Non-planar spin bits beyond two loops

TL;DR

This paper investigates higher-loop non-planar spin bit models in $ ext{N}=4$ SYM, deriving identities and proposing conjectures for the full non-planar, higher-loop Hamiltonian of the $su(2)$ spin-chain.

Contribution

It introduces a new set of non-planar identities and formulates conjectures for the non-planar, higher-loop Hamiltonian in the $su(2)$ spin-chain model.

Findings

Derived a tower of non-planar identities involving permutation operators.

Formulated planarly consistent, testable conjectures for the full non-planar Hamiltonian.

Enhanced understanding of non-planar dynamics beyond two loops.

Abstract

We study higher-loop orders of spin bit models underlying the non-planar dynamics of $\mathcal{N}=4$ SYM gauge theory. In particular, we derive a ''tower'' of non-planar identities involving products of site permutation operators. Such identities are then applied in the formulation of planarly consistent, testable conjectures for the full non-planar, higher-loop Hamiltonian of the $su(2)$ spin-chain.