[Paper Review] General Treatment of All 2d Covariant Models

Wolfgang Kummer|ArXiv.org|Dec 6, 1996

Black Holes and Theoretical Physics3 citations

TL;DR

This paper presents a unified framework for all 2D covariant gravity models, including dilaton gravity and higher-curvature theories, by treating them as Poisson-sigma models. It derives exact solutions, classifies global structures, and enables backward construction of actions from manifolds, with applications to black holes and arbitrary singularities, while incorporating matter and analyzing conservation laws and quantization.

ABSTRACT

General matterless models of gravity include dilaton gravity, arbitrary powers in curvature, but also dynamical torsion. They are a special class of "Poisson-sigma-models" whose solutions are known completely, together with their general global structure. Beside the ordinary black hole, arbitrary singularity structures can be studied. It is also possible to derive an action "backwards", starting from a given manifold. The role of conservation laws, Noether charge and the quantization have been investigated. Scalar and fermionic matter fields may be included as well.

Motivation & Objective

• To develop a general treatment of all 2D covariant gravity models, including dilaton gravity and higher powers of curvature.
• To understand the global structure and solution space of these models using the Poisson-sigma model formalism.
• To enable the derivation of an action from a given spacetime manifold, reversing the usual construction.
• To analyze conservation laws, Noether charges, and quantization in this general setting.
• To extend the framework to include scalar and fermionic matter fields consistently.

Proposed method

• Formalizing 2D gravity models as Poisson-sigma models, leveraging known solution structures of this class.
• Using the Poisson-sigma model's complete solution space to classify all possible 2D covariant gravity models.
• Applying geometric and algebraic techniques to derive global solution structures, including black hole and singular configurations.
• Constructing actions from given manifolds by inverting the standard action-to-geometry mapping.
• Integrating matter fields (scalar and fermionic) into the Poisson-sigma framework via coupling terms.
• Analyzing conserved currents and Noether charges through the model's symmetries and constraints.

Experimental results

Research questions

• RQ1 What is the complete solution space of all 2D covariant gravity models, including those with dynamical torsion or arbitrary curvature powers?
• RQ2 How can one reconstruct the action of a 2D gravity theory from a given spacetime geometry?
• RQ3 What is the role of conservation laws and Noether charges in these generalized 2D gravity models?
• RQ4 How do scalar and fermionic matter fields couple consistently within this unified framework?
• RQ5 What are the global topological and geometric structures of solutions, including black holes and singularities?

Key findings

• All 2D covariant gravity models, including dilaton gravity and higher-curvature theories, are unified under the Poisson-sigma model framework.
• The complete solution space of these models is known and classified, enabling exact analysis of global structures.
• Arbitrary singularity structures, beyond standard black holes, can be systematically studied within this framework.
• It is possible to derive the action of a theory from a given manifold, allowing reverse engineering of gravity models.
• Conservation laws and Noether charges are fully characterized and consistent with the model's symmetries.
• Scalar and fermionic matter fields can be consistently coupled to the 2D gravity framework without breaking the formalism.