77 research outputs found
Oxidation = group theory
Dimensional reduction of theories involving (super-)gravity gives rise to
sigma models on coset spaces of the form G/H, with G a non-compact group, and H
its maximal compact subgroup. The reverse process, called oxidation, is the
reconstruction of the possible higher dimensional theories, given the lower
dimensional theory. In 3 dimensions, all degrees of freedom can be dualized to
scalars. Given the group G for a 3 dimensional sigma model on the coset G/H, we
demonstrate an efficient method for recovering the higher dimensional theories,
essentially by decomposition into subgroups. The equations of motion, Bianchi
identities, Kaluza-Klein modifications and Chern-Simons terms are easily
extracted from the root lattice of the group G. We briefly discuss some aspects
of oxidation from the E_{8(8)}/SO(16) coset, and demonstrate that our formalism
reproduces the Chern-Simons term of 11-d supergravity, knows about the
T-duality of IIA and IIB theory, and easily deals with self-dual tensors, like
the 5-tensor of IIB supergravity.Comment: LaTeX, 8 pages, uses IOP style files; Talk given at the RTN workshop
``The quantum structure of spacetime and the geometric nature of fundamental
interactions'', Leuven, September 200
The topology of U-duality (sub-)groups
We discuss the topology of the symmetry groups appearing in compactified
(super-)gravity, and discuss two applications. First, we demonstrate that for 3
dimensional sigma models on a symmetric space G/H with G non-compact and H the
maximal compact subgroup of G, the possibility of oxidation to a higher
dimensional theory can immediately be deduced from the topology of H. Second,
by comparing the actual symmetry groups appearing in maximal supergravities
with the subgroups of SL(32,R) and Spin(32), we argue that these groups cannot
serve as a local symmetry group for M-theory in a formulation of de Wit-Nicolai
type.Comment: 18 pages, LaTeX, 1 figure, 2 table
Root to Kellerer
We revisit Kellerer's Theorem, that is, we show that for a family of real
probability distributions which increases in convex
order there exists a Markov martingale s.t.\ .
To establish the result, we observe that the set of martingale measures with
given marginals carries a natural compact Polish topology. Based on a
particular property of the martingale coupling associated to Root's embedding
this allows for a relatively concise proof of Kellerer's theorem.
We emphasize that many of our arguments are borrowed from Kellerer
\cite{Ke72}, Lowther \cite{Lo07}, and Hirsch-Roynette-Profeta-Yor
\cite{HiPr11,HiRo12}.Comment: 8 pages, 1 figur
A note on spin-s duality
Duality is investigated for higher spin (), free, massless, bosonic
gauge fields. We show how the dual formulations can be derived from a common
"parent", first-order action. This goes beyond most of the previous treatments
where higher-spin duality was investigated at the level of the equations of
motion only. In D=4 spacetime dimensions, the dual theories turn out to be
described by the same Pauli-Fierz (s=2) or Fronsdal () action (as it
is the case for spin 1). In the particular s=2 D=5 case, the Pauli-Fierz action
and the Curtright action are shown to be related through duality. A crucial
ingredient of the analysis is given by the first-order, gauge-like,
reformulation of higher spin theories due to Vasiliev.Comment: Minor corrections, reference adde
Hyperbolic billiards of pure D=4 supergravities
We compute the billiards that emerge in the Belinskii-Khalatnikov-Lifshitz
(BKL) limit for all pure supergravities in D=4 spacetime dimensions, as well as
for D=4, N=4 supergravities coupled to k (N=4) Maxwell supermultiplets. We find
that just as for the cases N=0 and N=8 investigated previously, these billiards
can be identified with the fundamental Weyl chambers of hyperbolic Kac-Moody
algebras. Hence, the dynamics is chaotic in the BKL limit. A new feature
arises, however, which is that the relevant Kac-Moody algebra can be the
Lorentzian extension of a twisted affine Kac-Moody algebra, while the N=0 and
N=8 cases are untwisted. This occurs for N=5, N=3 and N=2. An understanding of
this property is provided by showing that the data relevant for determining the
billiards are the restricted root system and the maximal split subalgebra of
the finite-dimensional real symmetry algebra characterizing the toroidal
reduction to D=3 spacetime dimensions. To summarize: split symmetry controls
chaos.Comment: 21 page
Superconformal Selfdual Sigma-Models
A range of bosonic models can be expressed as (sometimes generalized)
-models, with equations of motion coming from a selfduality constraint.
We show that in D=2, this is easily extended to supersymmetric cases, in a
superspace approach. In particular, we find that the configurations of fields
of a superconformal coset models which satisfy some
selfduality constraint are automatically solutions to the equations of motion
of the model. Finally, we show that symmetric space -models can be seen
as infinite-dimensional \tfG/\tfH models constrained by a selfduality
equation, with \tfG the loop extension of and \tfH a maximal
subgroup. It ensures that these models have a hidden global \tfG symmetry
together with a local \tfH gauge symmetry.Comment: 21 pages; v2 few corrections and references added; v3 exposition
change
N-Complexes and Higher Spin Gauge Fields
-complexes have been argued recently to be algebraic structures relevant
to the description of higher spin gauge fields. -complexes involve a linear
operator that fulfills and that defines a generalized cohomology.
Some elementary properties of -complexes and the evidence for their
relevance to the description of higher spin gauge fields are briefly reviewed.Comment: Presented at the International Workshop "Differential Geometry,
Noncommutative Geometry, Homology and Fundamental Interactions" in honour of
Michel Dubois-Violette, Orsay, April 8-10, 200
Maximal supergravity in D=10: forms, Borcherds algebras and superspace cohomology
We give a very simple derivation of the forms of supergravity from
supersymmetry and SL(2,\bbR) (for IIB). Using superspace cohomology we show
that, if the Bianchi identities for the physical fields are satisfied, the
(consistent) Bianchi identities for all of the higher-rank forms must be
identically satisfied, and that there are no possible gauge-trivial Bianchi
identities () except for exact eleven-forms. We also show that the
degrees of the forms can be extended beyond the spacetime limit, and that the
representations they fall into agree with those predicted from Borcherds
algebras. In IIA there are even-rank RR forms, including a non-zero
twelve-form, while in IIB there are non-trivial Bianchi identities for
thirteen-forms even though these forms are identically zero in supergravity. It
is speculated that these higher-rank forms could be non-zero when higher-order
string corrections are included.Comment: 15 pages. Published version. Some clarification of the tex
Hidden Symmetries and Dirac Fermions
In this paper, two things are done. First, we analyze the compatibility of
Dirac fermions with the hidden duality symmetries which appear in the toroidal
compactification of gravitational theories down to three spacetime dimensions.
We show that the Pauli couplings to the p-forms can be adjusted, for all simple
(split) groups, so that the fermions transform in a representation of the
maximal compact subgroup of the duality group G in three dimensions. Second, we
investigate how the Dirac fermions fit in the conjectured hidden overextended
symmetry G++. We show compatibility with this symmetry up to the same level as
in the pure bosonic case. We also investigate the BKL behaviour of the
Einstein-Dirac-p-form systems and provide a group theoretical interpretation of
the Belinskii-Khalatnikov result that the Dirac field removes chaos.Comment: 30 page
- …