Fermions on Non-Trivial Topologies

TL;DR

This paper derives an exact Green function for fermions on a manifold with non-trivial topology, revealing connections between topology, particle statistics, and potential mass generation in higher dimensions.

Contribution

It provides a novel exact expression for the fermionic Green function on manifolds with non-trivial topology, linking topology to particle statistics and mass generation mechanisms.

Findings

Exact Green function expression for fermions on $ e imes orus^{D-1}$

Topological measures related to particle statistics in 2D and 3D

Potential implications for mass generation in higher dimensions

Abstract

An exact expression for the Green function of a purely fermionic system moving on the manifold $\Re \times \Sigma^{D-1}$, where $\Sigma^{D-1}$ is a $(D-1)$-torus, is found. This expression involves the bosonic analog of $\chi_n = e^{in\theta}$ corresponding to the irreducible representation for the n-th class of homotopy and in the fermionic case for D=2 and 3, $\chi_n$ is a measure of the statistics of the particles. For higher dimensions ($D \geq 4$), there is no analogue interpretation however this could, presumably, indicate a generation of mass as in quantum field theories at finite temperature.