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 $\mathbb{Z}_2$ global symmetry. For the leading $\mathbb{Z}_2$-odd operator $\sigma$ and $\mathbb{Z}_2$-even operator $\epsilon$, we obtain numerical constraints on the allowed dimensions $(\Delta_\sigma, \Delta_\epsilon)$ assuming that $\sigma$ and $\epsilon$ are the only relevant scalars in the theory. These constraints yield a small closed region in $(\Delta_\sigma, \Delta_\epsilon)$ 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 $\phi_i \times \phi_j$ OPE, where $\phi_i$ 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

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 $O(N)$ 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

Bootstrapping the Minimal 3D SCFT

We study the conformal bootstrap constraints for 3D conformal field theories with a $\mathbb{Z}_2$ or parity symmetry, assuming a single relevant scalar operator $\epsilon$ that is invariant under the symmetry. When there is additionally a single relevant odd scalar $\sigma$, 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 $\sigma'$, we identify a feature in the allowed space compatible with 3D $\mathcal{N}=1$ superconformal symmetry and conjecture that it corresponds to the minimal $\mathcal{N}=1$ 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 $\mathcal{N}=1$ superconformal symmetry, we isolate this theory and use the numerical bootstrap to compute the leading scaling dimensions $\Delta_{\sigma} = \Delta_{\epsilon} - 1 = .58444(22)$ and three-point couplings $\lambda_{\sigma\sigma\epsilon} = 1.0721(2)$ and $\lambda_{\epsilon\epsilon\epsilon} = 1.67(1)$. 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 $\lambda_{\epsilon\epsilon\epsilon}/\lambda_{\sigma\sigma\epsilon} = \tan(1)$.Comment: 16 pages, 6 figures; V2: references adde

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