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On The Hamiltonian Circle Actions And Symplectic ReductionDemir, Ali Sait 01 January 2003 (has links) (PDF)
Given a symplectic manifold, it is of interest how Lie group actions, their orbit
spaces look like and what are some topological requirements on the existence of
such actions. In this thesis we present the work of Ono, giving some sufficient
conditions for non-existence of circle actions on symplectic manifolds and work
of Li, describing the fundamental groups of symplectic reductions of circle actions.
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Moduli spaces and deformation quantization in infinite dimensionsFedosov, Boris January 1998 (has links)
We construct a deformation quantization on an infinite-dimensional symplectic space of regular connections on an SU(2)-bundle over a Riemannian surface of genus g ≥ 2. The construction is based on the normal form thoerem representing the space of connections as a fibration over a finite-dimensional moduli space of flat connections whose fibre is a cotangent bundle of the infinite-dimensional gauge group. We study the reduction with respect to the gauge groupe both for classical and quantum cases and show that our quantization commutes with reduction.
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Singularité et théorie de Lie / Singularity and Lie TheoryCaradot, Antoine 14 June 2017 (has links)
Soit Γ un sous-groupe fini de SU2(ℂ). Alors le quotient ℂ2/Γ peut être plongé dans ℂ3 sous la forme d'une surface munie d'une singularité isolée. Le quotient ℂ2/Γ est appelé singularité de Klein, d'après F. Klein qui fut le premier à les décrire en 1884. A travers leurs résolutions minimales, ces singularités ont un lien étroit avec les diagrammes de Dynkin simplement lacés de types Ar, Dr et Er. Dans les années 1970, E. Brieskorn et P. Slodowy ont tiré profit de cette connection pour décrire les résolutions et les déformations de ces singularités à l'aide de la théorie de Lie. En 1998 P. Slodowy et H. Cassens ont construit les déformations semiuniverselles des ℂ2/Γ à l'aide de la théorie des carquois ainsi que des travaux de P.B. Kronheimer en géométrie symplectique datant de 1989. En théorie de Lie, la classification des algèbres de Lie simples divisent ces dernières en deux classes: les algèbres de Lie de types Ar, Dr et Er qui sont simplement lacées, et celles de types Br, Cr, F4 et G2 appelées non-homogènes. A l'aide d'un second sous-groupe fini Γ' de SU2(ℂ) tel que Γ ⊲ Γ', P. Slodowy a étendu en 1978 la notion de singularité de Klein aux algèbres de Lie non-homogènes en ajoutant à ℂ2/Γ le groupe d'automorphismes Ω= Γ'/Γ du diagramme de Dynkin associé à la singularité. L'objectif de cette thèse est de généraliser la construction de H. Cassens et P. Slodowy à ces singularités de types Br, Cr, F4 et G2. Il en résultera des constructions explicites des déformations semiuniverselles de types inhomogènes sur les fibres desquelles le groupe Ω agit. Le passage au quotient d'une telle application révèle alors une déformation d'une singularité de type ℂ2/Γ' / Let Γ be a finite subgroup of SU2(ℂ). Then the quotient ℂ2/Γ can be embedded in ℂ3 as a surface with an isolated singularity. The quotient ℂ2/Γ is called a Kleinian singularity, after F. Klein who studied them first in 1884. Through their minimal resolutions, these singularities have a deep connection with simply-laced Dynkin diagrams of types Ar, Dr and Er. In the 1970's E. Brieskorn and P. Slodowy took advantage of this connection to describe the resolutions and deformations of these singularities in terms of Lie theory. In 1998 P. Slodowy and H. Cassens constructed the semiuniversal deformations of the Kleinian singularities using quiver theory and work from 1989 by P.B. Kronheimer on symplectic geometry. In Lie theory, the classification of simple Lie algebras allows for a separation in two classes: those simply-laced of types Ar, Dr and Er, and those of types Br, Cr, F4 and G2 called inhomogeneous. With the use of a second finite subgroup Γ’ of SU2(ℂ) such that Γ ⊲ Γ’, P. Slodowy extended in 1978 the definition of a Kleinian singularity to the inhomogeneous types by adding to ℂ2/Γ the group of automorphisms Ω= Γ’/Γ of the Dynkin diagram associated to the singularity. The purpose of this thesis is to generalize H. Cassens' and P. Slodowy's construction to the singularities of types Br, Cr, F4 and G2. It will lead to explicit semiuniversal deformations of inhomogeneous types on the fibers of which the group Ω acts. By quotienting such a map we obtain a deformation of a singularity ℂ2/Γ’
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Normal Form of Equivariant Maps and Singular Symplectic Reduction in Infinite Dimensions with Applications to Gauge Field TheoryDiez, Tobias 02 September 2019 (has links)
Inspired by problems in gauge field theory, this thesis is concerned with various
aspects of infinite-dimensional differential geometry.
In the first part, a local normal form theorem for smooth equivariant maps
between tame Fréchet manifolds is established. Moreover, an elliptic version of
this theorem is obtained. The proof these normal form results is inspired by
the Lyapunov–Schmidt reduction for dynamical systems and by the Kuranishi
method for moduli spaces, and uses a slice theorem for Fréchet manifolds as
the main technical tool. As a consequence of this equivariant normal form
theorem, the abstract moduli space obtained by factorizing a level set of the
equivariant map with respect to the group action carries the structure of a
Kuranishi space, i.e., such moduli spaces are locally modeled on the quotient
by a compact group of the zero set of a smooth map.
In the second part of the thesis, the theory of singular symplectic reduction
is developed in the infinite-dimensional Fréchet setting. By refining the above
construction, a normal form for momentum maps similar to the classical
Marle–Guillemin–Sternberg normal form is established. Analogous to the
reasoning in finite dimensions, this normal form result is then used to show
that the reduced phase space decomposes into smooth manifolds each carrying
a natural symplectic structure.
Finally,the singular symplectic reduction scheme is further investigated in the
situation where the original phase space is an infinite-dimensional cotangent
bundle. The fibered structure of the cotangent bundle yields a refinement of
the usual orbit-momentum type strata into so-called seams. Using a suitable
normal form theorem, it is shown that these seams are manifolds. Taking
the harmonic oscillator as an example, the influence of the singular seams on
dynamics is illustrated.
The general results stated above are applied to various gauge theory models.
The moduli spaces of anti-self-dual connections in four dimensions and of
Yang–Mills connections in two dimensions is studied. Moreover, the stratified
structure of the reduced phase space of the Yang–Mills–Higgs theory is
investigated in a Hamiltonian formulation after a (3 + 1)-splitting.
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