On matter coupled to the higher spin square

TL;DR

This paper formulates linearized equations for matter fields coupled to the higher spin square in string theory, confirming the duality with the symmetric orbifold CFT and analyzing the spectrum through group theory.

Contribution

It introduces the linearized Vasiliev-type equations for matter coupled to the higher spin square and verifies the spectrum matches the untwisted sector of the dual CFT.

Findings

Spectrum of matter fields matches the untwisted sector of the dual CFT.

Pinpoints group-theoretic data determining matter spectra.

Revisits the $hs[1/2]$ algebra case for illustration.

Abstract

Gaberdiel and Gopakumar recently proposed that the tensionless limit of string theory on $AdS_3 \times S^3 \times T^4$ takes the form of a higher spin theory with a gauge algebra that is referred to as the higher spin square. In this note, we formulate the linearized Vasiliev-type equations which describe a matter field coupled to the higher spin square. We study the particle spectrum of this field and show that it accounts for the entire untwisted sector of the dual symmetric orbifold CFT, thereby confirming a conjecture by Gaberdiel and Gopakumar. In doing so, we pinpoint the group-theoretic data which determine the spectrum of a matter field coupled to a general higher spin algebra, which we illustrate by revisiting the theory based on the $hs[1/2]$ algebra.