Euclidean Wormholes and Connected Correlators

A Euclidean wormhole contributes to a connected multi-boundary amplitude by filling disconnected boundary components with one connected bulk. The contribution is weighted by its renormalized action, topology, moduli measure, zero modes, determinants, and contour coefficient. In JT gravity these ingredients can be organized explicitly; outside such controlled models, the existence of a classical wormhole does not determine the exact correlator.

Required background. Quantum Information supplies tensor products and probabilistic covariance.

Helpful background. Semiclassical Gravity supplies Euclidean saddle expansions; Quantum Fields in Curved Spacetime supplies partition functions and determinants; Black-Hole Information supplies replica wormholes; Quantum Gravity supplies nonperturbative contour and observable questions.

Connected amplitudes from the saddle sum

For two boundary conditions $B_1$ and $B_2$, define the connected gravitational amplitude by

$$Z_{\mathrm c}(B_1,B_2) := Z(B_1\sqcup B_2)-Z(B_1)Z(B_2).$$

At leading semiclassical order, disconnected bulk fillings cancel in this subtraction. A connected wormhole saddle $g_{12}$ gives

$$Z_{\mathrm c}(B_1,B_2) \simeq \mathfrak n_{12} e^{-I_{\mathrm{ren}}[g_{12}]} \frac{\det{}'\mathcal O_{\mathrm{gh}}} {\sqrt{\det{}'\mathcal O_{\mathrm{bos}}}} \int_{\mathcal M_{12}}d\mu +\cdots .$$

$\mathfrak n_{12}$ is the contour or thimble coefficient, primes omit zero modes, and $\mathcal M_{12}$ is the remaining moduli space with its induced measure. Fermionic determinants, matter factors, and phases must be added when present. The formula is schematic but displays why comparing only classical actions is incomplete.

For source-dependent boundaries, derivatives of $W=\log Z$ give connected correlators. One must not confuse $Z_{\mathrm c}$ with a normalized connected observable such as $Z_{\mathrm c}/[Z(B_1)Z(B_2)]$; their topology weights differ by the disk normalizations.

JT gluing and topology counting

In JT gravity an orientable surface of genus $g$ with $b$ asymptotic boundaries has Euler characteristic

$$\chi=2-2g-b$$

and topological weight $e^{S_0\chi}$. Two disks have total $\chi=2$, while the connected genus-zero cylinder has $\chi=0$. Its normalized connected contribution is therefore suppressed by $e^{-2S_0}$ before boundary wavefunctions and moduli integrals are included.

The cylinder can be cut along a geodesic of length $\ell$ into two trumpets. Its amplitude has the gluing structure

$$Z_{0,2}(\beta_1,\beta_2) = \int_0^\infty \ell\,d\ell\, Z_{\mathrm T}(\beta_1,\ell) Z_{\mathrm T}(\beta_2,\ell).$$

The measure $\ell\,d\ell$ incorporates the twist modulus: gluing two geodesics allows a relative twist ranging over an interval of length $\ell$. This construction, together with higher-genus Weil–Petersson volumes, is derived in Saad, Shenker, and Stanford 2019, §3. It gives a concrete connected correlator rather than relying on a drawing of a wormhole.

For example, in the normalization $Z_{\mathrm T}(\beta,\ell)=\sqrt{C/(2\pi\beta)}\,e^{-C\ell^2/(2\beta)}$, with positive boundary coupling $C$, the Gaussian integral gives

$$Z_{0,2}(\beta_1,\beta_2) =\frac{\sqrt{\beta_1\beta_2}}{2\pi(\beta_1+\beta_2)}.$$

Here geodesic lengths are in AdS-radius units and $C/\beta$ is dimensionless. To check the normalization, use $\int_0^\infty\ell e^{-a\ell^2}d\ell=1/(2a)$. The coupling cancels; the answer is symmetric in the two boundaries and equals $1/(4\pi)$ for equal positive $\beta_i$. It is the unnormalized connected cylinder amplitude, not yet divided by the two disk amplitudes.

Zero modes, weights, and dominance

Gauge zero modes are removed from determinants and replaced by collective-coordinate integrals divided by the gauge-group volume. Genuine moduli are integrated rather than treated as Gaussian directions. A negative mode can signal that the real saddle is not on a convergent steepest-descent cycle; a complex phase can produce cancellation with another saddle.

The determinant power follows from the Gaussian integral. Expanding a bosonic field as $\Phi=\Phi_*+\eta$ gives $I=I_*+\tfrac12\langle\eta,\mathcal O\eta\rangle+\cdots$, so each nonzero real mode contributes the inverse square root of its eigenvalue. Grassmann ghost integration contributes its determinant in the numerator. This derivation also exposes a failure mode: treating a zero eigenvalue as an ordinary Gaussian makes the expression divergent instead of replacing it by the correct collective coordinate and Jacobian.

Let $I_{12}-I_1-I_2=\Delta I$. After normalizing by the one-boundary amplitudes, the connected magnitude scales schematically as

$$\frac{Z_{\mathrm c}}{Z_1Z_2} \sim e^{-\operatorname{Re}\Delta I} \times \text{determinants and moduli}.$$

If $\operatorname{Re}\Delta I$ is parametrically large and positive, the wormhole is perturbatively small. Near degeneracy, the omitted factors and additional topologies decide the result. An infinite topology sum can be asymptotic, so term-by-term control does not by itself define the nonperturbative answer.

Multi-boundary Euclidean solutions in asymptotically AdS settings are described in Maldacena and Maoz 2004, §§2–4. Their existence is a solution-level statement, distinct from contour inclusion or dominance.

Interpretation and independent checks

In the JT matrix-integral dictionary, $Z_{\mathrm c}(\beta_1,\beta_2)$ equals an ensemble connected correlator

$$\left\langle Z_H(\beta_1)Z_H(\beta_2)\right\rangle_{\mathrm c}.$$

This is compatible with nonzero covariance. For one fixed Hamiltonian, however, the product of its two numerical partition functions factorizes exactly. The wormhole calculation therefore needs an ensemble, conditional sector, or nonperturbative fixed-theory interpretation before it can be equated with a boundary observable.

Topology check. Adding a handle changes $\chi$ by $-2$ and hence supplies an $e^{-2S_0}$ suppression before moduli growth.

Cluster check. If a proposed normalized connected correlator remains order one when boundary data enter a regime where the declared boundary theory has exact independence, either the observable is averaged, the saddle approximation is incomplete, or the boundary premises differ.

This page evaluates the semiclassical contribution; it does not settle the single-theory factorization problem or rederive generic QFT path integrals.

A final normalization check is obtained by removing the connected topology from the declared saddle set. The subtraction then gives $Z_{\mathrm c}=0$ at this order, as it must. If a residual remains, it comes from inconsistent one-boundary normalization, another connected saddle, or shared boundary data rather than the cylinder that was removed.

Exercises

Suppose each disk amplitude is $Z_i=e^{S_0}A_i$, while the cylinder amplitude is $Z_{0,2}=C$. Compute the normalized connected ratio. How does one additional handle change it before moduli factors?

Solution

The cylinder ratio is $C/(e^{S_0}A_1e^{S_0}A_2)=e^{-2S_0}C/(A_1A_2)$, displaying the expected Euler-characteristic suppression. An additional handle reduces $\chi$ by two, so it contributes another factor $e^{-2S_0}$ before its own moduli volume and determinant are included. Those factors can affect coefficients but do not change the topological exponent.

References

• Maldacena and Maoz 2004, Juan Maldacena and Liat Maoz, “Wormholes in AdS,” Journal of High Energy Physics 02 (2004) 053.
• Saad, Shenker, and Stanford 2019, Phil Saad, Stephen H. Shenker, and Douglas Stanford, “JT Gravity as a Matrix Integral,” arXiv:1903.11115 (2019).