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17
LOCALIZATION OF THE RIEMANN-ROCH CHARACTER
, 2005
"... We present a K-theoretic approach to the Guillemin-Sternberg conjecture [17], about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken [28, 29] and Tian-Zhang [35]. Besides providing a new proof of this conjecture for the full non-abelian group act ..."
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Cited by 42 (11 self)
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We present a K-theoretic approach to the Guillemin-Sternberg conjecture [17], about the commutativity of geometric quantization and symplectic reduction, which was proved by Meinrenken [28, 29] and Tian-Zhang [35]. Besides providing a new proof of this conjecture for the full non-abelian group action case, our methods lead to a generalization for compact Lie group actions on manifolds that are not symplectic; these manifolds carry an invariant almost complex structure and an abstract moment map.
Singular reduction and quantization
, 1996
"... Abstract. Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The “quantization commutes with reduction ” theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic qu ..."
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Cited by 34 (3 self)
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Abstract. Consider a compact prequantizable symplectic manifold M on which a compact Lie group G acts in a Hamiltonian fashion. The “quantization commutes with reduction ” theorem asserts that the G-invariant part of the equivariant index of M is equal to the Riemann-Roch number of the symplectic quotient of M, provided the quotient is nonsingular. We extend this result to singular symplectic quotients, using partial desingularizations of the symplectic quotient to define its Riemann-Roch number. By similar methods we also compute multiplicities for the equivariant index of the dual of a prequantum bundle, and furthermore show that the arithmetic genus of a Hamiltonian G-manifold is invariant under symplectic reduction.
ON THE UNFOLDING OF FOLDED SYMPLECTIC STRUCTURES
"... Abstract. A folded symplectic structure is a closed 2-form which is nondegenerate except on a hypersurface, and whose restriction to that hypersurface has maximal rank. We show how a compact manifold equipped with a folded symplectic structure can sometimes be broken apart, or “unfolded”, into hones ..."
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Cited by 7 (0 self)
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Abstract. A folded symplectic structure is a closed 2-form which is nondegenerate except on a hypersurface, and whose restriction to that hypersurface has maximal rank. We show how a compact manifold equipped with a folded symplectic structure can sometimes be broken apart, or “unfolded”, into honest compact symplectic orbifolds. A folded symplectic structure induces a spin-c structure which is canonical (up to homotopy). We describe how the index of the spin-c Dirac operator behaves with respect to unfolding. 1.
Holomorphic Morse inequalities and symplectic reduction, Topology 38
, 1999
"... Abstract. We introduce Morse-type inequalities for a holomorphic circle action on a holomorphic vector bundle over a compact Kähler manifold. Our inequalities produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomology in terms of the data of the fixed points and of ..."
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Cited by 4 (1 self)
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Abstract. We introduce Morse-type inequalities for a holomorphic circle action on a holomorphic vector bundle over a compact Kähler manifold. Our inequalities produce bounds on the multiplicities of weights occurring in the twisted Dolbeault cohomology in terms of the data of the fixed points and of the symplectic reduction. This result generalizes both Wu-Zhang extension of Witten’s holomorphic Morse inequalities and Tian-Zhang Morse-type inequalities for symplectic reduction. As an application we get a new proof of the Tian-Zhang relative index theorem for symplectic quotients. 1.
Spin c-Quantization and the K-multiplicities of the discrete series
, 2002
"... We express the K-multiplicities of a representation of the discrete series associated to a coadjoint orbit O in terms of Spin c-index on symplectic reductions of O. ..."
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Cited by 4 (2 self)
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We express the K-multiplicities of a representation of the discrete series associated to a coadjoint orbit O in terms of Spin c-index on symplectic reductions of O.
COHOMOLOGICAL LOCALIZATION FOR MANIFOLDS WITH BOUNDARY
, 1999
"... Abstract. For an S 1-manifold with boundary, we prove a localization formula applying to any equivariant cohomology theory satisfying a certain algebraic condition. We show how the localization result of Kalkman and a case of the quantization commutes with reduction theorem follow easily from the lo ..."
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Cited by 2 (0 self)
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Abstract. For an S 1-manifold with boundary, we prove a localization formula applying to any equivariant cohomology theory satisfying a certain algebraic condition. We show how the localization result of Kalkman and a case of the quantization commutes with reduction theorem follow easily from the localization formula. 1.
4 L∞-ALGEBRAS OF LOCAL OBSERVABLES FROM HIGHER PREQUANTUM BUNDLES
"... The following full text is a preprint version which may differ from the publisher's version. For additional information about this publication click this link. ..."
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The following full text is a preprint version which may differ from the publisher's version. For additional information about this publication click this link.
COHOMOLOGIE ÉQUIVARIANTE ET QUANTIFICATION GÉOMÉTRIQUE
"... In this paper I summarize the work I have done to realize the program of Witten called non abelian localization. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternberg problem-Quantization commutes with reduction- ..."
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In this paper I summarize the work I have done to realize the program of Witten called non abelian localization. This work deals first with problems of localization in equivariant cohomology. The second part of this paper concerns the Guillemin-Sternberg problem-Quantization commutes with reduction- in the geometric quantization procedure.
