Complex Saddles and Contour Questions

A complex solution of the field equations does not contribute merely because its on-shell action is small. Contribution is a property of a specified integration contour in complexified field space: the saddle’s downward Lefschetz thimble must occur with nonzero coefficient in the homology class of that contour. Gravity adds gauge redundancy, the Euclidean conformal-factor problem, boundary terms, and possible singular geometries. At the 2026-08-19 evidence cutoff, useful contour constructions exist in perturbation theory and reduced models, but a universal nonperturbative contour for gravitational thermodynamic partition functions remains open.

Required background. Gravitational Actions and Boundary Terms supplies the holomorphically continued action and fixed sources; Variational and Symplectic Geometry supplies gauge reduction and saddle fluctuations.

Helpful background. Quantum Fields in Curved Spacetime supplies analytic continuation and thermal traces; Einstein Equations supplies the complexified saddle equations; and Black-Hole Geometry and Mechanics supplies the boundary charges and horizon regularity conditions.

A saddle contributes only relative to a cycle

Schematically, a thermodynamic path integral is

$$Z(\beta,\Omega,\Phi) =\int_{\mathcal C}\mathcal Dg\,\mathcal D\psi\, \exp\!\left[-I_E(g,\psi;\beta,\Omega,\Phi)\right].$$

The boundary metric and proper thermal circle fix $\beta$; angular and gauge sources fix $\Omega$ and $\Phi$ in the chosen ensemble. Counterterms or reference subtraction must be part of the same action before it is complexified. A critical point $p_\sigma$ satisfies the complexified field equations, but its semiclassical coefficient is

$$n_\sigma =\langle\mathcal C,\mathcal K_\sigma\rangle,$$

the intersection number of the original cycle $\mathcal C$ with the upward cycle $\mathcal K_\sigma$. If $n_\sigma=0$, even an exponentially favorable saddle is absent. Across a Stokes line, the thimble decomposition can jump while the original integral remains continuous.

This is the decisive answer to the page’s question: the contributing contour must be a convergent middle-dimensional cycle derived from the intended thermal trace or another physical definition, with nonzero intersection number for the proposed saddle. Allowability, regularity, and a plausible thermodynamic action are necessary checks in many proposals, but none alone computes $n_\sigma$.

Results as of 2026-08-19

The status below distinguishes full functional-integral claims from perturbative or symmetry-reduced evidence.

Evidence or formulation	Result for contour selection	Scope and limitation
Perturbative conformal rotation	Rotating the unstable conformal Gaussian makes the local one-loop integral convergent Gibbons, Hawking, and Perry 1978, pp. 141–150	A neighborhood of a chosen real saddle; it does not define a global cycle or its saddle coefficients
Allowable complex metrics	Positivity of real parts of matter kinetic actions excludes many pathological complex metrics and retains several useful saddles Witten 2022, §§2–4, PDF	Explicitly a proposed filter; §5 finds no general gradient-flow cycle that is both allowable and convergent
Lorentzian definition of thermal traces	A real Lorentzian contour enlarged by controlled codimension-two defects yields evidence that positive-specific-heat black-hole saddles have nonzero weight Marolf 2022, §§2–4.2 and §5, PDF	Semiclassical construction with stated singularities and deformation steps; not a complete nonperturbative measure over all metrics
Reduced AdS black-hole integrals	Picard–Lefschetz analysis of one- and two-dimensional radius/charge integrals selects real and complex saddles by ensemble and tests KSW allowability Ailiga, Mallik, and Narain 2026, §§II–IV, PDF	Peer-reviewed but symmetry reduced; homology averaging and the reduced starting integral are not derived as the unique full metric contour

The 2026 charged-AdS calculation is stronger evidence than a bare saddle list: within its reduced integrals, it tracks thimbles, Stokes changes, ensembles, and allowability. It should nevertheless not be promoted to a theorem for the full gravitational functional integral. Its v2 was published in JHEP before the cutoff, but independent full-metric reproduction is still absent from the evidence assembled here.

Competing positions

Trace-first position. Begin with a Hilbert-space thermal trace or a real Lorentzian path integral whose contour has physical meaning. Complex saddles enter only through justified deformation. This best protects unitarity and fixes intersection numbers, but presently covers restricted classes of geometries and singularities.

Allowability-first position. Admit complex metrics on which ordinary matter actions have acceptable real parts, then search for convergent gradient-flow cycles inside that domain. Witten shows that the filter removes bad examples and retains useful saddles, but also explains why the desired global gravitational cycle is not settled Witten 2022, §§3–5, PDF.

Thermodynamic-consistency position. Retain saddles that reproduce known entropy, first-law, and dual-field-theory behavior, using negative modes and Stokes phenomena to organize metastability. This is productive evidence in controlled models, but it risks circularity: agreement with an expected phase diagram cannot itself define the contour used to compute that diagram.

The conservative conclusion is not that complex saddles are forbidden. Rotating black holes already make complex continuation natural. It is that a proposed complex saddle remains conditional until the ensemble, original cycle, deformation, gauge treatment, and renormalized action are all given.

Gaussian check and an adverse example

The finite-dimensional Gaussian exposes the logic. Let $a=\lvert a\rvert e^{i\theta}$ and consider

$$Z(a)=\int_{\mathcal C}dz\,e^{-az^2/2}.$$

The only saddle is $z=0$, independent of $\mathcal C$. Choose $z=e^{-i\theta/2}y$ with real $y$; then

$$-\frac{az^2}{2}=-\frac{\lvert a\rvert y^2}{2},$$

so this steepest-descent contour converges and gives the analytic continuation of the ordinary Gaussian. Rotating it into a wedge where $\operatorname{Re}(az^2)\lt0$ makes the integral diverge, although the saddle and its action have not changed. The saddle equation therefore cannot decide contribution.

Gravity’s conformal factor supplies the adverse analogue. Along a real Euclidean Weyl fluctuation, the Einstein action is unbounded below. A formal real-metric integral diverges; perturbatively rotating that mode makes its Gaussian converge, but different global continuations can encounter different saddles and Stokes walls. Witten’s analysis of gradient flow, gauge symmetry, allowability, and convergence makes this obstruction explicit Witten 2022, §5, PDF.

Limitations of present thermodynamic uses

Several boundaries should accompany any claimed complex contribution:

• a solution of complexified Einstein equations may be degenerate, non-allowable, singular, or unreachable from the physical cycle;
• fixed potential, fixed charge, and fixed angular momentum define different cycles and Hessians;
• a negative mode requires a contour prescription and does not automatically imply a Lorentzian instability;
• local counterterms and finite boundary terms must be continued consistently so that action comparisons remain in one scheme;
• a reduced lapse, area, radius, or minisuperspace integral may omit full metric directions that change intersection numbers;
• one-loop dominance can fail near a degenerate saddle or Stokes wall, where a uniform approximation is required.

These limitations leave the general contour status unresolved, even though specific real and complex saddles can be reliable within a stated construction.

Exercise

For the Gaussian above with $a=e^{i\pi/3}$, find a straight steepest-descent contour through $z=0$ and evaluate the integral by analytic continuation from positive real $a$. Why would quoting only $I(0)=0$ be insufficient?

Solution

Take $z=e^{-i\pi/6}y$ with $y$ real. Then $az^2=e^{i\pi/3}e^{-i\pi/3}y^2=y^2$ and $dz=e^{-i\pi/6}dy$. Therefore

$$Z =e^{-i\pi/6}\int_{-\infty}^{\infty}dy\,e^{-y^2/2} =e^{-i\pi/6}\sqrt{2\pi}.$$

This equals $\sqrt{2\pi/a}$ on the branch reached continuously from positive real $a$. The saddle value $I(0)=0$ contains neither the phase from the contour Jacobian nor the convergence wedges nor the intersection number. A different straight line through the same saddle can diverge, so the critical point alone does not define the integral.

What could change these conclusions

The conclusions could change after any decisive derivation of a gauge-fixed nonperturbative gravitational cycle, an independent full-metric reproduction or counterexample to the 2026 reduced AdS thimble results, a correction that changes Witten allowability, a Lorentzian construction that fixes rotating or charged intersection numbers, or a demonstrated Stokes transition that changes which thermodynamic saddle contributes.

References

• Ailiga, M., S. Mallik, and G. Narain (2026), “Complex Saddles of Charged-AdS Gravitational Partition Function,” Journal of High Energy Physics 2026(2), 54. Open PDF.
• Gibbons, G. W., S. W. Hawking, and M. J. Perry (1978), “Path Integrals and the Indefiniteness of the Gravitational Action,” Nuclear Physics B 138, 141–150. DOI.
• Marolf, D. (2022), “Gravitational Thermodynamics without the Conformal Factor Problem: Partition Functions and Euclidean Saddles from Lorentzian Path Integrals,” Journal of High Energy Physics 2022(7), 108. Open PDF.
• Witten, E. (2022), “A Note on Complex Spacetime Metrics,” arXiv:2111.06514v2. Open PDF.