144 research outputs found
The twistor spinors of generic 2- and 3-distributions
Generic distributions on 5- and 6-manifolds give rise to conformal structures
that were discovered by P. Nurowski resp. R. Bryant. We describe both as
Fefferman-type constructions and show that for orientable distributions one
obtains conformal spin structures. The resulting conformal spin geometries are
then characterized by their conformal holonomy and equivalently by the
existence of a twistor spinor which satisfies a genericity condition. Moreover,
we show that given such a twistor spinor we can decompose a conformal Killing
field of the structure. We obtain explicit formulas relating conformal Killing
fields, almost Einstein structures and twistor spinors.Comment: 26 page
Supergeometry in locally covariant quantum field theory
In this paper we analyze supergeometric locally covariant quantum field theories. We develop suitable categories SLoc of super-Cartan supermanifolds, which generalize Lorentz manifolds in ordinary quantum field theory, and show that, starting from a few representation theoretic and geometric data, one can construct a functor A : SLoc --> S*Alg to the category of super-*-algebras which can be interpreted as a non-interacting super-quantum field theory. This construction turns out to disregard supersymmetry transformations as the morphism sets in the above categories are too small. We then solve this problem by using techniques from enriched category theory, which allows us to replace the morphism sets by suitable morphism supersets that contain supersymmetry transformations as their higher superpoints. We construct super-quantum field theories in terms of enriched functors eA : eSLoc --> eS*Alg between the enriched categories and show that supersymmetry transformations are appropriately described within the enriched framework. As examples we analyze the superparticle in 1|1-dimensions and the free Wess-Zumino model in 3|2-dimensions
Going chiral: overlap versus twisted mass fermions
We compare the behavior of overlap fermions, which are chirally invariant,
and of Wilson twisted mass fermions at full twist in the approach to the chiral
limit. Our quenched simulations reveal that with both formulations of lattice
fermions pion masses of O(250 MeV) can be reached in practical applications.
Our comparison is done at a fixed value of the lattice spacing a=0.123 fm. A
number of quantities are measured such as hadron masses, pseudoscalar decay
constants and quark masses obtained from Ward identities. We also determine the
axial vector renormalization constants in the case of overlap fermions.Comment: 22 pages, 10 figure
Grafting and Poisson structure in (2+1)-gravity with vanishing cosmological constant
We relate the geometrical construction of (2+1)-spacetimes via grafting to
phase space and Poisson structure in the Chern-Simons formulation of
(2+1)-dimensional gravity with vanishing cosmological constant on manifolds of
topology , where is an orientable two-surface of genus
. We show how grafting along simple closed geodesics \lambda is
implemented in the Chern-Simons formalism and derive explicit expressions for
its action on the holonomies of general closed curves on S_g. We prove that
this action is generated via the Poisson bracket by a gauge invariant
observable associated to the holonomy of . We deduce a symmetry
relation between the Poisson brackets of observables associated to the Lorentz
and translational components of the holonomies of general closed curves on S_g
and discuss its physical interpretation. Finally, we relate the action of
grafting on the phase space to the action of Dehn twists and show that grafting
can be viewed as a Dehn twist with a formal parameter satisfying
.Comment: 43 pages, 10 .eps figures; minor modifications: 2 figures added,
explanations added, typos correcte
Geometrical (2+1)-gravity and the Chern-Simons formulation: Grafting, Dehn twists, Wilson loop observables and the cosmological constant
We relate the geometrical and the Chern-Simons description of
(2+1)-dimensional gravity for spacetimes of topology , where
is an oriented two-surface of genus , for Lorentzian signature and general
cosmological constant and the Euclidean case with negative cosmological
constant. We show how the variables parametrising the phase space in the
Chern-Simons formalism are obtained from the geometrical description and how
the geometrical construction of (2+1)-spacetimes via grafting along closed,
simple geodesics gives rise to transformations on the phase space. We
demonstrate that these transformations are generated via the Poisson bracket by
one of the two canonical Wilson loop observables associated to the geodesic,
while the other acts as the Hamiltonian for infinitesimal Dehn twists. For
spacetimes with Lorentzian signature, we discuss the role of the cosmological
constant as a deformation parameter in the geometrical and the Chern-Simons
formulation of the theory. In particular, we show that the Lie algebras of the
Chern-Simons gauge groups can be identified with the (2+1)-dimensional Lorentz
algebra over a commutative ring, characterised by a formal parameter
whose square is minus the cosmological constant. In this
framework, the Wilson loop observables that generate grafting and Dehn twists
are obtained as the real and the -component of a Wilson loop
observable with values in the ring, and the grafting transformations can be
viewed as infinitesimal Dehn twists with the parameter .Comment: 50 pages, 6 eps figure
Constant Curvature Coefficients and Exact Solutions in Fractional Gravity and Geometric Mechanics
We study fractional configurations in gravity theories and Lagrange
mechanics. The approach is based on Caputo fractional derivative which gives
zero for actions on constants. We elaborate fractional geometric models of
physical interactions and we formulate a method of nonholonomic deformations to
other types of fractional derivatives. The main result of this paper consists
in a proof that for corresponding classes of nonholonomic distributions a large
class of physical theories are modelled as nonholonomic manifolds with constant
matrix curvature. This allows us to encode the fractional dynamics of
interactions and constraints into the geometry of curve flows and solitonic
hierarchies.Comment: latex2e, 11pt, 27 pages, the variant accepted to CEJP; added and
up-dated reference
Is there a Jordan geometry underlying quantum physics?
There have been several propositions for a geometric and essentially
non-linear formulation of quantum mechanics. From a purely mathematical point
of view, the point of view of Jordan algebra theory might give new strength to
such approaches: there is a ``Jordan geometry'' belonging to the Jordan part of
the algebra of observables, in the same way as Lie groups belong to the Lie
part. Both the Lie geometry and the Jordan geometry are well-adapted to
describe certain features of quantum theory. We concentrate here on the
mathematical description of the Jordan geometry and raise some questions
concerning possible relations with foundational issues of quantum theory.Comment: 30 page
Polyakov soldering and second order frames : the role of the Cartan connection
The so-called "soldering" procedure performed by A.M. Polyakov in [1] for a
SL(2,R)-gauge theory is geometrically explained in terms of a Cartan connection
on second order frames of the projective space RP^1. The relationship between a
Cartan connection and the usual (Ehresmann) connection on a principal bundle
allows to gain an appropriate insight into the derivation of the genuine "
diffeomorphisms out of gauge transformations" given by Polyakov himself.Comment: Accept\'e pour publication dans Lett. Math. Phy
On post-Lie algebras, Lie--Butcher series and moving frames
Pre-Lie (or Vinberg) algebras arise from flat and torsion-free connections on
differential manifolds. They have been studied extensively in recent years,
both from algebraic operadic points of view and through numerous applications
in numerical analysis, control theory, stochastic differential equations and
renormalization. Butcher series are formal power series founded on pre-Lie
algebras, used in numerical analysis to study geometric properties of flows on
euclidean spaces. Motivated by the analysis of flows on manifolds and
homogeneous spaces, we investigate algebras arising from flat connections with
constant torsion, leading to the definition of post-Lie algebras, a
generalization of pre-Lie algebras. Whereas pre-Lie algebras are intimately
associated with euclidean geometry, post-Lie algebras occur naturally in the
differential geometry of homogeneous spaces, and are also closely related to
Cartan's method of moving frames. Lie--Butcher series combine Butcher series
with Lie series and are used to analyze flows on manifolds. In this paper we
show that Lie--Butcher series are founded on post-Lie algebras. The functorial
relations between post-Lie algebras and their enveloping algebras, called
D-algebras, are explored. Furthermore, we develop new formulas for computations
in free post-Lie algebras and D-algebras, based on recursions in a magma, and
we show that Lie--Butcher series are related to invariants of curves described
by moving frames.Comment: added discussion of post-Lie algebroid
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