Single Particle Operators and their Correlators in Free $\mathcal{N}=4$ SYM

TL;DR

This paper constructs explicit single-particle operators in free $ abla$4 super Yang-Mills theory, analyzing their properties, orthogonality, and correlators, revealing their interpolation between single-trace and giant graviton states at large N.

Contribution

It provides explicit formulas for single-particle operators, explores their orthogonality, and analyzes their role in correlators and the operator product expansion in free $ abla$4 SYM.

Findings

Single-particle operators are explicitly constructed.

Orthogonality leads to vanishing near-extremal correlators.

Operators interpolate between single-trace and giant graviton states.

Abstract

We consider a set of half-BPS operators in $\mathcal{N}=4$ super Yang-Mills theory which are appropriate for describing single-particle states of superstring theory on AdS${}_5 \times S^5$. These single-particle operators are defined to have vanishing two-point functions with all multi-trace operators and therefore correspond to admixtures of single- and multi-traces. We find explicit formulae for all single-particle operators and for their two-point function normalisation. We show that single-particle $U(N)$ operators belong to the $SU(N)$ subspace, thus for length greater than one they are simply the $SU(N)$ single-particle operators. Then, we point out that at large $N$, as the length of the operator increases, the single-particle operator naturally interpolates between the single-trace and the $S^3$ giant graviton. At finite $N$, the multi-particle basis, obtained by taking products of the single-particle operators, gives a new basis for all half-BPS states, and this new basis naturally cuts off when the length of any of the single-particle operators exceeds the number of colours. From the two-point function orthogonality we prove a multipoint orthogonality theorem which implies vanishing of all near-extremal correlators. We then compute all maximally and next-to-maximally extremal free correlators, and we discuss features of the correlators when the extremality is lowered. Finally, we describe a half-BPS projection of the operator product expansion on the multi-particle basis which provides an alternative construction of four- and higher-point functions in the free theory.