Finite Temperature and Large Gauge Invariance

Gauge theories are beautiful theories which describe physical forces in a natural manner and because of their rich structure, the study of gauge theories at nite temperature [1] is quite interesting in itself. However, to avoid getting into technicalities, we will not discuss the intricacies of such theories either at zero temperature or at nite temperature. Rather, we would study a particular puzzle that arises when a fermion interacts with an external gauge eld in odd space-time dimensions. To motivate, let us note that gauge invariance is realized as an internal symmetry in quantum mechanical systems. Consequently, we do not expect a macroscopic external surrounding, such as a heat bath, to modify gauge invariance. This is more or less what is also found by explicit computations at nite temperature, namely, that gauge invariance and Ward identities continue to hold even at nite temperature [2]. This is certainly the case when one is talking about small gauge transformations for which the parameters of transformation vanish at in nity. However, there is a second class of gauge transformations, commonly known as large gauge transformations, where the parameters do not vanish at in nity. In odd space-time dimensions, invariance under large gauge transformations leads to some interesting features in physical theories. Let us note that, in odd space-time dimensions, one can, in addition to the usual Maxwell (Yang-Mils) term, have a topological term in the gauge Lagrangian known as the Chern-Simons term. For example, in 2+1 dimensions, a fermion interacting with a non-Abelian gauge eld can be described by a Lagrangian density of the form