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A Note on Hamiltonian Lie Group Actions and Massey Products

Zofia Stępień, Aleksy Tralle (2004)

Bulletin of the Polish Academy of Sciences. Mathematics

We show that the property of having only vanishing triple Massey products in equivariant cohomology is inherited by the set of fixed points of hamiltonian circle actions on closed symplectic manifolds. This result can be considered in a more general context of characterizing homotopic properties of Lie group actions. In particular it can be viewed as a partial answer to a question posed by Allday and Puppe about finding conditions ensuring the "formality" of G-actions.

A quantum Duistermaat-Heckamn formula?

Alberto Ibort (2003)


Some aspects of Duistermaat-Heckman formula in finite dimensions are reviewed. We especulate with some of its possible extensions to infinite dimensions. In particular we review the localization principle and the geometry of loop spaces following Witten and Atiyah?s insight.

Collective geodesic flows

Léo T. Butler, Gabriel P. Paternain (2003)

Annales de l’institut Fourier

We show that most compact semi-simple Lie groups carry many left invariant metrics with positive topological entropy. We also show that many homogeneous spaces admit collective Riemannian metrics arbitrarily close to the bi-invariant metric and whose geodesic flow has positive topological entropy. Other properties of collective geodesic flows are also discussed.

Harder-Narasimhan filtrations and optimal destabilizing vectors in complex geometry

Laurent Bruasse, Andrei Teleman (2005)

Annales de l’institut Fourier

We give here a generalization of the theory of optimal destabilizing 1-parameter subgroups to non algebraic complex geometry : we consider holomorphic actions of a complex reductive Lie group on a finite dimensional (possibly non compact) Kähler manifold. In a second part we show how these results may extend in the gauge theoretical framework and we discuss the relation between the Harder-Narasimhan filtration and the optimal detstabilizing vectors of a non semistable object....

Higher symmetries of the Laplacian via quantization

Jean-Philippe Michel (2014)

Annales de l’institut Fourier

We develop a new approach, based on quantization methods, to study higher symmetries of invariant differential operators. We focus here on conformally invariant powers of the Laplacian over a conformally flat manifold and recover results of Eastwood, Leistner, Gover and Šilhan. In particular, conformally equivariant quantization establishes a correspondence between the algebra of Hamiltonian symmetries of the null geodesic flow and the algebra of higher symmetries of the conformal Laplacian. Combined...

Invariants symplectiques et semi-classiques des systèmes intégrables avec singularités

San Vũ Ngọc (2000/2001)

Séminaire Équations aux dérivées partielles

On définit les notions de feuilletages classiques et semi-classiques pour les systèmes complètement intégrables avec singularités. Les résultats de classification standard (telles les coordonnées actions-angles semi-classiques) sont rappelés. Le cas du feuilletage classique de type foyer-foyer est examiné en détail, où des nouveaux invariants semi-globaux apparaissent. Ces invariants sont identifiés dans les conditions de Bohr-Sommerfeld singulières qui donnent le spectre conjoint au voisinage d’une...

Isospectrality for quantum toric integrable systems

Laurent Charles, Álvaro Pelayo, San Vũ Ngoc (2013)

Annales scientifiques de l'École Normale Supérieure

We give a full description of the semiclassical spectral theory of quantum toric integrable systems using microlocal analysis for Toeplitz operators. This allows us to settle affirmatively the isospectral problem for quantum toric integrable systems: the semiclassical joint spectrum of the system, given by a sequence of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the classical integrable system given by the symplectic manifold and commuting Hamiltonians. This type of...

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