Single particle operators and their correlators in free $$ \mathcal{N} $$ = 4 SYM
F. Aprile, J. M. Drummond, P. Heslop, H. Paul, F. Sanfilippo, M. Santagata, A. Stewart
Abstract
A bstract We consider a set of half-BPS operators in $$ \mathcal{N} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>N</mml:mi> </mml:math> = 4 super Yang-Mills theory which are appropriate for describing single-particle states of superstring theory on AdS 5 × 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.