Euclidean Gravitational Path Integrals, Topology Sums, and Boundary Conditions

A Euclidean gravitational path integral is not one universal object. It is a family of conditional prescriptions indexed by the boundary preparation, field and bundle content, gauge quotient, action and renormalization scheme, integration cycle, allowed topologies, relative weights, and observable normalization. This page makes that list operational for asymptotically locally AdS$_{d+1}$ Einstein gravity. Its central check compares thermal AdS and Euclidean AdS–Schwarzschild: they fill the same boundary, but they are different bulk topologies, so a topology policy can change the partition function without changing the printed symbol $Z_E$.

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.

Boundary data at conformal infinity

An asymptotically locally AdS metric is most cleanly described by compactifying the bulk. Near conformal infinity, choose a defining coordinate $\rho$ and write the metric in Fefferman–Graham form,

$$\mathrm ds^2 =L^2\left[ \frac{\mathrm d\rho^2}{4\rho^2} +\frac{1}{\rho}\, g_{ij}(x,\rho)\,\mathrm dx^i\mathrm dx^j \right], \qquad g_{ij}(x,\rho)=g_{(0)ij}(x)+\cdots .$$

The conformal boundary $\Sigma$ at $\rho=0$, a chosen representative $g_{(0)}$ of its conformal class, and the induced metric

$$h_{\epsilon,ij} =\frac{L^2}{\epsilon}\,g_{ij}(x,\epsilon)$$

on the regulator surface $\Sigma_\epsilon$ are three different objects. The boundary-value problem fixes $g_{(0)}$ and the leading matter sources $J_{(0)}$; the cutoff metric $h_\epsilon$ is an intermediate field used to renormalize the action. In even boundary dimension, the Weyl anomaly makes the choice of representative and renormalization scale consequential. The full asymptotic expansion and response map are developed on Asymptotically Locally AdS Boundary Data and Metric Counterterms and the Boundary Stress Tensor, following de Haro, Skenderis, and Solodukhin 2001, §§2–3 and appendix B.

For this page, a marked boundary datum is abbreviated by

$$B=\bigl( \Sigma,\ g_{(0)},\ J_{(0)};\ \text{labels, spin and bundle data, preparation} \bigr).$$

The marking says which boundary component is which. “Preparation” says whether the functional is a thermal trace, a kernel between boundary configurations, or a wavefunctional obtained by integrating over part of a manifold. Fixing local sources does not by itself choose among those objects.

The observable must also be declared. Here $Z_E[B]$ is an unnormalized fixed-source Euclidean functional, and $W_E[B]=\log Z_E[B]$ is its connected generator wherever the logarithm is defined. With the local source convention

$$I_E[J]=I_E[0] -\int_\Sigma\sqrt{g_{(0)}}\,J_{(0)}\mathcal O,$$

one has $\delta W_E/\delta J_{(0)}=\sqrt{g_{(0)}}\,\langle\mathcal O\rangle_B$. A normalized insertion is instead

$$\langle X\rangle_B =\frac{1}{Z_E[B]} \int X\,e^{-I_E},$$

with the same domain and measure in numerator and denominator. Neither expression is automatically a probability: gravitational saddle weights can carry signs or phases.

Before calculating, the following contract should have an entry in every row.

Contract entry	What must be declared
Boundary problem	Marked components, $g_{(0)}$, sources, spin and bundle data, and state preparation
Configuration space	Bulk dimension, orientation, fields, internal boundaries, defects, and allowed singularities
Gauge quotient	Transformations trivial on the fixed data, large-diffeomorphism policy, stabilizers, and boundary symmetries
Action	Bulk effective action, boundary and joint terms, radial counterterms, and finite scheme
Integration cycle	Real or complex fields, convergence prescription, and how the cycle is inherited or defined
Topology policy	Allowed diffeomorphism classes, auxiliary bundle sectors, disconnected and closed components, and weights
Observable	Generating functional, kernel, trace, insertion, normalization, and source-derivative convention
Approximation	Saddles retained, loop and derivative orders, moduli treatment, and error estimate

The fixed-topology Dirichlet amplitude

Work with a positive-definite Euclidean metric and the weight $e^{-I_E}$. The curvature convention is

$$[\nabla_\mu,\nabla_\nu]V^\rho =R^\rho{}_{\sigma\mu\nu}V^\sigma .$$

For

$$\Lambda=-\frac{d(d-1)}{2L^2}, \qquad K=h^{ij}\nabla_i n_j,$$

where $n^\mu$ points outward from the regulated region $M_\epsilon$, the classical Dirichlet action is

$$\begin{aligned} I_{E,D}^{(\epsilon)} ={}&-\frac{1}{16\pi G_{d+1}} \int_{M_\epsilon}\sqrt g\,(R-2\Lambda)\\ &-\frac{1}{8\pi G_{d+1}} \int_{\Sigma_\epsilon}\sqrt h\,K +I_{\mathrm{hol.ct}}[h] +I_{\mathrm{matter,bulk}}[g,\Phi] +I_{\mathrm{matter,bdy}} +I_{\mathrm{joint}} . \end{aligned}$$

The Gibbons–Hawking–York term cancels normal derivatives of $\delta g_{\mu\nu}$, so the classical variational problem is well posed when the induced boundary metric is fixed Gibbons and Hawking 1977, pp. 2752–2754. Matter fields, branes, nonsmooth boundaries, and alternative boundary ensembles require their own boundary or joint terms. After adding local radial counterterms,

$$I_{E,\mathrm{ren}} =\lim_{\epsilon\to0}I_{E,D}^{(\epsilon)},$$

and the response coefficients may be defined by

$$\left.\delta I_{E,\mathrm{ren}}\right|_{\mathrm{EOM}} \equiv \frac12\int_\Sigma\sqrt{g_{(0)}}\, T_{\mathrm{ren}}^{ij}\,\delta g_{(0)ij} +\int_\Sigma\sqrt{g_{(0)}}\, \mathcal O_{\mathrm{ren}}\,\delta J_{(0)} .$$

The variation vanishes under the declared Dirichlet variations. A finite local counterterm can shift $I_{E,\mathrm{ren}}$, $W_E$, and contact terms, but it shifts two saddles with identical boundary data equally; their action difference is unchanged.

Do not confuse $I_{\mathrm{hol.ct}}$, which removes the radial AdS cutoff, with quantum effective-field-theory renormalization. At loop order the bulk action also contains higher-curvature operators and their boundary completions, with renormalized coefficients fixed by the ultraviolet theory or by measurement. This is the gravitational effective-field-theory distinction emphasized by Donoghue 1994, §3.

A sign and cutoff check

For Euclidean Poincaré AdS,

$$\mathrm ds^2 =\frac{L^2}{z^2} \left(\mathrm dz^2+\delta_{ij}\mathrm dx^i\mathrm dx^j\right), \qquad z\ge\epsilon ,$$

let $V_d$ be a finite coordinate volume and define

$$A_\epsilon =\frac{L^{d-1}V_d} {8\pi G_{d+1}\epsilon^d}.$$

At $z=\epsilon$, the outward normal is $n=-(z/L)\partial_z$, so $K=d/L$. The leading counterterm is

$$I_{\mathrm{hol.ct}}^{(0)} =\frac{d-1}{8\pi G_{d+1}L} \int_{\Sigma_\epsilon}\sqrt h .$$

The three divergent contributions are then

$$I_{\mathrm{bulk}}=A_\epsilon, \qquad I_{\mathrm{GHY}}=-dA_\epsilon, \qquad I_{\mathrm{hol.ct}}^{(0)}=(d-1)A_\epsilon .$$

They cancel exactly. Thus the flat-boundary vacuum has $I_{E,\mathrm{ren}}=0$ in this scheme. This small calculation checks the sign of $\Lambda$, the outward-normal convention, the GHY coefficient, the leading counterterm, and the cutoff power independently.

From one topology to a declared topology sum

Let $\eta$ collect orientation, spin structure, gauge bundle, brane data, and other discrete choices. For a fixed pair $(M,\eta)$, divide by the gauge transformations connected to the identity that act trivially on the marked boundary data; call this group $\mathcal G_\partial^0(M,\eta;B)$. The remaining mapping-class group needs a declared fundamental domain or an explicit sector sum. Continuous stabilizers produce zero modes or group volumes, while discrete stabilizers produce automorphism factors. Transformations that change the marked boundary sources are not gauge transformations within this boundary-value problem.

A schematic topology-summed functional is

$$\begin{aligned} Z_{\mathfrak T,E}[B] &=\sum_{[M,\eta]\in\mathfrak T(B)} \frac{w(M,\eta)} {\lvert\operatorname{Aut}(M,\eta;B)\rvert}\, Z_{M,\eta,E}[B],\\ Z_{M,\eta,E}[B] &\stackrel{\mathrm{formal}}{=} \int_{\mathcal C_{M,\eta}/\mathcal G_\partial^0} [\mathcal Dg\,\mathcal D\Phi]_{\mathrm{gf}}\, e^{-I_{E,\mathrm{EFT}}^{\mathrm{ren}}[g,\Phi]} . \end{aligned}$$

Here $\mathfrak T(B)$ is the declared set of allowed diffeomorphism classes with boundary $B$, and $w(M,\eta)$ may contain topological couplings or model-specific phases. The gauge-fixed measure in the second line represents the quotient; it is not an additional division by the same group. On this page, boundary components are labelled and closed vacuum components are excluded. A different convention must say whether such components are divided out by $Z_E[\varnothing]$. In two-dimensional gravity, the topology label can be genus and the weight can involve $e^{S_0\chi(M)}$; genus is not a classification of general $(d+1)$-dimensional manifolds.

The second line is still formal. Gauge fixing is analogous to choosing one representative from each orbit in a finite-dimensional integral, but gravity also has stabilizers, zero modes, mapping-class identifications, and a nontrivial functional measure. Most importantly, the real Euclidean conformal factor makes the Einstein action unbounded below Gibbons, Hawking, and Perry 1978, pp. 143–147. A contour $\mathcal C_{M,\eta}$ in complexified field space is therefore part of the definition, not a late convergence trick.

If regular saddles $\sigma$ exist on an admitted topology, a conditional semiclassical expansion has the form

$$Z_{\mathfrak T,E}[B] \sim \sum_{[M,\eta]\in\mathfrak T(B)} \sum_{\sigma\in\operatorname{Crit}(M,\eta;B)} \frac{w(M,\eta)\,n_\sigma} {\lvert\operatorname{Aut}(M,\eta;B)\rvert}\, \mathcal A_\sigma\,e^{-I_{E,\mathrm{ren}}[\sigma]} .$$

The intersection number $n_\sigma$ says whether the defining contour actually reaches the saddle. The prefactor $\mathcal A_\sigma$ abbreviates the regulated gauge-fixed Hessian, Faddeev–Popov ghosts, matter fluctuations, zero-mode Jacobians, stabilizer volume, and any phase from rotating unstable directions. Writing a universal ratio of “physical” and ghost determinants would hide choices and can double-count modes. An admitted topology can have no regular saddle, and a regular saddle with $n_\sigma=0$ contributes nothing.

There are therefore two distinct questions:

1. Is $(M,\eta)$ in the integration domain $\mathfrak T(B)$?
2. Does the cycle on that domain give a particular saddle a nonzero $n_\sigma$?

The next two articles separate the conformal-factor and contour problem from negative modes and steepest-descent cycles.

Worked application: two fillings of one thermal boundary

Fix the canonical boundary

$$B_\beta =S^1_\beta\times S^{d-1}_L, \qquad d>2,$$

including the same metric representative, thermal spin structure, sources, and counterterm scheme in both sectors. There are two familiar smooth Einstein fillings:

Filling	Bulk topology	Contractible cycle
Thermal AdS	$S^1\times B^d$	The spatial $S^{d-1}$ shrinks at the center; $S^1_\beta$ does not
Euclidean AdS–Schwarzschild	$B^2\times S^{d-1}$	$S^1_\beta$ shrinks smoothly at the horizon

For the black-hole filling,

$$\mathrm ds^2 =f(r)\,\mathrm d\tau^2 +\frac{\mathrm dr^2}{f(r)} +r^2\mathrm d\Omega_{d-1}^2,$$

with

$$f(r)=1+\frac{r^2}{L^2} -\frac{\mu}{r^{d-2}}, \qquad \mu=r_h^{d-2} \left(1+\frac{r_h^2}{L^2}\right).$$

Smoothness of the $(r,\tau)$ cigar fixes the period:

$$\beta =\frac{4\pi L^2r_h} {d r_h^2+(d-2)L^2}.$$

To compare actions, match the induced thermal-circle length at a common radial cutoff and use the same GHY term and counterterm scheme. The finite result is

$$\Delta I_E \equiv I_{E,\mathrm{BH}}-I_{E,\mathrm{AdS}} =\beta\, \frac{\Omega_{d-1}r_h^{d-2}} {16\pi G_{d+1}} \left(1-\frac{r_h^2}{L^2}\right).$$

This is the general-dimensional Hawking–Page comparison in Witten 1998, §§2.3–2.4, pp. 7–10; the original four-dimensional result is Hawking and Page 1983, §§II–III. It also passes a thermodynamic check:

$$\Delta I_E =\beta\,(M-TS), \quad M=\frac{(d-1)\Omega_{d-1}\mu} {16\pi G_{d+1}}, \quad S=\frac{\Omega_{d-1}r_h^{d-1}} {4G_{d+1}}.$$

The classical crossing occurs at

$$r_h=L, \qquad T_{\mathrm{HP}}=\frac{d-1}{2\pi L}.$$

That conclusion depends on the domain. If $\mathfrak T_1(B_\beta)$ contains only thermal AdS, there is no Hawking–Page competition in $Z_{\mathfrak T_1,E}$. If $\mathfrak T_2(B_\beta)$ contains both fillings, then

$$\begin{aligned} Z_{\mathfrak T_2,E}[B_\beta] \sim{}& n_{\mathrm{AdS}}\mathcal A_{\mathrm{AdS}} e^{-I_{E,\mathrm{AdS}}}\\ &+n_{\mathrm{BH}}\mathcal A_{\mathrm{BH}} e^{-I_{E,\mathrm{BH}}} +\cdots . \end{aligned}$$

When both coefficients are nonzero and their phases permit a real-positive thermodynamic interpretation, the black hole dominates at classical order for $r_h>L$. Near $\Delta I_E=0$, both saddles must be kept; a large action separation is required only to discard one saddle, not to trust a controlled two-saddle expansion. One-loop prefactors can shift the finite-coupling crossing.

This is the required adversarial test. The fixed-topology on-shell actions, smoothness condition, and action difference survive when $\mathfrak T_2$ is replaced by $\mathfrak T_1$. The failed hypothesis is that the two occurrences of $Z_E[B_\beta]$ integrate over the same configuration space. A second change is equally sharp: a nonbounding spin structure around $S^1_\beta$ excludes a smooth filling in which that circle contracts, even though the boundary metric is unchanged.

The dedicated Hawking–Page article develops the branch structure and boundary thermodynamics. Here the pair serves a different purpose: it proves that the integration domain is part of the observable.

Same symbol, inequivalent amplitudes

Several changes that are casually called “changing the ensemble” are actually different operations.

Printed object	Held fixed	Changed domain or preparation	Formula-level consequence
$Z_E[B_\beta]$	Thermal boundary data	Admit the black-hole topology	Add its sector and any saddles with nonzero contour coefficient
$Z_E[B_\beta]$	Boundary metric and sources	Choose a nonbounding thermal-circle spin structure	Exclude smooth fillings where that circle contracts
$Z_E[B_1\sqcup B_2]$	Labelled local boundary data	Use a nonseparable boundary wavefunctional	Endpoint integrations need not factorize
$Z_E[B_1\sqcup B_2]$	Independent boundary preparations	Admit a connected filling $M_{12}$	Add a connected sector only if a solution exists and its contour coefficient is nonzero
$Z_E[B_1\sqcup B_2]$	Local boundary shapes	Integrate a shared modulus $\lambda$	Replace a product by $\int\mathrm d\mu(\lambda)\,Z_1(\lambda)Z_2(\lambda)$
$Z_E[B]$ with a brane	Outer boundary data	Add brane embeddings and conditions	Change the configuration space, action, and measure

For a labelled disconnected filling $M_1\sqcup M_2$,

$$Z_{M_1\sqcup M_2,E}[B_1,B_2] =Z_{M_1,E}[B_1]\,Z_{M_2,E}[B_2]$$

only if the action, preparation, gauge quotient, regulator, measure, contour, and collective-coordinate integrals all split, with no shared constraint or modulus. Disconnected boundary, disconnected bulk, product preparation, and factorized amplitude are four different statements.

When the one-boundary denominators are nonzero and use the same closed-component normalization, the diagnostic

$$\mathcal R_{12} =\frac{Z_E[B_1\sqcup B_2]} {Z_E[B_1]Z_E[B_2]}-1$$

vanishes for the factorized prescription. A connected saddle can make it nonzero, but so can a shared modulus or a nonfactorizing boundary preparation. Thus $\mathcal R_{12}\neq0$ is neither a proof of a bulk wormhole nor a probability covariance without further input.

There is also an existence ceiling. Under the Witten–Yau hypotheses, a complete conformally compact Einstein manifold with a positive-scalar-curvature conformal-boundary component has connected conformal boundary. Consequently, two positive-Yamabe boundary components do not admit the assumed smooth connected pure-Einstein filling Witten and Yau 1999, theorem 3.2, pp. 16–17. Matter support, a different boundary Yamabe class, or relaxed geometric hypotheses can evade that obstruction; those are new domain data, as explicit examples illustrate Maldacena and Maoz 2004, abstract and §2.

Control parameters and evidence boundary

A useful semiclassical result reports separate sources of control and uncertainty.

• Bulk loops: require $G_{d+1}/L_{\mathrm{curv}}^{d-1}\ll1$ after treating zero modes and collective coordinates.
• Derivative expansion: require $(\ell_{\mathrm{UV}}/L_{\mathrm{curv}})^p\ll1$ for the first omitted higher-derivative operator; in a string regime this includes $\alpha'/L_{\mathrm{curv}}^2\ll1$.
• Saddle truncation: an omitted saddle with $\Delta\operatorname{Re}I_E>0$ is suppressed as $e^{-\Delta\operatorname{Re}I_E}$ only when its prefactor and moduli integral are controlled.
• Topology truncation: no generic small parameter exists. JT gravity has an $e^{-S_0}$ expansion, but higher-dimensional Einstein gravity does not inherit it.
• Contour and measure: negative modes, endpoints of moduli space, large-diffeomorphism quotients, regulator removal, and complex phases remain separate checks.

Evidence cutoff: 29 August 2026. GHY terms and holographic counterterms make a fixed-topology asymptotically AdS Dirichlet on-shell action finite; they do not construct the metric measure, choose a nonperturbative contour, or prescribe a topology sum. Lorentzian-contour arguments give evidence for nonzero weights of certain positive-specific-heat black-hole saddles, but do not furnish a general Euclidean definition Marolf 2022, §§1, 4.1, and 5. Restrictions on admissible complex metrics remain proposals rather than a completed measure Witten 2021, abstract. Even a recent AdS$_3$ construction under statistical-boundary assumptions finds many possible topology choices rather than a unique universal prescription Belin et al. 2026, abstract and §1.

The strongest general claim is therefore a conditional semiclassical expansion of a declared domain. It is not a nonperturbative definition of higher-dimensional quantum gravity. It also does not establish an ensemble interpretation, Lorentzian traversability, or the reflection positivity needed for Osterwalder–Schrader reconstruction; compare Reflection Positivity and Osterwalder–Schrader Reconstruction.

Continue to Euclidean Wormholes and Connected Boundary Amplitudes for the multi-boundary interpretation and then to Factorization, Ensembles, and the Gravitational Path Integral for the fixed-theory test.

Common pitfalls

Treating the cutoff metric as the boundary source. $h_\epsilon$ diverges as the cutoff is removed; $g_{(0)}$ is the finite boundary representative. Mixing them obscures both the variational problem and the anomaly.

Equating an allowed topology with a contribution. Domain membership, existence of a regular solution, and a nonzero contour coefficient are three separate gates.

Calling every nonfactorizing result a wormhole or an ensemble. A shared modulus or entangled preparation can correlate two boundary factors on a disconnected bulk. A connected gravitational term becomes an ensemble covariance only after an averaging measure has been independently defined.

Using “sum over all genera” in general dimension. Genus classifies two-dimensional surfaces. Higher-dimensional topology sums require a declared class of manifolds, bundles, singularities, weights, and degeneration rules.

Exercises

1. Reproduce the Hawking–Page free energy

Using the displayed $M$, $S$, and $T=1/\beta$, show that $\Delta F=M-TS$ has the sign of $1-r_h^2/L^2$. Locate the crossing.

Solution

Insert

$$T=\frac{d r_h^2+(d-2)L^2} {4\pi L^2r_h}$$

and $\mu=r_h^{d-2}(1+r_h^2/L^2)$. Then

$$\begin{aligned} M-TS &=\frac{\Omega_{d-1}r_h^{d-2}} {16\pi G_{d+1}}\\ &\quad\times \left[ (d-1)\left(1+\frac{r_h^2}{L^2}\right) -\left(d\frac{r_h^2}{L^2}+d-2\right) \right]\\ &=\frac{\Omega_{d-1}r_h^{d-2}} {16\pi G_{d+1}} \left(1-\frac{r_h^2}{L^2}\right). \end{aligned}$$

Hence $\Delta F=0$ at $r_h=L$. Substitution into $T(r_h)$ gives $T_{\mathrm{HP}}=(d-1)/(2\pi L)$.

2. Test scheme independence of relative dominance

Add the same finite local counterterm $C[B]$ to two saddle actions with identical boundary data. Determine what happens to their weight ratio and to $W_E$.

Solution

Each action changes as $I_\sigma\mapsto I_\sigma+C[B]$, so

$$\frac{e^{-I_1-C[B]}}{e^{-I_2-C[B]}} =e^{-(I_1-I_2)}.$$

The action difference and relative dominance are unchanged. The full partition function acquires the common factor $e^{-C[B]}$, so $W_E=\log Z_E$ shifts by $-C[B]$. Source derivatives of $C[B]$ can therefore shift local contact terms even though the saddle comparison is invariant.

3. Separate topology admission from contour membership

For one candidate saddle, write its contribution as

$$\mathbf 1_{\mathfrak T}(M)\, n_\sigma\mathcal A_\sigma e^{-I_\sigma}.$$

Evaluate the term when the topology is excluded, when it is admitted but no regular saddle exists, when $n_\sigma=0$, and when all three gates pass.

Solution

• If $M\notin\mathfrak T(B)$, the indicator is zero.
• If $M\in\mathfrak T(B)$ but no regular critical point exists, there is no $\sigma$ to sum.
• If a regular saddle exists but $n_\sigma=0$, the chosen contour gives zero contribution.
• Only an admitted topology with a regular saddle and nonzero $n_\sigma$ yields the displayed term.

Thus solving the Euclidean field equations is necessary for a saddle contribution but is not sufficient.

4. Produce nonfactorization without a connected bulk

Let a shared real modulus $\lambda$ have a centered Gaussian measure with variance $\sigma^2$, and let $Z_i(\lambda)=e^{a_i\lambda}$. Compare $\mathbb E[Z_1Z_2]$ with $\mathbb E[Z_1]\mathbb E[Z_2]$.

Solution

The Gaussian moment-generating function gives

$$\mathbb E[Z_1Z_2] =\exp\!\left[ \frac{\sigma^2}{2}(a_1+a_2)^2 \right],$$

whereas

$$\mathbb E[Z_1]\mathbb E[Z_2] =\exp\!\left[ \frac{\sigma^2}{2}(a_1^2+a_2^2) \right].$$

Their ratio is $e^{\sigma^2a_1a_2}$, which is generally not one. The correlation arose from integrating a common modulus, not from adding a connected bulk manifold.

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

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• Donoghue, John F. “General Relativity as an Effective Field Theory: The Leading Quantum Corrections.” Physical Review D 50 (1994): 3874–3888. DOI; arXiv:gr-qc/9405057.
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• 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.
• 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.
• Maldacena, Juan, and Liat Maoz. “Wormholes in AdS.” Journal of High Energy Physics 2004, 2 (2004): 053. DOI; arXiv:hep-th/0401024.
• Marolf, Donald. “Gravitational Thermodynamics without the Conformal Factor Problem: Partition Functions and Euclidean Saddles from Lorentzian Path Integrals.” Journal of High Energy Physics 2022, 7 (2022): 108. DOI; arXiv:2203.07421.
• Witten, Edward. “Anti-de Sitter Space, Thermal Phase Transition, and Confinement in Gauge Theories.” Advances in Theoretical and Mathematical Physics 2 (1998): 505–532. DOI; arXiv:hep-th/9803131.
• Witten, Edward. “A Note on Complex Spacetime Metrics.” arXiv:2111.06514.
• Witten, Edward, and Shing-Tung Yau. “Connectedness of the Boundary in the AdS/CFT Correspondence.” Advances in Theoretical and Mathematical Physics 3 (1999): 1635–1655. arXiv:hep-th/9910245.