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

TL;DR

This paper investigates the strong coupling behavior of certain operators in ${ m extbf{N}}=4$ super Yang-Mills theory using a connection with a twisted Hubbard model, providing explicit calculations and new formulas for various operator lengths.

Contribution

It introduces a detailed analysis of the strong coupling limit of operators in ${ m extbf{N}}=4$ SYM via the Hubbard model, including explicit diagonalizations and a new general formula for NLO expansions.

Findings

Explicit strong coupling expansions for lengths up to 32.

Identification of asymptotic fermion configurations.

Proposed general formula for NLO expansion for specific lengths.

Abstract

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 $L=4(2k+1)$, $k\in\mathbb{N}$.