Euclidean AdS3 Saddles and Handlebody Sums

A Euclidean AdS3 path integral begins with boundary data, not with a preferred bulk topology. For a fixed conformal structure on a boundary Riemann surface, several smooth locally hyperbolic fillings can exist. Handlebody sums provide a computable semiclassical sector, but a declared set of saddles, measure, regulator, and contour is required before the result can be interpreted.

Required background. BTZ Black Holes and Modular CFT Thermodynamics explains the torus fillings, and Multi-Saddle Sums and Dilute Ensembles supplies the asymptotic logic of adding saddles.

Helpful background. Chiral Blocks, Sewing, and Modular Invariance supplies the boundary moduli, while Complex Saddles, Lefschetz Thimbles, and Integration Cycles explains why a solution of the complexified equations contributes only when its thimble belongs to the chosen contour.

Fillings of a boundary Riemann surface

For a torus boundary with modulus $\tau$, a smooth solid-torus filling is specified by which primitive boundary cycle contracts in the bulk. Thermal AdS and Euclidean BTZ are the two familiar representatives. More generally, modular images label choices related by

$$\tau\longmapsto\gamma\tau =\frac{a\tau+b}{c\tau+d}, \qquad \gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix} \in SL(2,\mathbb Z),$$

modulo the subgroup that leaves the chosen contractible cycle unchanged.

At genus $g>1$, a handlebody filling chooses $g$ independent cycles to contract. A Schottky presentation makes its classical geometry and one-loop determinant accessible. Yet “handlebody” is a topological restriction, not a theorem that all relevant solutions have this form; non-handlebody real or complex saddles and different integration cycles can contribute.

What a saddle sum computes

A torus handlebody prescription has the schematic form

$$Z_{\rm hb}(\tau,\bar\tau) =\sum_{\gamma\in\Gamma_\infty\backslash SL(2,\mathbb Z)} Z_{\rm vac}(\gamma\tau,\gamma\bar\tau),$$

where $Z_{\rm vac}$ includes the classical action and perturbative boundary-graviton determinant. The Poincaré series is not absolutely convergent as written, so its regulator is part of the definition. Different subtractions may alter finite terms, and modular covariance alone does not ensure a discrete positive spectrum.

For any multi-saddle approximation, the observable is

$$Z\sim\sum_s n_s\,e^{-I_s}\,Z_s^{\text{1-loop}}\left(1+O\!\left(c^{-1}\right)\right),$$

where $n_s$ is fixed by the integration cycle rather than by the mere existence of the saddle. Negative modes, the Euclidean conformal-factor problem, and complex saddles therefore cannot be hidden inside an unspecified “sum over geometries.”

First application. Sum the thermal-AdS and BTZ modular images for a torus boundary with a declared measure and regulator. The output should identify the coset representatives, stabilize the regulated answer under the chosen subtraction, and separate the classical $O(c)$ action from $O(c^0)$ one-loop factors. Its modular behavior can then be tested without assuming a CFT interpretation.

Completeness and factorization tests

A candidate gravitational answer must withstand more than genus-one modular covariance. Its spectral density should be discrete when one fixed compact CFT is claimed, with nonnegative integer degeneracies. Higher-genus amplitudes must factorize on degenerations with coefficients compatible with one set of OPE data. A sum that behaves like an ensemble average may obey different factorization rules from a fixed theory.

Adversarial control. Add a known class of omitted fillings or permit a complex saddle whose thimble intersects the contour. If its exponential weight competes, the handlebody truncation is not complete. Independently inverse-Laplace-transform the regulated torus result: a continuous density or negative contribution prevents interpreting it as the exact partition function of a single unitary compact CFT, even though the original saddle calculation remains valid as a semiclassical construction.

Evidence ceiling

Handlebody sums organize a well-defined chosen sector of the semiclassical expansion once the measure, regulator, and contour are stated. They do not establish that the sector is complete, that the result has a fixed-theory Hilbert-space interpretation, or that all nonperturbative ambiguities have been resolved.

The standard modular sum and its spectral difficulties are analyzed by Maloney and Witten 2010; this supports the saddle-sum calculation, not the existence of a complete unitary pure-gravity theory.

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

• 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.
• 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.
• Yin, Xi. “Partition Functions of Three-Dimensional Pure Gravity.” Communications in Number Theory and Physics 2 (2008): 285–324. DOI; Open PDF.