39 research outputs found
Arbitrage-Free Pricing Before and Beyond Probabilities
"Fundamental theorem of asset pricing" roughly states that absence of
arbitrage opportunity in a market is equivalent to the existence of a
risk-neutral probability. We give a simple counterexample to this
oversimplified statement. Prices are given by linear forms which do not always
correspond to probabilities. We give examples of such cases. We also show that
arbitrage freedom is equivalent to the continuity of the pricing linear form in
the relevant topology. Finally we analyze the possible loss of martingality of
asset prices with lognormal stochastic volatility. For positive correlation
martingality is lost when the financial process is modelled through standard
probability theory. We show how to recover martingality using the appropriate
mathematical tools.Comment: 5 page
Selfduality of d=2 Reduction of Gravity Coupled to a Sigma-Model
Dimensional reduction in two dimensions of gravity in higher dimension, or
more generally of d=3 gravity coupled to a sigma-model on a symmetric space, is
known to possess an infinite number of symmetries. We show that such a
bidimensional model can be embedded in a covariant way into a sigma-model on an
infinite symmetric space, built on the semidirect product of an affine group by
the Witt group. The finite theory is the solution of a covariant selfduality
constraint on the infinite model. It has therefore the symmetries of the
infinite symmetric space. (We give explicit transformations of the gauge
algebra.) The usual physical fields are recovered in a triangular gauge, in
which the equations take the form of the usual linear systems which exhibit the
integrable structure of the models. Moreover, we derive the constraint equation
for the conformal factor, which is associated to the central term of the affine
group involved.Comment: 7 page
Cosmological billiards and oxidation
We show how the properties of the cosmological billiards provide useful
information (spacetime dimension and -form spectrum) on the oxidation
endpoint of the oxidation sequence of gravitational theories. We compare this
approach to the other available methods: subgroups and the
superalgebras of dualities.Comment: To appear in the Proceedings of the 27th Johns Hopkins Workshop and
in the Proceedings of the 36th International Symposium Ahrenshoop; v2: minor
error correcte
Borcherds symmetries in M-theory
It is well known but rather mysterious that root spaces of the Lie
groups appear in the second integral cohomology of regular, complex, compact,
del Pezzo surfaces. The corresponding groups act on the scalar fields (0-forms)
of toroidal compactifications of M theory. Their Borel subgroups are actually
subgroups of supergroups of finite dimension over the Grassmann algebra of
differential forms on spacetime that have been shown to preserve the
self-duality equation obeyed by all bosonic form-fields of the theory. We show
here that the corresponding duality superalgebras are nothing but Borcherds
superalgebras truncated by the above choice of Grassmann coefficients. The full
Borcherds' root lattices are the second integral cohomology of the del Pezzo
surfaces. Our choice of simple roots uses the anti-canonical form and its known
orthogonal complement. Another result is the determination of del Pezzo
surfaces associated to other string and field theory models. Dimensional
reduction on corresponds to blow-up of points in general position
with respect to each other. All theories of the Magic triangle that reduce to
the sigma model in three dimensions correspond to singular del Pezzo
surfaces with (normal) singularity at a point. The case of type I and
heterotic theories if one drops their gauge sector corresponds to non-normal
(singular along a curve) del Pezzo's. We comment on previous encounters with
Borcherds algebras at the end of the paper.Comment: 30 pages. Besides expository improvements, we exclude by hand real
fermionic simple roots when they would naively aris
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
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
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