Chiral Gravity in Three Dimensions (0801.4566v2)

Abstract: Three dimensional Einstein gravity with negative cosmological constant -1/\ell^2 deformed by a gravitational Chern-Simons action with coefficient 1/\mu is studied in an asymptotically AdS_3 spacetime. It is argued to violate unitary or positivity for generic \mu due to negative-energy massive gravitons. However at the critical value \mu\ell=1, the massive gravitons disappear and BTZ black holes all have mass and angular momentum related by \ell M=J. The corresponding chiral quantum theory of gravity is conjectured to exist and be dual to a purely right-moving boundary CFT with central charges (c_L,c_R)=(0,3\ell /G).

• The paper presents a novel deformation of 3D Einstein gravity by adding a gravitational Chern-Simons term, leading to topologically massive gravity with distinctive chiral properties.
• It demonstrates that at the critical ℓ=1 point, the central charges conform (cL=0, cR=3ℓ/G) and both massless and massive graviton modes become pure gauge, ensuring theoretically consistent dynamics.
• The study confirms non-negative energy across all modes and proposes a duality with a holomorphic boundary CFT, paving the way for insights into quantum black hole entropy.

Overview of Chiral Gravity in Three Dimensions

The paper "Chiral Gravity in Three Dimensions" by Wei Li, Wei Song, and Andrew Strominger explores the properties and implications of a deformation of three-dimensional Einstein gravity with a negative cosmological constant. This deformation involves the addition of a gravitational Chern-Simons action, resulting in what is known as topologically massive gravity (TMG). The paper is conducted in an asymptotically anti-de Sitter (AdS) spacetime, focusing on critical parametrical relationships which allow the formulation of chiral gravity, a distinct theoretical proposal.

Theoretical Framework and Key Results

Initially, the paper addresses the foundational aspects of three-dimensional gravity, emphasizing that the theory, while seemingly trivial due to the lack of local degrees of freedom, generates asymptotically AdS black holes with thermodynamic characteristics. The addition of a gravitational Chern-Simons term introduces local, massive propagating degrees of freedom, transforming the existing framework into TMG.

An intriguing point of paper is the critical value ($\ell = 1$) where chiral gravity can emerge. At this juncture, several notable phenomena occur:

1. The central charges of the boundary conformal field theory (CFT) obey specific constraints, $c_L = 0$ and $c_R = \frac{3\ell}{G}$, indicating a purely right-moving theory.
2. Massless boundary and massive gravitons share the same conformal weights and wave functions, both becoming pure gauge in nature but distinct as they do not vanish at infinity.
3. The angular momentum and mass are linearly related through $J = M\ell$, allowing for a stable and consistent theory configuration.

Positivity and Energy Considerations

Central to the analysis is the energy consideration of the system. The authors explored BTZ black holes and graviton modes, revealing that negative-energy states can emerge under certain conditions, potentially threatening the stability and consistency of the theory. However, for $\ell = 1$, the energy remains non-negative across all modes, signifying a potential resolution to these concerns.

Implications and Future Directions

The paper posits that, for the appropriate boundary conditions, chiral gravity might constitute a stable quantum theory when dual to a holomorphic boundary CFT. The self-dual nature could allow the formulation of a microscopic origin of black hole entropy using the Cardy formula.

Further exploration into the validity and viability of chiral gravity poses theoretical challenges, such as establishing boundary conditions that correctly discard zero-energy excitations as pure gauge, demonstrating positive energy at the non-linear level, and ensuring the renormalizability of the quantum theory. As such, research pivoting from this paper could notably focus on identifying a consistent unitary dual CFT and developing insights from holomorphic factorization and partition functions.

Conclusion

In conclusion, this paper on chiral gravity enriches the discourse on three-dimensional gravity in two main ways: by providing a theoretically sustainable framework for TMG when $\ell = 1$, and by opening channels toward understanding gravitational dynamics in lower dimensions through new concepts in duality and CFT descriptions. The advancement of this avenue of research may well contribute to a deeper comprehension of gravitational theories in higher dimensions and expand our insight into the holographic principles underlying quantum gravity.