[Paper Review] Three-Dimensional Gravity Revisited

Edward Witten|ArXiv.org|Jun 22, 2007

Black Holes and Theoretical Physics54 references417 citations

TL;DR

This paper proposes that three-dimensional quantum gravity with negative cosmological constant is dual to a holomorphically factorized 2D CFT, with exact spectrum determined by the BTZ black hole's minimum mass and the $c$-theorem. It identifies the monster CFT at $k^*=4$ and suggests a discrete series of CFTs dual to gravity, with the baby monster group symmetry at $k^*=6$, supported by partition function coefficients matching group representation dimensions.

ABSTRACT

We consider the problem of identifying the CFT's that may be dual to pure gravity in three dimensions with negative cosmological constant. The c-theorem indicates that three-dimensional pure gravity is consistent only at certain values of the coupling constant, and the relation to Chern-Simons gauge theory hints that these may be the values at which the dual CFT can be holomorphically factorized. If so, and one takes at face value the minimum mass of a BTZ black hole, then the energy spectrum of three-dimensional gravity with negative cosmological constant can be determined exactly. At the most negative possible value of the cosmological constant, the dual CFT is very likely the monster theory of Frenkel, Lepowsky, and Meurman. The monster theory may be the first in a discrete series of CFT's that are dual to three-dimensional gravity. The partition function of the second theory in the sequence can be determined on a hyperelliptic Riemann surface of any genus. We also make a similar analysis of supergravity.

Motivation & Objective

• To identify consistent quantum theories of 3D gravity with negative cosmological constant by leveraging the $c$-theorem and holographic duality.
• To determine the exact energy spectrum of 3D pure gravity using the minimum mass of the BTZ black hole and the Chern-Simons formulation.
• To explore a discrete series of 2D CFTs dual to 3D gravity, with increasing symmetry groups such as the monster and baby monster groups.
• To analyze supergravity partition functions on hyperelliptic Riemann surfaces and test for sporadic group symmetries at specific levels $k^*$.

Proposed method

• Using the Chern-Simons gauge theory formulation of 3D gravity, which makes perturbation theory finite and avoids counterterms.
• Applying the $c$-theorem to constrain coupling constants, implying consistency only at discrete values of the cosmological constant.
• Analyzing the partition function of the dual CFT on hyperelliptic Riemann surfaces to extract spectrum and symmetry data.
• Matching coefficients in the modular $q$-expansion of the partition function to dimensions of irreducible representations of sporadic groups like the monster and baby monster.
• Using the minimum mass of the BTZ black hole to fix the spectrum of the dual CFT, ensuring consistency with unitarity and modular invariance.
• Testing whether the leading coefficients in the Ramond and NS sectors of the partition function can be expressed as positive integer combinations of group representation dimensions.

Experimental results

Research questions

• RQ1 Which values of the coupling constant allow a consistent quantum theory of 3D gravity with negative cosmological constant?
• RQ2 Can the energy spectrum of 3D pure gravity be determined exactly using the BTZ black hole's minimum mass and holographic duality?
• RQ3 Is the dual CFT at the most negative cosmological constant the monster CFT of Frenkel, Lepowsky, and Meurman?
• RQ4 Does a discrete series of CFTs with increasing sporadic group symmetry (e.g., baby monster) arise as duals to 3D gravity at higher $k^*$ levels?
• RQ5 Can the partition function of the dual CFT on a hyperelliptic Riemann surface of genus $g$ be computed exactly for all $g$?

Key findings

• At $k^*=4$, the partition function coefficients match the dimensions of irreducible representations of the monster group, strongly suggesting monster symmetry in the dual CFT.
• At $k^*=6$, the first non-trivial NS coefficient (3,724,378) and the leading Ramond coefficient (1) are consistent with a positive integer linear combination of baby monster group representation dimensions.
• The leading coefficient in $H_6$ is 2,589,372,416, and $h_1/2 = f_1 - f_{1/2}$ can be expressed as a positive linear combination of baby monster representation dimensions.
• For $k^*=7$ to $k^*=10$, the partition functions $F_k$ and $H_k$ are computed explicitly, with $H_k$ showing negative leading coefficients requiring integer shifts.
• The coefficient $f_1 = 1,298,410,586$ at $k^*=6$ matches $14r_1 + 16r_2 + 7r_3 + 8r_4 + 4r_6 + 3r_7 + r_8 + 2r_9$, where $r_i$ are baby monster representation dimensions.
• The analysis suggests a discrete series of CFTs dual to 3D gravity, with increasing symmetry groups and exact partition functions computable on hyperelliptic Riemann surfaces.