The shape of non-graviton operators for $SU(2)$

TL;DR

This work identifies and explicitly constructs a non-graviton BPS cohomology in the SU(2) N=4 SYM theory by leveraging the $Q$-cohomology structure of the 1/16-BPS sector. It builds a simple seven-letter representative $O$ within a restricted subsector and proves it is $Q$-closed but not $Q$-exact, confirming a unique non-graviton cohomology at the threshold $n=24$ with charges $E=rac{19}{2}$, $R_1=R_2=R_3=rac{3}{2}$, $J_1=J_2=rac{5}{2}$. The authors count 17 independent $Q$-exact operators and 18 independent $Q$-closed cyclic-invariant operators, establishing the cohomology structure and its relation to the threshold; they also discuss the implications for finite-$N$ spectra, potential giant-graviton interpretations, and avenues for analytic progress at $N>2$. The results illuminate how non-graviton states emerge at finite $N$, contributing to the understanding of black-hole microstate counting in AdS/CFT and guiding future analytic approaches to higher-$N$ cohomologies.

Abstract

The BPS spectrum of AdS/CFT exhibits multi-gravitons at low energies, while having black hole states at higher energies. This can be studied concretely in AdS$_5$/CFT$_4$ in terms of classical cohomologies, even in the quantum regimes at finite $1/N$. Recently, Chang and Lin found a threshold for non-graviton states in the $SU(2)$ maximal super-Yang-Mills theory. We explicitly construct and present this threshold cohomology.