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Relative periods are governed by target periods plus a defect homomorphism, admissible prequantization levels always form a cyclic group, and relative comoment maps are automatically unique, strict, and equivariant.

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Relative periods are governed by target periods plus a defect homomorphism, admissible prequantization levels always form a cyclic group, and relative comoment maps are automatically unique, strict, and equivariant.

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arxiv 2607.07149 v2 pith:OXB64KNE submitted 2026-07-08 math.SG

Periods, prequantization, and rigidity in relative multisymplectic geometry

Djounvouna Dinamo This is my paper
classification math.SG
keywords relativetheorembulk--boundarycriteriondifferentialfinitelyformforms
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The pith

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The reading

Ordinary multisymplectic geometry studies closed forms on one manifold. This paper studies a pair: a closed form ω on a manifold N and a form η on a second manifold M whose derivative equals the pullback of ω along a smooth map F:M→N. Together (ω,η) is a cocycle in the mapping cone of F. The new bookkeeping treats bulk and boundary, or target and source, as two halves of a single object.

The paper's first main result reduces 'is the pair integral?' to a finite check: integrate ω over cycles in N and compute a defect homomorphism on homology classes of M that die in N. A consequence is structural: the levels k at which k(ω,η) is integral always form a cyclic group k0Z, with k0 computable from finitely many integrals. This explains why level quantization in quasi-Hamiltonian geometry always looks like multiples of one integer.

The rest applies the machinery. A bulk-boundary action becomes the pairing of the relative cocycle with a relative cycle, so closed cocycles are exactly homotopy-invariant topological terms. A relative line bundle with a prescribed trivialization over M prequantizes (ω,η) exactly when the pair is relatively integral; for a Lagrangian submanifold this is the Bohr-Sommerfeld condition. A Noether identity splits conserved charges into bulk minus boundary. For n=1, if a comoment map exists it is unique, strict, and equivariant, because the only d_F-closed function on a connected target with nonempty source is zero—the Kostant-Souriau cocycle disappears. The paper closes by checking both degenerate cases (empty source, point target) and separating weak from strong nondegeneracy.

Core claim

The central load-bearing assertion is the relative period criterion and its consequences: a closed relative form ϖ=(ω,η) is relatively integral exactly when the absolute periods of ω are integral and the defect homomorphism Θϖ on ker F_* is integral (Theorem 3.8); admissible levels therefore always form a cyclic group k0Z (Proposition 3.13, Theorem 3.15); and relative prequantization exists iff ϖ is relatively integral (Theorem 5.3). The rigidity theorem (Theorem 7.1) adds that any existing comoment map is unique, strict and equivariant because H^0(Ω(F))=0. If the paper is correct, these statements are all proven as stated.

Load-bearing premise

The entire framework—relative Cartan calculus, L∞-algebra of observables, relative homotopy moment maps, and obstruction-vanishing—is imported from the author's own companion papers [19,20,21]. Section 2 says 'proofs may be found there', and all applications in §§4–8 depend on that machinery; if any companion result contains an error, the present theorems inherit it. A second fragile point is the Čech identification in Theorem 5.3, where the class of the constructed cocycle is asserted to equal [ϖ] via a standard double-complex zig-zag without displaying the comparison.

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Axiom & Free-Parameter Ledger

0 free parameters · 9 axioms · 0 invented entities

No empirical free parameters appear; the settings are mathematical hypotheses (connectedness, finite type, nonempty source). The axioms are a mix of standard homological algebra and the companion framework [19,20,21], which is assumed rather than proved here. Invented entities: none beyond standard mathematical constructions like the defect homomorphism, which is a new function but not a postulated physical entity.

axioms (9)
  • standard math The mapping cone complex (Ω•(F),d_F) and its chain counterpart (C•(F),∂_F) compute relative de Rham/singular cohomology of the mapping cone; the relative integration pairing satisfies Stokes (Prop 3.4).
    Used to define relative cycles, periods, and dualities; derived in §3 from standard homological algebra.
  • domain assumption The relative Cartan calculus identities, including the magic formula and master identity (2.4), hold for the relative de Rham cone.
    Imported from companion [20] (Prop 3.5, Lem 3.7); proofs not reproduced, and used in Theorem 6.2 and Theorem 7.1.
  • domain assumption The L∞-algebra L∞(F,ϖ) of relative observables exists as described, and the relative homotopy moment map component equations (2.6) and one-step extension formula (2.7) are valid.
    Imported from companions [19,20]; used in §6 and §8.
  • domain assumption H^0(Ω(F))=0 when N is connected and M≠∅.
    Stated in (2.5) as [20, Rem. 3.2]; the computation d_F h=(dh,F*h) is elementary but the vanishing is load-bearing for Theorem 7.1.
  • standard math For finite type M,N, relative integrality of [ϖ] is equivalent to [ϖ] lying in the image of H^{n+1}(F;Z)→H^{n+1}(F;R).
    Remark 3.6; used to phrase Theorem 5.3 and Theorem 3.15.
  • standard math Good covers and refinement maps exist for smooth paracompact manifolds, and Čech cohomology computes singular cohomology of the cone; locally constant cochains suffice.
    Used in the sufficiency proof of Theorem 5.3.
  • domain assumption A quasi-Hamiltonian G-space is equivalent to a relative 2-plectic map with one-step extension (2.8)-(2.9).
    Imported from [20, Thms 8.4-8.5] and [2]; used throughout §8.
  • domain assumption Relative gerbes are classified by H^3(F;Z) and correspond to relatively integral 3-forms.
    Used in §5.3 for degree-three prequantization; cited to Shahbazi [52].
  • domain assumption In places, every tangent vector of M extends to an F-pair (e.g., F an immersion, submersion, or M a point) to deduce weak from strong nondegeneracy.
    Theorem 9.3(1); a technical condition on F, stated in the theorem.

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Pith. "Pith review of Periods, prequantization, and rigidity in relative multisymplectic geometry." pith.science (2026). https://pith.science/paper/OXB64KNE

@misc{pith2026260707149,
  author       = {Pith},
  title        = {Pith review of: Periods, prequantization, and rigidity in relative multisymplectic geometry},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OXB64KNE}},
  note         = {Machine review of arXiv:2607.07149}
}
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read the original abstract

Relative multisymplectic geometry replaces differential forms on a single manifold by cocycles in the mapping cone of a smooth map $F\colon M\to N$. Building on the relative Cartan calculus, the Lie $n$-algebras of relative observables, and the relative homotopy moment maps developed in companion work, we establish a range of applications showing that the framework is a working tool rather than a formal generalization. We first construct the integration pairing between relative differential forms and smooth relative chains, and prove two structural results that make it usable: a period criterion, reducing relative integrality to the periods of the target form together with a defect homomorphism on the classes killed by $F_*$, and a functoriality theorem for morphisms of arrows. The criterion yields a structure theorem for levels: the integers $k$ at which $k\varpi$ is relatively integral always form a cyclic group $k_0\mathbb{Z}$, with no hypothesis on $F$, and $k_0$ is computable from finitely many integrals whenever the relevant homology is finitely generated. Together these tools yield a characterization of homotopy-invariant bulk--boundary action functionals, hence a precise treatment of Wess--Zumino terms; a relative Weil--Kostant theorem, whose specialization to a Lagrangian submanifold is the Bohr--Sommerfeld condition of geometric quantization; a relative Noether identity, with a bulk--boundary splitting of the conserved charges; and a rigidity theorem making comoment maps unique, strict and equivariant, so that the Kostant--Souriau cocycle disappears. Two closing sections analyse the degenerate edges $M=\varnothing$ and $N=\mathrm{pt}$, at which the absolute theories are recovered, and separate weak from strong nondegeneracy, determining which results require which.

Figures

Figures reproduced from arXiv: 2607.07149 by Djounvouna Dinamo.

Figure 1
Figure 1. Figure 1: The relative integration theory of Section 3 – cycles, Stokes, the period criterion and functoriality – is the shared engine; the five applications are logically independent branches and may be read in any order. 1.2. Relation to the literature. The integration theory of Section 3 is classical homological algebra adapted to the smooth mapping cone; its specializations recover Poincar´e–Lefschetz￾type pairi… view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.