Abstract
In this Letter, we take the first step towards construction of nontrivial Lagrangian theories of higher-spin gravity in a metriclike formulation in three dimensions. The crucial feature of a metriclike formulation is that it is known how to incorporate matter interactions into the description. We derive a complete classification of cubic interactions for arbitrary triples s_{1}, s_{2}, s_{3} of massless fields, which are the building blocks of any interacting theory with massless higher spins. We find that there is, at most, one vertex for any given triple of spins in 3D (with one exception, s_{1}=s_{2}=s_{3}=1, which allows for two vertices). Remarkably, there are no vertices for spin values that do not respect strict triangle inequalities and contain at least two spins greater than one. This translates into selection rules for three-point functions of higher-spin conserved currents in two dimensional conformal field theory. Furthermore, universal coupling to gravity for any spin is derived. Last, we argue that this classification persists in arbitrary Einstein backgrounds.
Highlights
In the quest for a simple model of quantum gravity, particular optimism is related to higher-spin (HS) gravity theories in three dimensions. This is related to the fact that these theories bypass all the no-go theorems that put stringent constraints on HS gravity theories in D ≥ 4
Do they allow for Lagrangian formulation [3], finite HS spectrum [4], Minkowski background [5], and color decoration [6], which are problematic for known theories in higher dimensions, but they allow for nontrivial solutions [7] and possess infinite-dimensional asymptotic symmetries [4,8] familiar from two dimensional conformal field theories (CFTs)
Despite all of these simplifying properties, one important but basic problem still remains to be understood in these theories. It is the compatibility of matter coupling with the local Lagrangian formulation
Summary
Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Am Mühlenberg 1, 14476 Golm, Germany (Received 9 January 2018; revised manuscript received 30 March 2018; published 29 May 2018). The question of the Lagrangian formulation of HS theories is a long standing puzzle, addressed, in particular, through attempts for perturbative constructions of the action, in the spirit of the so-called Fronsdal program [13], that resulted in full classification [14,15] of cubic interactions in dimensions greater than three (see, [16,17,18] for generating functions and discussion about possible relation to string theory) This classification is in one-to-one correspondence with that of conformal threepoint correlators of conserved currents in dimensions lower by one via the AdS=CFT dictionary [15,19].
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