Induced Connections in Field Theory: The Odd Dimensional Yang-Mills Case

TL;DR

This paper explores the topological properties of SU(N) Yang-Mills theories in odd dimensions with fermions, revealing a non-trivial U(1)-connection and its Chern-class, which is linked to the flavor number, through both Lagrangian and Hamiltonian analyses.

Contribution

It introduces the concept of induced U(1)-connections in odd-dimensional Yang-Mills theories and computes their Chern-classes in the infinite fermion mass limit, highlighting their topological significance.

Findings

The wave functional inherits a non-trivial U(1)-connection.

The Chern-class of this connection is half the flavor number.

Topological origin explained via Lagrangian and Hamiltonian frameworks.

Abstract

We consider SU(N) Yang-Mills theories in 2n+1 Euclidean dimensions coupled to an even flavour-number of Dirac fermions. After integrating out the fermions the wave functional for the effective Yang-Mills theory inherits a non-trivial U(1)-connection which is computed in the limit of infinite fermion mass (adiabatic connection). Its Chern-class turns out to be just half the flavour number. Its topological origin is explained in the Lagrangean and Hamiltonian picture.