Abstract
We propose a non-abelian higher-spin theory in two dimensions for an infinite multiplet of massive scalar fields and infinitely many topological higher-spin gauge fields together with their dilaton-like partners. The spectrum includes local degrees of freedom although the field equations take the form of flatness and covariant constancy conditions because fields take values in a suitable extension of the infinite-dimensional higher-spin algebra \U0001d525\U0001d530[λ]. The corresponding action functional is of BF-type and generalizes the known topological higher-spin Jackiw-Teitelboim gravity.
Highlights
Let us stress that, in three dimensions, the inclusion of matter is known to reinstate the same level of intricacy as in four dimensions
The BF-type action associated to this extended HS algebra provides a natural extension of the HS Jackiw-Teitelboim gravity, the linearization of which reproduces the correct equations of motion for topological and local modes dictated by symmetries
The corresponding equations of motion describe an infinite tower of scalar fields with fine-tuned increasing masses (3.6), coupled to the topological gauge fields of HS Jackiw-Teitelboim gravity
Summary
The kinematics of HS gravity theories in two dimensions is entirely governed by the oneparameter family of Lie algebras hs[λ] and representations thereof. The Lie algebra hs[N ] contains an infinite-dimensional ideal JN to be factored out and the corresponding quotient is finite-dimensional, hs[N ]/JN ∼= sl(N, R) [31, 32]. The other algebras in the upper half of table 1 are useful auxiliary tools (e.g. the associative algebras) or illustrative toy models (e.g. the finite-dimensional algebras) but the kinematics of pure HS gravity theories in two dimensions is determined by the one-parameter family of Lie algebras hs[λ] and representations thereof. Specifying y ∈ hs[λ] defines the twisted-adjoint action of the higher-spin algebra hs[λ] on the linear space of gl[λ]. They can be decomposed into irreducible submodules of so(2, 1) which are finite-dimensional (“Killing”) modules for the adjoint action and infinite-dimensional (“Weyl”) modules for the twisted-adjoint action The latter modules are Verma modules of so(2, 1) with running weights expressed in terms of λ (see [33] for details)
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