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On the variation of the Poisson structures of certain moduli spaces  [PDF]
Johannes Huebschmann
Mathematics , 1997,
Abstract: Given a Lie group G whose Lie algebra is endowed with a nondegenerate invariant symmetric bilinear form, we construct a Poisson algebra of continuous functions on a certain open subspace R of the space of representations in G of the fundamental group of a compact connected orientable topological surface with finitely many boundary circles; when G is compact and connected, R may be taken dense in the space of all representations. The space R contains spaces of representations where the values of those generators of the fundamental group which correspond to the boundary circles are constrained to lie in fixed conjugacy classes and, on these representation spaces, the Poisson algebra restricts to stratified symplectic Poisson algebras constructed elsewhere earlier. Hence the Poisson algebra on R gives a description of the variation of the stratified symplectic Poisson structures on the smaller representation spaces as the chosen conjugacy classes move.
Poisson structures on certain moduli spaces for bundles on a surface  [PDF]
Johannes Huebschmann
Mathematics , 1994,
Abstract: Let $\Sigma$ be a closed surface, $G$ a compact Lie group, with Lie algebra $g$, and $\xi \colon P \to \Sigma$ a principal $G$-bundle. In earlier work we have shown that the moduli space $N(\xi)$ of central Yang- Mills connections, for appropriate additional data, is stratified by smooth symplectic manifolds and that the holonomy yields a diffeomorphism from $N(\xi)$ onto a certain representation space $\roman{Rep}_{\xi}(\Gamma,G)$, with reference to suitable smooth structures $C^{\infty}(N(\xi))$ and $C^{\infty}(\roman{Rep}_{\xi}(\Gamma,G))$ where $\Gamma$ denotes the universal central extension of the fundamental group of $\Sigma$. Given an invariant symmetric bilinear form on $g^*$, we construct here Poisson structures on $C^{\infty}(N(\xi))$ and $C^{\infty}(\roman{Rep}_{\xi}(\Gamma,G))$ in such a way that the mentioned diffeomorphism identifies them. When the form on $g^*$ is non-degenerate the Poisson structures are compatible with the stratifications where $\roman{Rep}_{\xi}(\Gamma,G)$ is endowed with the corresponding stratification and, furthermore, yield structures of a {\it stratified symplectic space\/}, preserved by the induced action of the mapping class group of $\Sigma$.
Symplectic structures on moduli spaces of sheaves via the Atiyah class  [PDF]
A. Kuznetsov,D. Markushevich
Mathematics , 2007,
Abstract: Several situations are known when a holomorphic 2-form on a moduli space of sheaves over some base S is induced by a holomorphic 2-form on S. Moreover, the closedness of the 2-form on the base implies the closedness on the moduli space, which provides a stock of symplectic structures on moduli spaces (Mukai, Kobayashi, O'Grady, Tyurin, Huybrechts--Lehn). A parallel theory was developed for bivector fields and Poisson structures (Bottacin). However, there exist symplectic moduli spaces of sheaves over bases that have no holomorphic forms at all. A well-known example (Beauville--Donagi) is the family of lines on the cubic 4-fold Y, which can be thought of as the moduli space parameterizing the structure sheaves of lines in Y. The paper produces a general construction of closed 2-forms on the moduli spaces of sheaves, using the Atiyah class of the sheaves, and proves that this construction provides symplectic structures in 2 examples: the first one is the family of lines on Y, and the second one is the moduli space of sheaves which are supported on the hyperplane sections of Y and are cokernels of the Pfaffian representations of those hyperplane sections.
Nonlinearizability of certain Poisson structures near a symplectic leaf  [PDF]
Benjamin Lent Davis,Aissa Wade
Mathematics , 2004,
Abstract: We give an intrinsic proof that Vorobjev's first approximation of a Poisson manifold near a symplectic leaf is a Poisson manifold. We also show that Conn's linearization results cannot be extended in Vorobjev's setting.
Symplectic geometry on moduli spaces of holomorphic bundles over complex surfaces  [PDF]
Boris Khesin,Alexei Rosly
Mathematics , 2000,
Abstract: We give a comparative description of the Poisson structures on the moduli spaces of flat connections on real surfaces and holomorphic Poisson structures on the moduli spaces of holomorphic bundles on complex surfaces. The symplectic leaves of the latter are classified by restrictions of the bundles to certain divisors. This can be regarded as fixing a "complex analogue of the holonomy" of a connection along a "complex analogue of the boundary" in analogy with the real case.
Symplectic and Poisson geometry of the moduli spaces of flat connections over quilted surfaces  [PDF]
David Li-Bland,Pavol Severa
Mathematics , 2013,
Abstract: In this paper we study the symplectic and Poisson geometry of moduli spaces of flat connections over quilted surfaces. These are surfaces where the structure group varies from region to region in the surface, and where a reduction (or relation) of structure occurs along the boundaries of the regions. Our main theoretical tool is a new form moment-map reduction in the context of Dirac geometry. This reduction framework allows us to use very general relations of structure groups, and to investigate both the symplectic and Poisson geometry of the resulting moduli spaces from a unified perspective. The moduli spaces we construct in this way include a number of important examples, including Poisson Lie groups and their Homogeneous spaces, moduli spaces for meromorphic connections over Riemann surfaces (following the work of Philip Boalch), and various symplectic groupoids. Realizing these examples as moduli spaces for quilted surfaces provides new insights into their geometry.
Generalized classical dynamical Yang-Baxter equations and moduli spaces of flat connections on surfaces  [PDF]
Xiaomeng Xu
Mathematics , 2014,
Abstract: In this paper, we explain how generalized dynamical r-matrices can be obtained by (quasi-)Poisson reduction. New examples of Poisson structures, Poisson $G$-spaces and Poisson groupoid actions naturally appear in this setting. As an application, we use a generalized dynamical r-matrix induced by the gauge fixing procedure to give a new finite dimensional description of the Atiyah-Bott symplectic structure on the moduli space of flat connections on a surface.
Shifted Poisson and symplectic structures on derived N-stacks  [PDF]
J. P. Pridham
Mathematics , 2015,
Abstract: We show that on a derived Artin N-stack, there is a canonical equivalence between the spaces of n-shifted symplectic structures and non-degenerate n-shifted Poisson structures.
Shifted Poisson Structures and Deformation Quantization  [PDF]
D. Calaque,T. Pantev,B. Toen,M. Vaquie,G. Vezzosi
Mathematics , 2015,
Abstract: This paper is the sequel to [PTVV] (IHES Vol. 117, 2013). We develop a general and flexible context for differential calculus in derived geometry, including the de Rham algebra and polyvector fields. We then introduce the formalism of formal derived stacks and prove formal localization and gluing results. These allow us to define shifted Poisson structures on general derived Artin stacks, and prove that the non-degenerate Poisson structures correspond exactly to shifted symplectic forms. Shifted deformation quantization for a derived Artin stack endowed with a shifted Poisson structure is discussed in the last section. This paves the way for shifted deformation quantization of many interesting derived moduli spaces, like those studied in [PTVV] and probably many others.
Symplectic geometry of the moduli space of projective structures in homological coordinates  [PDF]
Marco Bertola,Dmitry Korotkin,Chaya Norton
Mathematics , 2015,
Abstract: We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure coincides with the canonical Poisson structure on the cotangent bundle of the moduli space of Riemann surfaces, and therefore the homological coordinates provide a new system of Darboux coordinates. We define a natural family of commuting "homological flows" on the moduli space of quadratic differentials and find the corresponding action-angle variables. The space of projective structures over the moduli space can be identified with the cotangent bundle upon selection of a reference projective connection that varies holomorphically and thus can be naturally endowed with a symplectic structure. Different choices of projective connections of this kind (Bergman, Schottky, Wirtinger) give rise to equivalent symplectic structures on the space of projective connections but different symplectic polarizations: the corresponding generating functions are found. We also study the monodromy representation of the Schwarzian equation associated with a projective connection, and we show that the natural symplectic structure on the the space of projective connections induces the Goldman Poisson structure on the character variety. Combined with results of Kawai, this result shows the symplectic equivalence between the embeddings of the cotangent bundle into the space of projective structures given by the Bers and Bergman projective connections.
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