Wormholes and the imaginary distance bound

Abstract: Some of the simplest wormhole solutions involve massless scalar fields that take imaginary values. Massless fields can be interpreted as coupling constants in asymptotically flat or asymptotically AdS gravity theories. We argue that wormhole effects imply an imaginary distance bound, an upper limit for the analytic continuation of the theory to imaginary values of these couplings. In string theory examples, we find explicit effects that render the low-energy theory invalid either before or precisely at this wormhole limit. We argue that the existence of such effects enforcing the distance bound is a general feature of string theories containing wormholes. In some cases, the bounds we discuss coincide with the weak gravity conjecture, and with the Kontsevich-Segal-Witten condition on complex metrics.

• The paper shows that Euclidean wormhole solutions supported by imaginary scalar fields impose an intrinsic upper bound on analytic continuation of coupling constants in quantum gravity.
• It employs rigorous gravitational analysis using string theory compactifications and dimensional reductions to demonstrate the breakdown of effective field theory at the IDB.
• The study unifies the weak gravity conjecture and the KSW criterion, suggesting a universal swampland constraint on moduli space dynamics.

Wormholes, Scalar Fields, and the Imaginary Distance Bound in Quantum Gravity

Overview and Motivation

This work rigorously analyzes a class of Euclidean gravitational wormhole solutions supported by massless scalar fields configured along imaginary directions in moduli space. The central thesis is that the existence of these wormhole solutions, which connect asymptotic regions differing by a specific imaginary displacement in scalar vevs (interpreted as coupling constants), imposes an intrinsic upper bound on possible analytic continuations of these couplings. This Imaginary Distance Bound (IDB) is argued to be a general feature of consistent quantum gravity theories and is substantiated by direct analysis in string theory compactifications, reductions, and their associated low-energy limits. The paper further relates the IDB to the weak gravity conjecture (WGC) and the Kontsevich-Segal-Witten (KSW) criterion, suggesting a unifying principle governing bounds on couplings and moduli distances in the Swampland context.

Scalar Field Wormholes and Analytic Continuation

The authors construct explicit Euclidean wormhole solutions in Einstein gravity coupled to real, massless scalar fields $\varphi_a$, focusing on solutions where the scalar vevs are continued to imaginary values. In this context, the scalar displacement across the wormhole, $\Delta \varphi_{\infty}$, becomes purely imaginary and supports a nontrivial wormhole geometry. The asymptotic values of these scalars can be interpreted as coupling constants in the associated gravity or gauge theory—particularly in AdS/CFT duality, they directly correspond to dual CFT couplings.

The key geometric property is that the wormhole exists only when the difference in asymptotic scalar values is precisely a critical imaginary displacement:

$$-i \Delta \varphi_\infty = \pi \sqrt{\frac{2(D-1)}{D-2}}$$

In flat space, the action of these solutions vanishes for this critical value, and more generally, in AdS, the wormholes exist over a finite range of imaginary displacements, with the action increasing with separation. As soon as the displacement exceeds this bound, the semiclassical solution ceases to exist or becomes pathologically singular.

Figure 1: Analytic continuation in the imaginary direction for scalar vevs $\varphi_\infty$; the appearance of an on-shell flat space wormhole (at a finite imaginary distance) sets the upper bound for analytic continuation in any consistent theory.

Figure 2: Schematic depiction of a wormhole geometry supported by imaginary-valued scalar fields.

The Imaginary Distance Bound (IDB) and Its Physical Origin

The authors conjecture that, in any UV-complete theory of gravity, an intrinsic mechanism always “enforces” the IDB: attempting to set boundary conditions for the scalars corresponding to an imaginary displacement beyond the bound leads to a breakdown of effective field theory (EFT) or the emergence of unsuppressed corrections that destroy semiclassical control.

This is explained physically: generic quantum gravity corrections, often of the type $c_n e^{i n \varphi}$ for some $n$, remain controlled for real $\varphi$ but can become exponentially large for imaginary $\varphi$ (e.g., instanton actions becoming negative or vanishing). Thus, nonperturbative or higher-derivative effects ensure the breakdown of the low-energy EFT precisely when the IDB is reached, or potentially before, depending on the model (as demonstrated in specific string theory compactifications later in the text).

Connections with Swampland Conjectures and KSW Criterion

Swampland Distance and Weak Gravity Conjectures

Through a series of dimensional reduction examples, the IDB is shown to coincide with conditions imposed by the WGC. For instance, in Kaluza-Klein reductions, the requirement that charged superextremal particles become relevant before the wormhole solution is achieved ensures that instanton-like corrections dominate, preventing semiclassical wormhole formation. In these setups, the radius of the complexified moduli space geodesic is precisely set by the WGC bound.

Figure 3: Log-log plot of the action for AdS$_5$ wormholes, including the asymptotic small and large $a_0$ behaviors, illustrating growth in action with wormhole size.

Figure 4: Scalar field trajectories as geodesics in a complexified moduli space (e.g., AdS$\Delta \varphi_{\infty}$0), showing the portion traversed by the wormhole depends on the overall moduli space radius.

Kontsevich-Segal-Witten (KSW) Condition

For toroidal compactifications, the IDB is shown to align with the KSW criterion, which constrains the analytic continuation of metrics in the gravitational path integral. Specifically, the maximum allowed imaginary distance in the scalar moduli space matches the KSW requirement that all metric eigenvalues stay within an allowed sector in the complex plane (i.e., the argument of the determinant does not exceed $\Delta \varphi_{\infty}$1). Thus, the imaginary displacement bound from wormholes is argued to be a generalization of the KSW criterion, applicable to all geometric and non-geometric moduli.

Examples from String Theory: Breakdown at or Before the Bound

By analyzing IIB string theory compactifications, AdS/CFT duality, and toroidal reductions, the paper shows that the effective theory is rendered invalid at or before the IDB due to stringy corrections, instanton effects, or the appearance of tachyonic modes. For instance, in AdS$\Delta \varphi_{\infty}$2 (dual to $\Delta \varphi_{\infty}$3 SYM), the scalar moduli space action develops divergent corrections (e.g., in the $\Delta \varphi_{\infty}$4 coupling) as instanton corrections become unsuppressed exactly at the IDB.

Figure 5: Moduli space slice showing the red wormhole trajectory crossing loci (yellow region boundaries) where instanton actions change sign and higher-derivative corrections become uncontrolled.

The analysis further demonstrates that in certain reductions (e.g., $\Delta \varphi_{\infty}$5 compactification), worldsheet instanton corrections become actionless before the total geodesic distance used by a diagonal wormhole solution is reached, thus invalidating the wormhole from UV data. In all explicit string constructions presented, dangerous corrections align with or precede the IDB, supporting the universality of the bound.

Figure 6: Contours in the complex $\Delta \varphi_{\infty}$6-plane; SSS double cone solution and the real-$\Delta \varphi_{\infty}$7 contour, illustrating wormhole/black hole analytic structure relevant to the IDB.

Figure 7: Multi-scalar moduli space chart (case of $\Delta \varphi_{\infty}$8 reduction): red denotes wormhole path, green denotes appearance of tachyonic (massless) modes marking breakdown of EFT before the full wormhole distance.

Figure 8: Moduli space in global coordinates, showing that geodesics of a given length always cross the envelope where corrections or tachyons appear, independently of path.

Implications and Theoretical Synthesis

Formulating the IDB unifies various heuristic and technical observations in the literature about the boundaries of reliable EFT in quantum gravity. The central implication is that complexification in moduli space is sharply limited, not solely in axion/periodic directions but for general scalar moduli. This unifies the WGC, KSW, and string instanton arguments under a single imaginary displacement restriction, with direct relevance for Swampland-type criteria.

Practically, this means:

• Path integral computations or analytic continuations in couplings are subject to absolute limits defined by quantum gravity microphysics.
• UV-complete theories “enforce” breakdown before dangerous wormholes destabilize the theory, much as the WGC ensures that extremal black holes are unstable.
• Analyses of holographic CFTs, string compactifications, and AdS/CFT setups must respect these analytic continuation boundaries.
• The conditions derived here may serve as new Swampland criteria, constraining EFT constructions even along unphysical, non-unitary (imaginary) directions in parameter space.

Theoretically, the data suggest the SWGC, KSW, and IDB may be different aspects of a deeper principle relating UV consistency, quantum gravity geometry, and analytic structure of physical couplings.

Conclusions

The existence of scalar field-supported wormhole solutions in gravity theories, requiring specific imaginary shifts in massless scalar fields (interpreted as couplings), implies an upper bound—the Imaginary Distance Bound—on the analytic continuation of those couplings. Detailed analysis in string theory, including explicit AdS/CFT and toroidal reductions, demonstrates that nonperturbative or higher-derivative corrections generically invalidate the low-energy effective theory at or before this bound. The IDB is shown to coincide with both the WGC and the KSW criterion in relevant limits. These results provide strong evidence for a universal quantum gravity swampland constraint on the extent of analytic continuation in scalar field moduli space, significantly constraining the landscape of consistent EFTs.

Figure 1: Analytic structure of scalar field space with the emergent wormhole boundary marking the upper limit for imaginary continuation in any consistent quantum gravity theory.