Relation between large dimension operators and oscillator algebra of Young diagrams

TL;DR

This paper explores the connection between large dimension operators labeled by Young diagrams and oscillator algebra, showing how the algebra emerges from contractions of $u(p)$ and may indicate persistent integrability at higher loops.

Contribution

It demonstrates the oscillator algebra structure in large dimension operators and suggests its role in maintaining integrability beyond the planar limit.

Findings

Oscillator algebra arises from $u(2)$ contraction within $u(p)$.

Evidence suggests integrability persists at higher loops.

Oscillator algebra remains robust under loop corrections.

Abstract

The operators with large scaling dimensions can be labelled by Young diagrams. Among other bases, the operators using restricted Schur polynomials have been known to have a large $N$ but nonplanar limit under which they map to states of a system of harmonic oscillators. We analyze the oscillator algebra acting on pairs of long rows or long columns in the Young diagrams of the operators. The oscillator algebra can be reached by a Inonu-Wigner contraction of the $u(2)$ algebra inside of the $u(p)$ algebra of $p$ giant gravitons. We present evidences that integrability in this case can persist at higher loops due to the presence of the oscillator algebra which is expected to be robust under loop corrections in the nonplanar large $N$ limit.