  We consider possible discretizations for a gauge-fixed Green-Schwarz action
of Type IIB superstring. We use them for measuring the action, from which we
extract the cusp anomalous dimension of planar $\mathcal{N}=4$ SYM as derived
from AdS/CFT, as well as the mass of the two $AdS$ excitations transverse to
the relevant null cusp classical string solution. We perform lattice
simulations employing a Rational Hybrid Monte Carlo (RHMC) algorithm and two
Wilson-like fermion discretizations, one of which preserves the global $SO(6)$
symmetry of the model. We compare our results with the expected behavior at
various values of $g=\frac{\sqrt{\lambda}}{4\pi}$. For both the observables, we
find a good agreement for large $g$, which is the perturbative regime of the
sigma-model. For smaller values of $g$, the expectation value of the action
exhibits a deviation compatible with the presence of quadratic divergences.
After their non-perturbative subtraction the continuum limit can be taken, and
suggests a qualitative agreement with the non-perturbative expectation from
AdS/CFT. Furthermore, we detect a phase in the fermion determinant, whose
origin we explain, that for very small $g$ leads to a sign problem not
treatable via standard reweigthing. The continuum extrapolations of the
observables in the two different discretizations agree within errors, which is
strongly suggesting that they lead to the same continuum limit. Part of the
results discussed here were presented earlier in arXiv:1601.04670.
