Imaginary Distance Bound in Quantum Gravity

• Imaginary Distance Bound is defined as the maximal analytic continuation of couplings into imaginary directions before a quantum gravity EFT breaks down, based on Euclidean wormhole analyses.
• It is derived from explicit wormhole solutions in flat and AdS spaces, where bounds depend on spacetime dimensions and signal where UV effects or nonperturbative corrections must intervene.
• The IDB connects to broader concepts like the Weak Gravity Conjecture and the KSW condition, offering insights into when higher-derivative corrections, towers of states, or instanton effects become significant.

Searching arXiv for recent and foundational papers relevant to "Imaginary Distance Bound." First, searching for the exact phrase and closely related wormhole-based work. Now searching for foundational distance-conjecture and information-theoretic distance papers that clarify what is and is not meant by “imaginary distance.” The Imaginary Distance Bound (IDB) is a proposed upper limit on how far a quantum-gravity effective field theory can be analytically continued along imaginary directions in coupling or moduli space before the low-energy description breaks down. In the formulation based on Euclidean wormholes, massless scalars parameterize a manifold with metric $h_{ab}(\phi)$ and proper distance $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$; after continuation $\phi \to i\phi_I$, wormhole solutions imply a maximal proper imaginary displacement $\tau$ compatible with the validity of Einstein gravity coupled to massless scalars (Maldacena et al., 6 May 2026). This notion is specific to complexified couplings in quantum gravity. It does not mean that the metric distance itself becomes imaginary: in information geometry and in standard Swampland-distance formulations, the relevant distances are real and non-negative (Stout, 2021, Etheredge et al., 2022).

1. Definition and kinematic setting

In the effective gravity description, massless scalar fields $\phi^a$ parameterize a moduli or coupling-constant manifold with metric $h_{ab}(\phi)$. The field-space distance between configurations is

$$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$

Along a classical wormhole solution, the scalars follow a geodesic, and the proper distance along the path is denoted $\tau$. In the IDB construction, the scalars are analytically continued to imaginary values, $\phi \to i\phi_I$, and $\tau$ is the corresponding proper imaginary distance (Maldacena et al., 6 May 2026).

The IDB is then an upper limit on the amount one can analytically continue the couplings along imaginary directions while the low-energy EFT of gravity, consisting of Einstein gravity plus massless scalars with standard-sign kinetic terms, remains valid. The proposed bounds depend on the asymptotics of the bulk spacetime. For asymptotically flat space,

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$0

while for AdS wormholes with flat slicing,

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$1

Here $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$2 is the bulk spacetime dimension, and the bounds are derived from on-shell wormholes with imaginary scalars (Maldacena et al., 6 May 2026).

A central point is that the bound is not merely geometric. It is an EFT-validity statement: in a UV-complete theory, some effect intrinsic to the theory—nonperturbative corrections, instantons, towers of states, or higher-derivative terms—must invalidate the low-energy description either before or precisely at these $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$3-thresholds. This suggests that the IDB is a constraint on complexified couplings rather than a reformulation of ordinary real-distance conjectures.

2. Wormhole origin of the bound

For Euclidean gravity coupled to real massless scalars with canonical kinetic terms, the flat-space wormhole ansatz is

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$4

with solution

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$5

The throat radius is $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$6, and the scalar undergoes a jump in imaginary value between the two asymptotic regions. The total two-sided imaginary displacement is

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$7

which is independent of $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$8 (Maldacena et al., 6 May 2026).

In flat space, the Dirichlet on-shell action vanishes,

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$9

and for fixed wormhole charge $\phi \to i\phi_I$0 one has

$\phi \to i\phi_I$1

If the asymptotic boundary values are shifted away from the critical value $\phi \to i\phi_I$2, the off-shell variation behaves as

$\phi \to i\phi_I$3

This identifies the critical imaginary displacement as a marginal point: infinitesimal deviations drive the throat radius either to zero or to infinity (Maldacena et al., 6 May 2026).

The AdS analysis is more structured. For spherical slicing, the renormalized on-shell action is positive, whereas flat-slicing AdS wormholes have zero action and support a smaller universal displacement,

$\phi \to i\phi_I$4

with

$\phi \to i\phi_I$5

This is why the AdS bound is stronger than the flat-space bound in the flat-slicing channel (Maldacena et al., 6 May 2026).

The single-sided interpretation is obtained by taking half of the two-sided displacement. In that interpretation, analytic continuation by more than $\phi \to i\phi_I$6 in flat space, or $\phi \to i\phi_I$7 in AdS, produces divergences in coarse-grained observables or makes previously small terms such as $\phi \to i\phi_I$8 exponentially large after $\phi \to i\phi_I$9. This motivates the claim that a UV-complete theory must cease to be described by the original EFT at or before the IDB.

3. String-theoretic mechanisms that enforce the bound

The wormhole argument becomes concrete in string examples because identifiable UV effects invalidate the EFT before or at the wormhole threshold. In the type IIB axion-dilaton system, the moduli metric is

$\tau$0

With

$\tau$1

the moduli space becomes Lorentzian $\tau$2 in $\tau$3. The D$\tau$4-instanton actions are

$\tau$5

Along timelike geodesics symmetric around $\tau$6, the trajectory crosses the null lines $\tau$7, where the instanton action vanishes and the instanton sum becomes non-suppressed. The $\tau$8 coupling is controlled by an $\tau$9 Eisenstein series, and this coefficient diverges at $\phi^a$0 or $\phi^a$1, so the EFT is invalid before reaching $\phi^a$2 (Maldacena et al., 6 May 2026).

In torus compactifications, the same pattern reappears through worldsheet instantons and momentum-winding states. In type II on $\phi^a$3, a family of worldsheet instantons acquires zero action at

$\phi^a$4

which is strictly below

$\phi^a$5

for $\phi^a$6. In a Poisson-dual frame, the corresponding mass formula is

$\phi^a$7

and an explicit synchronized-modulus trajectory yields

$\phi^a$8

so states become massless at $\phi^a$9 and tachyonic beyond. In the AdS$h_{ab}(\phi)$0 example discussed there, this breakdown point equals $h_{ab}(\phi)$1 (Maldacena et al., 6 May 2026).

These examples support the stronger claim that the IDB is not only a wormhole criterion but also a diagnostic of when nonperturbative sectors, higher-derivative corrections, or towers of states become unavoidable. A plausible implication is that the bound is realized by the earliest UV obstruction available in a given compactification, rather than by a single universal mechanism.

4. Relation to the Weak Gravity Conjecture, the KSW condition, and axiverse instantons

A striking feature of the IDB proposal is that, in several settings, it coincides with previously known consistency bounds. In dimensional reduction of Einstein–Maxwell theory from $h_{ab}(\phi)$2 to $h_{ab}(\phi)$3 dimensions, the radion $h_{ab}(\phi)$4 and Wilson-line scalar $h_{ab}(\phi)$5 span an $h_{ab}(\phi)$6 moduli space with

$h_{ab}(\phi)$7

After continuing $h_{ab}(\phi)$8, the proper imaginary distance from the real slice to the Poincaré horizon is exactly $h_{ab}(\phi)$9. The wrapped-particle action vanishes on

$$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$0

for particles satisfying $$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$1. Since the Weak Gravity Conjecture requires

$$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$2

superextremal particles destabilize the wormholes before the forbidden region is crossed. With multiple $$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$3 factors, the IDB sphere in $$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$4-space reproduces the convex hull condition (Maldacena et al., 6 May 2026).

The IDB also matches the Kontsevich–Segal–Witten condition for complex metrics in torus compactifications. KSW requires

$$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$5

for the eigenvalues $$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$6 of the complex metric. For an internal torus metric $$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$7 with eigenphases $$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$8, one has

$$d=\int ds, \qquad ds^2 = h_{ab}(\phi)\, d\phi^a d\phi^b.$$9

and combining this with $\tau$0 gives

$\tau$1

exactly. In this sector, the flat-space IDB and KSW are therefore identical (Maldacena et al., 6 May 2026).

A four-dimensional axionic version sharpens the picture further. In multi-axion axiverse EFTs, wormholes imply a universal half-displacement magnitude

$\tau$2

and the critical displacement satisfies

$\tau$3

The divergence of the fixed-boundary wormhole sum then implies that instanton corrections of the form

$\tau$4

must become order one before the bound is reached. This yields the minimal condition

$\tau$5

the AWGC-type inequality

$\tau$6

and a convex-hull formulation in terms of charge-to-action vectors. In $\tau$7 axiverses, the same logic suggests towers or sublattices of EFT instantons, and for homogeneous degree $\tau$8 it leads to infinitely many superpotential terms

$\tau$9

(Licciardello et al., 24 Jun 2026).

5. Distinction from real-distance notions in information geometry and the Swampland program

The phrase imaginary distance can be misleading because several nearby literatures use real distances on spaces of states or moduli. In classical and quantum information geometry, the Fisher information metric and quantum information metric are positive semidefinite. The line element

$\phi \to i\phi_I$0

is therefore real and non-negative, and there is no notion of an imaginary-valued information distance. Even when parameters are complexified to define Kähler structures, the physical metric is obtained by restricting to the real submanifold. In that framework, infinite distance means hyper-distinguishability, not imaginary distance (Stout, 2021).

The sharpened Distance Conjecture is likewise formulated in terms of proper geodesic distance in a real moduli space. For scalar sector

$\phi \to i\phi_I$1

the proper distance is

$\phi \to i\phi_I$2

and the conjectured tower scaling is

$\phi \to i\phi_I$3

for the lightest tower in an infinite-distance limit (Etheredge et al., 2022). No imaginary-valued distance is introduced there.

A related but distinct usage appears in the study of locally symmetric moduli spaces. There, “imaginary directions” such as $\phi \to i\phi_I$4 denote Iwasawa $\phi \to i\phi_I$5-flows toward rational-parabolic cusps. For $\phi \to i\phi_I$6, the Poincaré metric is

$\phi \to i\phi_I$7

and the distance to the cusp behaves as

$\phi \to i\phi_I$8

which is again a real quantity. The exponential mass rate is determined by

$\phi \to i\phi_I$9

not by any imaginary metric length (Baines et al., 25 Aug 2025). This distinction is essential: the wormhole IDB concerns analytic continuation into complexified coupling space, whereas ordinary distance-conjecture statements concern real geodesic motion toward infinite-distance loci.

6. Terminological variants and unrelated uses of the phrase

The expression imaginary distance bound also appears in unrelated literatures, where it denotes a different bounded quantity rather than the wormhole constraint just described.

Context	Quantity bounded	Meaning
Wormholes and complexified moduli	$\tau$0, $\tau$1, $\tau$2	Maximal analytic continuation of couplings before EFT breakdown
Heavy-quark potential in plasma	$\tau$3	Minimal quark–antiquark separation where $\tau$4 first becomes nonzero
Imaginary-time quantum dynamics	$\tau$5	Geometric quantum speed limit for non-unitary evolution

In holographic heavy-quarkonium, the “Imaginary Distance Bound” is the minimal inter-quark separation $\tau$6 at which the imaginary part of the heavy-quark potential first becomes nonzero and negative. The threshold depends on the turning point conditions

$\tau$7

and $\tau$8 decreases with increasing velocity, with the decrease stronger for motion perpendicular to the plasma wind (Ali-Akbari et al., 2014). In a related classical-statistical study, the imaginary part of the static heavy-quark potential satisfies

$\tau$9

while in continuum HTL

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$00

(Boguslavski et al., 2022). These are bounds on an imaginary potential, not on imaginary excursions in moduli space.

A second distinct usage appears in imaginary-time quantum dynamics, where the “imaginary-distance bound” is a geometric quantum speed limit. For normalized evolving state $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$01, angular distance $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$02, and time-averaged energy dispersion $ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$03, one has

$ds^2 = h_{ab}(\phi)\,d\phi^a d\phi^b$04

Here the bounded quantity is the runtime required to traverse a given projective Hilbert-space angle under non-unitary Schrödinger evolution, not a complexified-field-space displacement (Kobayashi, 14 Aug 2025).

Taken together, these variants show that the wormhole-based IDB is a specialized term of art. Its distinctive content is the claim that Euclidean wormholes, and the UV phenomena associated with them, place an absolute upper limit on analytic continuation into imaginary coupling directions in quantum gravity.