Casey Blacker
We extend the Marsden-Weinstein-Meyer symplectic reduction theorem to the setting of multisymplectic manifolds. In this context, we investigate the dependence of the reduced space on the reduction parameters. With respect to a distinguished class of multisymplectic moment maps, an exact stationary phase approximation and nonabelian localization theorem are also obtained.
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Coisotropic reduction in Multisymplectic Geometry
该文研究多辛几何的余迷向约化，将哈密顿多向量场解释为拉格朗日子流形并证明k-余迷向子流形诱导李子代数，还将辛几何中拉格朗日子流形投影的经典结果推广到具多辛结构的形式丛。
Homotopy reduction of multisymplectic structures in Lagrangian field theory
本文针对Lagrangian场论，基于Hamilton形式的$L_\infty$-代数与同伦动量映射，提出局部同伦动量映射的同伦约化方法。
Geometry of bundle-valued multisymplectic structures with Lie algebroids
本文从哈密顿对称性角度研究带联络向量丛上的多重辛结构，引入丛值 n-重辛结构及同伦动量截面，并推广 Marsden-Weinstein-Meyer 约化定理，构造向量值 1-重辛流形的两类约化。
We give a detailed, self-contained proof of Geoffrey Martin's normal form theorem for Lagrangian submanifolds of standard multisymplectic manifolds (that generalises Alan Weinstein's famous normal form theorem in symplectic geometry), providing also complete proofs for the necessary results in foliated differential topology, i.e., a foliated tubular neighborhood theorem and a foliated relative Poincaré lemma.
Multisymplectic geometry admits an operation that has no counterpart in symplectic geometry, namely, taking the product of two multisymplectic manifolds endowed with the wedge product of the multisymplectic forms. We show that there is an L-infinity-embedding of the L-infinity-algebra of observables of the individual factors into the observables of the product, and that homotopy moment maps for the individual factors induce a homotopy moment map for the product. As a by-product, we associate to every multisymplectic form a curved L-infinity-algebra, whose curvature is the multisymplectic form …
We introduce the notion of a weak (homotopy) moment map associated to a Lie group action on a multisymplectic manifold. We show that the existence/uniqueness theory governing these maps is a direct generalization from symplectic geometry. We use weak moment maps to extend Noether's theorem from Hamiltonian mechanics by exhibiting a correspondence between multisymplectic conserved quantities and continuous symmetries on a multi-Hamiltonian system. We find that a weak moment map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We define a multisympl…
We introduce a notion of a homotopy momentum section on a Lie algebroid over a pre-multisymplectic manifold. A homotopy momentum section is a generalization of the momentum map with a Lie group action and the momentum section on a pre-symplectic manifold, and is also regarded as a generalization of the homotopy momentum map on a multisymplectic manifold. We show that a gauged nonlinear sigma model with Wess-Zumino term with Lie algebroid gauging has the homotopy momentum section structure.
We utilize Spin(7) identities to prove that the Cayley four-form associated to a torsion-free Spin(7)-Structure is non-degenerate in the sense of multisymplectic geometry. Therefore, Spin(7) geometry may be treated as a special case of multisymplectic geometry. We then capitalize on this relationship to make statements about Hamiltonian multivector fields and differential forms associated to torsion-free Spin(7)-Structures.
An invitation to multisymplectic geometry
本文系统介绍多辛几何，阐述经典场论的多辛重述、正规形与 Darboux 型定理、微分同胚分类，以及对称性和守恒量的最新进展。
Reduction of relative multisymplectic manifolds
该研究将多辛约化理论扩展到相对多辛几何，引入相关代数和矩映射，证明约化定理，还证明了动力学约化等的相对类似物及相关变分公式，最后得出相对局部化约束等成果。
Multisymplectic observable reduction using constraint triples
本文提出一种基于Gerstenhaber代数、BV-模和约束三元组的完全代数形式化方法，用于构造和约化受多辛几何启发的可观测量$L_\infty$-代数，并重构解释了arXiv:2206.03137(3)的结果。
Density-valued symplectic forms from a multisymplectic viewpoint
本文从多辛视角出发，研究密度值辛形式的内在特性、达布定理证明及其对称性分析。
Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds
本文在（预）多重辛流形上的李代数胚中引入相容 E-n-形式，统一刻画扭曲泊松结构、动量映射及其同伦推广等结构。
This Master Thesis is devoted to the study of $n$-plectic manifolds and the Strongly Homotopy Lie algebras, also called $L_{\infty}$-algebras, that can be associated to them. Since multisymplectic geometry and $L_{\infty}$-algebras are relevant in Theoretical Physics, and in particular in String Theory, we introduce the relevant background material in order to make the exposition accessible to non-experts, perhaps interested physicists. The background material includes graded and homological algebra theory, fibre bundles, basics of group actions on manifolds and symplectic geometry. We give an…
Given a vector field on a manifold M, we define a globally conserved quantity to be a differential form whose Lie derivative is exact. Integrals of conserved quantities over suitable submanifolds are constant under time evolution, the Kelvin circulation theorem being a well-known special case. More generally, conserved quantities are well-behaved under transgression to spaces of maps into M. We focus on the case of multisymplectic manifolds and Hamiltonian vector fields. We show that in the presence of a Lie group of symmetries admitting a homotopy co-momentum map, one obtains a whole family o…
Reduction of $L_\infty$-Algebras of Observables on Multisymplectic Manifolds
本文提出预多辛流形上可观测量 $L_\infty$-代数的约化方案，可在辛情形重现 Marsden-Weinstein-Meyer 约化的 Poisson 代数，并通过多余切丛等例子展示其与既有约化理论的区别及场论应用。
A manifold is multisymplectic, or more specifically n-plectic, if it is equipped with a closed nondegenerate differential form of degree n+1. In our previous work with Baez and Hoffnung, we described how the `higher analogs' of the algebraic and geometric structures found in symplectic geometry should naturally arise in 2-plectic geometry. In particular, we showed that just as a symplectic manifold gives a Poisson algebra of functions, any 2-plectic manifold gives a Lie 2-algebra of 1-forms and functions. Lie n-algebras are examples of L-infinity algebras: graded vector spaces which are equipp…
We extend Noether's theorem to the setting of multisymplectic geometry by exhibiting a correspondence between conserved quantities and continuous symmetries on a multi-Hamiltonian system. We show that a homotopy co-momentum map interacts with this correspondence in a way analogous to the moment map in symplectic geometry. We apply our results to generalize the theory of the classical momentum and position functions from the phase space of a given physical system to the multisymplectic phase space. We also apply our results to manifolds with a torsion-free $G_2$ structure.
Observables on multisymplectic manifolds and higher Courant algebroids
本文研究了多变重流形上由闭非退化微分形式关联的可观测量的 $L_{\infty}$-代数与扭曲的高阶 Courant algebroid 截面的 $L_{\infty}$-代数，证明了前者的 $L_{\infty}$-嵌入，给出了涉及伯努利数的显式公式，并证明其与规范变换相容。
Floer sections in multisymplectic geometry
本文在多辛几何中引入了新的框架，将辛几何中的椭圆方法推广，并证明了伪Fueter曲线的存在。
A hydrodynamical homotopy co-momentum map and a multisymplectic interpretation of higher order linking numbers
本文构造并推广了流体动力学同伦余动量映射，探讨了理想流体 Euler 演化的协变相空间解释，并在多辛框架下将高阶环绕数重新解释为守恒量。
There is a well-established procedure of assigning a strong homotopy Lie algebra of local observables to a multisymplectic manifold which can be regarded as part of a categorified Poisson structure. For a 2-plectic manifold, the resulting Lie 2-algebra is isomorphic to a sub Lie 2-algebra of a natural Lie 2-algebra structure on an exact Courant algebroid. We generalize this statement to arbitrary n-plectic manifolds and study automorphisms on the arising Lie n-algebras. Our observations may be useful in studying the quantization problem on multisymplectic manifolds.
Periods, prequantization, and rigidity in relative multisymplectic geometry
本文研究相对多辛几何，通过构建相对积分理论，证明了关于闭相对形式、相对Weil - Kostant定理等多个结论，并展示了拟哈密顿\(G\)空间矩映射理论在融合下的典范函子性，还对相关领域进行了展望。
The multisymplectic formalism of field theories developed by many mathematicians over the last fifty years is extended in this work to deal with manifolds that have boundaries. In particular, we develop a multisymplectic framework for first order covariant Hamiltonian field theories on manifolds with boundaries. This work is a geometric fulfillment of Fock's characterization of field theories as it appears in recent work by Cattaneo, Mnev and Reshetikhin [Ca14]. This framework leads to a true geometric understanding of conventional choices for boundary conditions. For example, the boundary con…
A zoo of coisotropic embeddings
本文将预辛流形的余迷向嵌入定理推广到余辛、切触、多辛等七类更一般的几何流形，采用通用方法推导，为奇异拉格朗日系统正则化问题的应用奠定基础。
Reduction of coKähler and 3-cosymplectic manifolds
本文研究了coKähler流形与3-cosymplectic流形的约化定理，并证明了其与Kähler和hyperKähler约化在柱构造及映射环面构造下的相容性。
Reduction of symplectic groupoids and quotients of quasi-Poisson manifolds
本文通过范畴化辛群胚约化方法，统一了拟泊松流形商的可积性理论，并构造了积分规范群胚上泊松结构的双重辛群胚。
We present a general framework for reduction of symplectic Q-manifolds via graded group actions. In this framework, the homological structure on the acting group is a multiplicative multivector field.
We extend the Falceto-Zambon version of Marsden-Ratiu Poisson reduction to Poisson quasi-Nijenhuis structures with background on manifolds. We define gauge transformations of Poisson quasi-Nijenhuis structures with background, study some of their properties and show that they are compatible with reduction procedure. We use gauge transformations to construct Poisson quasi-Nijenhuis structures with background.
We first recall some basic definitions and facts about Jacobi manifolds, generalized Lie bialgebroids, generalized Courant algebroids and Dirac structures. We establish an one-one correspondence between reducible Dirac structures of the generalized Lie bialgebroid of a Jacobi manifold $(M,Λ,E)$ for which 1 is an admissible function and Jacobi quotient manifolds of $M$. We study Jacobi reductions from the point of view of Dirac structures theory and we present some examples and applications.
Reduction of Poisson manifolds with Hamiltonian Lie algebroids
本文提出了具有哈密顿李代数胚的Poisson流形的约化定理，通过引入相容动量截面概念并证明其为李代数同态，得出其零水平集商空间为Poisson流形。
We define reduction of locally conformal Kaehler manifolds, considered as conformal Hermitian manifolds, and we show its equivalence with an unpublished construction given by Biquard and Gauduchon. We show the compatibility between this reduction and Kaehler reduction of the universal cover. By a recent result of Kamishima and the second author, in the Vaisman case (that is, when a metric in the conformal class has parallel Lee form) if the manifold is compact its universal cover comes equipped with the structure of Kaehler cone over a Sasaki compact manifold. We show the compatibility between…
Reduction theory for singular symplectic manifolds and singular forms on moduli spaces
本文填补了一般对称性下奇异辛流形约化理论的空白，研究了Marsden-Weinstein约化理论，提出了考虑微分形式奇异性的更大容许哈密顿函数集，并构建了新的奇异quasi-Hamiltonian空间。
We compute explicitly the equivariant Hirzebruch $χ_y$-characteristic of an equivariant complex line bundle over a toric manifold and state a weighted version of the quantization commutes with reduction principle in symplectic geometry. Then, we give a weighted decomposition formula for any simple polytope in $\R^n$. This formula generalizes a polytope decomposition due to Lawrence [10] and Varchenko [14] and extends a previous weighted version obtained by Karshon, Sternberg and Weitsman [9].
For a Hamiltonian, proper and free action of a Lie group $G$ on a Dirac manifold $(M,L)$, with a regular moment map $μ:M\to \mathfrak{g}^*$, the manifolds $M/G$, $μ^{-1}(0)$ and $μ^{-1}(0)/G$ all have natural induced Dirac structures. If $(M,L)$ is an integrable Dirac structure, we show that $M/G$ is always integrable, but $μ^{-1}(0)$ and $μ^{-1}(0)/G$ may fail to be integrable, and we describe the obstructions to their integrability.
We consider locally conformal Kaehler geometry as an equivariant (homothetic) Kaehler geometry: a locally conformal Kaehler manifold is, up to equivalence, a pair (K,Γ) where K is a Kaehler manifold and Γa discrete Lie group of biholomorphic homotheties acting freely and properly discontinuously. We define a new invariant of a locally conformal Kaehler manifold (K,Γ) as the rank of a natural quotient of Γ, and prove its invariance under reduction. This equivariant point of view leads to a proof that locally conformal Kaehler reduction of compact Vaisman manifolds produces Vaisman manifolds and…
We recall the presentation of the generalized, complex structures by classical tensor fields, while noticing that one has a similar presentation and the same integrability conditions for generalized, paracomplex and subtangent structures. This presentation shows that the generalized, complex, paracomplex and subtangent structures belong to the realm of Poisson geometry. Then, we prove geometric reduction theorems of Marsden-Ratiu and Marsden-Weinstein type for the mentioned generalized structures and give the characterization of the submanifolds that inherit an induced structure via the corres…
Given the Euclidean space $\R^{2n+2}$ endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen have defined a reduction procedure which yields a symplectic manifold endowed with a Ricci-type connection. We observe that any symplectic manifold of dimension greater than 2 endowed with a symplectic connection of Ricci-type is locally given by a local version of such a reduction. We also consider the reverse of this reduction procedure, an induction procedure: we construct globally on a symplectic manifold end…
In this paper, we develop results in the direction of an analogue of Sjamaar and Lerman's singular reduction of Hamiltonian symplectic manifolds in the context of reduction of Hamiltonian generalized complex manifolds (in the sense of Lin and Tolman). Specifically, we prove that if a compact Lie group acts on a generalized complex manifold in a Hamiltonian fashion, then the partition of the global quotient by orbit types induces a partition of the Lin-Tolman quotient into generalized complex manifolds. This result holds also for reduction of Hamiltonian generalized Kähler manifolds.
Reduction of exact symplectic manifolds and energy hypersurfaces
本文提出两种精确辛流形上哈密顿系统的约简方法，并通过修改的马尔登-梅尔-韦instein定理证明其等价性。
For a possibly singular subset of a regular Poisson manifold we construct a deformation quantization of its algebra of Whitney functions. We then extend the construction of a deformation quantization to the case where the underlying set is a subset of a not necessarily regular Poisson manifold which can be written as the quotient of a regular Poisson manifold on which a compact Lie group acts freely by Poisson maps. Finally, if the quotient Poisson manifold is regular as well, we show a "quantization commutes with reduction" type result. For the proofs, we use methods stemming from both singul…
We propose a generalization of the reduction of Poisson manifolds by distributions introduced by Marsden and Ratiu. Our proposal overcomes some of the restrictions of the original procedure, and makes the reduced Poisson structure effectively dependent on the distribution. Different applications are discussed, as well as the algebraic interpretation of the procedure and its formulation in terms of Dirac structures.
We introduce a new method to perform reduction of contact manifolds that extends Willett's (math.SG/0104080) and Albert's results. To carry out our reduction procedure all we need is a complete Jacobi map $J$ from a contact manifold $M$ to a Jacobi manifold $Γ_0$. This naturally generates the action of the contact groupoid of $Γ_0$ on $M$, and we show that the quotients of fibers of $J$ by suitable Lie subgroups are either contact or locally conformal symplectic manifolds with structures induced by the one on $M$. We show that Willett's reduced spaces are prequantizations of our reduced spaces…
We study reduction of generalized complex structures. More precisely, we investigate the following question. Let $J$ be a generalized complex structure on a manifold $M$, which admits an action of a Lie group $G$ preserving $J$. Assume that $M_0$ is a $G$-invariant smooth submanifold and the $G$-action on $M_0$ is proper and free so that $M_G:=M_0/G$ is a smooth manifold. Under what condition does $J$ descend to a generalized complex structure on $M_G$? We describe a sufficient condition for the reduction to hold, which includes the Marsden-Weinstein reduction of symplectic manifolds and the r…
This paper develops a generalized formulation of Lagrangian mechanics on fibered manifolds, together with a reduction theory for symmetries corresponding to Lie groupoid actions. As special cases, this theory includes not only Lagrangian reduction (including reduction by stages) for Lie group actions, but also classical Routh reduction, which we show is naturally posed in this fibered setting. Along the way, we also develop some new results for Lagrangian mechanics on Lie algebroids, most notably a new, coordinate-free formulation of the equations of motion. Finally, we extend the foregoing to…
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