1,246 research outputs found
Integrable probability: From representation theory to Macdonald processes
These are lecture notes for a mini-course given at the Cornell Probability
Summer School in July 2013. Topics include lozenge tilings of polygons and
their representation theoretic interpretation, the (q,t)-deformation of those
leading to the Macdonald processes, nearest neighbor dynamics on Macdonald
processes, their limit to semi-discrete Brownian polymers, and large time
asymptotic analysis of polymer's partition function.Comment: 49 pages; 22 figures; v3, v4: minor typos fixe
Symmetries in quantum field theory and quantum gravity
In this paper we use the AdS/CFT correspondence to refine and then establish
a set of old conjectures about symmetries in quantum gravity. We first show
that any global symmetry, discrete or continuous, in a bulk quantum gravity
theory with a CFT dual would lead to an inconsistency in that CFT, and thus
that there are no bulk global symmetries in AdS/CFT. We then argue that any
"long-range" bulk gauge symmetry leads to a global symmetry in the boundary
CFT, whose consistency requires the existence of bulk dynamical objects which
transform in all finite-dimensional irreducible representations of the bulk
gauge group. We mostly assume that all internal symmetry groups are compact,
but we also give a general condition on CFTs, which we expect to be true quite
broadly, which implies this. We extend all of these results to the case of
higher-form symmetries. Finally we extend a recently proposed new motivation
for the weak gravity conjecture to more general gauge groups, reproducing the
"convex hull condition" of Cheung and Remmen.
An essential point, which we dwell on at length, is precisely defining what
we mean by gauge and global symmetries in the bulk and boundary. Quantum field
theory results we meet while assembling the necessary tools include continuous
global symmetries without Noether currents, new perspectives on spontaneous
symmetry-breaking and 't Hooft anomalies, a new order parameter for confinement
which works in the presence of fundamental quarks, a Hamiltonian lattice
formulation of gauge theories with arbitrary discrete gauge groups, an
extension of the Coleman-Mandula theorem to discrete symmetries, and an
improved explanation of the decay in the standard model
of particle physics. We also describe new black hole solutions of the Einstein
equation in dimensions with horizon topology .Comment: You probably don't want to print this single-sided. v2: minor
corrections and clarifications throughout, references adde
Isospectral deformations of the Dirac operator
We give more details about an integrable system in which the Dirac operator
D=d+d^* on a finite simple graph G or Riemannian manifold M is deformed using a
Hamiltonian system D'=[B,h(D)] with B=d-d^* + i b. The deformed operator D(t) =
d(t) + b(t) + d(t)^* defines a new exterior derivative d(t) and a new Dirac
operator C(t) = d(t) + d(t)^* and Laplacian M(t) = d(t) d(t)^* + d(t)* d(t) and
so a new distance on G or a new metric on M.Comment: 32 pages, 8 figure
On Exact Solutions and Perturbative Schemes in Higher Spin Theory
We review various methods for finding exact solutions of higher spin theory
in four dimensions, and survey the known exact solutions of (non)minimal
Vasiliev's equations. These include instanton-like and black hole-like
solutions in (A)dS and Kleinian spacetimes. A perturbative construction of
solutions with the symmetries of a domain wall is described as well.
Furthermore, we review two proposed perturbative schemes: one based on
perturbative treatment of the twistor space field equations followed by
inverting Fronsdal kinetic terms using standard Green's functions; and an
alternative scheme based on solving the twistor space field equations exactly
followed by introducing the spacetime dependence using perturbatively defined
gauge functions. Motivated by the need to provide a higher spin invariant
characterization of the exact solutions, aspects of a proposal for a geometric
description of Vasiliev's equation involving an infinite dimensional
generalization of anti de Sitter space is revisited and improved.Comment: 45 pages. Clarifying remarks and references are added. Version
published in Universe, Special Issue on Higher Spin Gauge Theorie
Morse theory applied to semilinear problems
We use Morse theoretic arguments to obtain nontrivial solutions of semilinear elliptic boundary value problems where the nonlinearity interacts with the spectrum, without assuming that the asymptotic limits at zero and infinity exist
Chern-Simons-Ghost Theories and de Sitter Space
We explore Chern-Simons theories coupled to fundamental ghost-like matter in
the large limit at 't Hooft coupling . These theories have been
conjectured to be holographically dual to parity-violating, asymptotically
dS universes with a tower of light higher-spin fields. On ,
to all orders in large- perturbation theory, we show that Chern-Simons-ghost
theories are related to ordinary Chern-Simons-matter theories by mapping and keeping fixed. Consequently, the bosonization
duality of ordinary Chern-Simons-matter theories extends to a bosonization
duality of Chern-Simons-ghost theories on . On ,
in the small- limit, neither nor bosonization hold, as
we show by extensively studying large- saddles of the theories with both
ghost and ordinary matter. The partition functions we compute along the way can
be viewed as pieces of the late-time Hartle-Hawking wavefunction for the bulk
dS gravity theories.Comment: 27 pages + appendices, 6 figure
Localized States and Dynamics in the Nonlinear Schroedinger / Gross-Pitaevskii Equation
This article is a review of results on the nonlinear Schroedinger /
Gross-Pitaevskii equation (NLS / GP). Nonlinear bound states and aspects of
their stability theory are discussed from variational and bifurcation
perspectives. Nonlinear bound states, in the particular cases where the
potential is a single-well, double-well potential and periodic potential, are
discussed in detail. We then discuss particle-like dynamics of solitary wave
solutions interacting with a potential over a large, but finite, time interval.
Finally we turn to the very long time behavior of solutions. We focus on the
important resonant radiation damping mechanism that drives the relaxation of
the system to a nonlinear ground state and underlies the phenomena of {\it
Ground State Selection} and {\it Energy Equipartition}. We also analyze linear
and nonlinear "toy minimal models", which illustrate these mechanisms.
Regarding the overall style of this article, we seek to emphasize the key ideas
and therefore do not present all technical details, leaving that to the
references.Comment: To appear in Frontiers in Applied Dynamics: Reviews and Tutorials,
Volume 3 (Springer, 2015) 40 pages, 9 figure
Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs
We seek solutions to the semilinear elliptic partial difference
equation , where is the matrix corresponding to the
Laplacian operator on a graph and is a one-parameter family of
nonlinear functions. This article combines the ideas introduced by the authors
in two papers: a) {\it Nonlinear Elliptic Partial Difference Equations on
Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and
numerical techniques for solving such equations, and b) {\it Symmetry and
Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region}
wherein we present some of our recent advances concerning symmetry,
bifurcation, and automation fo
We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in
order to obtain better initial guesses for Newton's method, create informative
graphics, and be in the underlying variational structure. We use two modified
implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and
Swift) to follow bifurcation branches in a robust way. By handling difficulties
that arise when encountering accidental degeneracies and higher-dimension we
can find many solutions of many symmetry types to the discrete nonlinear
system. We present a selection of experimental results which demonstrate our
algorithm's capability to automatically generate bifurcation diagrams and
solution graphics starting with only an edgelis of a graph. We highlight
interesting symmetry and variational phenomena
Moduli Spaces of Singular Yamabe Metrics
Complete, conformally flat metrics of constant positive scalar curvature on
the complement of points in the -sphere, , , were
constructed by R\. Schoen [S2]. We consider the problem of determining the
moduli space of all such metrics. All such metrics are asymptotically periodic,
and we develop the linear analysis necessary to understand the nonlinear
problem. This includes a Fredholm theory and asymptotic regularity theory for
the Laplacian on asymptotically periodic manifolds, which is of independent
interest. The main result is that the moduli space is a locally real analytic
variety of dimension . For a generic set of nearby conformal classes the
moduli space is shown to be a dimensional real analytic manifold. The
structure as a real analytic variety is obtained by writing the space as an
intersection of a Fredholm pair of infinite dimensional real analytic
manifolds.Comment: 45 pages, AMS-TeX. MSRI Preprint No. 019-94 (Dec. 1993
Variational and linearly-implicit integrators, with applications
We show that symplectic and linearly-implicit integrators proposed by [Zhang
and Skeel, 1997] are variational linearizations of Newmark methods. When used
in conjunction with penalty methods (i.e., methods that replace constraints by
stiff potentials), these integrators permit coarse time-stepping of
holonomically constrained mechanical systems and bypass the resolution of
nonlinear systems. Although penalty methods are widely employed, an explicit
link to Lagrange multiplier approaches appears to be lacking; such a link is
now provided (in the context of two-scale flow convergence [Tao, Owhadi and
Marsden, 2010]). The variational formulation also allows efficient simulations
of mechanical systems on Lie groups
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