1,758 research outputs found
A bootstrap method for sum-of-poles approximations
A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138–1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples
Bootstrapping the O(N) Vector Models
We study the conformal bootstrap for 3D CFTs with O(N) global symmetry. We
obtain rigorous upper bounds on the scaling dimensions of the first O(N)
singlet and symmetric tensor operators appearing in the
OPE, where is a fundamental of O(N). Comparing these bounds to
previous determinations of critical exponents in the O(N) vector models, we
find strong numerical evidence that the O(N) vector models saturate the
bootstrap constraints at all values of N. We also compute general lower bounds
on the central charge, giving numerical predictions for the values realized in
the O(N) vector models. We compare our predictions to previous computations in
the 1/N expansion, finding precise agreement at large values of N.Comment: 26 pages, 5 figures; V2: typos correcte
Bootstrapping Mixed Correlators in the 3D Ising Model
We study the conformal bootstrap for systems of correlators involving
non-identical operators. The constraints of crossing symmetry and unitarity for
such mixed correlators can be phrased in the language of semidefinite
programming. We apply this formalism to the simplest system of mixed
correlators in 3D CFTs with a global symmetry. For the leading
-odd operator and -even operator
, we obtain numerical constraints on the allowed dimensions
assuming that and are
the only relevant scalars in the theory. These constraints yield a small closed
region in space compatible with the known
values in the 3D Ising CFT.Comment: 39 pages, 6 figure
A new handle on three-point coefficients: OPE asymptotics from genus two modular invariance
We derive an asymptotic formula for operator product expansion coefficients
of heavy operators in two dimensional conformal field theory. This follows from
modular invariance of the genus two partition function, and generalises the
asymptotic formula for the density of states from torus modular invariance. The
resulting formula is universal, depending only on the central charge, but
involves the asymptotic behaviour of genus two conformal blocks. We use
monodromy techniques to compute the asymptotics of the relevant blocks at large
central charge to determine the behaviour explicitly.Comment: 32 pages, 2 figures, 1 appendix, 2 moose, a bear and an o
Conformal Bootstrap in the Regge Limit
We analytically solve the conformal bootstrap equations in the Regge limit
for large N conformal field theories. For theories with a parametrically large
gap, the amplitude is dominated by spin-2 exchanges and we show how the
crossing equations naturally lead to the construction of AdS exchange Witten
diagrams. We also show how this is encoded in the anomalous dimensions of
double-trace operators of large spin and large twist. We use the chaos bound to
prove that the anomalous dimensions are negative. Extending these results to
correlators containing two scalars and two conserved currents, we show how to
reproduce the CEMZ constraint that the three-point function between two
currents and one stress tensor only contains the structure given by
Einstein-Maxwell theory in AdS, up to small corrections. Finally, we consider
the case where operators of unbounded spin contribute to the Regge amplitude,
whose net effect is captured by summing the leading Regge trajectory. We
compute the resulting anomalous dimensions and corrections to OPE coefficients
in the crossed channel and use the chaos bound to show that both are negative.Comment: 40 pages, 1 figure; V2: Small corrections and clarification
The Analytic Bootstrap and AdS Superhorizon Locality
We take an analytic approach to the CFT bootstrap, studying the 4-pt
correlators of d > 2 dimensional CFTs in an Eikonal-type limit, where the
conformal cross ratios satisfy |u| << |v| < 1. We prove that every CFT with a
scalar operator \phi must contain infinite sequences of operators O_{\tau,l}
with twist approaching \tau -> 2\Delta_\phi + 2n for each integer n as l ->
infinity. We show how the rate of approach is controlled by the twist and OPE
coefficient of the leading twist operator in the \phi x \phi OPE, and we
discuss SCFTs and the 3d Ising Model as examples. Additionally, we show that
the OPE coefficients of other large spin operators appearing in the OPE are
bounded as l -> infinity. We interpret these results as a statement about
superhorizon locality in AdS for general CFTs.Comment: 33 pages, no figures; V2 citations adde
Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
We use the conformal bootstrap to perform a precision study of the operator
spectrum of the critical 3d Ising model. We conjecture that the 3d Ising
spectrum minimizes the central charge c in the space of unitary solutions to
crossing symmetry. Because extremal solutions to crossing symmetry are uniquely
determined, we are able to precisely reconstruct the first several Z2-even
operator dimensions and their OPE coefficients. We observe that a sharp
transition in the operator spectrum occurs at the 3d Ising dimension
Delta_sigma=0.518154(15), and find strong numerical evidence that operators
decouple from the spectrum as one approaches the 3d Ising point. We compare
this behavior to the analogous situation in 2d, where the disappearance of
operators can be understood in terms of degenerate Virasoro representations.Comment: 55 pages, many figures; v2 - refs and comments added, to appear in a
special issue of J.Stat.Phys. in memory of Kenneth Wilso
A Numerical Approach to Virasoro Blocks and the Information Paradox
We chart the breakdown of semiclassical gravity by analyzing the Virasoro
conformal blocks to high numerical precision, focusing on the heavy-light limit
corresponding to a light probe propagating in a BTZ black hole background. In
the Lorentzian regime, we find empirically that the initial exponential
time-dependence of the blocks transitions to a universal
power-law decay. For the vacuum block the transition occurs at , confirming analytic predictions. In the Euclidean regime,
due to Stokes phenomena the naive semiclassical approximation fails completely
in a finite region enclosing the `forbidden singularities'. We emphasize that
limitations on the reconstruction of a local bulk should ultimately stem from
distinctions between semiclassical and exact correlators.Comment: 45 pages, 23 figure
- …