  We study the Polyakov loop and the correlator of two Polyakov loops at finite
temperature in the weak-coupling regime. We calculate the Polyakov loop at
order g^4. The calculation of the correlator of two Polyakov loops is performed
at distances shorter than the inverse of the temperature and for electric
screening masses larger than the Coulomb potential. In this regime, it is
accurate up to order g^6. We also evaluate the Polyakov-loop correlator in an
effective field theory framework that takes advantage of the hierarchy of
energy scales in the problem and makes explicit the bound-state dynamics. In
the effective field theory framework, we show that the Polyakov-loop correlator
is at leading order in the multipole expansion the sum of a colour-singlet and
a colour-octet quark-antiquark correlator, which are gauge invariant, and
compute the corresponding colour-singlet and colour-octet free energies.
