Conformal-Factor Problem, Integration Contours, and Resurgent-Completion Proposals

The Euclidean Einstein action is unbounded below along rapidly varying conformal rescalings, so integration over real positive-definite metrics cannot define the amplitude. A usable prescription deforms the conformal mode onto a convergent complex cycle; a resurgent completion is stronger only if it fixes the transseries parameters and cycle globally, including their Stokes jumps.

Required background. Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions defines the amplitude being completed. Complex Saddles, Lefschetz Thimbles, and Integration Cycles supplies the relative-homology language.

Helpful background. Stationary Phase, Coalescing Saddles, and Stokes Geometry explains saddle rearrangement. Complex Saddles, Quasi-Zero Modes, and Hidden Phases develops the corresponding transseries data.

Why the conformal direction diverges

Write in $D=d+1>2$ Euclidean dimensions

$$g_{\mu\nu}=e^{2\sigma}\bar g_{\mu\nu},\qquad \sqrt g\,R =\sqrt{\bar g}\,e^{(D-2)\sigma} \left[\bar R+(D-1)(D-2)(\bar\nabla\sigma)^2\right]$$

after integrating the $\bar\nabla^2\sigma$ term by parts. Because the Euclidean action carries an overall minus sign, its conformal kinetic contribution is

$$I_E[\sigma]\supset -\frac{(D-1)(D-2)}{16\pi G} \int\!\sqrt{\bar g}\,e^{(D-2)\sigma}(\bar\nabla\sigma)^2 .$$

Choosing $\sigma$ with short wavelength drives $I_E\to-\infty$ and $e^{-I_E}\to\infty$. Gauge fixing does not by itself remove this physical integration-direction problem Gibbons, Hawking, and Perry 1978.

A one-mode contour calculation

Near a saddle, isolate a conformal eigenmode $q$ whose quadratic action is $I^{(2)}=-\lambda q^2/2$ with $\lambda>0$. The real integral diverges. Two convergent rays are

$$\mathcal C_\pm:\quad q=\pm i y,\quad y\in\mathbb R,$$

for which

$$\int_{\mathcal C_\pm}dq\,e^{+\lambda q^2/2} =\pm i\int_{-\infty}^{\infty}dy\,e^{-\lambda y^2/2} =\pm i\sqrt{\frac{2\pi}{\lambda}}.$$

They share the same absolute determinant and perturbative power series but differ by orientation and phase. More general cycles may add thimbles of other complex saddles with integer intersection numbers. Thus “rotate the conformal factor” is incomplete unless the direction, orientation, endpoints, and continuation from a better-defined contour are specified.

In minisuperspace a lapse integral often takes the schematic form

$$\Psi(a_f)=\int_{\mathcal C_N}dN\,\mu(N)\exp\!\left[\frac{i}{\hbar}S_{\mathrm{red}}(N;a_f)\right].$$

Different admissible $\mathcal C_N$ can select distinct saddle combinations and hence different wavefunctions. Picard–Lefschetz theory analyzes a declared finite-dimensional integral; it does not choose the gravitational integration cycle on physical grounds Feldbrugge, Lehners, and Turok 2017.

What a resurgent completion must add

Suppose perturbation theory about saddle $s$ gives

$$Z_s(\hbar)\sim e^{-I_s/\hbar}\sum_{k\ge0}a_{s,k}\hbar^k.$$

Large-order relations can reveal action differences $I_{s'}-I_s$ and Stokes constants. A proposed completion has the transseries form

$$Z(\hbar)=\sum_s n_s(\theta)\,e^{-I_s/\hbar}\, \mathcal S_\theta\Phi_s(\hbar),$$

where lateral resummation $\mathcal S_\theta$ and jumps of $n_s$ cancel ambiguities. The perturbative coefficients do not determine the initial $n_s$, the allowed topology sectors, or the physical contour. Those are boundary data for the completion.

Adversarial Stokes test

Vary a boundary source until $\operatorname{Im}(I_s-I_{s'})/\hbar=0$. Across this Stokes wall the thimble basis jumps. The full cycle must remain the same relative-homology class even though its saddle decomposition changes. If a proposal holds $n_s$ fixed and develops a discontinuity, it has specified a local expansion, not a global amplitude. Likewise, adding a homologically inequivalent contour that has the same perturbative coefficients but a different exponentially small term demonstrates nonuniqueness.

Limits and handoff

The conformal rotation is a semiclassical contour prescription, not a constructive measure on geometries. It must coexist with gauge fixing, matter instabilities, topology policy, and boundary reality conditions. The physical spectrum around each selected saddle is analyzed next in Saddles, Negative Modes, and Steepest-Descent Cycles.

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

• Feldbrugge, J., J.-L. Lehners, and N. Turok. “Lorentzian Quantum Cosmology.” Physical Review D 95 (2017): 103508. DOI.
• Gibbons, G. W., S. W. Hawking, and M. J. Perry. “Path Integrals and the Indefiniteness of the Gravitational Action.” Nuclear Physics B 138 (1978): 141–150. DOI.
• Witten, E. “Analytic Continuation of Chern–Simons Theory.” In Chern–Simons Gauge Theory: 20 Years After, 347–446. AMS/IP Studies in Advanced Mathematics 50 (2011). arXiv:1001.2933.