Gauge invariant effective action for the Polyakov line in the SU(N)   Yang--Mills theory at high temperatures
  We integrate out fast varying quantum fluctuations around static A_4 and A_i fields for the SU(N) gauge group. By assuming that the gluon fields are slowly varying but allowing for an arbitrary amplitude of A_4 we obtain two variants of the effective high-temperature theory for the Polyakov line. One is the effective action for the gauge-invariant eigenvalues of the Polyakov line, and it is explicitly Z(N) symmetric. The other is the effective action for the Polyakov line itself as an element of the SU(N). In this case the theory necessarily includes the spatial components A_i to ensure its gauge invariance under spatial gauge transformations. We derive the 1-loop effective action in the `electric' and `magnetic' sectors, summing up all powers of A_4. 
Dmitri Diakonov and Michaela Oswald