Generalised Symmetries and Swampland-Type Constraints from Charge Quantisation via Rational Homotopy Theory

Abstract: Sati and Schreiber [arXiv:2402.18473, arXiv:2512.12431] have proposed that charge quantisation in quantum field theory and string theory is governed by a homotopy type $\mathcal A$. We provide a refinement of this postulate, incorporating other currents including matter, connecting it to adjustments in higher gauge theory and providing a prescription for determining $\mathcal A$, and show that, while the homotopy groups of $\mathcal A$ classify the possible brane charges, the homology groups of $\mathcal A$ classify the invertible higher-form symmetries. Furthermore, we show that the charge-quantisation postulate implies a number of non-trivial constraints on quantum field theories similar to those implied by swampland conjectures; in particular, it rules out noncompact gauge groups and one-form field strengths that form a non-nilpotent Lie algebra. Finally, we argue that for theories of quantum gravity the space $\mathcal A$ must be contractible, in accordance with the swampland conjectures on the absence of global generalised symmetries and the completeness of the spectrum of charges, and explain how this explicitly arises in the case of Type I string theory.

• The paper introduces a rigorous charge quantisation framework using rational homotopy theory to classify brane charges and higher-form symmetries.
• It employs L-infinity algebras, Sullivan models, and Whitehead towers to extract gauge-invariant fluxes and incorporate non-Abelian and matter current couplings.
• The work derives swampland-type constraints that forbid noncompact gauge symmetries and require a contractible classifying space in ultraviolet-complete quantum gravity.

Generalised Symmetries and Swampland Constraints from Charge Quantisation via Rational Homotopy Theory

Charge Quantisation Framework and Rational Homotopy Theory

The paper develops a mathematically rigorous refinement of the Sati–Schreiber postulate, which posits that charge quantisation in QFT and string theory is fundamentally governed by a homotopy type $\mathcal{A}$. The authors introduce rational homotopy theory as a central tool, allowing the classification of brane charges via homotopy groups $\pi_\bullet(\mathcal{A})$ and invertible higher-form symmetries via homology groups $H_\bullet(\mathcal{A})$. The technical apparatus combines $L_\infty$-algebras, Sullivan models, and Whitehead towers, providing a prescription for extracting $\mathcal{A}$ directly from physical field content and Bianchi/flux laws. The paper systematically formalises how local flux data, encoded in higher Gauss laws and adjusted higher-gauge theoretic structures, is globally refined to discrete charge sectors.

A crucial refinement is extending the charge quantisation framework beyond Abelian Maxwell-type theories to non-Abelian gauge theories and theories with matter currents. The authors employ adjusted inner-derivation $L_\infty$-algebras, allowing field strengths and magnetic currents to be encapsulated within a unified algebraic structure that describes both gauge and matter sectors. The underlying graded vector space and the associated Chevalley–Eilenberg algebra encode all Bianchi identities and invariant local observables. The global charge-quantisation law is then a quasi-isomorphism $\mathfrak{a} \simeq \mathfrak{l}(\mathcal{A})$, where $\mathfrak{a}$ is the algebra of gauge-invariant fluxes and $\mathfrak{l}(\mathcal{A})$ is the Whitehead $L_\infty$-algebra of the classifying space $\pi_\bullet(\mathcal{A})$0.

Interpretation: Brane Charges and Higher-Form Symmetries

The homotopy-theoretic perspective yields concrete classification results:

• Brane Charge Classification: The homotopy groups $\pi_\bullet(\mathcal{A})$1 enumerate allowed (possibly torsional) $\pi_\bullet(\mathcal{A})$2-brane charges. The triviality or non-triviality of these groups signals the presence or absence of stable topological charges for defects.
• Higher-Form Symmetry Realisation: The Pontryagin duals of homology groups $\pi_\bullet(\mathcal{A})$3 classify invertible Abelian $\pi_\bullet(\mathcal{A})$4-form symmetries. These arise from pulling back $\pi_\bullet(\mathcal{A})$5-valued cohomology classes from $\pi_\bullet(\mathcal{A})$6 to spacetime, generating topological operators in the path integral.

The framework reconciling brane charges and higher-form symmetries via the same classifying space $\pi_\bullet(\mathcal{A})$7 unifies several known symmetry structures in gauge and string theory. In particular, discrete electric and magnetic one-form symmetries in Yang–Mills theory and their roles in confinement are shown to correspond to homological invariants of $\pi_\bullet(\mathcal{A})$8. Moreover, the formalism yields symmetry phases for Abelian two-bundles, dyonic branes, and nontrivial torsion charges in higher gauge theories.

Swampland-Type Constraints from Charge Quantisation

A significant contribution of the paper is the derivation of nontrivial constraints on permissible quantum field theories—paralleling the swampland program:

• No Noncompact Gauge or 't Hooft-Anomaly-Free Global Symmetries: The rational homotopy classification forbids noncompact gauge groups and global symmetries, as gauging discrete non-torsion subgroups of such symmetries would contradict the requirement that $\pi_\bullet(\mathcal{A})$9 is torsion.
• Nilpotency of One-Form Flux Algebras: The algebraic constraints from Bianchi identities imply that only nilpotent one-form flux algebras are consistent. Simple Lie algebras are excluded from being allowable algebras for gauge-invariant one-form currents under this charge-quantisation proposal.
• Quantum Gravity Constraint—Contractibility of $H_\bullet(\mathcal{A})$0: The approach implies that in any ultraviolet-complete quantum gravity, the classifying space $H_\bullet(\mathcal{A})$1 must be contractible. This encodes the absence of global generalised symmetries and the completeness of the spectrum (i.e., all possible brane charges are realized dynamically rather than as defect charges). The contractibility arises explicitly in Type I string theory via the Whitehead tower construction, where each nontrivial homotopy group is trivialised by corresponding BPS and non-BPS branes.

These constraints formalise aspects commonly found in the swampland literature, such as the absence of global symmetries and the completeness of charged spectra, within a homotopy-theoretic and charge-quantisation context. The results have implications for the admissibility of symmetry structures in QFTs and string backgrounds and for the classification of possible topological sectors in quantum theories with gravity.

Methodological Applications and Examples

The paper systematically applies the refined charge-quantisation framework to several archetypal cases:

• Nonlinear Sigma Models: The defect brane charges correspond to the homotopy groups of the target space, while higher-form symmetries are classified by its homology.
• Abelian Higher Gauge Theory: Demonstrated with explicit computation of homology and cohomology for $H_\bullet(\mathcal{A})$2, identifying both one-form symmetries and their characteristic topological operators.
• Yang–Mills Theory: Both in four and higher dimensions, including effects of S-duality, electric–magnetic duality, and centre symmetries, with nontrivial cohomological and homotopical classification.
• String Theory with 16 Supercharges: Explicit construction via adjusted $H_\bullet(\mathcal{A})$3-algebra and Whitehead tower, realizing the trivialisation of all brane charges and symmetries in the classifying space—a concrete realisation of the quantum gravity constraints.

The paper further discusses ramifications in M-theory and Type II theories, noting subtleties in the inclusion of charged matter and the necessity for adjusted Bianchi identities.

Theoretical and Practical Implications

• Unified Symmetry Structure: The framework provides a unifying algebraic and topological classification of brane charges and higher-form symmetries in QFT, gauge theory, and string theory, linking local equations of motion, adjusted higher gauge theory, and global topology.
• Constraints on Theories: The derived swampland-type constraints cut across fields, restricting the spectrum of allowable gauge and symmetry structures in non-gravitational and gravitational theories.
• Rational Homotopy as Classification Tool: The methodological utilisation of $H_\bullet(\mathcal{A})$4-algebras, Sullivan models, and rational homotopy theory is general and robust, offering systematic computational prescriptions for many physical models.

The approach highlights how the inclusion of matter, adjustment of gauge structure, and the requirements imposed by gravity can all be captured in the same mathematical framework, predicting which symmetries can exist and which charges are possible.

Speculations for Future Developments

This homotopy-theoretic charge quantisation framework is expected to influence several areas:

• Symmetry TFTs and Extended Higher-Form Symmetries: Expansion to non-invertible, non-Abelian, and more general symmetry TFTs—currently only invertible Abelian symmetries are classified by the approach.
• Quantum Gravity and Holography: More precise formulations of swampland constraints in string theory backgrounds, potentially linking with K-theoretic spectra of brane charges and defect operators.
• Algorithmic Classification of QFTs: Implementation of these algebraic and topological tools may allow algorithmic inventorying of admissible quantum field theories according to their charge spectrum and symmetry content.

Methodologically, future work may more deeply connect with the formalism of stacks, higher categories, and twisted homotopy structures, especially in gravitational backgrounds with nontrivial bundle data.

Conclusion

The paper provides a rigorous algebraic-topological framework for charge quantisation in quantum field theory and string theory, encompassing brane charges and higher-form symmetries via rational homotopy theory. It formalises and refines the Sati–Schreiber charge quantisation postulate, derives swampland-type constraints forbidding noncompact symmetries and enforcing completeness in quantum gravity, and elaborates concrete prescriptions for constructing the classifying space $H_\bullet(\mathcal{A})$5 from physical data. The implications span theoretical, practical, and computational aspects, establishing a robust bridge between local field-theoretic dynamics and global topological structure. The approach promises further insight into symmetry structures, the classification of quantum field theories, and the mathematical foundations underpinning consistent quantum gravity theories.