Quantum cohomology via vicious and osculating walkers

TL;DR

This paper connects quantum cohomology of the Grassmannian with solvable lattice models, revealing new algebraic structures and explicit formulas for Gromov-Witten invariants through lattice path enumeration.

Contribution

It establishes a novel link between quantum cohomology, integrable lattice models, and algebraic combinatorics, providing explicit solutions and interpretations.

Findings

Partition functions expressed via Postnikov's toric Schur functions

Eigenvectors yield idempotents of Verlinde algebra

Models relate to Gromov-Witten invariants

Abstract

We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang-Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged u(n)-WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov's toric Schur functions and can be interpreted as generating functions for Gromov-Witten invariants.