# Periodicity of gauge field in finite temperature field theory

I am unable to understand why the gauge field must be periodic along the compactified direction in finite temperature field theory. I get that whatever quantity that is measurable and gauge invariant along that direction must be periodic. But that means that the gauge field must be periodic only upto a gauge transformation and not strictly periodic. This is actually the condition used by 't Hooft when formulating gauge theory on $$T^4$$, where all the four directions are compactified. So, why is the condition different when one direction is compactified?

In terms of equations, I'm saying that the boundary condition must be $$A_{\mu}(\vec{x},\tau + \beta)=\omega A_{\mu}(\vec{x},\tau)\omega^{-1}+\omega\partial_{\mu}\omega^{-1}$$ instead of $$A_{\mu}(\vec{x},\tau +\beta)=A_{\mu}(\vec{x},\tau)$$ where $$\omega$$ belongs to the gauge group under consideration, say $$SU(2)$$. It would be great if someone could help me clear up this confusion.

For reference, I am studying from the paper by Harrington and Shepard called "Periodic Euclidean solution and finite temperature Yang-Mills gas". They also have another paper called "Euclidean solutions and finite temperature gauge theory" that talks about this aspect more.

't Hooft, in "A property of electric and magnetic flux in non-Abelian gauge theories'', explains how to compute the free energy of a configuration with $$(n,m)$$ units of electric/magnetic flux by considering twisted boundary conditions.
In thermal field theory, we wish to compute the partition function $$Z = \mathrm{Tr}[\exp(-\beta H)]$$ in the zero flux sector (in particular, with Gauss law imposed on the states in the trace). As explained, for example, in the classic paper by Gross, Pisarski, and Yaffe: "QCD and instantons at finite temperature", this corresponds to strictly periodic gauge fields.
• @ChiralAnomaly Sort of (that's waht I say if I teach a course), but it's a little more subtle. To compute Z we need to fix the gauge and construct H. Say we pick temporal gauge, so the "coordinates" are $\vec{A}$. Then indeed $\vec{A}$ should be periodic, but what about $A_0$? The answer is that $A_0$ appears as the lagrange multiplier for implementing Gauss law, and to see that it has to be periodic we really have to write this down carefully. Apr 10, 2020 at 17:21