Scattering amplitudes at strong coupling for 4 K gluons

In this paper we study the scattering amplitudes in mathcal{N} = 4 SYM at strong coupling for the case where the number of gluons is a multiple of four. This is an important missing piece in [41]. The tricky point for n = 4 K is that there is some accidental degeneracy in such case. We explain this point in detail and show that a non-trivial monodromy around infinity was developed by the world-sheet coordinate transformation appearing in the computation. It turns out that besides solving the Y system, we also need to calculate T functions to compute the full amplitudes. We show that the T functions can be derived by taking a limit of Y functions of a higher-point case. As a check, we obtain the known result of eight-point in AdS ₃ in [39].