Higher Spin Symmetry and N=4 SYM

TL;DR

The paper reorganizes the free $\mathcal{N}=4$ SYM spectrum into irreps of the higher-spin algebra $\mathfrak{hs}(2,2|4)$, showing that HS multiplets (YT-pletons) are classified by Young tableaux built from singleton letters and constrained by trace cyclicity. It computes the decomposition of tripletons into ${\mathfrak{psu}}(2,2|4)$ multiplets, and derives their one-loop anomalous dimensions at large $N$ via planar spin chains in relevant sectors; it also derives a compact partition function for BPS and semishort primaries by applying a semishort sieve. Upon turning on interactions, HS symmetry breaks to the superconformal group, causing HS multiplets to decompose into infinite towers of ${\mathfrak{psu}}(2,2|4)$ multiplets and massless HS fields to acquire masses consistent with the Grande Bouffe mechanism. The twist-3 anomalous dimensions obey an integrable spin-chain pattern with $\mathfrak{sl}(2)$ symmetry, suggesting an underlying integrable structure linking HS symmetry, operator mixing, and perturbative dynamics in $\mathcal{N}=4$ SYM.

Abstract

We assemble the spectrum of single-trace operators in free N=4 SU(N) SYM theory into irreducible representations of the Higher Spin symmetry algebra hs(2,2|4). Higher Spin representations or YT-pletons are associated to Young tableaux (YT) corresponding to representations of the symmetric group compatible with the cyclicity of color traces. After turning on interactions, YT-pletons decompose into infinite towers of representations of the superconformal algebra PSU(2,2|4) and anomalous dimensions are generated. We work out the decompositions of tripletons with respect to the N=4 superconformal algebra PSU(2,2|4) and compute their one anomalous dimensions at large N. We then focus on operators/states sitting in semishort superconformal multiplets. By passing them through a semishort-sieve that removes superdescendants, we derive compact expressions for the partition function of semishort primaries.