# Deconfinement at high T $\leftrightarrow$ spontaneous breaking of the center of the gauge group

I am reading Witten's "Anti-de Sitter Space, Thermal Phase Transitions, And Confinement In Gauge Theories" (see here), in which he connects the confinement-deconfinement transition in $\mathcal{N}=4$ Super Yang-Mills theory with gauge group $SU(N)$ with the Hawking-Page transition via AdS/CFT.

My question is a purely field theoretical one, it is how I should understand the following statement:
"Deconfinement at high temperatures can be usefully described, in a certain sense, in terms of spontaneous breaking of the center of the gauge group" (p.5 of the above mentioned paper).

The center of $SU(N)$ is $\mathbb{Z}_N$ and is composed by the matrices $e^{2\pi i/n}\,\mathbb{I}$, where $\mathbb{I}$ is the unit matrix. As I understand, at high temperatures the vacuum state does not preserve this $\mathbb{Z}_N$-symmetry. How can one see this? What is the significance of the center of the gauge group in general? Why is it important that it is broken?

In the deconfined phase, single test quarks can exist, while in the confined phase, there are only colour singlets, i.e. there are no free test quarks. This statement can be reformulated using free energy: in the deconfined phase, the free energy of a single static test quark is finite, while in the confined phase, it is infinite. The free energy $F$ of such a test quark can be described in terms of a Polyakov loop $\Phi$ (a Wilson loop around the Euclidean time direction). Its thermal expectation value is given by $\langle\Phi\rangle=\exp(-F/T),$ where $T$ corresponds to temperature. Finite free energy implies $\langle\Phi\rangle\neq0$, while infinite free energy corresponds to $\langle\Phi\rangle=0$. The action of the theory is center-symmetric at all temperatures, while the expectation value of the Polyakov loop is only invariant under center symmetry transformations if it is zero. Hence, the center symmetry is spontaneously broken in the deconfined phase, where $\langle\Phi\rangle$ acts as an order parameter for the phase transition.