2,025 research outputs found
Entry and Return times distribution
This is a review article on the distributions of entry and return times in
dynamical systems which discusses recent results for systems of positive
entropy.Comment: To appear in "Dynamical Systems: An International Journal dedicated
to the Statistical Properties of Dynamical Systems
Limiting distribution and error terms for the number of visits to balls in non-uniformly hyperbolic dynamical systems
We show that for systems that allow a Young tower construction with
polynomially decaying correlations the return times to metric balls are in the
limit Poisson distributed. We also provide error terms which are powers of
logarithm of the radius. In order to get those uniform rates of convergence the
balls centres have to avoid a set whose size is estimated to be of similar
order. This result can be applied to non-uniformly hyperbolic maps and to any
invariant measure that satisfies a weak regularity condition. In particular it
shows that the return times to balls is Poissonian for SRB measures on
attractors.Comment: 28 page
Gradient Representations and Affine Structures in AE(n)
We study the indefinite Kac-Moody algebras AE(n), arising in the reduction of
Einstein's theory from (n+1) space-time dimensions to one (time) dimension, and
their distinguished maximal regular subalgebras sl(n) and affine A_{n-2}^{(1)}.
The interplay between these two subalgebras is used, for n=3, to determine the
commutation relations of the `gradient generators' within AE(3). The low level
truncation of the geodesic sigma-model over the coset space AE(n)/K(AE(n)) is
shown to map to a suitably truncated version of the SL(n)/SO(n) non-linear
sigma-model resulting from the reduction Einstein's equations in (n+1)
dimensions to (1+1) dimensions. A further truncation to diagonal solutions can
be exploited to define a one-to-one correspondence between such solutions, and
null geodesic trajectories on the infinite-dimensional coset space H/K(H),
where H is the (extended) Heisenberg group, and K(H) its maximal compact
subgroup. We clarify the relation between H and the corresponding subgroup of
the Geroch group.Comment: 43 page
K(E10), Supergravity and Fermions
We study the fermionic extension of the E10/K(E10) coset model and its
relation to eleven-dimensional supergravity. Finite-dimensional spinor
representations of the compact subgroup K(E10) of E(10,R) are studied and the
supergravity equations are rewritten using the resulting algebraic variables.
The canonical bosonic and fermionic constraints are also analysed in this way,
and the compatibility of supersymmetry with local K(E10) is investigated. We
find that all structures involving A9 levels 0,1 and 2 nicely agree with
expectations, and provide many non-trivial consistency checks of the existence
of a supersymmetric extension of the E10/K(E10) coset model, as well as a new
derivation of the `bosonic dictionary' between supergravity and coset
variables. However, there are also definite discrepancies in some terms
involving level 3, which suggest the need for an extension of the model to
infinite-dimensional faithful representations of the fermionic degrees of
freedom.Comment: 50 page
An E9 multiplet of BPS states
We construct an infinite E9 multiplet of BPS states for 11D supergravity. For
each positive real root of E9 we obtain a BPS solution of 11D supergravity, or
of its exotic counterparts, depending on two non-compact transverse space
variables. All these solutions are related by U-dualities realised via E9 Weyl
transformations in the regular embedding of E9 in E10, E10 in E11. In this way
we recover the basic BPS solutions, namely the KK-wave, the M2 brane, the M5
brane and the KK6-monopole, as well as other solutions admitting eight
longitudinal space dimensions. A novel technique of combining Weyl reflexions
with compensating transformations allows the construction of many new BPS
solutions, each of which can be mapped to a solution of a dual effective action
of gravity coupled to a certain higher rank tensor field. For real roots of E10
which are not roots of E9, we obtain additional BPS solutions transcending 11D
supergravity (as exemplified by the lowest level solution corresponding to the
M9 brane). The relation between the dual formulation and the one in terms of
the original 11D supergravity fields has significance beyond the realm of BPS
solutions. We establish the link with the Geroch group of general relativity,
and explain how the E9 duality transformations generalize the standard Hodge
dualities to an infinite set of `non-closing dualities'.Comment: 76 pages, 6 figure
E10 and SO(9,9) invariant supergravity
We show that (massive) D=10 type IIA supergravity possesses a hidden rigid
SO(9,9) symmetry and a hidden local SO(9) x SO(9) symmetry upon dimensional
reduction to one (time-like) dimension. We explicitly construct the associated
locally supersymmetric Lagrangian in one dimension, and show that its bosonic
sector, including the mass term, can be equivalently described by a truncation
of an E10/K(E10) non-linear sigma-model to the level \ell<=2 sector in a
decomposition of E10 under its so(9,9) subalgebra. This decomposition is
presented up to level 10, and the even and odd level sectors are identified
tentatively with the Neveu--Schwarz and Ramond sectors, respectively. Further
truncation to the level \ell=0 sector yields a model related to the reduction
of D=10 type I supergravity. The hyperbolic Kac--Moody algebra DE10, associated
to the latter, is shown to be a proper subalgebra of E10, in accord with the
embedding of type I into type IIA supergravity. The corresponding decomposition
of DE10 under so(9,9) is presented up to level 5.Comment: 1+39 pages LaTeX2e, 2 figures, 2 tables, extended tables obtainable
by downloading sourc
Domain walls in three dimensional gauged supergravity
We explicitly construct two Chern-Simons gauged supergravities in three
dimensions with N=4 and N=8 supersymmetries and non-semisimple gauge groups.
The N=4 theory has scalar manifold with the gauge
group . The theory describes
(1,0) six dimensional supergravity reduced on an SU(2) group manifold. The
equivalent Yang-Mills type gauged supergravity has SO(3) gauge group coupled to
three massive vector fields. The N=8 theory is described by
scalar manifold, and the gauge group is given by
. The theory is a truncation of the gauged N=16 theory with scalar manifold and
can be obtained by an S^7 compactification of type I theory in ten dimensions.
Domain wall solutions of both gauged supergravities are analytically found and
can be uplifted to higher dimensions. These provide domain wall vacua in the
three dimensional gauged supergravity framework which might be useful for the
study of Domain Wall/QFT correspondence.Comment: 19 pages, no figures, typoes and a mistake in a sign corrected,
clarifications on the notations adde
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