952 research outputs found
Topological Signals of Singularities in Ricci Flow
We implement methods from computational homology to obtain a topological
signal of singularity formation in a selection of geometries evolved
numerically by Ricci flow. Our approach, based on persistent homology, produces
precise, quantitative measures describing the behavior of an entire collection
of data across a discrete sample of times. We analyze the topological signals
of geometric criticality obtained numerically from the application of
persistent homology to models manifesting singularities under Ricci flow. The
results we obtain for these numerical models suggest that the topological
signals distinguish global singularity formation (collapse to a round point)
from local singularity formation (neckpinch). Finally, we discuss the
interpretation and implication of these results and future applications.Comment: 24 pages, 14 figure
The dS/CFT Correspondence and the Big Smash
Recent observations suggest that the cosmological equation-of-state parameter
w is close to -1. To say this is to imply that w could be slightly less than
-1, which leads to R.Caldwell's "Phantom cosmologies". These often have the
property that they end in a "Big Smash", a final singularity in which the
Universe is destroyed in a finite proper time by excessive *expansion*. We show
that, classically, this fate is not inevitable: there exist Smash-free Phantom
cosmologies, obtained by a suitable perturbation of the deSitter equation of
state, in which the spacetime is in fact asymptotically deSitter. [Contrary to
popular belief, such cosmologies, which violate the Dominant Energy Condition,
do not necessarily violate causality.] We also argue, however, that the
physical interpretation of these classically acceptable spacetimes is radically
altered by ``holography'', as manifested in the dS/CFT correspondence. It is
shown that, if the boundary CFTs have conventional properties, then recent
ideas on "time as an inverse renormalization group flow" can be used to rule
out these cosmologies. Very recently, however, it has been argued that the CFTs
in dS/CFT are of a radically unconventional form, and this opens up the
possibility that Smash-free Phantom spacetimes offer a simple model of a
"bouncing" cosmology in which the quantum-mechanical entanglement of the field
theories in the infinite past and future plays an essential role.Comment: 22 pages, clarification of triple analytic continuation, additional
Comments added in the light of hep-th/020724
Ricci Flow Approach to the Very Early Universe
A framework of quantum spacetime reference frame is proposed and reviewed, in
which the quantum spacetime at the Gaussian approximation is deformed by the
Ricci flow. At sufficient large scale, the Ricci flow not only smooths out the
local small irregularities and makes the universe a homogeneous and isotropic
Friedman-Robertson-Walker metric, but also develops a local singularity at the
physical-time origin. Due to the phenomenological suppression of the
non-Gaussian primordial perturbations, we assume the validity of the Ricci flow
applying to the high curvature region near the local singularity of the very
early universe. The no-local-collapsing theorem of Perelman ensures the
existence of a canonical neighborhood around the large curvature pinching
point, which resembles a gradient shrinking Ricci soliton (GSRS) solution of
the Ricci flow. The GSRS reproduces an exact inflationary deSitter universe
near the singularity at the leading order. The slow roll parameters are
calculated by a small deviation from the singular flow-time via the Ricci flow,
and primordial perturbations are also studied on the GSRS background, the power
spectrum of the scalar perturbation agrees with present observations, and the
one of the tensor perturbation is predicted too small to be detected than the
standard inflation.Comment: 20 page
Hamiltonian dynamics and geometry of phase transitions in classical XY models
The Hamiltonian dynamics associated to classical, planar, Heisenberg XY
models is investigated for two- and three-dimensional lattices. Besides the
conventional signatures of phase transitions, here obtained through time
averages of thermodynamical observables in place of ensemble averages,
qualitatively new information is derived from the temperature dependence of
Lyapunov exponents. A Riemannian geometrization of newtonian dynamics suggests
to consider other observables of geometric meaning tightly related with the
largest Lyapunov exponent. The numerical computation of these observables -
unusual in the study of phase transitions - sheds a new light on the
microscopic dynamical counterpart of thermodynamics also pointing to the
existence of some major change in the geometry of the mechanical manifolds at
the thermodynamical transition. Through the microcanonical definition of the
entropy, a relationship between thermodynamics and the extrinsic geometry of
the constant energy surfaces of phase space can be naturally
established. In this framework, an approximate formula is worked out,
determining a highly non-trivial relationship between temperature and topology
of the . Whence it can be understood that the appearance of a phase
transition must be tightly related to a suitable major topology change of the
. This contributes to the understanding of the origin of phase
transitions in the microcanonical ensemble.Comment: in press on Physical Review E, 43 pages, LaTeX (uses revtex), 22
PostScript figure
Non-BPS D8-branes and Dynamic Domain Walls in Massive IIA Supergravities
We study the D8-branes of the Romans massive IIA supergravity theory using
the coupled supergravity and worldvolume actions. D8 branes can be regarded as
domain walls with the jump in the extrinsic curvature at the brane given by the
Israel matching conditions. We examine the restrictions that these conditions
place on extreme and non-extreme solutions and find that they rule out some of
the supersymmetric solutions given by Bergshoeff {\em et al}. We consider what
happens when the dilaton varies on the worldvolume of the brane, which implies
that the brane is no longer static. We obtain a family of D8-brane solutions
parametrized by a non-extremality term on each side of the brane and the
asymptotic values of the 10-form field. The non-extremality parameters can be
related to the velocity of the brane. We also study 8-brane solutions of a
massive IIA supergravity theory introduced by Howe, Lambert and West. This
theory also admits a 10-form formulation, but the 10-form is not a R-R sector
field and so these 8-branes are not D-branes.Comment: 23 pages REVTeX, 2 .eps figures. This paper completely replaces and
extends an earlier paper (hep-th/9712112) by Chamblin and Perr
Universal consistent truncation for 6d/7d gauge/gravity duals
Recently, AdS_7 solutions of IIA supergravity have been classified; there are
infinitely many of them, whose expression is known analytically, and with
internal space of S^3 topology. Their field theory duals are six-dimensional
(1,0) SCFT's. In this paper we show that for each of these AdS_7 solutions
there exists a consistent truncation from massive IIA supergravity to minimal
gauged supergravity in seven dimensions. This theory has an SU(2) gauge group,
and a single scalar, whose value is related to a certain distortion of the
internal S^3. This explains the universality observed in recent work on AdS_5
and AdS_4 solutions dual to compactifications of the (1,0) SCFT_6's. Thanks to
previous work on the minimal gauged supergravity, the truncation also implies
the existence of holographic RG-flows connecting those solutions to the AdS_7
vacuum, as well as new classes of IIA AdS_3 solutions.Comment: 23 pages; v2: references added, minor changes, additions and
correction
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