Schubert Calculus via Fermionic Variables

TL;DR

This paper develops a fermionic variable approach to compute intersection numbers of Schubert cycles on Grassmannians, building on previous physical models and cohomology representations.

Contribution

It introduces a method to compute intersection numbers using only fermionic variables, simplifying previous approaches that involved path integrals.

Findings

Successfully computed intersection numbers of Schubert cycles.

Represented cohomology ring using fermionic variables.

Connected physical models with algebraic geometry computations.

Abstract

Imanishi, Jinzenji and Kuwata provided a recipe for computing Euler number of Grassmann manifold $G(k,N)$ using physical model and its path-integral [S.Imanishi, M.Jinzenji and K.Kuwata, Journal of Geometry and Physics, Volume 180, October 2022, 104623]. They demonstrated that the cohomology ring of $G(k,N)$ is represented by fermionic variables. In this study, using only fermionic variables, we computed an integral of the Chern classes of the dual bundle of the tautological bundle on $G(k,N)$. In other words, the intersection number of the Schubert cycles is obtained using the fermion integral.