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Groupvalued implosion and parabolic structures
 Amer. J. Math
"... ABSTRACT. The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasiHamiltonian Kmanifolds, where K is a simply connected compact Lie group. The imploded crosssection of the double K × K turns out to be universal in a suitable sense. It is a ..."
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ABSTRACT. The purpose of this paper is twofold. First we extend the notion of symplectic implosion to the category of quasiHamiltonian Kmanifolds, where K is a simply connected compact Lie group. The imploded crosssection of the double K × K turns out to be universal in a suitable sense. It is a singular space, but some of its strata have a nonsingular closure. This observation leads to interesting new examples of quasiHamiltonian Kmanifolds, such as the “spinning 2nsphere ” for K = SU(n). Secondly we construct a universal (“master”) moduli space of parabolic bundles with structure group K over a marked Riemann surface. The master moduli space carries a natural action of a maximal torus of K and a torusinvariant stratification into manifolds, each of which has a symplectic structure. An essential ingredient in the construction is the universal implosion. Paradoxically, although the universal implosion has no complex structure (it is the foursphere for K = SU(2)), the master moduli space turns out to be a complex algebraic variety.
Poisson geometry and the KashiwaraVergne conjecture
 C. R. Math. Acad. Sci. Paris
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CONNECTED COMPONENTS OF THE SPACE OF SURFACE GROUP REPRESENTATIONS
, 2003
"... Abstract. Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface Σ of genus l> 1, the order of the group H 2 (Σ, π1(G)) is equal to the number of connected components of the space Hom(π1(Σ), G)/G which can also be identified with the moduli s ..."
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Abstract. Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface Σ of genus l> 1, the order of the group H 2 (Σ, π1(G)) is equal to the number of connected components of the space Hom(π1(Σ), G)/G which can also be identified with the moduli space of gauge equivalence classes of flat Gbundles over Σ. We show that the same statement for a closed compact nonorientable surface which is homeomorphic to the connected sum of k copies of the real projective plane, where k ̸ = 1,2, 4, can be easily derived from a result in A. Alekseev, A.Malkin and E. Meinrenken’s recent work on Lie group valued moment maps. 1. introduction Let Σ be a closed compact orientable surface of genus l> 1. In [Go], W.M. Goldman conjectured that for any connected complex semisimple Lie group G, there is a bijection π0(Hom(π1(Σ),G)) → H 2 (Σ;π1(G)) ∼ = π1(G). In [Li], J. Li proved the above conjecture by Goldman as well as the following: Theorem 1 ([Li, Theorem 0.5]). Let Σ be a closed orientable Riemann surface of genus l> 1. Let G be a connected, compact, semisimple Lie group. Then there is a bijection π0(Hom(π1(Σ),G)/G) → H 2 (Σ;π1(G)) ∼ = π1(G), where G acts on Hom(π1(Σ),G) by conjugation. Geometrically, Hom(π1(Σ),G)/G can be identified with the moduli space of gauge equivalence classes of flat Gbundles over Σ, where a flat Gbundle is a principal Gbundle together with a flat connection. It is known that there is a onetoone correspondence between topological principal Gbundles over Σ and elements in H 2 (Σ;π1(G)) ∼ = π1(G). In [Li], J. Li combined the argument in [Ra] and [AB] to prove that the moduli space of gauge equivalence classes of flat connections on a fixed underlying topological principal Gbundle over Σ is nonempty and connected, where Σ and G are as in Theorem 1.
DECOMPOSABLE REPRESENTATIONS AND LAGRANGIAN SUBMANIFOLDS OF MODULI SPACES ASSOCIATED TO SURFACE Groups
, 2008
"... The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as [9, 25]: given a 3manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the AtiyahFloer conjecture relating the symplectic Floer homology of the re ..."
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The importance of explicit examples of Lagrangian submanifolds of moduli spaces is revealed by papers such as [9, 25]: given a 3manifold M with boundary ∂M = Σ, Dostoglou and Salamon use such examples to obtain a proof of the AtiyahFloer conjecture relating the symplectic Floer homology of the representation space Hom(π1(Σ = ∂M), U)/U (associated to an explicit pair of Lagrangian submanifolds of this representation space) and the instanton homology of the 3manifold M. In the present paper, we construct a Lagrangian submanifold of the space of representations Mg,l:= HomC(πg,l, U)/U of the fundamental group πg,l of a punctured Riemann surface Σg,l into an arbitrary compact connected Lie group U. This Lagrangian submanifold is obtained as the fixedpoint set of an antisymplectic involution ˆ β defined on Mg,l. We show that the involution ˆ β is induced by a formreversing involution β defined on the quasiHamiltonian space (U × U) g × C1 × · · · × Cl. The fact that ˆ β has a nonempty fixedpoint set is a consequence of the real convexity theorem for groupvalued momentum maps proved in [28]. The notion of decomposable representation provides a geometric interpretation of the Lagrangian submanifold thus obtained.
DIRAC STRUCTURES AND DIXMIERDOUADY BUNDLES
, 907
"... Abstract. A Dirac structure on a vector bundle V is a maximal isotropic subbundle E of the direct sum V ⊕ V ∗. We show how to associate to any Dirac structure a DixmierDouady bundle AE, that is, a Z2graded bundle of C ∗algebras with typical fiber the compact operators on a Hilbert space. The cons ..."
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Abstract. A Dirac structure on a vector bundle V is a maximal isotropic subbundle E of the direct sum V ⊕ V ∗. We show how to associate to any Dirac structure a DixmierDouady bundle AE, that is, a Z2graded bundle of C ∗algebras with typical fiber the compact operators on a Hilbert space. The construction has good functorial properties, relative to Morita morphisms of DixmierDouady bundles. As applications, we → G over a compact, connected Lie group (as constructed by AtiyahSegal) is multiplicative, and we obtain a canonical ‘twisted Spincstructure ’ on spaces with group valued moment maps. show that the DixmierDouady bundle A Spin
ON THE CONNECTEDNESS OF MODULI SPACES OF FLAT CONNECTIONS OVER COMPACT SURFACES
, 2004
"... Abstract. We study the connectedness of the moduli space of gauge equivalence classes of flat Gconnections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and ..."
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Abstract. We study the connectedness of the moduli space of gauge equivalence classes of flat Gconnections on a compact orientable surface or a compact nonorientable surface for a class of compact connected Lie groups. This class includes all the compact, connected, simply connected Lie groups, and some nonsemisimple classical groups. 1. introduction Given a compact Lie group G and a compact surface Σ, let M(Σ,G) denote the moduli space of gauge equivalence classes of flat Gconnections on Σ. We know that M(Σ,G) can be identified with Hom(π1(Σ),G)/G, where G acts on Hom(π1(Σ),G) by conjugate action of G on itself (see e.g. [G]). It is known that if G is compact, connected, simply connected (in particular, G is semisimple), and Σ is orientable, then M(Σ,G) is nonempty and connected (see e.g. [Li], [AMM], [AMW]). Thus it is natural to ask about the connectedness of M(Σ,G) for nonorientable Σ. From classification of compact surfaces, all nonorientable compact surfaces are homeomorphic to the connected sum of the real projective planes RP 2. Recall that we have the following structure theorem of compact connected Lie groups [K, Theorem 4.29]:
PREQUANTIZATION OF QUASIHAMILTONIAN SPACES
, 2005
"... Abstract. This paper develops the prequantization of Lie groupvalued moment maps, and establishes its equivalence with the prequantization of infinitedimensional Hamiltonian loop group spaces. 1. ..."
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Abstract. This paper develops the prequantization of Lie groupvalued moment maps, and establishes its equivalence with the prequantization of infinitedimensional Hamiltonian loop group spaces. 1.
Floer field theory for tangles
"... Abstract. We construct functorvalued invariants invariants for cobordisms possibly containing tangles and certain trivalent graphs. The latter gives gaugetheoretic invariants similar to KhovanovRozansky homology [12]. Contents ..."
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Cited by 4 (4 self)
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Abstract. We construct functorvalued invariants invariants for cobordisms possibly containing tangles and certain trivalent graphs. The latter gives gaugetheoretic invariants similar to KhovanovRozansky homology [12]. Contents
THE VOLUME OF THE MODULI SPACE OF FLAT CONNECTIONS ON A NONORIENTABLE
, 2003
"... Abstract. We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2manifold, for a natural class of metrics. We also show that Witten’s volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten’s volume formula fo ..."
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Abstract. We compute the Riemannian volume on the moduli space of flat connections on a nonorientable 2manifold, for a natural class of metrics. We also show that Witten’s volume formula for these moduli spaces may be derived using Haar measure, and we give a new proof of Witten’s volume formula for the moduli space of flat connections on an orientable surface using Haar measure. 1.
A NEW PROOF OF FORMULAS FOR INTERSECTION NUMBERS IN QHAMILTONIAN REDUCED SPACES
, 2008
"... Abstract. Jeffrey and Kirwan [21] gave expressions for intersection pairings on the reduced space µ −1 (0)/G of a particular Hamiltonian Gspace M in terms of iterated residues. The definition of quasiHamiltonian spaces was introduced in [2]. In [4] a localization formula for equivariant de Rham co ..."
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Abstract. Jeffrey and Kirwan [21] gave expressions for intersection pairings on the reduced space µ −1 (0)/G of a particular Hamiltonian Gspace M in terms of iterated residues. The definition of quasiHamiltonian spaces was introduced in [2]. In [4] a localization formula for equivariant de Rham cohomology of a compact qHamiltonian Gspace was proved. In this paper we prove a residue formula for intersection pairings of reduced spaces of certain quasiHamiltonian Gspaces, by constructing the corresponding Hamiltonian Gspace. We show that the result agrees with that in [4]. In this article we rely heavily on the methods of [21]; for the more general class of compact Lie groups G treated in [4], we rely on results of Szenes and BrionVergne concerning diagonal bases.