Strong coupling anomalous dimensions of ${\cal N} = 4$ super Yang-Mills

We study the strong coupling behaviour of fixed length single trace operators in the scalar SU(2) sector of ${\cal N} = 4$ SYM. We assume the recently proposed connection with a twisted half-filled Hubbard model. By explicit direct diagonalization of operators with length $L=4, 6, 8$ we study the full {\em perturbative multiplet} of those lattice states which have a clear correspondence with gauge theory composite operators. For this multiplet, we follow the weak-strong coupling flow to free fermion states and identify in particular the precise asymptotic fermion configuration. Next, we analyze the Lieb-Wu equations of the twisted Hubbard model. For the antiferromagnetic state we derive its strong coupling expansion working at $L$ up to 32. We also study the lightest state in the perturbative multiplet. This state is non trivial since involves complex solutions of the Lieb-Wu equations. It is particularly interesting for $AdS_5\times S^5$ duality since it is dual to the folded string semiclassical solution in the thermodynamical limit. We are able to perform the full analysis and compute the next-to-next-to leading terms in the strong coupling expansion for the non trivial lengths L=12 and L=20. A general formula is proposed for the NLO expansion for any $L=4(2k+1)$, $k\in\mathbb{N}$.