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

A bstract 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 MHV n , NMHV 4 and NMHV 5 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 g l 4 | 4 super spin chain.