Non-planar spin bits 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.

30 pages, no figures, PACS numbers: 11.15.Pg; 11.15.-q; 11.25.Hf; extensive revisions in Section 4, including complete parameter fixing; conclusions improved; moved subsection on spin-edge differences Ansaetze in appendix III; a few references added