2,111 research outputs found
Detours and Paths: BRST Complexes and Worldline Formalism
We construct detour complexes from the BRST quantization of worldline
diffeomorphism invariant systems. This yields a method to efficiently extract
physical quantum field theories from particle models with first class
constraint algebras. As an example, we show how to obtain the Maxwell detour
complex by gauging N=2 supersymmetric quantum mechanics in curved space. Then
we concentrate on first class algebras belonging to a class of recently
introduced orthosymplectic quantum mechanical models and give generating
functions for detour complexes describing higher spins of arbitrary symmetry
types. The first quantized approach facilitates quantum calculations and we
employ it to compute the number of physical degrees of freedom associated to
the second quantized, field theoretical actions.Comment: 1+35 pages, 1 figure; typos corrected and references added, published
versio
Holomorphic Simplicity Constraints for 4d Spinfoam Models
Within the framework of spinfoam models, we revisit the simplicity
constraints reducing topological BF theory to 4d Riemannian gravity. We use the
reformulation of SU(2) intertwiners and spin networks in term of spinors, which
has come out from both the recently developed U(N) framework for SU(2)
intertwiners and the twisted geometry approach to spin networks and spinfoam
boundary states. Using these tools, we are able to perform a
holomorphic/anti-holomorphic splitting of the simplicity constraints and define
a new set of holomorphic simplicity constraints, which are equivalent to the
standard ones at the classical level and which can be imposed strongly on
intertwiners at the quantum level. We then show how to solve these new
holomorphic simplicity constraints using coherent intertwiner states. We
further define the corresponding coherent spin network functionals and
introduce a new spinfoam model for 4d Riemannian gravity based on these
holomorphic simplicity constraints and whose amplitudes are defined from the
evaluation of the new coherent spin networks.Comment: 27 page
Spin Foams and Noncommutative Geometry
We extend the formalism of embedded spin networks and spin foams to include
topological data that encode the underlying three-manifold or four-manifold as
a branched cover. These data are expressed as monodromies, in a way similar to
the encoding of the gravitational field via holonomies. We then describe
convolution algebras of spin networks and spin foams, based on the different
ways in which the same topology can be realized as a branched covering via
covering moves, and on possible composition operations on spin foams. We
illustrate the case of the groupoid algebra of the equivalence relation
determined by covering moves and a 2-semigroupoid algebra arising from a
2-category of spin foams with composition operations corresponding to a fibered
product of the branched coverings and the gluing of cobordisms. The spin foam
amplitudes then give rise to dynamical flows on these algebras, and the
existence of low temperature equilibrium states of Gibbs form is related to
questions on the existence of topological invariants of embedded graphs and
embedded two-complexes with given properties. We end by sketching a possible
approach to combining the spin network and spin foam formalism with matter
within the framework of spectral triples in noncommutative geometry.Comment: 48 pages LaTeX, 30 PDF figure
Holographic entropy relations
We develop a framework for the derivation of new information theoretic
quantities which are natural from a holographic perspective. We demonstrate the
utility of our techniques by deriving the tripartite information (the quantity
associated to monogamy of mutual information) using a set of abstract arguments
involving bulk extremal surfaces. Our arguments rely on formal manipulations of
surfaces and not on local surgery or explicit computation of entropies through
the holographic entanglement entropy prescriptions. As an application, we show
how to derive a family of similar information quantities for an arbitrary
number of parties. The present work establishes the foundation of a broader
program that aims at the understanding of the entanglement structures of
geometric states for an arbitrary number of parties. We stress that our method
is completely democratic with respect to bulk geometries and is equally valid
in static and dynamical situations. While rooted in holography, we expect that
our construction will provide a useful characterization of multipartite
correlations in quantum field theories.Comment: v1: 58 pages, 1 pdf figur
Noncommutative geometry on trees and buildings
We describe the construction of theta summable and finitely summable spectral
triples associated to Mumford curves and some classes of higher dimensional
buildings. The finitely summable case is constructed by considering the
stabilization of the algebra of the dual graph of the special fiber of the
Mumford curve and a variant of the Antonescu-Christensen spectral geometries
for AF algebras. The information on the Schottky uniformization is encoded in
the spectral geometry through the Patterson-Sullivan measure on the limit set.
Some higher rank cases are obtained by adapting the construction for trees.Comment: 23 pages, LaTeX, 2 eps figures, contributed to a proceedings volum
B^F Theory and Flat Spacetimes
We propose a reduced constrained Hamiltonian formalism for the exactly
soluble theory of flat connections and closed two-forms over
manifolds with topology . The reduced phase space
variables are the holonomies of a flat connection for loops which form a basis
of the first homotopy group , and elements of the second
cohomology group of with value in the Lie algebra . When
, and if the two-form can be expressed as , for some
vierbein field , then the variables represent a flat spacetime. This is not
always possible: We show that the solutions of the theory generally represent
spacetimes with ``global torsion''. We describe the dynamical evolution of
spacetimes with and without global torsion, and classify the flat spacetimes
which admit a locally homogeneous foliation, following Thurston's
classification of geometric structures.Comment: 21 pp., Mexico Preprint ICN-UNAM-93-1
Enumeration of points, lines, planes, etc
One of the earliest results in enumerative combinatorial geometry is the
following theorem of de Bruijn and Erd\H{o}s: Every set of points in a
projective plane determines at least lines, unless all the points are
contained in a line. Motzkin and others extended the result to higher
dimensions, who showed that every set of points in a projective space
determines at least hyperplanes, unless all the points are contained in a
hyperplane. Let be a spanning subset of a -dimensional vector space. We
show that, in the partially ordered set of subspaces spanned by subsets of ,
there are at least as many -dimensional subspaces as there are
-dimensional subspaces, for every at most . This confirms the
"top-heavy" conjecture of Dowling and Wilson for all matroids realizable over
some field. The proof relies on the decomposition theorem package for
-adic intersection complexes.Comment: 18 pages, major revisio
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