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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:35217 |
| Date | 02 September 2019 |
| Creators | Diez, Tobias |
| Contributors | Universität Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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