1,972 research outputs found
Group field theories for all loop quantum gravity
Group field theories represent a 2nd quantized reformulation of the loop
quantum gravity state space and a completion of the spin foam formalism. States
of the canonical theory, in the traditional continuum setting, have support on
graphs of arbitrary valence. On the other hand, group field theories have
usually been defined in a simplicial context, thus dealing with a restricted
set of graphs. In this paper, we generalize the combinatorics of group field
theories to cover all the loop quantum gravity state space. As an explicit
example, we describe the GFT formulation of the KKL spin foam model, as well as
a particular modified version. We show that the use of tensor model tools
allows for the most effective construction. In order to clarify the
mathematical basis of our construction and of the formalisms with which we
deal, we also give an exhaustive description of the combinatorial structures
entering spin foam models and group field theories, both at the level of the
boundary states and of the quantum amplitudes.Comment: version published in New Journal of Physic
On the existence of 0/1 polytopes with high semidefinite extension complexity
In Rothvo\ss{} it was shown that there exists a 0/1 polytope (a polytope
whose vertices are in \{0,1\}^{n}) such that any higher-dimensional polytope
projecting to it must have 2^{\Omega(n)} facets, i.e., its linear extension
complexity is exponential. The question whether there exists a 0/1 polytope
with high PSD extension complexity was left open. We answer this question in
the affirmative by showing that there is a 0/1 polytope such that any
spectrahedron projecting to it must be the intersection of a semidefinite cone
of dimension~2^{\Omega(n)} and an affine space. Our proof relies on a new
technique to rescale semidefinite factorizations
Orbitopes
An orbitope is the convex hull of an orbit of a compact group acting linearly
on a vector space. These highly symmetric convex bodies lie at the crossroads
of several fields, in particular convex geometry, optimization, and algebraic
geometry. We present a self-contained theory of orbitopes, with particular
emphasis on instances arising from the groups SO(n) and O(n). These include
Schur-Horn orbitopes, tautological orbitopes, Caratheodory orbitopes, Veronese
orbitopes and Grassmann orbitopes. We study their face lattices, their
algebraic boundary hypersurfaces, and representations as spectrahedra or
projected spectrahedra.Comment: 37 pages. minor revisions of origina
Entropic lattice Boltzmann methods
We present a general methodology for constructing lattice Boltzmann models of
hydrodynamics with certain desired features of statistical physics and kinetic
theory. We show how a methodology of linear programming theory, known as
Fourier-Motzkin elimination, provides an important tool for visualizing the
state space of lattice Boltzmann algorithms that conserve a given set of
moments of the distribution function. We show how such models can be endowed
with a Lyapunov functional, analogous to Boltzmann's H, resulting in
unconditional numerical stability. Using the Chapman-Enskog analysis and
numerical simulation, we demonstrate that such entropically stabilized lattice
Boltzmann algorithms, while fully explicit and perfectly conservative, may
achieve remarkably low values for transport coefficients, such as viscosity.
Indeed, the lowest such attainable values are limited only by considerations of
accuracy, rather than stability. The method thus holds promise for
high-Reynolds number simulations of the Navier-Stokes equations.Comment: 54 pages, 16 figures. Proc. R. Soc. London A (in press
The Steinberg group of a monoid ring, nilpotence, and algorithms
For a regular ring R and an affine monoid M the homotheties of M act
nilpotently on the Milnor unstable groups of R[M]. This strengthens the K_2
part of the main result of [G5] in two ways: the coefficient field of
characteristic 0 is extended to any regular ring and the stable K_2-group is
substituted by the unstable ones. The proof is based on a
polyhedral/combinatorial techniques, computations in Steinberg groups, and a
substantially corrected version of an old result on elementary matrices by
Mushkudiani [Mu]. A similar stronger nilpotence result for K_1 and algorithmic
consequences for factorization of high Frobenius powers of invertible matrices
are also derived.Comment: final version, to appear in Journal of Algebr
Morse matchings on polytopes
We show how to construct homology bases for certain CW complexes in terms of
discrete Morse theory and cellular homology. We apply this technique to study
certain subcomplexes of the half cube polytope studied in previous works. This
involves constructing explicit complete acyclic Morse matchings on the face
lattice of the half cube; this procedure may be of independent interest for
other highly symmetric polytopes
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