Higher Spin Interactions in Four Dimensions: Vasiliev vs. Fronsdal

TL;DR

This paper explicitly derives cubic interaction vertices in four-dimensional Higher-Spin Theory from Vasiliev's equations, revealing new vertices and corrections to Fronsdal equations, and discusses challenges in correlator computations within HS AdS/CFT.

Contribution

It provides the first explicit derivation of all cubic interaction vertices from Vasiliev's equations in 4D Higher-Spin Theory, including those not fixed by algebra structure constants.

Findings

Derived all cubic vertices from Vasiliev's equations.

Identified corrections to Fronsdal equations involving arbitrary derivatives.

Discovered infinities in certain vertices affecting correlation function calculations.

Abstract

We consider four-dimensional Higher-Spin Theory at the first nontrivial order corresponding to the cubic action. All Higher-Spin interaction vertices are explicitly obtained from Vasiliev's equations. In particular, we obtain the vertices that are not determined solely by the Higher-Spin algebra structure constants. The dictionary between the Fronsdal fields and Higher-Spin connections is found and the corrections to the Fronsdal equations are derived. These corrections turn out to involve derivatives of arbitrary order. We observe that the vertices not determined by the Higher-Spin algebra produce naked infinities, when decomposed into the minimal derivative vertices and improvements. Therefore, standard methods can only be used to check a rather limited number of correlation functions within the HS AdS/CFT duality. A possible resolution of the puzzle is discussed.