Axions on de Sitter space

Abstract: We study a massless minimally coupled compact scalar, or axion, on global $D$-dimensional de Sitter space (dS$<em>D$). We quantise the theory canonically, determine the quantum dS charges, and find that the axion zero mode supplies a quantum-mechanical factor beyond the oscillator Fock space, $\mathcal{F}$. The full Hilbert space is $\mathcal{H}=L<sup>2(S<sup>1)\otimes\mathcal{F}$, with the integer quantum-mechanical momentum on $L<sup>2(S<sup>1)$ identified with the conserved $\mathrm{U}(1)$ shift charge. The 1-particle unitary irreducible representation (UIR) of the dS group, $\mathrm{SO}(D,1)$, captures the oscillator sector, but misses the zero mode. We find that the neutral 0-particle state is dS-invariant and normalisable. Charged 0-particle states are normalisable, but only $\mathrm{SO}(D)$ invariant. This implies that geodesic observers related by dS boosts do not agree on the particle number in a charged sector, an effect absent in QFTs equipped only with the standard Bunch-Davies vacuum. We compute field-strength Wightman 2-point functions in charged sectors and find that they are Hadamard. For non-zero charge they are not dS-invariant at finite global times, but they are asymptotically so at early and late times. We complement this analysis with a Euclidean perspective. The ordinary $D$-sphere path integral, $Z</em>{S<sup>D}$, written in terms of Harish-Chandra characters, has access only to the neutral sector. Charged sectors require vertex-operator insertions, and summing over them gives a decorated sphere path integral, $$\widehat{Z}<em>{S^D}=Z</em>\text{QM}\,Z_{S<sup>D}$$, that captures the entire Hilbert space, with $Z_\text{QM}$ denoting the partition function of a quantum rotor at a dimension-dependent effective temperature. Finally, in dS$_3$, we use the duality between an axion and a photon to translate our results to electromagnetism, where the axion zero mode gives rise to magnetic monopoles.

• The paper conducts a rigorous canonical and path-integral quantization of a compact axion field on global de Sitter, distinguishing zero and oscillator modes and isolating a unique de Sitter-invariant vacuum.
• It shows that the interplay between zero-modes and oscillator excitations leads to observer-dependent charged sectors, challenging traditional de Sitter QFT by breaking full de Sitter symmetry.
• The study establishes a duality with higher-form gauge theories in dS3, offering practical insights for axion cosmology, quantum gravity, and the role of symmetry in curved spacetimes.

Axions on de Sitter Space: Quantum Theory of Compact Scalars in $dS_D$

(2606.28858)

Overview

The paper conducts a rigorous canonical and path-integral quantization of a massless, minimally coupled compact scalar (“axion”) field in global $D$-dimensional de Sitter spacetime ($dS_D$). Focusing on the compactness of the target space ($S^1$), the authors contrast the Hilbert space, symmetry, and observable structure of compact versus non-compact scalar QFT on de Sitter. Notably, the work analyzes the interplay between zero modes and oscillator modes, Euclidean/Lorentzian perspectives, and the duality with higher-form theories (including electromagnetism in $dS_3$). The study clarifies representation-theoretic aspects and their implications for de Sitter QFT, demonstrating non-trivial observer dependence and explicit calculation of functional determinants, path integrals, and partition functions.

Hilbert Space Structure and De Sitter Symmetries

The canonical quantization in Lorentzian global $dS_D$ reveals that the full quantum Hilbert space is

$$\mathcal{H} = L^2(S^1) \otimes \mathcal{F}_{\mathrm{osc}}$$

where $L^2(S^1)$ describes the quantum mechanics of the spatial zero-mode (the “particle on a circle”) and $\mathcal{F}_{\mathrm{osc}}$ is the usual Fock space of non-zero oscillator modes. The quantization unambiguously identifies the discrete $U(1)$ “shift” charge as the eigenvalue of integer momentum on $D$0.

The key structure is that the single-particle sector associated with the Fock space realizes the unitary irreducible representation (UIR) of $D$1, classified by the principal or exceptional series based on $D$2. However, this UIR fails to capture the zero-mode sector; the latter is inert under $D$3 in the neutral state and transforms non-trivially in charged sectors.

A crucial result is that the set of normalizable zero-particle states $D$4 are all $D$5-invariant, but only $D$6 is invariant under the full de Sitter group $D$7. This sharply separates the neutral vacuum (unique, normalizable, de Sitter invariant) from the infinite tower of normalizable charged “vacua” which are physically inequivalent under boosts and exhibit a non-trivial interplay between zero-modes and particle number.

Observer Dependence and Particle Content

The analysis exposes an observer-dependent notion of “particles” in the charged sectors. Geodesic observers related by de Sitter boosts generally disagree on particle number in $D$8. Under such boosts, zero-mode charge is converted into oscillator occupation, breaking the de Sitter invariance of the corresponding particle number operator. This effect is absent in standard QFTs on $D$9 equipped with only the Bunch–Davies vacuum and has no analog in strictly non-compact QFTs where nontrivial charged vacua are not normalizable.

Physically, this means that in QFTs where compactness plays a role and $dS_D$0 shift sectors are superselected, global de Sitter symmetry is genuinely "broken" in charged sectors even in the free theory: inertial observers cannot agree on the notion of vacuum. This is reinforced by explicit calculations of field strength Wightman functions, which are Hadamard for all $dS_D$1 but only de Sitter-invariant asymptotically at early/late times, not at general times for $dS_D$2.

Path Integral and Euclidean Formulation

From the Euclidean perspective, the authors show that the standard $dS_D$3 path integral $dS_D$4 corresponds to summing only over the neutral sector. The inclusion of nontrivial charge sectors requires the insertion of local vertex operators $dS_D$5, corresponding to "decorating" the path integral. The full partition function is then:

$dS_D$6

where $dS_D$7 is the partition function of a quantum rotor (free particle on $dS_D$8) at a dimension-dependent effective temperature, and $dS_D$9 captures the oscillator sector via Harish-Chandra characters of the relevant $S^1$0 representation.

An explicit relation is drawn between sphere partition functions, operator insertions, and the characterization of the Hilbert space in the operator formalism, with careful attention paid to analytic continuation from Euclidean to Lorentzian signature and the role of symmetry in state preparation.

Duality and Gauge Theory Connections

In $S^1$1, the duality between a compact scalar and a $S^1$2 photon (i.e., a 1-form gauge field) is elucidated in both Hilbert space and operator terms. The oscillator sector maps directly to the photonic UIR, while the axion zero modes correspond to quantized magnetic monopole sectors permitted by the topology of global $S^1$3. The analysis clarifies how the quantum mechanical zero-mode (and nonlocal disorder operators) are faithfully realized in the electromagnetic theory, including their gravitational and topological implications.

Theoretical and Practical Implications

Theoretical Implications:

This framework provides a comprehensive and rigorous quantization of compact scalar fields on maximally symmetric spacetimes with positive cosmological constant, filling long-standing gaps in the literature. The observer dependence, representation-theoretic separation, and explicit Euclidean-Lorentzian matching have implications for axion cosmology, the role of global modes in inflationary and current-epoch de Sitter phases, and the emergence (or absence) of spontaneous symmetry breaking in compact spacetimes. The explicit identification with quantum rotor partition functions hints at deeper algebraic structures—possibly infinite-dimensional current algebras—governing the zero mode sector, even in higher dimensions. The duality with higher form gauge theories concretely maps the topological complexity of de Sitter spacetime to quantized observable sectors.

Practical Implications and Future Developments:

• For cosmology: The multiplicity and ambiguity of de Sitter “vacua” in compact scalar QFTs impact initial conditions, particle production, and perhaps observable features in primordial perturbations, especially if axion-like fields influence the early universe or dark energy dynamics.
• For quantum gravity and holography: The path integral construction suggests that standard $S^1$4 partition functions miss important superselection sectors, which could be relevant in gravitational amplitudes or in proposals for Euclidean quantum gravity in de Sitter.
• For algebraic QFT: The explicit identification of zero-mode partition functions as $S^1$5-functions (quantum rotor partition functions) points to potentially fruitful connections with modular invariance, current algebra representations, and the classification of superselection sectors.
• For gauge theory dualities: The correspondence between compact scalars and monopole sectors in three-dimensional de Sitter space should influence calculations of electromagnetic and dual higher-form observables in cosmological settings.

Future directions include the analysis of interacting or explicitly symmetry-breaking potentials, higher-form symmetry realizations, systematic treatment of gravitational backreaction (including effects like wormholes and new Euclidean de Sitter saddles), and exploration of state/observable algebras organizing the zero-mode sectors non-perturbatively.

Conclusion

By providing formal operator and path-integral quantization of compact scalars (“axions”) in global $S^1$6, this work rigorously exposes the necessity of accounting for compactness-induced zero-mode sectors, the observer-dependence of “particles” in such sectors, and the full characterization of de Sitter-invariant states. The connection to gauge theory duality, the careful treatment of Hilbert space superselection, and the explicit correspondence between Euclidean decorated partition functions and Lorentzian charged sectors deliver a systematic foundation for further investigations in quantum field theory and quantum gravity in cosmological spacetimes.