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Chiral Gravity, Log Gravity and Extremal CFT (0903.4573v1)

Published 26 Mar 2009 in hep-th

Abstract: We show that the linearization of all exact solutions of classical chiral gravity around the AdS3 vacuum have positive energy. Non-chiral and negative-energy solutions of the linearized equations are infrared divergent at second order, and so are removed from the spectrum. In other words, chirality is confined and the equations of motion have linearization instabilities. We prove that the only stationary, axially symmetric solutions of chiral gravity are BTZ black holes, which have positive energy. It is further shown that classical log gravity-- the theory with logarithmically relaxed boundary conditions --has finite asymptotic symmetry generators but is not chiral and hence may be dual at the quantum level to a logarithmic CFT. Moreover we show that log gravity contains chiral gravity within it as a decoupled charge superselection sector. We normally evaluate the Euclidean sum over geometries of chiral gravity and show that it gives precisely the holomorphic extremal CFT partition function. The modular invariance and integrality of the expansion coefficients of this partition function are consistent with the existence of an exact quantum theory of chiral gravity. We argue that the problem of quantizing chiral gravity is the holographic dual of the problem of constructing an extremal CFT, while quantizing log gravity is dual to the problem of constructing a logarithmic extremal CFT.

Citations (221)

Summary

  • The paper demonstrates that linearized chiral gravity eliminates non-chiral, negative-energy modes, affirming perturbative stability within topologically massive gravity.
  • It shows that log gravity, with relaxed boundary conditions, introduces left-moving modes linked to logarithmic CFTs and decoupled chiral sectors.
  • The study derives a modular invariant quantum partition function that supports extremal CFT predictions and reinforces the consistency of the gravitational models.

Overview of "Chiral Gravity, Log Gravity and Extremal CFT"

This paper explores the complexities of three-dimensional gravity theories, particularly focusing on chiral gravity, log gravity, and their connections to extremal conformal field theories (CFTs). Chiral gravity is positioned within the framework of topologically massive gravity (TMG) with specific coupling constants, exhibiting asymptotically anti-de Sitter (AdS) boundary conditions. The paper explores linearization, perturbative stability, stationary solutions, and quantum implications within these gravitational theories.

The authors begin by revisiting prior conjectures about chiral gravity, namely chirality and positivity of energy, by examining solutions to the equations of motion. It is shown that the linearization of solutions in chiral gravity respects these aspects: non-chiral and negative-energy modes are excluded from the physical spectrum due to linearization instabilities. Only the Banados-Teitelboim-Zanelli (BTZ) black holes are identified as stationary, axially symmetric solutions that preserve these properties.

Log gravity, an alternative to chiral gravity characterized by weaker boundary conditions, introduces modes with nonzero left-moving charges, previously considered inapplicable within strict chiral regimes. The spectrum of log gravity allows for more intricate solutions, potentially connected to logarithmic CFTs, with chiral gravity existing as a decoupled sector within it. The connection to extremal CFTs hinges on the calculated partition function aligning with the conjectured extremal partition functions initially proposed by Witten.

Strong Numerical Results and Theoretical Implications

The implications of the positive energy theorem at a perturbative level, evidenced by the BTZ black holes being the exclusive solutions in chiral gravity under axial symmetry, are profound. Furthermore, the rigorous derivation of the quantum partition function featuring modular invariance and quantized coefficients indicates a potential path toward identifying a discrete spectrum aligned with extremal CFT predictions. Notably, these results affirm prior conjectures about the stability and mathematical consistency of chiral gravity's solutions in the large-dimensional landscape of quantum gravity.

Contradictory Claims and Future Speculations

The discourse on log gravity introduces a divergence from chiral gravity principally due to its relaxed boundary conditions and connection to non-chiral extremal CFTs. The novel superselection sector containing chiral gravity posits intriguing perspectives on quantum consistency and the absence of extensions beyond known extremal CFTs — a domain still rife with challenges, including the absence of proven constructions of these theories at large coupling constants.

Concluding Remarks

Overall, the paper reinforces the viability of chiral gravity within a constrained framework, mandates reconsideration of its mathematical constitution through log gravity, and presents compelling insights leading towards a deepened understanding of quantum three-dimensional gravities. The research offers a promising frontier in high-energy theoretical physics, hinging on the corroboration and exploration of extremal CFTs and their log counterparts, presenting fertile ground for future theoretical developments.