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Metric-Like Formalism for Matter Fields Coupled to 3D Higher Spin Gravity
High Energy Physics - Theory · Submitted 30 Apr 2013
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The spin connection is eliminated by solving the eq of motion for the total action, and in the resulting metric-like formalism, $(BC)^2$ interaction terms are induced because of the torsion.
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The abstract, sentence by sentence
Action integral for a matter system composed of 0- and 2-forms, $C$ and $B_{\mu\nu}$, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. The field $C$ satisfies an eq of motion, $\partial_{\mu} \, C+A_{\mu} \, C-C \, \bar{A}_{\mu}=0$. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of $B_{\mu\nu}$, say $B_{\phi r}$, remains non-vanishing and satisfies an equation similar to that for $C$ with $A_{\mu}$ and $\bar{A}_{\mu}$ interchanged. The spin connection is eliminated by solving the eq of motion for the total action, and in the resulting metric-like formalism, $(BC)^2$ interaction terms are induced because of the torsion. The world-volume components of the matter field, $C^0$, $C^{\mu}$ and $C^{(\mu\nu)}$, are introduced by contracting the local-frame index of $C$ with those of the inverse vielbeins, $E_a^{\mu}$ and $E_a^{(\mu\nu)}$, which were defined by the present authors in ArXiv:1209.0894 [hep-th]. The metric-like fields, as well as the new connections and the generalized curvature tensors, introduced in the above mentioned paper, are explicitly expressed in terms of the metric $g_{\mu\nu}$ and the spin-3 field $\phi_{\mu\nu\lambda}$ by means of the $\phi$-expansion. The action integral for the pure spin-3 gravity in the metric-like formalism up to ${\cal O}(\phi^2)$, obtained before in the literature, is re-derived. Then the matter action is re-expressed in terms of $g_{\mu\nu}$, $\phi_{\mu\nu\rho}$ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Action integral for a matter system composed of 0- and 2-forms, $C$ and $B_{\mu\nu}$, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. | contribution 0.77 | 0.61 | 0.05 | asserted 0.97 |
| s1 | The field $C$ satisfies an eq of motion, $\partial_{\mu} \, C+A_{\mu} \, C-C \, \bar{A}_{\mu}=0$. | result 0.68 | 0.59 | 0.04 | asserted 1.00 |
| s2 | With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of $B_{\mu\nu}$, say $B_{\phi r}$, remains non-vanishing and satisfies an equation similar to that for $C$ with $A_{\mu}$ and $\bar{A}_{\mu}$ interchanged. | method 0.57 | 0.83 | 0.05 | asserted 0.99 |
| s3 | The spin connection is eliminated by solving the eq of motion for the total action, and in the resulting metric-like formalism, $(BC)^2$ interaction terms are induced because of the torsion. | method 0.51 | 0.85 | 0.20 | asserted 1.00 |
| s4 | The world-volume components of the matter field, $C^0$, $C^{\mu}$ and $C^{(\mu\nu)}$, are introduced by contracting the local-frame index of $C$ with those of the inverse vielbeins, $E_a^{\mu}$ and $E_a^{(\mu\nu)}$, which were defined by the present authors in ArXiv:1209.0894 [hep-th]. | method 0.64 | 0.77 | 0.05 | asserted 1.00 |
| s5 | The metric-like fields, as well as the new connections and the generalized curvature tensors, introduced in the above mentioned paper, are explicitly expressed in terms of the metric $g_{\mu\nu}$ and the spin-3 field $\phi_{\mu\nu\lambda}$ by means of the $\phi$-expansion. | method 0.48 | 0.81 | 0.07 | asserted 1.00 |
| s6 | The action integral for the pure spin-3 gravity in the metric-like formalism up to ${\cal O}(\phi^2)$, obtained before in the literature, is re-derived. | result 0.42 | 0.72 | 0.09 | asserted 1.00 |
| s7 | Then the matter action is re-expressed in terms of $g_{\mu\nu}$, $\phi_{\mu\nu\rho}$ and the covariant derivatives for spin-3 geometry. | contribution 0.68 | 0.89 | 0.12 | asserted 1.00 |
| s8 | Spin-3 gauge transformation is extended to the matter fields. | contribution 0.93 | 0.85 | 0.13 | asserted 0.99 |
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The paper
Original abstract
Action integral for a matter system composed of 0- and 2-forms, $C$ and $B_{\mu\nu}$, topologically coupled to 3D spin-3 gravity is considered first in the frame-like formalism. The field $C$ satisfies an eq of motion, $\partial_{\mu} \, C+A_{\mu} \, C-C \, \bar{A}_{\mu}=0$. With a suitable gauge fixing of a new local symmetry and diffeomorphism, only one component of $B_{\mu\nu}$, say $B_{\phi r}$, remains non-vanishing and satisfies an equation similar to that for $C$ with $A_{\mu}$ and $\bar{A}_{\mu}$ interchanged. The spin connection is eliminated by solving the eq of motion for the total action, and in the resulting metric-like formalism, $(BC)^2$ interaction terms are induced because of the torsion. The world-volume components of the matter field, $C^0$, $C^{\mu}$ and $C^{(\mu\nu)}$, are introduced by contracting the local-frame index of $C$ with those of the inverse vielbeins, $E_a^{\mu}$ and $E_a^{(\mu\nu)}$, which were defined by the present authors in ArXiv:1209.0894 [hep-th]. The metric-like fields, as well as the new connections and the generalized curvature tensors, introduced in the above mentioned paper, are explicitly expressed in terms of the metric $g_{\mu\nu}$ and the spin-3 field $\phi_{\mu\nu\lambda}$ by means of the $\phi$-expansion. The action integral for the pure spin-3 gravity in the metric-like formalism up to ${\cal O}(\phi^2)$, obtained before in the literature, is re-derived. Then the matter action is re-expressed in terms of $g_{\mu\nu}$, $\phi_{\mu\nu\rho}$ and the covariant derivatives for spin-3 geometry. Spin-3 gauge transformation is extended to the matter fields.
Human review
References
How the literature uses it
7 arXiv papers cite this in 16 sentences
Papers citing it · 9 sentences without a confident label
Uses its method or tool 1 paper
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We will follow here a different approach which was started also in [ref] .[For an alternative ansatz for a metric-like description see [ref] .]
Metric- and frame-like higher-spin gauge theories in three dimensions · Introduction · 0.86
Compares with it 2 papers
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The field equations of the HS field resemble the form of the matter field equations introduced in [ref] .
Generalized Chern-Simons higher-spin gravity theories in three dimensions · Conclusions · 0.97
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The model of [ref] is similar to the proposal above, but lacks any higher vertices in the equations of motion; its main part is $∫B∧DC$, where $B$ is a two-form. ]
Matter-coupled higher spin gravities in 3d: no- and yes-go results · Perturbation theory · 0.99
Cites it as background 2 papers
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In this section we will review the model of 3d higher spin gravity coupled to matter fields, which was introduced in [ref] .
AdS/CFT for 3D Higher-Spin Gravity Coupled to Matter Fields · Model · 0.89
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The relation between the Chern-Simons and metric formulations for the $SL(3,ℝ) ×SL(3,ℝ)$ Chern-Simons theory was studied in [ref] .
Connection vs metric description for non-AdS solutions in higher spin theories · Introduction · 0.89
What it cites · 23 sentences without a confident label
Contests 1 paper
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As was shown in [ref] , the second-order action for spin-3 gravity obtained by substituting ([ref]) into the Chern-Simons action is given by [equation] Here $k=ℓ/4G$ and $G$ is the 3D gravitational constant, and $ℓ$ is the cosmological length related to the cosmological constant by $Λ=-2/ℓ^2$.[The normalization of the action (4.30) in [ref] is not appropriate.
Second-Order Formalism for 3D Spin-3 Gravity · Action Integral for pure spin-3 Gravity · contest 0.86
Uses the method or tool of 2 papers
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The equation [equation] with $(Γ̃^M_μν-Γ̃^M_νμ) e_M=-T_μν$ can be solved for $Γ̃^M_μν$, by the same procedure as in sec.3 of [ref] and then $Γ̃^M_μ(νλ)$ is determined in terms of them.
Second-Order Formalism for 3D Spin-3 Gravity · Spin-3 Gravity with Torsion · 0.95
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[equation] We use the same three-dimensional representation as in [ref] with the parameter $σ=-1$.
Asymptotic symmetries of three-dimensional gravity coupled to higher-spin fields · Notations for sl(3,R) algebra · 0.81
Compares with 1 paper
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The action for the spin-3 field turned out to agree with the result of [ref] .
Towards metric-like higher-spin gauge theories in three dimensions · Introduction · 1.00
Cites as background 19 papers
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In [ref] we defined a subsidiary field $e^a_(μν)$ in terms of the vielbein field $e^a_μ$.
Second-Order Formalism for 3D Spin-3 Gravity · Brief Review of the Formalism · 0.81
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[ref] [ref] [ref] [ref] [ref] [ref] [ref] [ref] [ref] In this frame-like approach the gravity and the higher-spin fields are described in terms of the vielbein $e^a_μ (a=1,..,N^2-1)$ and the spin connection $ω^a_μ$ and the action integral is first order in the derivatives of these fields.
Black holes in three dimensional higher spin gravity: A review · Introduction · 0.90
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[ref] [ref] [ref] [ref] [ref] [ref] [ref] [ref] [ref] In this frame-like approach the gravity and the higher-spin fields are described in terms of the vielbein $e^a_μ (a=1,..,N^2-1)$ and the spin connection $ω^a_μ$ and the action integral is first order in the derivatives of these fields.
Black holes in Truncated Higher Spin AdS$_3$ Gravity · Introduction · 0.90
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