arXiv subjects

Djounvouna Dinamo

Publications and source records attributed to Djounvouna Dinamo.

4 recordsLinked to original sources

arXiv2026

Periods, prequantization, and rigidity in relative multisymplectic geometry

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.

math.SG↗

arXiv2026

Belyi's theorem: coverings, dessins, and fields of definition

Belyi's theorem asserts that a smooth projective curve over $\mathbb{C}$ is defined over a number field if and only if it admits a non-constant morphism to $\mathbb{P}^1$ ramified over at most three points. This article gives a complete, self-contained account of the theorem -- both implications, with all descent machinery proved rather than quoted -- together with a study of the objects the proof produces. Beyond the exposition it contains original results, proved in full. The principal one is a structure theorem for the Belyi polynomials $P_{m,n}(z)=\frac{(m+n)^{m+n}}{m^{m}n^{n}}z^{m}(1-z)^{n}$: their dessins are double stars, they satisfy the exact identity $P_{m,n}=π_{k}\circ P_{m/k,n/k}$ with $k=\gcd(m,n)$ and $π_{k}(w)=w^{k}$, and their monodromy is the wreath product $\mathfrak{S}_{(m+n)/k}\wr\mathbb{Z}/k$, the full symmetric group precisely when $\gcd(m,n)=1$. Around the sharp lower bounds $d\geq 2g+1$ (all Belyi maps) and $d\geq 4g$ (clean maps) we study the extremal maps attaining $d=2g+1$: their monodromy lies in the alternating group, they need not be cyclic -- the smallest non-cyclic one has degree $5$, genus $2$, monodromy $A_{5}$, and is defined over $\mathbb{Q}$ -- and they satisfy the exact mass formula $\sum 1/|\mathrm{Aut}|=2(d-1)!/d(d+1)$, obtained from Boccara's cycle-factorization count -- reproved by a self-contained Frobenius computation -- and verified by complete enumeration for $d\leq7$. We also bound the degree of the rationalization step of Belyi's algorithm by $N!$ in the number $N$ of irrational branch values. The theory is illustrated by fully computed examples, and by an explicit $G_{\mathbb{Q}}$-orbit of three plane trees whose fields of moduli are the three conjugate embeddings of the non-Galois cubic field $\mathbb{Q}(\sqrt[3]{2})$.

math.AG↗

arXiv2026

Reduction of relative multisymplectic manifolds

We extend the multisymplectic reduction theory of Blacker -- itself the extension of Marsden--Weinstein--Meyer reduction to $k$-plectic manifolds -- to the setting of \emph{relative} multisymplectic geometry, in which a smooth map $F\colon M\to N$ carries a closed nondegenerate relative $(k{+}1)$-form $\varpi=(ω,η)$ in the mapping-cone complex of $F$. We introduce the Leibniz algebra of relative Hamiltonian pairs and the associated relative moment maps $μ\inΩ^{k-1}(F, \mathfrak{g}^{*})$, and prove a relative multisymplectic reduction theorem: for a closed equivariant level $ϕ\inΩ^{k-1}(F, \mathfrak{g}^{*})$, the level pair of $μ$ descends to a reduced smooth map $F_ϕ\colon M_ϕ\to N_ϕ$ carrying a unique reduced closed relative form $\varpi_ϕ$; remarkably, the horizontality of the trivializing component is forced by the relative closedness of the level. We further prove relative analogues of the reduction of dynamics, of the structure theory of split moment maps $μ=ν\cdotκ$ -- for which the splitting datum is a one-step cocycle in the relative Cartan model, so that split relative Hamiltonian $G$-spaces carry canonical relative homotopy moment maps -- and of the Duistermaat--Heckman-type variation formula: with respect to suitable conjugate distributions, the reduced relative class satisfies $\partial_λ[\varpi_ψ]=\langle c, λ\rangle \cdot[κ_ψ]$, where $c$ is the Chern form of the target model bundle acting through the natural $Ω(N)$-module structure of the mapping cone. Finally, we transport the exact stationary phase approximation to the target component and prove a genuinely relative localization constraint: the fixed-point contributions of the pulled-back split package on the source cancel identically.

math.SG↗

arXiv2026

Relative homotopy moment maps

Associated to any smooth map $F\colon M\to N$ equipped with a closed, nondegenerate relative $(n+1)$-form $\varpi$ -- a \emph{relative $n$-plectic structure} -- is an $L_\infty$-algebra of relative observables $L_{\infty}(F,\varpi)$, constructed by the author in earlier work. In this article we develop the corresponding theory of moment maps: for a Lie group $G$ acting compatibly on $M$ and $N$ and preserving $\varpi$, we define a \emph{relative homotopy moment map} as an $L_\infty$-morphism from $\mathfrak{g}$ into $L_{\infty}(F,\varpi)$ lifting the infinitesimal action, thereby providing a full relative generalization of the homotopy moment maps of Callies, Frégier, Rogers and Zambon. We characterize such morphisms by explicit component equations, show that a relative homotopy moment map is equivalent to a homotopy moment map on the target $N$ together with a coherent trivialization of its pullback to $M$, and relate relative moment maps to a relative Cartan model computing relative equivariant de Rham cohomology. Every cocycle in the relative Cartan model extending $\varpi$ induces a relative homotopy moment map via explicit formulas, and we prove the one-step case in full detail. In the existence theory a new phenomenon appears: under a mild connectivity hypothesis the Lie-algebra-cohomology obstruction present in the absolute theory vanishes identically in the relative setting. Finally, we show that quasi-Hamiltonian $G$-spaces with group-valued moment map $μ\colon M\to G$ fit into this framework: the pair $(η,ω)$ built from the Cartan $3$-form is a relative $2$-plectic structure whose Alekseev--Malkin--Meinrenken axioms amount precisely to a canonical one-step cocycle in the relative Cartan model, and hence every quasi-Hamiltonian $G$-space carries a canonical relative homotopy moment map, which we compute explicitly.

math.SG↗