385 research outputs found
Slice theorem and orbit type stratification in infinite dimensions
We establish a general slice theorem for the action of a locally convex Lie
group on a locally convex manifold, which generalizes the classical slice
theorem of Palais to infinite dimensions.
We discuss two important settings under which the assumptions of this theorem
are fulfilled. First, using Gl\"ockner's inverse function theorem, we show that
the linear action of a compact Lie group on a Fr\'echet space admits a slice.
Second, using the Nash--Moser theorem, we establish a slice theorem for the
tame action of a tame Fr\'echet Lie group on a tame Fr\'echet manifold. For
this purpose, we develop the concept of a graded Riemannian metric, which
allows the construction of a path-length metric compatible with the manifold
topology and of a local addition.
Finally, generalizing a classical result in finite dimensions, we prove that
the existence of a slice implies that the decomposition of the manifold into
orbit types of the group action is a stratification
Exact solutions in Einstein-Yang-Mills-Dirac systems
We present exact solutions in Einstein-Yang-Mills-Dirac theories with gauge
groups SU(2) and SU(4) in Robertson-Walker space-time , which
are symmetric under the action of the group SO(4) of spatial rotations. Our
approach is based on the dimensional reduction method for gauge and
gravitational fields and relates symmetric solutions in EYMD theory to certain
solutions of an effective dynamical system.
We interpret our solutions as cosmological solutions with an oscillating
Yang-Mills field passing between topologically distinct vacua. The explicit
form of the solution for spinor field shows that its energy changes the sign
during the evolution of the Yang-Mills field from one vacuum to the other,
which can be considered as production or annihilation of fermions.
Among the obtained solutions there is also a static sphaleron-like solution,
which is a cosmological analogue of the first Bartnik-McKinnon solution in the
presence of fermions.Comment: 18 pages, LaTeX 2
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