology.

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.