A Combinatorial Derivation of the Racah-Speiser Algorithm for Gromov-Witten invariants

TL;DR

This paper introduces a new combinatorial approach using Clifford algebra to compute Gromov-Witten invariants in quantum cohomology, providing explicit formulas and a q-deformation of the Racah-Speiser algorithm.

Contribution

It presents a novel combinatorial product formula for quantum cohomology of Grassmannians and a simple projection formula onto the Verlinde algebra, linking geometric and algebraic structures.

Findings

Expressed Gromov-Witten invariants via Clifford algebra elements

Developed a q-deformation of the Racah-Speiser algorithm

Provided an explicit combinatorial projection formula

Abstract

Using a finite-dimensional Clifford algebra a new combinatorial product formula for the small quantum cohomology ring of the complex Grassmannian is presented. In particular, Gromov-Witten invariants can be expressed through certain elements in the Clifford algebra, this leads to a q-deformation of the Racah-Speiser algorithm allowing for their computation in terms of Kostka numbers. The second main result is a simple and explicit combinatorial formula for projecting product expansions in the quantum cohomology ring onto the sl(n) Verlinde algebra. This projection is non-trivial and amounts to an identity between numbers of rational curves intersecting Schubert varieties and dimensions of moduli spaces of generalised theta-functions.