[Paper Review] On the local symmetries of gravity and supergravity models

Igor Bandos, J. A. de Azcárraga+1 more|ArXiv.org|Jan 10, 2002

Black Holes and Theoretical Physics4 references3 citations

TL;DR

This paper provides a comprehensive analysis of local symmetries in D-dimensional supergravity using both component and superfield formulations, emphasizing a field-space democracy perspective. It establishes that $κ$-symmetry of the supermembrane in 11D supergravity arises as the preserved part of full local supersymmetry when coupled to the superbrane, with the system retaining only half of the original local supersymmetries due to partial breaking by the brane's presence.

ABSTRACT

We present here a detailed analysis of the local symmetries of supergravity in an arbitrary dimension D, both in the component and superfield approaches, using a field-space democracy point of view. As an application, we discuss briefly how a complete description of the local gauge symmetries clarifies the properties of the supergravity-superbrane coupled system in the standard background superfield approximation for supergravity.

Motivation & Objective

• To systematically analyze the local gauge symmetries of supergravity in arbitrary dimensions using both component and superfield formalisms.
• To clarify the role of diffeomorphism invariance and its variational formulation in the context of supergravity–superbrane coupling.
• To establish the precise relationship between local supersymmetry and $κ$-symmetry in the supermembrane action within a 11D superfield supergravity background.
• To demonstrate that the superbrane action breaks half of the local supersymmetries of free supergravity, preserving only the $κ$-symmetry.

Proposed method

• Formulates D-dimensional gravity using differential forms, with vielbeins $e^a$ and spin connections $w^{ab}$, and derives the Einstein–Hilbert action in terms of curvature $R^{ab}$ and vielbein forms.
• Applies the second Noether theorem to identify Noether identities associated with local symmetries, including diffeomorphisms and local Lorentz transformations.
• Analyzes the component formulation of supergravity, identifying its local symmetries and associated Noether identities through the action principle.
• Uses the on-shell superfield formulation of supergravity, where constraints on superspace torsion encode the dynamics, and identifies the full set of gauge symmetries, including superspace general coordinate invariance.
• Applies the variational version of local symmetries to show that fermionic variations of the supermembrane action correspond to pullbacks of superspace general coordinate transformations.
• Demonstrates that the supermembrane action is invariant under $κ$-symmetry by showing that the fermionic variation vanishes when projected via $(1 + \bar{\gamma})/2$, confirming the Noether identity for $κ$-symmetry.

Experimental results

Research questions

• RQ1 How do the local symmetries of D-dimensional supergravity, particularly local supersymmetry, relate to the $κ$-symmetry of the supermembrane in 11D supergravity?
• RQ2 What is the precise mechanism by which the supermembrane action breaks half of the local supersymmetries of free supergravity?
• RQ3 How does the variational formulation of diffeomorphisms and local Lorentz symmetry relate to the dynamics of the supergravity–superbrane coupled system?
• RQ4 In what way does the superfield formulation of supergravity encode the full set of gauge symmetries, including those related to $κ$-symmetry?
• RQ5 Can the $κ$-symmetry of the supermembrane be understood as a residual part of the full local supersymmetry after coupling to supergravity?

Key findings

• The $κ$-symmetry of the supermembrane in 11D supergravity is identified as the part of the full local supersymmetry that remains unbroken when the supermembrane couples to the supergravity background.
• The coupled supergravity–superbrane system breaks half of the local supersymmetries of free supergravity, preserving only $1/2$ of the original local supersymmetry.
• The fermionic variation of the supermembrane action, $δ_f S_{11,2}$, is equivalent to the pullback of a superspace general coordinate transformation, $φ^*(\delta_{sgcf}) = \delta_f$, showing that $κ$-symmetry arises from a restricted subset of local supersymmetry.
• The Noether identity for $κ$-symmetry is confirmed by showing that the equation of motion $\hat{\Xi}_{\underline{\beta}}$ vanishes when projected via $(1 + \bar{\gamma})$, i.e., $\hat{\Xi}_{\underline{\beta}}(1 + \bar{\gamma}) \equiv 0$.
• The supermembrane action is invariant under the $κ$-symmetry transformation, as demonstrated by the vanishing of the variation $\delta_{\kappa} S_{11,2} = 0$ when the fermionic parameter is restricted to $\epsilon^{\underline{\alpha}} = (1 - \bar{\gamma})^{\underline{\alpha}}{}_{\underline{\beta}} \kappa^{\underline{\beta}}$.
• The complete coupled system of supergravity and superbrane preserves superdiffeomorphism symmetry $\delta_{sdiff}$, and the action is invariant under the variational version of local supersymmetry $\tilde{\delta}_{sgcf}$, which corresponds to the physical symmetry of the system.