  We propose a new approach to the study of the inter-quark potential in
Lattice Gauge Theories. Instead of looking at the expectation value of Polyakov
loop correlators we study the modifications induced in the chromoelectric flux
by the presence of the Polyakov loops. In abelian LGTs, thanks to duality, this
study can be performed in a very efficient way, allowing to reach high
precision with a reasonable CPU cost. The major advantage of this numerical
strategy is that it allows to eliminate the dominant effective string
correction to the inter-quark potential (the Luscher term) thus giving an
unique opportunity to test higher order corrections. Performing a set of
simulations in the 3d gauge Ising model we were thus able to precisely identify
and measure both the quartic and the sextic effective string corrections to the
inter-quark potential. While the quartic term perfectly agrees with the
Nambu-Goto one the sextic term is definitely different. Our result seems to
disagree with the recent proof by Aharony and Karzbrun of the universality of
the sextic correction. We discuss a few possible explanations of this
disagreement. The numerical approach described above can also be applied to the
study of Wilson loops. In this case, the numerical results are precise enough
to test the two-loop prediction of the Nambu-Goto action. The two-loop NG
result computed time ago by by Dietz and Filk is incompatible with the data;
however, after correcting some mistakes in their expression, compatibility is
restored. The viability of a first-order, operatorial description of the Wilson
loop is also pointed out.
