1,726 research outputs found
The Spacetime of Double Field Theory: Review, Remarks, and Outlook
We review double field theory (DFT) with emphasis on the doubled spacetime
and its generalized coordinate transformations, which unify diffeomorphisms and
b-field gauge transformations. We illustrate how the composition of generalized
coordinate transformations fails to associate. Moreover, in dimensional
reduction, the O(d,d) T-duality transformations of fields can be obtained as
generalized diffeomorphisms. Restricted to a half-dimensional subspace, DFT
includes `generalized geometry', but is more general in that local patches of
the doubled space may be glued together with generalized coordinate
transformations. Indeed, we show that for certain T-fold backgrounds with
non-geometric fluxes, there are generalized coordinate transformations that
induce, as gauge symmetries of DFT, the requisite O(d,d;Z) monodromy
transformations. Finally we review recent results on the \alpha' extension of
DFT which, reduced to the half-dimensional subspace, yields intriguing
modifications of the basic structures of generalized geometry.Comment: 50 pages, v2: minor corrections, version to be published in
Fortschritte der Physik, v3: refs. added, discussion of non-geometric
backgrounds extende
Operations between sets in geometry
An investigation is launched into the fundamental characteristics of
operations on and between sets, with a focus on compact convex sets and star
sets (compact sets star-shaped with respect to the origin) in -dimensional
Euclidean space . For example, it is proved that if , with three
trivial exceptions, an operation between origin-symmetric compact convex sets
is continuous in the Hausdorff metric, GL(n) covariant, and associative if and
only if it is addition for some . It is also
demonstrated that if , an operation * between compact convex sets is
continuous in the Hausdorff metric, GL(n) covariant, and has the identity
property (i.e., for all compact convex sets , where
denotes the origin) if and only if it is Minkowski addition. Some analogous
results for operations between star sets are obtained. An operation called
-addition is generalized and systematically studied for the first time.
Geometric-analytic formulas that characterize continuous and GL(n)-covariant
operations between compact convex sets in terms of -addition are
established. The term "polynomial volume" is introduced for the property of
operations * between compact convex or star sets that the volume of ,
, is a polynomial in the variables and . It is proved that if
, with three trivial exceptions, an operation between origin-symmetric
compact convex sets is continuous in the Hausdorff metric, GL(n) covariant,
associative, and has polynomial volume if and only if it is Minkowski addition
Unitary representations of the Galilean line group: Quantum mechanical principle of equivalence
We present a formalism of Galilean quantum mechanics in non-inertial
reference frames and discuss its implications for the equivalence principle.
This extension of quantum mechanics rests on the Galilean line group, the
semidirect product of the real line and the group of analytic functions from
the real line to the Euclidean group in three dimensions. This group provides
transformations between all inertial and non-inertial reference frames and
contains the Galilei group as a subgroup. We construct a certain class of
unitary representations of the Galilean line group and show that these
representations determine the structure of quantum mechanics in non-inertial
reference frames. Our representations of the Galilean line group contain the
usual unitary projective representations of the Galilei group, but have a more
intricate cocycle structure. The transformation formula for the Hamiltonian
under the Galilean line group shows that in a non-inertial reference frame it
acquires a fictitious potential energy term that is proportional to the
inertial mass, suggesting the equivalence of inertial mass and gravitational
mass in quantum mechanics
On a Subposet of the Tamari Lattice
We explore some of the properties of a subposet of the Tamari lattice
introduced by Pallo, which we call the comb poset. We show that three binary
functions that are not well-behaved in the Tamari lattice are remarkably
well-behaved within an interval of the comb poset: rotation distance, meets and
joins, and the common parse words function for a pair of trees. We relate this
poset to a partial order on the symmetric group studied by Edelman.Comment: 21 page
Chiral-Yang-Mills theory, non commutative differential geometry, and the need for a Lie super-algebra
In Yang-Mills theory, the charges of the left and right massless Fermions are
independent of each other. We propose a new paradigm where we remove this
freedom and densify the algebraic structure of Yang-Mills theory by integrating
the scalar Higgs field into a new gauge-chiral 1-form which connects Fermions
of opposite chiralities. Using the Bianchi identity, we prove that the
corresponding covariant differential is associative if and only if we gauge a
Lie-Kac super-algebra. In this model, spontaneous symmetry breakdown naturally
occurs along an odd generator of the super-algebra and induces a representation
of the Connes-Lott non commutative differential geometry of the 2-point finite
space.Comment: 17 pages, no figur
Stacky Lie groups
Presentations of smooth symmetry groups of differentiable stacks are studied
within the framework of the weak 2-category of Lie groupoids, smooth principal
bibundles, and smooth biequivariant maps. It is shown that principality of
bibundles is a categorical property which is sufficient and necessary for the
existence of products. Stacky Lie groups are defined as group objects in this
weak 2-category. Introducing a graphic notation, it is shown that for every
stacky Lie monoid there is a natural morphism, called the preinverse, which is
a Morita equivalence if and only if the monoid is a stacky Lie group. As
example we describe explicitly the stacky Lie group structure of the irrational
Kronecker foliation of the torus.Comment: 40 pages; definition of group objects in higher categories added;
coherence relations for groups in 2-categories given (section 4
- …