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

TL;DR

This paper studies odd-dimensional Yang-Mills theories coupled with fermions, revealing a topological $U(1)$-connection in the wave functional that arises from integrating out fermions, with implications for the theory's bundle structure.

Contribution

It introduces the concept of induced $U(1)$-connections in the wave functional of odd-dimensional Yang-Mills theories with fermions, highlighting their topological origin and computing their Chern-class.

Findings

Wave functional inherits a non-trivial $U(1)$-connection.

Chern-class is proportional to half the flavor number.

Wave functional becomes a section of a non-trivial line bundle.

Abstract

We consider $SU(N)$ Yang-Mills theories in $(2n+1)$-dimensional Euclidean spacetime, where $N\geq n+1$, coupled to an even flavour number of Dirac fermions. After integration over the fermionic degrees of freedom the wave functional for the gauge field inherits a non-trivial $U(1)$-connection which we compute in the limit of infinite fermion mass. Its Chern-class turns out to be just half the flavour number so that the wave functional now becomes a section in a non-trivial complex line bundle. The topological origin of this phenomenon is explained in both the Lagrangean and the Hamiltonian picture.