⚛️ high-energy theory

Non-invertible symmetries in the axiverse, and the imaginary wormholes

This paper investigates the invertible and non-invertible generalized symmetries in four-dimensional axiverse effective field theories, demonstrating how quantum-gravitational effects like wormholes and the Imaginary Distance Bound break non-invertible axion shift symmetries and, in N=1 models, induce towers of BPS instantons that generate infinite superpotential terms.

Original authors: Daniele Licciardello, Luca Martucci

Published 2026-06-25

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Daniele Licciardello, Luca Martucci

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: A Universe Full of Invisible Strings

Imagine our universe isn't just made of particles like electrons and protons, but also filled with invisible, vibrating "strings" of energy called axions. In this paper, the authors study a theoretical version of the universe called the Axiverse, which might contain hundreds or even thousands of these axion fields.

Think of these axions like the knobs on a giant, cosmic radio. Turning a knob changes the setting of the universe slightly. In physics, we often look for "symmetries"—rules that say, "If I turn this knob, nothing fundamental changes." Usually, we think of these rules as perfect, like a lock that can only be opened with a specific key (an invertible symmetry).

However, this paper discovers that in the Axiverse, the rules are much stranger. The "locks" are non-invertible. Imagine a lock that, when you turn it, doesn't just open or stay closed, but actually changes the shape of the door itself. You can't simply "undo" the turn to get back to exactly where you started. These are the non-invertible symmetries the authors study.

The Cast of Characters

To understand how these strange locks work, the authors introduce three types of "defects" or "glitches" in the fabric of space:

Winding Strings (The Vortex): Imagine wrapping a rubber band around a pole. If you have a field of axions, you can have "strings" where the axion field twists around a line. These are the vortex operators.
Magnetic Monopoles (The 't Hooft Lines): Think of these as isolated North or South poles (which we don't see in normal magnets). These are magnetic lines that thread through space.
Electric Charges (The Wilson Lines): These are the standard electric charges, like electrons, moving through space.

The paper shows that in the Axiverse, these three characters are deeply entangled. If you try to break the symmetry of one (like by introducing a magnetic monopole), it forces the other symmetries to break in a specific, hierarchical way. It's like a game of Jenga: pulling out one block (breaking a symmetry) causes the whole tower to wobble in a predictable pattern.

The Plot Twist: Imaginary Wormholes

The most exciting part of the paper involves wormholes. In science fiction, a wormhole is a tunnel connecting two distant points in space. In this paper, the authors look at a very specific, weird kind of wormhole called an Imaginary Wormhole.

Here is the analogy:

Imagine you are walking on a flat road (our normal universe).
Now, imagine you start walking "sideways" into a dimension that doesn't exist in real life, but exists in math. This is the "imaginary" direction.
The authors propose that if you try to walk too far in this imaginary direction, the math breaks down. The path becomes infinitely long or diverges.

They call this the Imaginary Distance Bound (IDB). It's like a "Do Not Enter" sign on the map of the universe. The paper argues that the universe must have a mechanism to stop you from crossing this line, or else the theory falls apart.

The Solution: Instantons as Roadblocks

So, what stops you from walking too far into the imaginary dimension? The paper suggests that Instantons act as the roadblocks.

Instantons are like sudden, tiny "bursts" of activity in the quantum foam. They are fleeting events that happen and disappear.
The authors argue that the universe is filled with an infinite tower of these instantons.
As you try to walk further into the "imaginary" wormhole, you eventually hit a wall of these instantons. They generate new forces (potentials) that push you back, ensuring you never cross the Imaginary Distance Bound.

In the specific case of Supersymmetric models (a more complex version of the universe where every particle has a "super-partner"), these instantons are special. They are called EFT Instantons. The paper claims these specific instantons are the "guardians" that enforce the rules of the universe, generating a "superpotential" (a kind of energy map) that keeps the axions stable.

The Hierarchy of Breaking

The paper also explains a "pecking order" of how these symmetries break:

The Strings: First, the winding strings (axion strings) might break the symmetry.
The Monopoles: If strings aren't enough, magnetic monopoles step in.
The Instantons: Finally, if the universe is still too "symmetric," the instantons (the quantum bursts) step in to break the symmetry completely.

The authors show that these layers are connected. You can't have the instantons breaking the symmetry without the strings and monopoles having already done their part. It's a chain reaction.

The Conclusion: Why This Matters

The paper concludes that the universe is not as "symmetric" as we might hope. The existence of these non-invertible symmetries and the Imaginary Distance Bound tells us that:

Symmetry is approximate: Global symmetries (perfect rules) don't exist in a universe with gravity. They are always broken by quantum effects.
The Universe is self-correcting: The "Imaginary Wormholes" act as a warning system. If the universe tries to become too symmetric (by letting axions shift too far), the wormholes signal that new physics (instantons) must kick in to fix it.
A New Tool for Physicists: By understanding these "non-invertible" rules, physicists can better predict which theories of the universe are possible and which are impossible (a concept known as the "Swampland").

In short, the paper uses the strange geometry of imaginary wormholes to prove that the universe is filled with an infinite army of instantons that constantly tweak the laws of physics to keep everything stable, preventing the universe from falling into a mathematical abyss.

Technical Summary: Non-invertible symmetries in the axiverse, and the imaginary wormholes

Problem Statement
This paper investigates the symmetry structure of four-dimensional effective field theories (EFTs) describing the "axiverse"—models containing a large number ( $n_A \gg 1$ ) of axion-like fields coupled to abelian gauge sectors. While global symmetries are generally expected to be broken in quantum gravity (the "No Global Symmetry Conjecture"), understanding the precise structure of these symmetries and their breaking mechanisms is crucial for constraining low-energy dynamics, particularly regarding the naturalness of axion potentials. The authors aim to:

Systematically identify the maximal global symmetry structure (both invertible and non-invertible) of general axiverse models, including their $N=1$ supersymmetric extensions.
Analyze the mechanisms by which quantum gravity breaks these symmetries, specifically utilizing the recent "Imaginary Distance Bound" (IDB) framework proposed in [46, 47]. This framework posits that the existence of wormhole saddles with imaginary axion profiles signals a breakdown of the EFT, necessitating symmetry-breaking corrections.

Methodology
The authors employ a bottom-up EFT approach, constructing general Lagrangians for axions coupled to $U(1)$ gauge fields, including Gauss-Bonnet and Pontryagin curvature-squared terms. They utilize the modern framework of generalized symmetries, where symmetries are encoded by topological operators of various codimensions.

Symmetry Construction: The authors construct topological defects for:
- Winding two-form symmetries (associated with axion strings).
- Magnetic one-form symmetries (associated with 't Hooft lines).
- Electric one-form symmetries and axion shift zero-form symmetries. Crucially, they demonstrate that while continuous axion shift symmetries are broken by couplings to gauge fields and gravity, a dense subset of non-invertible (categorical) symmetries survives. These are constructed using "half higher gauging" techniques and involve condensation defects that fuse non-trivially.
Supersymmetric Extension: The analysis is extended to $N=1$ models where axions combine with saxions into chiral multiplets. The authors focus on a specific class of models characterized by a homogeneous Kähler potential with degree $k$ , allowing for a well-defined large saxion regime.
Wormhole Analysis: The paper analyzes Euclidean wormhole solutions (Giddings-Strominger type) connecting two asymptotically flat regions. Following [46, 47], the authors analytically continue the asymptotic boundary conditions to imaginary values. They argue that the divergence of the path integral at a critical imaginary distance (the IDB) implies the necessity of non-perturbative corrections (instantons) to restore consistency.
Dual Formulations: The analysis frequently utilizes the dual description where axions are replaced by two-form potentials, and in the supersymmetric case, by linear multiplets, to clarify the interplay between instantons, strings, and wormholes.

Key Contributions and Results

Classification of Non-Invertible Symmetries:
The authors provide a general construction of non-invertible topological operators for axion shift symmetries and electric one-form symmetries in the presence of arbitrary axion-gauge couplings ( $K_{iIJ}$ ). They show that these symmetries are not independent; the existence of non-invertible axion shift symmetries relies on the presence of specific subgroups of magnetic one-form symmetries.
Hierarchy of Symmetry Breaking Scales:
By analyzing the condensation defects associated with non-invertible symmetries, the authors derive strict hierarchies between symmetry-breaking scales. Specifically, the breaking scale of a non-invertible electric one-form symmetry ( $E^{(e)}_\alpha$ ) is bounded by the masses of the lightest monopoles and axion strings that break the associated magnetic and winding symmetries ( $E^{(m)}_\beta$ and $E^{(w)}_\beta$ ). This establishes a "self-consistency" check where the breaking of one symmetry necessitates the breaking of others at specific scales.
Imaginary Wormholes and the IDB:
Applying the IDB framework to axiverse models, the authors demonstrate that the divergence of the wormhole partition function at a critical imaginary distance $D_{IDB}$ implies the existence of fundamental instanton corrections.
- Non-Supersymmetric Case: The IDB implies an axion Weak Gravity Conjecture (WGC) bound, requiring a tower of instantons to satisfy a convex hull condition in charge space.
- Supersymmetric Case ( $N=1$ ): The analysis reveals a distinction based on the homogeneity degree $k$ $k$ of the Kähler potential:
  - For $k \leq 3$ : The imaginary wormhole saddle is dominated by EFT instantons (BPS instantons associated with "movable" branes). These instantons generate an infinite tower of superpotential terms ( $F$ -terms). For the specific case $k=3$ , the authors show that the wormhole geometry preserves half of the supersymmetry at long distances, directly linking the wormhole divergence to the generation of a non-vanishing superpotential.
  - For $k > 3$ : BPS instantons are insufficient to satisfy the IDB. The authors argue that non-BPS instantons must exist to realize the bound. These instantons are predicted to generate $D$ -term corrections (modifying the Kähler potential) and must have actions smaller than their BPS counterparts ( $S_{non-BPS} < S_{BPS}$ ), a phenomenon referred to as an "inverse BPS bound."
Role of Gauss-Bonnet Terms:
The paper investigates the impact of the Gauss-Bonnet term (coupled to saxions) on wormhole solutions. For $k \geq 3$ , this term provides a finite suppression that does not alter the main conclusions. However, for $k < 3$ , the term leads to divergences that require careful regularization (e.g., principal value), suggesting that the interpretation of wormholes in these regimes is more subtle and that the Gauss-Bonnet term cannot be treated as a negligible perturbation.

Significance and Claims
The authors claim that their work provides a unified framework for understanding the symmetry structure of axiverse models, extending previous analyses of the axion-Maxwell system to general multi-axion and supersymmetric settings.

Naturalness and Symmetry Breaking: The paper argues that the non-invertible nature of axion shift symmetries imposes strong constraints on the generation of axion potentials. The breaking of these symmetries is not arbitrary but is tied to the mass scales of monopoles and strings via the derived hierarchies.
Quantum Gravity Constraints: By adopting the IDB perspective, the paper offers a bottom-up derivation of the necessity for infinite towers of instantons in quantum gravity. It specifically identifies EFT instantons as the mechanism realizing the IDB in supersymmetric sectors with $k \leq 3$ , providing a concrete realization of the "Supersymmetric Genericity Conjecture" (that superpotentials are generically generated).
Distinction between BPS and Non-BPS: A significant result is the prediction that for $k > 3$ , the IDB forces the existence of non-BPS instantons that violate the standard BPS bound, suggesting a rich non-perturbative structure in the UV completion of these theories.

The paper concludes that while many open questions remain (such as the precise microscopic realization of non-BPS instantons or the full implications for the strong CP problem in the presence of an axiverse), the interplay between non-invertible symmetries and imaginary wormholes serves as a powerful organizing principle for decoding quantum gravity effects in low-energy effective theories.