On the conformal higher spin unfolded equation for a three-dimensional self-interacting scalar field

Abstract

Abstract:We propose field equations for the conformal higher spin system in three dimensions coupled to a conformal scalar field with a sixth order potential. Both the higher spin equation and the unfolded equation for the scalar field have source terms and are based on a conformal higher spin algebra which we treat as an expansion in multi-commutators. Explicit expressions for the source terms are suggested and subjected to some simple tests. We also discuss a cascading relation between the Chern-Simons action for the higher spin gauge theory and an action containing a term for each spin that generalizes the spin 2 Chern-Simons action in terms of the spin connection expressed in terms of the frame field. This cascading property is demonstrated in the free theory for spin 3 but should work also in the complete higher spin theory.

Topologically massive higher spin gauge theories

“…An intriguing question is: Do nonlinear higher-spin extensions exist? Within the approach initiated in [70,71], Linander and Nilsson [72] constructed the full nonlinear spin-3 Cotton equation coupled to spin-2. They made use of the frame field description and the Chern-Simons formulation for 3D (super)conformal field theory due to Fradkin and Linetsky [33].…”

Conformal higher-spin gravity: linearized spectrum = symmetry algebra

“…The unfolded formulation on which Vasiliev's equations are based can be also applied to free CHS gauge fields (see e.g. [20][21][22][23][24][25][26]).…”

“…In the formal approach, the gauge symmetries (D. for unitary operatorsÔ = e iX . 26 On the one hand, it is not clear whether such invertible operators (D.4) form a group because the products of such operators may not be the 24 In mathematical terms, the vector space C ∞ is the direct limit of the N-filtration of vector spaces:…”

The Schouten tensor as a connection in the unfolding of 3D conformal higher-spin fields

“…Conformal higher-spin theory is a fascinating field of investigation in its own and in relation with studies of string theory in an unbroken, tensionless phase. As a result, it has received a lot of attention, as for example [1][2][3][4][5][6][7][8][9][10][11][12][13][14], and more recently [15][16][17][18][19][20][21][22][23][24] to cite but a few works on the subject. A reason for the importance it possesses is due to the fact that it combines two unifying symmetries encountered in field theory, namely higher-spin gauge symmetry and conformal symmetry, the latter being promoted to Weyl invariance when gravity is considered.…”

“…It is particularly convenient to study (super)conformal higher-spin theory in 2+1 dimensions [3,4] since the gauge sector can be described by a Chern-Simons action; see e.g. [18,[25][26][27][28] for recent related works. Particularly powerful indeed is the Cartan formulation of gauge systems, where gauge fields are incorporated into Lie algebra valued local p-forms on a base manifold and field equations are given by integrable constraints on generalised curvatures.…”