About the gauge formalism in statistical quantum field theory

I would like to understand a bit more the aspects of the gauge theory in statistical field theory. In particular, I would like to understand how the replacement $\tau \rightarrow it/\hbar$ is performed in a mathematically proper way, when $\tau$ is an inverse temperature. This replacement comes from the resemblance between the evolution operator $e^{-iHt/\hbar}$ in quantum field theory, and the statistical weight $e^{-H\tau}$ in statistical physics (one sees then that $\tau=\left(k_{B}T\right)^{-1}$ in case you wonder :-).

In principle it leads to a compact momentum space, when the frequencies become discrete, and called Matsubara frequencies $\omega_{n}=2\pi k_B T \left(n+1/2\right)$ and $\omega_{n}=2\pi n k_{B}T$ with $n$ an integer for fermions and bosons. I'm perfectly aware of the classic book

Methods of quantum field theory in statistical physics by Abrikosov, Gor'kov and Dzyalochinski - Dover Books on Physics

but I'm stuck on the gauge formalism. Can we do a gauge transformation in imaginary time-$\tau$ as we do with real time-$t$ ? Does the imaginary-time-covariant derivative $\partial_{\tau}+A_{\tau}$ make any sense? Are there some precautions to take?

Any comment, answer, indications or even good reference (or even just keywords) about this topic is warm welcome. I precise I'm a condensed matter physicist, so if you could adapt your parlance to me (for instance, please talk slowly and loudly), I would greatly appreciate :-)

EDIT: Clearly the keyword is thermal quantum field theory and there is an associated Wikipedia page with plenty good references. Anyway, any comment are still welcome, since I progress really slowly understanding this, especially what a gauge choice mean? Thanks in advance.