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

Abstract: We study the symmetry structure of four-dimensional axiverse effective field theories with multiple axions coupled to abelian gauge sectors, including their extensions to broad classes of N=1 models. We identify the invertible and non-invertible generalized symmetries, and discuss the associated symmetry-breaking mechanisms together with the resulting hierarchies of energy scales. In particular, we discuss the quantum-gravitational breaking of non-invertible axion shift symmetries predicted by the existence of wormholes and by the corresponding recently proposed Imaginary Distance Bound. In N=1 axiverses, these wormhole-based arguments imply that towers of BPS EFT instantons play a distinguished role and generate infinitely many superpotential terms.

• The paper introduces a systematic framework for classifying both invertible and non-invertible generalized symmetries in multi-axion effective field theories.
• It applies the Imaginary Distance Bound to analyze wormhole contributions, establishing hierarchy bounds and quantum-gravity constraints.
• In supersymmetric models, the study reveals an infinite tower of BPS instantons generating superpotential corrections essential for moduli stabilization.

Non-Invertible Symmetries in the Axiverse and the Imaginary Wormholes

Framework and Motivation

The paper "Non-invertible symmetries in the axiverse, and the imaginary wormholes" (2606.26004) introduces a rigorous analysis of symmetry structures in multi-axion effective field theories (EFTs) and their implications in quantum gravity, particularly focusing on the axiverse paradigm, where $n_A > 1$ axion fields couple to abelian gauge sectors. The first goal is to systematically characterize both invertible and non-invertible generalized symmetries—including zero-form and higher-form symmetries—relevant for axion phenomenology, inflation, and string compactifications. The second goal is to elucidate quantum-gravitational mechanisms for symmetry breaking, employing the recent Imaginary Distance Bound (IDB) derived from analyses of axion wormhole contributions to the gravitational path integral.

In the axiverse, global symmetries (including axion shift symmetries) play a pivotal role in constraining the low-energy dynamics. Although quantum gravity prohibits exact global symmetries (No Global Symmetry Conjecture), effective theories often exhibit approximate symmetries, which inform hierarchy generation and naturalness criteria. The paper deploys the formalism of generalized (categorical) symmetries, encoding symmetries via topological defects, and addresses the constraints imposed by non-invertible symmetry operators on EFTs.

Symmetry Structures in Axiverse EFTs

Axion-Gauge Couplings and Supersymmetric Extensions

The EFTs examined contain $n_A$ periodic axions and $n_y$ $U(1)$ gauge fields, with generalized kinetic matrices and curvature-squared couplings (Gauss-Bonnet and Pontryagin). Supersymmetric extensions (minimal $N=1$) organize axions and saxions into complex chiral multiplets with field-dependent Kähler potentials and gauge couplings, preserving shift symmetries at the perturbative level.

Explicit construction of topological operators and identification of charged sectors (vortex operators for winding symmetries, 't Hooft operators for magnetic one-form symmetries) reveal the non-trivial interplay between different symmetry types. Extended operators (codimension, conserved currents) act as symmetry generators, while anomaly inflow and world-volume degrees of freedom play central roles in ensuring consistency.

Hierarchies and Symmetry Breaking Mechanisms

The breaking of higher-form symmetries (windings, magnetic, electric) necessitates the emergence of physical charged objects: axion strings, monopoles, and charged particles, each associated with characteristic energy scales below which symmetry-breaking effects manifest. The non-invertibility of certain symmetries results in intricate hierarchies between the corresponding breaking scales, bounded by tension/mass of the lightest relevant objects and further constrained by the Weak Gravity Conjecture (WGC).

Notably, the electric one-form and axion shift symmetries—rendered non-invertible by axion-gauge couplings—require careful analysis. The condensation defects built from fusion rules (e.g., half higher gauging) depend on finite subgroups determined by matrix coprimeness and lattice structures. The breaking scales of these symmetries are correlated with the scales of winding and magnetic symmetry breaking via precisely constructed diagrams and inequalities.

In supersymmetric theories, BPS objects (strings and instantons) are proven to saturate WGC bounds, and the completeness hypothesis ensures coverage of the charge lattice.

Quantum Gravity Constraints and Imaginary Wormholes

Imaginary Distance Bound and Wormhole Analysis

A central claim is the quantum-gravitational breaking of non-invertible axion shift symmetries, quantifiable via the novel Imaginary Distance Bound. Euclidean axion wormholes, originally studied by Giddings-Strominger, interpolate between asymptotic regions with fixed axion boundary conditions. Analytical continuation to imaginary values of boundary axions exposes divergences in the path integral, signaling a breakdown of the effective description when the imaginary displacement exceeds the IDB.

The IDB is computed in terms of the kinetic metric’s norm for trajectories in axion field space, leading to $|\Delta a| \leq D_{IDB}$. For each rational direction in axion charge space, the existence of instanton corrections before reaching the IDB is required. This enforces convex hull conditions on instanton charge-to-action vectors, with the minimal requirement that they densely populate the charge lattice—an axion WGC sublattice/tower formulation.

Supersymmetric Axiverse: BPS Instantons and Superpotential Corrections

In $N=1$ supersymmetric axiverse models, wormholes and instantons are tied to saxionic cones determined by the Kähler potential. For sectors with homogeneity degree $k \leq 3$, wormhole solutions directly select BPS instanton charges within the dual saxionic cone, and corresponding instanton actions precisely realize the IDB. Homogeneous solutions lead to infinite towers of BPS EFT instantons, which generate superpotential terms in the low-energy theory.

For $k > 3$, the bound can only be realized by non-BPS instantons, which provide D-term corrections (modifying the Kähler potential rather than the superpotential). Detailed analysis of fermionic zero modes confirms the nature of generated terms: quartic fermionic couplings (for $k \geq 4$) or bilocal superpotentials (for $n_A$0), supporting the Supersymmetric Genericity Conjecture.

The inclusion of Gauss-Bonnet curvature corrections is shown to be negligible for $n_A$1 but potentially significant for $n_A$2, due to divergences along the wormhole trajectory.

Bold Numerical Results and Claims

• Convex hull conditions: For all directions in axion charge space, the IDB enforces that instanton actions must satisfy $n_A$3, with saturation by BPS EFT instantons in supersymmetric models.
• Tower requirement: The imaginary wormhole argument predicts that quantum gravity must generate infinitely many (sub)lattice instanton corrections populating all rational directions, strengthening the axion WGC.
• Hierarchy bounds: Symmetry-breaking scales for non-invertible symmetries are strictly bounded by the tensions/masses of corresponding BPS or non-BPS objects, consistent with the completeness hypothesis and WGC.
• Superpotential proliferation: In $n_A$4 models, the IDB implies a proliferation of superpotential terms generated by infinite towers of EFT instantons, with practical implications for moduli stabilization and genericity.

Implications and Outlook

The theoretical implications are manifold. The categorical structure of symmetries and their non-invertible nature provide sharp organizing principles for dissecting quantum gravity constraints on EFTs with axions. The IDB serves as a robust bottom-up tool for predicting the existence of instanton towers, deeply influencing naturalness, vacuum structure, and moduli dynamics—especially in string-theoretic UV completions.

Practically, these results dictate generic expectations for the axion sector in models addressing the strong CP problem, inflation, and dark matter. The proliferation of superpotential terms has ramifications for moduli stabilization and trans-Planckian field excursions. The proof of hierarchies between symmetry-breaking scales supports the construction of technically natural models.

Future avenues include the incorporation of non-abelian sectors (with direct implications for CP violation), refinement of categorical fusion rules, and exploration of the full consequences of wormhole-driven symmetry breaking for cosmological and particle physics observables. The categorical symmetry framework, complemented by wormhole path integral analyses, promises to serve as a unifying language for decoding quantum gravitational effects in low-energy physics.

Conclusion

This paper formulates a comprehensive approach to generalized symmetries and symmetry breaking in axiverse EFTs, leveraging non-invertible categorical structures, instanton-induced corrections, and wormhole-based path integral diagnostics. The rigorous deployment of the Imaginary Distance Bound provides a quantitative bridge from quantum gravity to phase structure in axion-rich models, particularly within supersymmetric contexts, where infinite towers of BPS instantons generate superpotential terms. These conclusions—rooted in both formal symmetry analysis and semiclassical gravitational techniques—establish precise constraints on the spectrum and dynamics of axion models in the quest for technically natural and quantum-gravity-consistent theories.