Schubert Calculus via Fermionic Variables

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.