Evaluation of Euler Number of Complex Grassmann Manifold G(k, N) via Mathai-Quillen Formalism

TL;DR

This paper develops a method to compute the Euler number of complex Grassmann manifolds using Mathai-Quillen formalism, linking topological quantum field theory techniques with geometric invariants.

Contribution

It introduces a path-integral representation for the Euler number of G(k,N) and constructs a free fermion realization of its cohomology ring, advancing topological and geometric analysis.

Findings

Path-integral representation of Euler number for G(k,N)

Connection between topological Yang-Mills theory and Grassmann manifolds

Explicit free fermion model for cohomology ring

Abstract

In this paper, we provide a recipe for computing Euler number of Grassmann manifold G(k,N) by using Mathai-Quillen formalism (MQ formalism) and Atiyah-Jeffrey construction. Especially, we construct path-integral representation of Euler number of G(k,N). Our model corresponds to a finite dimensional toy-model of topological Yang-Mills theory which motivated Atiyah-Jeffrey construction. As a by-product, we construct free fermion realization of cohomology ring of G(k,N).