Schubert calculus via fermionic variables. (English) Zbl 07854600
Summary: 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.
MSC:
| 81Q60 | Supersymmetry and quantum mechanics |
| 53C07 | Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
| 81V74 | Fermionic systems in quantum theory |
| 14F42 | Motivic cohomology; motivic homotopy theory |
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