Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions

A Euclidean gravitational path integral is a proposal only after one fixes its boundary fields, gauge quotient, measure, action and boundary terms, integration cycle, allowed topology class, and normalized observable. Here the working example is Einstein gravity on an asymptotically locally AdS$_{d+1}$ Euclidean manifold with Dirichlet conformal-boundary data; no sum over disconnected manifolds is assumed unless stated.

Required background. Boundaries and State Preparation supplies the relation between boundary conditions and states. Fixed-Theory, Ensemble, and Superselection Claims supplies the distinctions needed to interpret a multi-boundary result.

Helpful background. Multi-Saddle Sums and Dilute Ensembles gives ordinary semiclassical bookkeeping. JT Topological Expansion and Weil–Petersson Volumes is a controlled topology-sum example.

The Euclidean integration-domain contract

Let $B=(\partial M,[\gamma],J)$ denote a conformal boundary, a representative cutoff metric $\gamma_{ij}$, and matter sources $J$. For one declared manifold $M$, the formal Dirichlet amplitude is

$$Z_M[\gamma,J]=\int_{\mathcal C_M} \frac{\mathcal Dg\,\mathcal D\Phi}{\mathrm{Diff}_0(M;\partial M)} \exp\!\left[-I_E[g,\Phi]\right].$$

The quotient removes diffeomorphisms trivial at the boundary; asymptotic symmetries are not silently divided out. The contour $\mathcal C_M$ lies in a complexification of the fields because the real Euclidean metric integral is not convergent. Boundary conditions must also be supplied at any brane, defect, internal end, or corner.

For Einstein gravity with $\Lambda=-d(d-1)/(2L^2)$,

$$I_E=-\frac{1}{16\pi G}\int_M\!\sqrt g\,(R-2\Lambda) -\frac{1}{8\pi G}\int_{\partial M_\epsilon}\!\sqrt\gamma\,K +I_{\mathrm{ct}}[\gamma]+I_{\mathrm{matter}}.$$

The Gibbons–Hawking–York term makes the Dirichlet variation well posed; local counterterms cancel cutoff divergences and define a renormalization scheme. With outward normal $n^\mu$, its sign is tied to the displayed bulk-action convention. Changing from fixed boundary metric to fixed Brown–York stress tensor requires a boundary Legendre transform, not merely a relabeling Gibbons and Hawking 1977.

Application: two fixed asymptotic boundaries

Take $B=B_1\sqcup B_2$ and suppose the allowed domain initially contains only the disconnected topology $M_1\sqcup M_2$. At saddle order,

$$Z_{\mathrm{disc}}[B_1,B_2] \simeq \exp[-I_{\mathrm{ren}}(g_1)]\, \exp[-I_{\mathrm{ren}}(g_2)] \frac{(\det{}'\Delta_{\mathrm{gh},1})(\det{}'\Delta_{\mathrm{gh},2})} {\sqrt{\det{}'\Delta_{\mathrm{phys},1}\det{}'\Delta_{\mathrm{phys},2}}}.$$

Primes exclude zero modes, whose collective-coordinate measures must be supplied separately. If a connected manifold $M_{12}$ is admitted, it contributes an additional saddle rather than modifying either disconnected factor:

$$Z_{\mathfrak T}[B_1,B_2] =Z_{\mathrm{disc}}[B_1,B_2] +\chi_{\mathfrak T}(M_{12})\,Z_{M_{12}}[B_1,B_2]+\cdots .$$

Here $\mathfrak T$ is the declared topology policy and $\chi_{\mathfrak T}$ is zero or one. This notation makes the scientific choice visible. A sum over all genera would further require weights, moduli measures, rules for singular limits, and a completion of any divergent asymptotic series.

Boundary-condition and topology check

Keep the same symbol $Z[B_1\sqcup B_2]$ but change either the ensemble:

• allow $M_{12}$ while retaining Dirichlet data;
• keep only disconnected manifolds but prepare an entangled matter state across the two boundaries;
• impose a brane end-of-the-world condition;
• integrate over a boundary modulus that was previously fixed.

These are inequivalent amplitudes even if they share a leading classical action. The connected term may disappear, acquire a negative mode, or change normalization. Therefore a reported value without its topology policy and boundary preparation is not reproducible.

Controlled scope and handoff

The saddle expansion is controlled when curvatures are small in Planck units, the renormalized action separation is large enough to organize saddles, and all physical fluctuation modes and moduli are treated. This page does not define the contour or prove a nonperturbative sum. Continue to Conformal-Factor Problem, Integration Contours, and Resurgent-Completion Proposals for convergence and to Euclidean Wormholes and Connected Boundary Amplitudes for interpretation.

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

• Gibbons, G. W., and S. W. Hawking. “Action Integrals and Partition Functions in Quantum Gravity.” Physical Review D 15 (1977): 2752–2756. DOI.
• Hawking, S. W., and D. N. Page. “Thermodynamics of Black Holes in Anti-de Sitter Space.” Communications in Mathematical Physics 87 (1983): 577–588. DOI.
• Henningson, M., and K. Skenderis. “The Holographic Weyl Anomaly.” Journal of High Energy Physics 1998, 7 (1998): 023. DOI.
• York, J. W. “Role of Conformal Three-Geometry in the Dynamics of Gravitation.” Physical Review Letters 28 (1972): 1082–1085. DOI.