564 research outputs found
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
Carving Out the Space of 4D CFTs
We introduce a new numerical algorithm based on semidefinite programming to
efficiently compute bounds on operator dimensions, central charges, and OPE
coefficients in 4D conformal and N=1 superconformal field theories. Using our
algorithm, we dramatically improve previous bounds on a number of CFT
quantities, particularly for theories with global symmetries. In the case of
SO(4) or SU(2) symmetry, our bounds severely constrain models of conformal
technicolor. In N=1 superconformal theories, we place strong bounds on
dim(Phi*Phi), where Phi is a chiral operator. These bounds asymptote to the
line dim(Phi*Phi) <= 2 dim(Phi) near dim(Phi) ~ 1, forbidding positive
anomalous dimensions in this region. We also place novel upper and lower bounds
on OPE coefficients of protected operators in the Phi x Phi OPE. Finally, we
find examples of lower bounds on central charges and flavor current two-point
functions that scale with the size of global symmetry representations. In the
case of N=1 theories with an SU(N) flavor symmetry, our bounds on current
two-point functions lie within an O(1) factor of the values realized in
supersymmetric QCD in the conformal window.Comment: 60 pages, 22 figure
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
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 the Minimal 3D SCFT
We study the conformal bootstrap constraints for 3D conformal field theories
with a or parity symmetry, assuming a single relevant scalar
operator that is invariant under the symmetry. When there is
additionally a single relevant odd scalar , we map out the allowed
space of dimensions and three-point couplings of such "Ising-like" CFTs. If we
allow a second relevant odd scalar , we identify a feature in the
allowed space compatible with 3D superconformal symmetry and
conjecture that it corresponds to the minimal supersymmetric
extension of the Ising CFT. This model has appeared in previous numerical
bootstrap studies, as well as in proposals for emergent supersymmetry on the
boundaries of topological phases of matter. Adding further constraints from 3D
superconformal symmetry, we isolate this theory and use the
numerical bootstrap to compute the leading scaling dimensions and three-point couplings
and
. We additionally place bounds on
the central charge and use the extremal functional method to estimate the
dimensions of the next several operators in the spectrum. Based on our results
we observe the possible exact relation
.Comment: 16 pages, 6 figures; V2: references adde
The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
Conformal field theories have been long known to describe the fascinating
universal physics of scale invariant critical points. They describe continuous
phase transitions in fluids, magnets, and numerous other materials, while at
the same time sit at the heart of our modern understanding of quantum field
theory. For decades it has been a dream to study these intricate strongly
coupled theories nonperturbatively using symmetries and other consistency
conditions. This idea, called the conformal bootstrap, saw some successes in
two dimensions but it is only in the last ten years that it has been fully
realized in three, four, and other dimensions of interest. This renaissance has
been possible both due to significant analytical progress in understanding how
to set up the bootstrap equations and the development of numerical techniques
for finding or constraining their solutions. These developments have led to a
number of groundbreaking results, including world record determinations of
critical exponents and correlation function coefficients in the Ising and
models in three dimensions. This article will review these exciting
developments for newcomers to the bootstrap, giving an introduction to
conformal field theories and the theory of conformal blocks, describing
numerical techniques for the bootstrap based on convex optimization, and
summarizing in detail their applications to fixed points in three and four
dimensions with no or minimal supersymmetry.Comment: 81 pages, double column, 58 figures; v3: updated references, minor
typos correcte
Defining Geographic Communities
The purpose of this paper is to provide a guide to concepts, ideas, and measurements of geographic communities. The paper investigates the various concepts of geographic communities found in the literature and reviews existing studies to determine how researchers measure geographic communities in practice.Geographic communities, Local labour markets
Defining Geographic Communities
The purpose of this paper is to provide a guide to concepts, ideas, and measurements of geographic communities. The paper investigates the various concepts of geographic communities found in the literature and reviews existing studies to determine how researchers measure geographic communities in practice.Geographic communities, Local labour markets
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