Wilson lines, Grassmannians and gauge invariant off-shell amplitudes in N=4 SYM

In this paper we consider tree-level gauge invariant off - shell amplitudes (Wilson line form factors) in N = 4 SYM. For the off - shell amplitudes with one leg off - shell we present a conjecture for their Grassmannian integral representation in spinor helicity, twistor and momentum twistor parameterizations. The presented conjecture is successfully checked against BCFW results for MHVn, NMHV4 and NMHV5 off - shell amplitudes. We have also verified that our Grassmannian integral representation correctly reproduces soft (on-shell) limit for the off - shell gluon momentum. It is shown that the (deformed) off - shell amplitude expressions could be also obtained using quantum inverse scattering method for auxiliary gl (4 vertical bar 4) super spin chain.