Schubert calculus via fermionic variables

Abstract

Imanishi, Jinzenji and Kuwata provided a recipe for computing Euler number of Grassmann manifold $G\left(k,N\right)$ 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 $G\left(k,N\right)$ 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 $G\left(k,N\right)$. In other words, the intersection number of the Schubert cycles is obtained using the fermion integral.