Canonical prescription for the integration contour over complex metrics

Identify a canonical prescription for the integration contour in complexified metric space for the gravitational path integral, determining which complex metrics are admissible saddles and how to integrate over them in a way consistent with quantum gravity and holographic expectations.

Background

Including complex metrics appears necessary in Euclidean treatments of rotating solutions and in supersymmetric contexts. However, the Euclidean Einstein–Hilbert action’s lack of boundedness under conformal rescalings (the conformal factor problem) undermines naive convergence arguments, and presently no canonical contour prescription is available.

A principled contour choice would systematize the inclusion of complex saddles, reconcile Euclidean methods with Lorentzian physics, and align gravitational path integrals with boundary CFT partition functions under holography.

References

Even off-shell, the Einstein--Hilbert action remains unbounded below under conformal deformations, and no canonical prescription for the integration contour in complexified metric space is currently known (as remarked in Section \ref{subsec:4_Rotation}).

— Introduction to black hole thermodynamics  (2512.24929 - Genolini, 31 Dec 2025) in Section 4.3.2, Conformal factor problem (subsubsec:4_ConformalFactor)

However, it is not obvious that the saddle $(M,g_{ij})$ lies on the correct integration contour.

— Quantum State of a Gravitating Spacetime Region  (2609.10684 - Bousso et al., 9 Sep 2026) in Section 4.2, subsection “Success criterion”

Its precise relation to the Liouville holographic setup and integration contours considered here remains to be clarified.

— Stokes Phenomena between AdS/CFT and dS/CFT  (2609.10677 - Honda et al., 9 Sep 2026) in Footnote in Section 1, following the discussion of three-dimensional pure gravity and the Chern–Simons formulation

Several issues remain open. Our treatment of the conformal sector is based on a local Gaussian prescription and does not determine the full nonlinear gravitational integration cycle.

— The Origin and Fate of Rapid Negative Modes in Coleman--De Luccia Tunneling  (2609.37092 - Ookouchi, 29 Sep 2026) in Section Discussion and outlook

The reason we do not study the Prescription 3 is that it is not clear how to treat the brane in the complexified geometry. Thus, we leave the study of Prescription 3 for a future work.

— Testing holographic computation of entanglement pseudo-entropy in dS\textsubscript{3}/ICFT\textsubscript{2}  (2608.26997 - Li, 27 Aug 2026) in Section 4, “Pseudo-entropy in the dS/ICFT,” immediately before Section 4.1

Several qualifications remain. The BF action alone does not define Euclidean JT gravity: the integration cycle \Gamma_{\mathrm{grav}} selects the gravitational configurations and the appropriate contour for the BF scalar. A global construction of this cycle in the complexified field space remains open.

— Conical Defects in JT Gravity from BF Theory: Quantization, Fusion, and Weighted Moduli Spaces  (2609.03347 - Gu, 3 Sep 2026) in Section 6, Discussion and outlook

The status of the corresponding saddle points in AdS/CFT has been repeatedly scrutinized, and it remains an open question which of them should be retained in the path integral.

— Scalar Gauss-Bonnet Wormholes and the Imaginary Distance Bound  (2609.28456 - Kehagias et al., 23 Sep 2026) in Section 1, Introduction

It is also necessary to identify whether these complex solutions lie on integration cycles connected to the original gravitational path integral.

— Scalar Gauss-Bonnet Wormholes and the Imaginary Distance Bound  (2609.28456 - Kehagias et al., 23 Sep 2026) in Section 7, Conclusion

There remain, of course, fundamental questions concerning this prescription. These include whether the resulting wave function can be consistently normalized and endowed with a probabilistic interpretation, as well as the more basic issue of how the gravitational path integral and its integration contour should be defined. We do not attempt to resolve these questions in the present work.

— Effective Dynamics of Inflationary End-of-the-World Branes in AdS$_3$  (2609.11643 - Fujiki et al., 10 Sep 2026) in Section 1, Introduction and summary

However, Bishop's theorem does not apply to the complex saddles relevant to the no-boundary state, and the dominance of the sphere saddle in our context is not rigorously proven.

— Logarithmic gravity from a very slow roll limit of inflation  (2609.38052 - Cotler et al., 29 Sep 2026) in Section 4.2, footnote following the discussion of the de Sitter example

It remains unclear whether the presence of the unstable saddle points is problematic or not.

— Connected Geometries in Gravitational Path Integrals, and Normalized No-Boundary Probabilities  (2610.08596 - Lehners, 6 Oct 2026) in Section 3, discussion following Figure 2 and the Dirichlet-Dirichlet contour analysis

Indeed, regularized disconnected geometries always strongly violate this criterion (hence it is quite conceivable that a somewhat weaker condition would do -- this is an interesting question for future work).

— Connected Geometries in Gravitational Path Integrals, and Normalized No-Boundary Probabilities  (2610.08596 - Lehners, 6 Oct 2026) in Section 5, Discussion