AdS3/CFT2 and Three-Dimensional Gravity
Three-dimensional gravity has no local graviton in pure Einstein theory, yet its boundaries carry nontrivial dynamics. This chapter connects those boundary degrees of freedom to Virasoro symmetry, BTZ thermodynamics, and candidate quantum partition functions, while distinguishing a controlled sector from a complete dual CFT.
Enter this chapter
Section titled “Enter this chapter”Begin with AdS3/CFT2 and the Brown–Henneaux Central Charge to derive the asymptotic charge algebra and its central extension.
Helpful background. The Virasoro Algebra and the Stress Tensor and Chiral Blocks, Sewing, and Modular Invariance supply Virasoro representations and modular consistency.
Topics
Section titled “Topics”- AdS3/CFT2 and the Brown–Henneaux Central Charge — Derives the central extension of the asymptotic symmetry algebra.
- AdS3 Boundary Gravitons and Vacuum Characters — Counts boundary excitations in the vacuum representation.
- BTZ Black Holes and Modular CFT Thermodynamics — Compares black-hole thermodynamics with modular CFT asymptotics.
- Virasoro Symmetry, Vacuum Blocks, and Semiclassical Gravity — Relates semiclassical Virasoro blocks to gravitational configurations.
- Chern–Simons Gravity and Boundary Currents — Rewrites local gravitational equations using gauge connections.
- Euclidean AdS3 Saddles and Handlebody Sums — Examines which Euclidean fillings contribute to a boundary partition function.
- Wilson Lines, Geodesic Blocks, and Entanglement — Connects line observables and geodesics to conformal blocks and entropy.
- Pure AdS3 Gravity and Candidate CFT Constraints — Tests candidate pure-gravity spectra against CFT consistency.
- Conical Defects, Orbifolds, and Heavy States — Distinguishes heavy-state geometries through global identifications.
- Finite-c Corrections and Nonperturbative Questions — Investigates effects beyond the leading semiclassical expansion.
How the ideas fit together
Section titled “How the ideas fit together”Boundary gravitons determine a vacuum contribution, and modular transformations generate further candidate saddle contributions. This builds a valuable bridge from the asymptotic symmetry algebra to black-hole thermodynamics. It does not guarantee that the sum is the partition function of a unitary CFT: degeneracies must also be nonnegative integers and the spectrum must satisfy the remaining consistency conditions. The difficulties in the pure-gravity sum are exhibited by Maloney and Witten 2010, § 1. Thus a successful BTZ entropy match leaves finite-central-charge completion as a separate question.
Claim-domain comparison
Section titled “Claim-domain comparison”Compare the assumptions and strongest supported conclusion for each case.
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| Claim object | State, ensemble, and conventions | Approximation, status, and evidence timing | Uncertainty and counterevidence | Falsifier | Failure condition | Licensed conclusion |
|---|---|---|---|---|---|---|
| Brown-Henneaux symmetry | Declare boundary falloffs and charge normalization; use the volume conventions unless the page states a local replacement. | Conditional theorem or structural result. Control chain: CFT2 and global data → AdS3 boundary conditions → Virasoro or Chern-Simons structure → BTZ and modular checks → three-dimensional gravity claim. Sources are cited on the destination page; literature checked through 10 August 2026. | Track omitted corrections, alternate branches, and competing definitions. A failed “central extension and integrability” check is counterevidence to the promoted claim. | The proposed charges are nonintegrable under the declared falloffs, or their central term disagrees with the fixed action and normalization. | Inferring a complete Hilbert space from the asymptotic Virasoro algebra. | asymptotic Virasoro algebra |
| BTZ saddle | Declare ensemble, identifications, and spin; use the volume conventions unless the page states a local replacement. | Model-specific calculation or conditional result. Control chain: CFT2 and global data → AdS3 boundary conditions → Virasoro or Chern-Simons structure → BTZ and modular checks → three-dimensional gravity claim. Sources are cited on the destination page; literature checked through 10 August 2026. | Track omitted corrections, alternate branches, and competing definitions. A failed “thermodynamic and modular check” check is counterevidence to the promoted claim. | With charges and ensemble matched, the result violates the first law or disagrees with the Cardy entropy in the regime where Cardy asymptotics apply. | Identifying all heavy CFT states with individual BTZ geometries. | controlled black-hole sector |
| pure gravity proposal | Declare spectrum, topology sum, and global form; use the volume conventions unless the page states a local replacement. | Proposal or conditional construction. Control chain: CFT2 and global data → AdS3 boundary conditions → Virasoro or Chern-Simons structure → BTZ and modular checks → three-dimensional gravity claim. Sources are cited on the destination page; literature checked through 10 August 2026. | Track omitted corrections, alternate branches, and competing definitions. A failed “unitarity and modular consistency” check is counterevidence to the promoted claim. | The proposed partition function fails modular consistency, or has negative or noninteger degeneracies when a unitary fixed CFT with discrete spectrum is claimed. | Presenting a candidate partition function as an established nonperturbative theory. | a candidate partition function or sector |
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Where to go next
Section titled “Where to go next”Continue to AdS2, JT Gravity, SYK, and Random Matrices to compare another solvable gravitational sector with a boundary effective theory and a matrix ensemble.