# Polyakov Loop and Chemical Potential

I have read in a paper (http://arxiv.org/abs/1203.3556) that in a thermal field theory, the chemical potential is $\mu=T \ln P$ where $$T^{-1}=\int_{0}^{\beta} \sqrt{-\xi^2}dt,$$ $\xi$ is $\partial_t$, and $P$ is the Polyakov Loop: $$P=e^{\int_{0}^{\beta} A_a \xi^a dt}.$$ How chemical potential is related to the Polaykov Loop? I did not find anything related in the web that's why I asked.

• Comparing with thermodynamics, T must be the temperature and P must represent the pressure. Also, that "xi" over there should be a Killing vector associated with time translations, right? I don't understand the integral giving the temperature. What is beta? Is it the inverse temperature? If yes, then the integral may say something like T=T I guess which isn't very useful. I'm not even sure what you are integrating over there. Excuse my many questions, but I believe your question needs more clarification and explanation of the equations you wrote. – Panos C. May 2 '18 at 14:34
• @PanosC. Yes, that's right. The first equation is almost obvious but the second is my question. – mathvc_ May 2 '18 at 14:37

I think the second formula is also not very profound. This is just based on the fact that the chemical potential $$\Delta S = \int dt \, \mu Q$$ enters the action like the zeroth component of an (imaginary) $U(1)$ gauge field $A_\beta=(i\mu,\vec{0})$. Note that this is the Polyakov line for a $U(1)$ background field, not the Polyakov line of (for example) the dynamical gauge field in QCD.