Finite-c Corrections and Nonperturbative Questions

Finite central charge is where semiclassical AdS3 claims meet exact quantum consistency. Perturbative corrections organize powers of $1/c$, while additional saddles and topology can contribute as $e^{-O(c)}$. Exact Virasoro null relations, integral degeneracies, spectral discreteness, and higher-genus factorization are not optional small refinements: they decide whether a gravitational asymptotic expansion belongs to any quantum theory.

Required background. Pure AdS3 Gravity and Candidate CFT Constraints defines the completion tests, and Conical Defects, Orbifolds, and Heavy States identifies the finite-energy sectors whose classical interpretation can change.

Helpful background. Spin, Charges, and Extended Modular Sectors supplies exact sector-by-sector constraints, while Spectral Statistics, Form Factors, and Late-Time Evidence explains why smoothed semiclassical observables cannot determine late-time discreteness.

Three scales of correction

It is useful to separate three effects:

1. Perturbative bulk loops. Around a named classical saddle, $$\log Z=-I_{\rm cl}+\log Z_{\text{1-loop}}+O(c^{-1}),$$ with $I_{\rm cl}=O(c)$ and the one-loop boundary-graviton determinant $O(c^0)$.
2. Exact representation theory. At special finite $c$, null vectors remove descendants from a Verma module. This changes coefficients exactly and cannot be recovered by blindly evaluating the generic-$c$ product.
3. Nonperturbative completion. Other real or complex saddles, topology changes, and integration-cycle choices can scale as $e^{-O(c)}$. Such terms are invisible to every finite order in the loop expansion but can control discreteness, late times, and factorization.

Keeping these scales distinct prevents an exact character identity from being confused with a bulk loop calculation, or a successful loop calculation from being called nonperturbative completion.

A finite-c comparison

For generic $c>1$, the thermal-AdS one-loop factor reproduces

$$\prod_{n=2}^{\infty}\frac{1}{|1-q^n|^2},$$

the left–right vacuum-descendant count. At a degenerate Virasoro central charge, the exact vacuum character contains additional numerator factors that remove null modules. A gravity prescription claiming that exact CFT must explain those factors—through altered boundary degrees of freedom, additional gauge identifications, or a different theory—rather than calling the generic determinant exact.

Quantum corrections to BTZ observables offer a complementary check. One-loop determinants and zero modes generate logarithmic and finite contributions to entropy or free energy; an exact CFT partition function must reproduce the same correction in the same ensemble and charge sector. Agreement of a leading area term is not enough to select among inequivalent finite-$c$ spectra.

First application. Track the first finite-c correction to a vacuum character, BTZ observable, or Virasoro block and compare with exact CFT consistency. For the vacuum sector, separate $-I_{\rm cl}$ from the one-loop product, expand the first several coefficients, and compare with the exact character after null states are removed. This pairs an original gravity-side calculation with an independent representation-theory test.

Nonperturbative questions that remain

A completed theory must say whether the spectrum is discrete, what topologies are summed, which complex saddles lie on the contour, and whether multi-boundary amplitudes describe a fixed theory or an average. At finite entropy, exact correlators have recurrences that a single black-hole saddle misses. Modular invariance constrains the necessary completion but does not uniquely choose it.

Adversarial control. Use the generic vacuum product at a minimal-model value of $c$ and identify the first spurious null descendant. Separately take a smooth semiclassical spectral density to times comparable to the inverse level spacing: it cannot reproduce exact recurrences without nonperturbative information. A classical inequality or saddle-dominance statement that fails either test must be restated with its large-$c$ and time-domain limits.

Evidence ceiling

Perturbative determinants, exact characters, and modular reconstructions provide independent and highly constraining checks. They do not yet select a unique finite-$c$ completion of pure AdS3 Einstein gravity. Any current proposal must therefore state its added states, contour, topology, and fixed-theory or ensemble interpretation explicitly.

The perturbative one-loop contribution is calculable as in Giombi, Maloney, and Yin 2008, whereas modular completion and a positive finite-$c$ CFT spectrum are separate nonperturbative requirements Keller and Maloney 2015.

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 Francesco, Philippe, Pierre Mathieu, and David Sénéchal. Conformal Field Theory. New York: Springer, 1997. DOI.
• Giombi, Simone, Alexander Maloney, and Xi Yin. “One-Loop Partition Functions of 3D Gravity.” Journal of High Energy Physics 2008, no. 8 (2008): 007. 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.