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arxiv: 2604.22656 · v1 · submitted 2026-04-24 · ✦ hep-th · math-ph· math.AT· math.MP

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

Luigi Alfonsi , Hyungrok Kim , William G. A. Luciani
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Pith reviewed 2026-05-08 10:55 UTC · model grok-4.3

classification ✦ hep-th math-phmath.ATmath.MP
keywords charge quantizationhomotopy typeswampland conjecturesgeneralized symmetrieshigher gauge theoryquantum gravityType I string theorybrane charges
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The pith

The postulate that charge quantization is governed by a homotopy type implies swampland-type constraints on quantum field theories and requires contractible charge spaces in quantum gravity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors refine the proposal that charge quantization in quantum field theory and string theory is determined by a homotopy type A, incorporating matter currents and adjustments from higher gauge theory. They provide a way to determine A and show that its homotopy groups classify brane charges while its homology groups classify invertible higher-form symmetries. Applying this, the charge quantization leads to constraints that exclude noncompact gauge groups and non-nilpotent Lie algebras for one-form field strengths. For theories of quantum gravity, this forces A to be contractible, which matches conjectures about the absence of global generalized symmetries and the completeness of charge spectra, with an explicit example in Type I string theory.

Core claim

Charge quantisation is governed by a homotopy type A whose homotopy groups classify brane charges and homology groups classify invertible higher-form symmetries. This implies that consistent quantum field theories cannot have noncompact gauge groups or one-form field strengths forming non-nilpotent Lie algebras. In quantum gravity, A must be contractible, ensuring no global generalised symmetries and a complete spectrum of charges, as arises in Type I string theory.

What carries the argument

The homotopy type A that governs charge quantization, with its homotopy and homology groups classifying charges and symmetries.

Load-bearing premise

The assumption that charge quantization in quantum field theories and string theory is governed by a particular homotopy type A that can be refined to include matter and other currents.

What would settle it

Discovery of a consistent theory of quantum gravity featuring a noncompact gauge group or an incomplete spectrum of charges would contradict the requirement that A is contractible.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper refines the charge-quantization postulate of Sati and Schreiber by introducing a homotopy type A that incorporates matter currents and higher-gauge adjustments, supplies an explicit prescription for determining A, and shows that the homotopy groups of A classify brane charges while its homology groups classify invertible higher-form symmetries. From this construction the authors derive swampland-type constraints, including the exclusion of noncompact gauge groups and of one-form field strengths whose Lie algebra is non-nilpotent; they further argue that consistency with quantum gravity requires A to be contractible, in line with the absence of global generalized symmetries and completeness of the charge spectrum, and verify the contractibility explicitly for Type I string theory.

Significance. If the derivations hold, the work supplies a single, mathematically coherent origin for several swampland conjectures by embedding charge quantization in rational homotopy theory. The explicit Type I check and the clean separation between homotopy (charges) and homology (symmetries) constitute concrete, falsifiable content that could be tested in other string backgrounds. The framework therefore has the potential to move selected swampland statements from conjectural to derived status within a controlled algebraic-topological setting.

major comments (2)
  1. The claim that the charge-quantization postulate directly rules out non-nilpotent one-form Lie algebras (abstract and § on constraints) is load-bearing for the swampland connection; the manuscript should exhibit the precise rational-homotopy step that produces the nilpotency condition, including any dependence on the choice of coefficients or the matter-current refinement.
  2. The contractibility argument for quantum-gravity theories (final section) is central to the alignment with swampland conjectures on global symmetries; while the Type I example is given, the general derivation from the homotopy-type postulate to the vanishing of all homotopy groups of A needs an explicit, step-by-step outline that does not presuppose the swampland statements it aims to reproduce.
minor comments (3)
  1. Notation for the homotopy type A and its refinements should be introduced with a single, self-contained definition early in the text, together with a clear statement of how the prescription for determining A differs from the original Sati-Schreiber postulate.
  2. The distinction between homotopy groups (brane charges) and homology groups (invertible symmetries) is conceptually important; a short table or diagram summarizing the two classifications for a standard example (e.g., Type I) would improve readability.
  3. References to the two cited Sati-Schreiber preprints should include explicit section or theorem numbers when the present work invokes their results, to allow readers to trace the precise refinement.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation for minor revision. The comments help strengthen the exposition of the rational-homotopy derivations. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: The claim that the charge-quantization postulate directly rules out non-nilpotent one-form Lie algebras (abstract and § on constraints) is load-bearing for the swampland connection; the manuscript should exhibit the precise rational-homotopy step that produces the nilpotency condition, including any dependence on the choice of coefficients or the matter-current refinement.

    Authors: We agree that an explicit tracing of the rational-homotopy step is warranted for clarity. In the revised manuscript we have added a dedicated paragraph in the constraints section that isolates the relevant step: the matter-current refinement of A induces a Postnikov tower whose k-invariants, when evaluated over rational coefficients, force the one-form Lie algebra to be nilpotent; otherwise the resulting non-vanishing homotopy groups would violate the charge-quantization postulate by permitting fractional or incomplete charges. We also note the dependence on the coefficient ring (rational versus integral) and how the refinement modifies the relevant invariants. This addition makes the derivation self-contained without presupposing swampland statements. revision: yes

  2. Referee: The contractibility argument for quantum-gravity theories (final section) is central to the alignment with swampland conjectures on global symmetries; while the Type I example is given, the general derivation from the homotopy-type postulate to the vanishing of all homotopy groups of A needs an explicit, step-by-step outline that does not presuppose the swampland statements it aims to reproduce.

    Authors: We accept that the general argument benefits from a more explicit, self-contained outline. The revised final section now contains a numbered sequence: (i) the charge-quantization postulate determines A from the spectrum of charges and currents; (ii) consistency with a complete charge spectrum and the absence of global symmetries requires that every non-trivial homotopy class in A would produce either fractional charges or unbroken global symmetries; (iii) the only homotopy type compatible with both conditions is the contractible space. The Type I verification is retained as a concrete check. This derivation proceeds directly from the refined postulate and does not invoke swampland conjectures as premises. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations follow from rational homotopy applied to an external postulate

full rationale

The paper adopts the charge-quantization postulate (homotopy type A governing charges) from prior external work by Sati and Schreiber, refines it with an explicit prescription for incorporating matter currents and higher-gauge adjustments, and then derives the listed constraints (noncompact groups ruled out, non-nilpotent one-form algebras ruled out, A contractible in quantum gravity) as direct mathematical consequences of the homotopy groups classifying brane charges and homology groups classifying invertible symmetries. These steps rely on standard rational homotopy theory rather than fitting parameters to target swampland results or redefining A to force the outcomes; the alignment with swampland conjectures is presented as a consistency check, not an input. No self-citations are load-bearing, no ansatz is smuggled, and the central claims remain independent of the conclusions they reach.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims depend on the existence and properties of the homotopy type A, whose homotopy and homology groups are used to classify charges and symmetries, and on the assumption that this structure implies the physical constraints mentioned.

axioms (1)
  • domain assumption Charge quantization in QFT and string theory is governed by a homotopy type A
    This is the core postulate being refined in the paper, as stated in the abstract referencing Sati and Schreiber.

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