[Paper Review] Quantum Gravity in 2+1 Dimensions

Steven Carlip|arXiv (Cornell University)|Dec 19, 2023

Noncommutative and Quantum Gravity Theories66 citations

TL;DR

This paper surveys classical and quantum aspects of gravity in 2+1 dimensions, emphasizing holonomies, geometric structures, and boundary/edge degrees of freedom, with concrete examples like the torus universe and BTZ black hole, and discusses quantization approaches.

ABSTRACT

General relativity becomes vastly simpler in three spacetime dimensions: all vacuum solutions have constant curvature, and the moduli space of solutions can be almost completely characterized. As a result, this lower dimensional setting becomes an ideal test bed for a wide range of approaches to quantum gravity, from reduced phase phase space quantization to covariant canonical quantization to path integral methods to asymptotic quantization of "edge states." Here I review a variety of classical descriptions of the moduli space of solutions and a broad range of quantizations, with special attention to implications for realistic quantum gravity in four spacetime dimensions.

Motivation & Objective

• Explain why gravity in 2+1 dimensions is simpler yet rich for conceptual quantum gravity questions.
• Characterize classical solutions via holonomies and geometric structures across different Λ.
• Present the Chern-Simons and ADM formalisms as quantization avenues and their implications.
• Highlight the role of boundaries and large diffeomorphisms (edge modes) in dynamics and quantization.
• Illustrate key examples (torus universe and BTZ black hole) to connect classical structures with quantization concepts.

Proposed method

• Describe the geometric-structure approach to vacuum solutions as gluing constant-curvature patches with holonomies (2.4).
• Introduce Chern-Simons formulation with gauge group G_Λ and flat connection A and holonomies (2.10, 2.12, 2.13).
• Relate the ADM reduced phase space to Teichmüller space and its cotangent bundle (2.3, 2.5).
• Discuss large diffeomorphisms and mapping class group actions on moduli spaces (2.4).
• Compute asymptotic symmetries and central charges in open (Λ<0) AdS cases (2.5, 2.6).
• Outline quantization strategies: reduced phase space, Wheeler-DeWitt, and Chern-Simons/quantum group approaches (3.1–3.3).

Experimental results

Research questions

• RQ1 What is the structure of the solution space for vacuum 2+1 gravity with different cosmological constants?
• RQ2 How do holonomies and geometric structures classify spacetimes in 2+1 dimensions?
• RQ3 What is the appropriate quantization framework (reduced phase space, Dirac/ Wheeler-DeWitt, or Chern-Simons) for 2+1 gravity, and what are its implications?
• RQ4 How do boundary conditions and large diffeomorphisms modify the physical content and symmetries of the theory?
• RQ5 What do concrete examples like the torus universe and BTZ black hole reveal about classical-quantum correspondence in this model?

Key findings

• Vacuum 2+1 gravity has no local propagating degrees of freedom but exhibits global geometric degrees of freedom via holonomies and boundary modes.
• The space of solutions can be described as moduli spaces tied to Teichmüller space, with the reduced phase space often forming a cotangent bundle over Teichmüller space.
• Chern-Simons formulation yields flat connections whose holonomies encode the geometry, with a natural split into independent sectors for Λ≠0.
• Open/spatially infinite cases reveal asymptotic symmetries (AdS3) governed by a Virasoro algebra with central charge c = 3ℓ/(2G), and in the flat case, BMS3 with c = 3/G.
• Two illustrative examples—the torus universe and BTZ black hole—provide concrete realizations of holonomies, moduli, and their quantization behavior.
• Quantization approaches include reduced phase space quantization (leading to a Maass Laplacian structure for the torus case), Wheeler-DeWitt equations, and Chern-Simons/quantum group frameworks, highlighting the role of boundary degrees of freedom and potential quantum group structures.