P. Baguis, M. Cahen
We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with Ricci type curvature, which are not locally symmetric; the existence of such symplectic connections was unknown.
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Conditions are given under which an infinitesimal automorphism of a torsion-free connection preserving a symplectic form is necessarily a symplectic vector field. An example is given of a compact symplectic manifold admitting a flat symplectic connection and an infinitesimal automorphism that is not symplectic.
We propose a reduction procedure for symplectic connections with symmetry. This is applied to coadjoint orbits whose isotropy is reductive.
We study the irreducible decomposition under Sp(2n, R) of the space of torsion tensors of almost symplectic connections. Then a description of all symplectic quadratic invariants of torsion-like tensors is given. When applied to a manifold M with an almost symplectic structure, these instruments give preliminary insight for finding a preferred linear almost symplectic connection on M . We rediscover Ph. Tondeur's Theorem on almost symplectic connections. Properties of torsion of the vectorial kind are deduced.
In this note we give conditions which ensure the reduction of a symplectic connection in the process of a Marsden-Weinstein reduction and of the reduction of a presymplectic manifold.
We show, in this note, that on any symplectic supermanifold, even or odd, there exist an infinite dimensional affine space of symmetric connections, compatible to the symplectic form.
A Yang-Mills theory in a purely symplectic framework is developed. The corresponding Euler-Lagrange equations are derived and first integrals are given. We relate the results to the work of Bourgeois and Cahen on preferred symplectic connections.
The imploded cross-section of a symplectic manifold is a stratified space allowing for an abelianization of its symplectic reduction. After recalling symplectic and Poisson reduction and reviewing the basics of symplectic implosion, we prove a cross-section theorem for Poisson manifolds, generalizing the Guillemin-Sternberg theorem for symplectic manifolds, which constitutes a first step towards Poisson implosion. On our way, we find and fix a mistake in the proof of Guillemin-Sternberg's theorem, and we identify Poisson transversals as the right analogue to symplectic submanifolds in this con…
This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example,…
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.
This article addresses the problem of developing an extension of the Marsden- Weinstein reduction process to symplectic Lie algebroids, and in particular to the case of the symplectic cover of a fiberwise linear Poisson structure, whose reduction process is the analogue to cotangent bundle reduction in the context of Lie algebroids.
In a previous article, we introduced a reduction procedure for locally conformally symplectic manifolds at any regular value of the momentum mapping. We use this construction to prove an analogue of a well-known theorem in the symplectic setting about the reduction of cotangent bundles.
We obtain universal models for several types of locally conformal symplectic manifolds via pullback or reduction. The relation with recent embedding results for locally conformal Kähler manifolds is discussed.
The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is Hamiltonian and calculated the moment map. This is analogous to, but distinct from, the action of Hamiltonian diffeomorphisms on the space of compatible almost complex structures that motivates study of extremal Kähler metrics. In particular, moment constant connections are critical, where a symplectic connection is \textit{critical} if it is critical, with respect to arbitrary variations, for the $L^…
We consider analytic curves $\nabla^t$ of symplectic connections of Ricci type on the torus $T^{2n}$ with $\nabla^0$ the standard connection. We show, by a recursion argument, that if $\nabla^t$ is a formal curve of such connections then there exists a formal curve of symplectomorphisms $ψ_t$ such that $ψ_t\cdot\nabla^t$ is a formal curve of flat invariant symplectic connections and so $\nabla^t$ is flat for all $t$. Applying this result to the Taylor series of the analytic curve, it means that analytic curves of symplectic connections of Ricci type starting at $\nabla^0$ are also flat. The gr…
This encyclopedia article briefly reviews without proofs some of the main results in symplectic reduction. The article recalls most the necessary prerequisites to understand the main results, namely, group actions, momentum maps, and coadjoint orbits, among others.
Stacky Hamiltonian actions and symplectic reduction
本文引入了étale李群叠在étale辛叠上的哈密顿作用概念，并在该背景下证明了Kirwan凸性定理、Meyer-Marsden-Weinstein辛约化定理及Duistermaat-Heckman定理的版本。
Let (M,ω) be a symplectic manifold, and (Σ,σ) a closed connected symplectic 2-manifold. We construct a weakly symplectic form {ω^{D}}_{(Σ, σ)} on the space of immersions Σ\to M that is a special case of Donaldson's form. We show that the restriction of {ω^{D}}_{(Σ,σ)} to any orbit of the group of Hamiltonian symplectomorphisms through a symplectic embedding (Σ,σ) \to (M,ω) descends to a weakly symplectic form ω^D_{\red} on the quotient by Sympl(Σ,σ), and that the obtained symplectic space is a symplectic quotient of the subspace of symplectic embeddings S_{e}(Σ,σ) with respect to the Sympl(Σ,σ…
We propose a produre of reduction a locally conformal symplectic structure. This procedure of reduction can be applied to wide class of submanifolds. There are no local obstructions for this procedure. But there are global obstructions. We find a necessary and sufficient condition when this reduction holds in terms of the special kind of de Rham cohomology class (tangent to the characteristic foliation) of the Lee form.
From a view point of the moment map, we shall introduce the notion of Einstein-Hermitian generalized connections over a generalized Kähler manifold of symplectic type. We show that moduli spaces of Einstein-Hermitian generalized connections arise as the Kähler quotients. The deformation complex of Einstein-Hermitian generalized connections is an elliptic complex and it turns out that the smooth part of the moduli space is a finite dimensional Kähler manifold. The canonical line bundle over a generalized Kähler manifold of symplectic type has the canonical generalized connection and its curvatu…
Symplectic reduction and a Darboux-Moser-Weinstein theorem for Lie algebroids
本文将 Marsden-Weinstein 约化定理与 Darboux-Moser-Weinstein 定理推广至辛李代数胚，并提出了相应的余迷向嵌入定理。
We present a reduction procedure for locally conformally symplectic (LCS) manifolds with an action of a Lie group preserving the conformal structure, with respect to any regular value of the momentum mapping. Under certain conditions, this reduction is compatible with the existence of a locally conformally Kähler structure. As a special consequence, we obtain a compatible contact reduction with respect to any regular value of the contact momentum mapping.
We use symplectic reduction to give a new construction of the core $C$ of a symplectic double groupoid $D$ as the common leaf space of characteristic foliations associated to various coisotropic submanifolds of $D$. In the case of the cotangent double groupoid of a Lie groupoid $G$, the canonical relations arising from this process turn out to be cotangent lifts of structure maps associated to $G$. We also show that under this reduction procedure the double groupoid structure on $D$ descends to a groupoid structure on the leaf space above, recovering the core groupoid structure on $C$ of Brown…
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…
We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebras on the space of symplectic connections. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the star product trace of the curvature is a formal symplectic form on the space of symplectic connections. The action of Hamiltonian diffeomorphisms on symplectic connections preserves the formal symplectic structure and we show the star product trace …
Simultaneous linear symplectic reduction and orbit fibrations
本文建立线性辛同胚群轨道与辛向量空间子空间格拉斯曼流形之间的对应，借助同时线性辛约化计算一般情形的同伦型，并统一恢复 Arnold、Oh-Park 和 Lee-Leung 的已知结果。
We consider invariant symplectic connections $\nabla$ on homogeneous symplectic manifolds $(M,ω)$ with curvature of Ricci type. Such connections are solutions of a variational problem studied by Bourgeois and Cahen, and provide an integrable almost complex structure on the bundle of almost complex structures compatible with the symplectic structure. If $M$ is compact with finite fundamental group then $(M,ω)$ is symplectomorphic to $¶_n(\C)$ with a multiple of its Kähler form and $\nabla$ is affinely equivalent to the Levi-Civita connection.
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。
In this paper we present a construction of Ricci-flat connections through an induction procedure. Given a symplectic manifold $(M,ω)$ of dimension $2n$, we define induction as a way to construct a symplectic manifold $(P,μ)$ of dimension $2n+2$. Given any symplectic connection $\nabla$ on $(M,ω)$, we define an induced connection $\nabla^P$ which is a Ricci-flat symplectic connection on $(P,μ)$.
In this article, we introduce symplectic reduction in the framework of nonrational toric geometry. When we specialize to the rational case, we get symplectic reduction for the action of a general, not necessarily closed, Lie subgroup of the torus.
We study symplectic manifolds $(M^{2l},ω)$ equipped with a symplectic torsion-free affine (also called Fedosov) connection $\nabla$ and admitting a metaplectic structure. Let $\mathcal{S}$ be the so called symplectic spinor bundle and let $R^S$ be the curvature tensor field of the symplectic spinor covariant derivative $\nabla^S$ associated to the Fedosov connection $\nabla.$ It is known that the space of symplectic spinor valued exterior differential 2-forms, $Γ(M,\bigwedge^2T^*M\otimes {\mathcal{S}}),$ decomposes into three invariant spaces with respect to the structure group, which is the m…
Automorphisms of Lie groupoids and symplectic reduction on orbifolds
本文研究了李群体的自同构2-群BAut(X)以及其在orbifold上的辛约简，通过Kan纤维化理论建立了2-群作用与李群体的等价关系，并探讨了辛约简的性质和应用。
We generalize various symplectic reduction techniques to the context of the optimal momentum map. Our approach allows the construction of symplectic point and orbit reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a momentum map. We construct an orbit reduction procedure for canonical actions on a Poisson manifold that exhibits an interesting interplay with the von Neumann condition previously introduced by the author in his study of singular dual pairs. This condition ensures that the orbits in the momentum space of the optimal …
There exist three main approaches to reduction associated to canonical Lie group actions on a symplectic manifold, namely, foliation reduction, introduced by Cartan, Marsden-Weinstein reduction, and optimal reduction, introduced by the authors. When the action is free, proper, and admits a momentum map these three approaches coincide. The goal of this paper is to study the general case of a symplectic action that does not admit a momentum map and one needs to use its natural generalization, a cylinder valued momentum map introduced by Condevaux, Dazord, and Molino. In this case it will be show…
Abelian instances of nonabelian symplectic reduction
研究在哈密顿群作用下，非交换辛约化与交换子群辛约化等价的充要条件，并给出包括海森堡群在内的幂零李群族实例。
A (biased and incomplete) review of the status of the theory of symplectic connections on supermanifolds is presented. Also, some comments regarding Fedosov's technique of quantization are made.
We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading. We show that a structure of each of these types on a smooth manifold $M$ determines a canonical compatible linear connection on the tangent bundle $TM$. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant's theorem on Lie algebra cohomology. For each type, we give an …
We give a new characterization of flat affine manifolds in terms of an action of the Lie algebra of classical infinitesimal affine transformations on the bundle of linear frames. We characterize flat affine symplectic Lie groups using symplectic étale affine representations and as a consequence of this, we show that a flat affine symplectic Lie group with bi-invariant symplectic connection contains a nontrivial one parameter subgroup formed by central translations. We give two methods for constructing flat affine symplectic Lie groups, thus obtaining all those having bi-invariant symplectic co…
We study iteration maps of recurrence relations arising from mutation periodic quivers of arbitrary period. Combining tools from cluster algebra theory and (pre)symplectic geometry, we show that these cluster iteration maps can be reduced to symplectic maps on a lower dimensional submanifold, provided the matrix representing the quiver is singular. The reduced iteration map is explicitly computed for several new periodic quivers.
We prove that quasi-morphisms and quasi-states on a closed integral symplectic manifold descend under symplectic reduction to symplectic hyperplane sections. Along the way we show that quasi-morphisms that arise from spectral invariants are the Calabi homomorphism when restricted to Hamiltonians supported on stably displaceable sets.
This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying structure is conformally symplectic (PCS-structures). In that case, we obtained a close relation to parabolic contact structures via a concept of parabolic contactification. It was also shown that special symplectic connections (and thus all connections of exotic symplectic holonomy) …
In this paper we study Poisson actions of complete Poisson groups, without any connectivity assumption or requiring the existence of a momentum map. For any complete Poisson group $G$ with dual $G^\star$ we obtain a suitably connected integrating symplectic double groupoid $\calS$. As a consequence, the cotangent lift of a Poisson action on an integrable Poisson manifold $P$ can be integrated to a Poisson action of the symplectic groupoid $\poidd{\calS}{G^\star}$ on the symplectic groupoid for $P$. Finally, we show that the quotient Poisson manifold $P/G$ is also integrable, giving an explicit…
Let n>1 and G be the group SU(n) or Sp(n). This paper constructs compact symplectic manifolds whose symplectic quotient under a Hamiltonian G-action does not inherit the strong Lefschetz property.
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。
We introduce the process of symplectic reduction along a submanifold as a uniform approach to taking quotients in symplectic geometry. This construction holds in the categories of smooth manifolds, complex analytic spaces, and complex algebraic varieties, and has an interpretation in terms of derived stacks in shifted symplectic geometry. It also encompasses Marsden--Weinstein--Meyer reduction, Mikami--Weinstein reduction, the pre-images of Poisson transversals under moment maps, symplectic cutting, symplectic implosion, and the Ginzburg--Kazhdan construction of Moore--Tachikawa varieties in T…
Derived Symplectic Reduction in Differential Geometry
本文证明了Marsden-Weinstein-Meyer辛约减定理的导出版本，通过dg群oid建模辛商，构建群oid内的约减辛形式，并展示其满足导出非退化条件。
During the last thirty years, symplectic or Marsden--Weinstein reduction has been a major tool in the construction of new symplectic manifolds and in the study of mechanical systems with symmetry. This procedure has been traditionally associated to the canonical action of a Lie group on a symplectic manifold, in the presence of a momentum map. In this note we show that the symplectic reduction phenomenon has much deeper roots. More specifically, we will find symplectically reduced spaces purely within the Poisson category under hypotheses that do not necessarily imply the existence of a moment…
There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen-Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson st…
These notes give an introduction to Geometric Invariant Theory and symplectic reduction, with lots of pictures and simple examples. We describe their applications to moduli of bundles and varieties, and their infinite dimensional analogues in gauge theory and the theory of special metrics on algebraic varieties. Donaldson's "quantisation" link between the infinite and finite dimensional situations is described, as are surprisingly strong connections between the bundle and variety cases.
In a previous article, we proved that symplectic homeomorphisms preserving a coisotropic submanifold C, preserve its characteristic foliation as well. As a consequence, such symplectic homeomorphisms descend to the reduction of the coisotropic C. In this article we show that these reduced homeomorphisms continue to exhibit certain symplectic properties. In particular, in the specific setting where the symplectic manifold is a torus and the coisotropic is a standard subtorus, we prove that the reduced homeomorphism preserves spectral invariants and hence the spectral capacity. To prove our main…
We prove a theorem on singular symplectic cotangent bundle reduction in the Fréchet setting and apply it to Yang-Mills-Higgs theory with special emphasis on the Higgs sector of the Glashow-Weinberg-Salam model. For the latter model we give a detailed description of the reduced phase space and show that the singular structure is encoded in a finite-dimensional Lie group action.
The first part of this paper deals with electrical networks and symplectic reductions. We consider two operations on electrical networks (the "trace map" and the "gluing map") and show that they correspond to symplectic reductions. We also give several general properties about symplectic reductions, in particular we study the singularities of symplectic reductions when considered as rational maps on Lagrangian Grassmannians. This is motivated by [23] where a renormalization map was introduced in order to describe the spectral properties of self-similar lattices. In this text, we show that this…
Some incarnations of Hamiltonian reduction in symplectic geometry and geometric representation theory
本文介绍了辛几何与几何表示论中哈密顿约简的现代应用，包括交换化、通过辛群状的约简及摩尔-塔奇克瓦猜想。
We show that each triangular Poisson Lie group can be decomposed into Poisson submanifolds each of which is a quotient of a symplectic manifold. The Marsden-Weinstein-Meyer symplectic reduction technique is then used to give a complete description of the symplectic foliation of all triangular Poisson structures on Lie groups. The results are illustrated in detail for the generalized Jordanian Poisson structures on SL(n).
Some invariant connections on symplectic reductive homogeneous spaces
本文研究辛约化齐性空间上的自然辛联络，引入一族不变联络并证明存在唯一自然辛联络，计算其曲率张量，并验证Wallach旗流形的首选联络与之重合且Ricci平行。
Let M be the product of two compact Hamiltonian T-spaces X and Y. We present a formula for evaluating integrals on the symplectic reduction of M by the diagonal T action. At every regular value of the moment map for X x Y, the integral is the convolution of two distributions associated to the symplectic reductions of X by T and of Y by T. Several examples illustrate the computational strength of this relationship. We also prove a linear analogue which can be used to find cohomology pairings on toric orbifolds.
A canonical connection is attached to any k-symplectic manifold. We study the properties of this connection and its geometric applications to k-symplectic manifolds. In particular we prove that, under some natural assumption, any ksymplectic manifold admits an Ehresmann connection, discussing some corollaries of this result, and we find vanishing theorems for characteristic classes on a k-symplectic manifold.
We show how to construct a resolution of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularise generic symplectic quotients. Given a manifold with a Hamiltonian action of a compact Lie group, symplectic reduction at a coadjoint orbit which is transverse to the moment map produces a symplectic orbifold. If moreover the points of this coadjoint orbit are regular elements of the Lie coalgebra, that is, their stabiliser is a maximal torus, the result for torus quotients may be applied to obtain a desingularisation…
Let L->M be a Hermitian line bundle over a compact manifold. Write S for the space of all unitary connections in L whose curvatures define symplectic forms on M and G for the group of unitary bundle isometries of L, which acts on S by pull-back. The main observation of this note is that S carries a G-invariant symplectic structure, there is a moment map for the G-action and that this embeds the components of S as G_0-coadjoint orbits (where G_0 is the component of the identity). Restricting to the subgroup of G which covers the identity on M, we see that prescribing the volume form of a symple…
Let X be a four-manifold with boundary three manifold M. We shall describe (i) a pre-symplectic structure on the space of connections of the trivial SU(n)-bundle over X that comes from the canonical symplectic structure on the cotangent bundle of the connection space, and (ii) a pre-symplectic structure on the space of flat connections of the trivial SU(n)-bundle over M that have null charge. These two structures are related by the boundary restriction map. We discuss also the Hamiltonian feature of the space of connections with the action of the group of gauge transformations.
In this article we relate the construction of Ricci type symplectic connections by reduction to the construction of star product by reduction yielding rather explicit descriptions for the star product on the reduced space.
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