The non-linear coupled spin 2 - spin 3 Cotton equation in three dimensions

Abstract

In the context of three-dimensional conformal higher spin theory we derive, in the frame field formulation, the full non-linear spin 3 Cotton equation coupled to spin 2. This is done by solving the corresponding Chern-Simons gauge theory system of equations, that is, using $F=0$ to eliminate all auxiliary fields and thus expressing the Cotton equation in terms of just the spin 3 frame field and spin 2 covariant derivatives and tensors (Schouten). In this derivation we neglect the spin 4 and higher spin sectors and approximate the star product commutator by a Poisson bracket. The resulting spin 3 Cotton equation is complicated but can be related to linearized versions in the metric formulation obtained previously by other authors. The expected symmetry (spin 3 "translation", "Lorentz" and "dilatation") properties are verified for Cotton and other relevant tensors but some perhaps unexpected features emerge in the process, in particular in relation to the non-linear equations. We discuss the structure of this non-linear spin 3 Cotton equation but its explicit form is only presented here, in an exact but not completely refined version, in appended files obtained by computer algebra methods. Both the frame field and metric formulations are provided.

45 pages, 2 ancillary files, v.2 references and comments on higher spin invariant metric added

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• Off-shell massive N=1 supermultiplets in three dimensions
• New Conformal Higher Spin Gravities in $3d$
• Conformal geometry and (super)conformal higher-spin gauge theories
• Chiral Tensors of Mixed Young Symmetry
• Topologically massive higher spin gauge theories
• Twisted self-duality for higher spin gauge fields and prepotentials
• Cubic interactions of arbitrary spin fields in 3d flat space
• The Schouten tensor as a connection in the unfolding of 3D conformal higher-spin fields
• Spin projection operators and higher-spin Cotton tensors in three dimensions
• One-loop effective actions and higher spins. II
• The Action of the (free) $(4,0)$-theory
• Three-dimensional conformal geometry and prepotentials for four-dimensional fermionic higher-spin fields
• Higher-spin Cotton tensors and massive gauge-invariant actions in AdS${}_3$
• Aspects of electric-magnetic dualities in maximal supergravity