Pure AdS3 Gravity and Candidate CFT Constraints

“Pure AdS3 gravity” is a proposed quantum theory, not a single universally agreed definition. A useful minimal target contains the Einstein vacuum sector and BTZ states without additional light matter, and admits a unitary modular-invariant CFT2 description. Each word in that target becomes a test: the spectrum must be discrete and positive, multiplicities integral, the light sector sparse, and amplitudes mutually factorizing. No torus calculation alone settles all of them.

Required background. AdS3 Boundary Gravitons and Vacuum Characters supplies the perturbative seed, and Euclidean AdS3 Saddles and Handlebody Sums supplies the candidate modular sum.

Helpful background. Modular Crossing and Spectral Bounds supplies model-independent CFT tests, while Fixed-Theory, Ensemble, and Superselection Claims distinguishes one boundary theory from an averaged observable.

Turning “pure gravity” into testable data

For a compact unitary CFT with a unique vacuum, a torus partition function can be decomposed schematically as

$$Z(\tau,\bar\tau) =|\chi_{\rm vac}(\tau)|^2 +\sum_{h,\bar h}N_{h,\bar h} \chi_h(\tau)\bar\chi_{\bar h}(\bar\tau), \qquad N_{h,\bar h}\in\mathbb Z_{\ge0}.$$

A pure-gravity candidate normally asks that no unwanted light primary appear below the black-hole regime, subject to whatever precise threshold and spin conditions define “light.” It must then pass at least six independent checks:

• modular covariance on the torus;
• a discrete spectrum with nonnegative integer multiplicities;
• a unique normalizable vacuum and the intended low-lying gap;
• Cardy growth compatible with BTZ thermodynamics;
• crossing and unitarity of correlators, not just state counting;
• factorization on higher-genus degenerations with one consistent set of OPE data.

Passing a subset cannot be relabeled as passing the rest.

What modular completion does and does not do

Starting from the vacuum character and summing modular images produces a natural semiclassical object. The regulated Einstein saddle sum studied by Maloney and Witten is modular invariant, but its spectral interpretation has continuous and nonpositive features. Keller and Maloney showed that corrections subleading at large $c$ can repair those particular torus features. Such corrections are physically consequential: they add spectral data that were not fixed by the original pure-Einstein saddle prescription.

Holomorphically factorized “extremal” candidates provide a different, stronger ansatz. The Monster CFT realizes exceptional chiral data at $c=24$, but this example does not establish an infinite family of nonchiral pure-Einstein duals. More recently, Di Ubaldo and Perlmutter constructed a unitary modular-invariant spectrum with a large primary gap; two parametrically heavy string-like states are essential to its positivity. That result demonstrates how close modular consistency can come to a pure spectrum while also showing why the extra states cannot simply be ignored.

First application. Test a proposed pure-gravity partition function against modular invariance, nonnegative integer degeneracies, and the expected semiclassical saddle expansion. Expand it in characters, verify every claimed degeneracy rather than only a smoothed density, and compare its large-$c$ exponential orders with named bulk saddles. Then record separately any unresolved crossing or higher-genus test.

Obstructions are part of the result

Negative density, nonintegral multiplicity, an unplanned light state, or failed factorization is not a cosmetic defect. Adding states may produce a consistent new candidate, but it changes the claim being tested. Likewise, an ensemble-averaged partition function can have a controlled gravitational interpretation while failing the factorization expected of one fixed CFT; that is a distinction of observables, not a contradiction to be averaged away.

Adversarial control. Inverse-transform a modularly completed candidate at fixed spin and inspect the threshold region, where a coarse Cardy approximation is least reliable. If the result is continuous or negative, retain that feature as an obstruction. If a compensating primary is added, rerun the light-gap, twist-gap, correlator, and bulk-field-content tests under the amended spectrum.

Evidence ceiling

Existing calculations furnish strong necessary conditions, explicit obstructions, and instructive near-pure spectra. They do not provide a general existence or nonexistence theorem for finite-$c$ pure AdS3 Einstein gravity. A stable account therefore states the definition and tests; a live verdict belongs to a dated research assessment with its precise assumptions.

The negative-density problem in the Poincaré-series construction is part of the explicit analysis of Keller and Maloney 2015, while recent unitarity constraints continue to leave the pure-gravity completion unsettled Di Ubaldo and Perlmutter 2024.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Di Ubaldo, Gabriele, and Eric Perlmutter. “AdS3 Pure Gravity and Stringy Unitarity.” Physical Review Letters 132 (2024): 041602. DOI; Open PDF.
• Keller, Christoph A., and Alexander Maloney. “Poincaré Series, 3D Gravity and CFT Spectroscopy.” Journal of High Energy Physics 2015, no. 2 (2015): 080. DOI; Open PDF.
• Maloney, Alexander, and Edward Witten. “Quantum Gravity Partition Functions in Three Dimensions.” Journal of High Energy Physics 2010, no. 2 (2010): 029. DOI; Open PDF.
• Witten, Edward. “Three-Dimensional Gravity Revisited.” (2007). arXiv:0706.3359.