$$\mathcal {N} = 4$$ Polygonal Wilson Loops: Fermions

The contributions of scalars and fermions to the null polygonal bosonic Wilson loops/gluon MHV scattering amplitudes in \(\mathcal {N} = 4\) SYM are considered. We first examine the re-summation of scalars at strong coupling. Then, we disentangle the form of the fermion contribution and show its strong coupling expansion. In particular, we derive the leading order with the appearance of a fermion-anti-fermion bound state first and then effective multiple bound states thereof. This reproduces the string minimal area result and also applies to the Nekrasov instanton partition function \(\mathcal {Z}\) of the \(\mathcal {N}=2\) theories. Especially, in the latter case the method appears to be suitable for a systematic expansion.