There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

第12次 아시아ㆍ太平洋 矯政局長會議 參加記

We study the constraints of crossing symmetry and unitarity in general 3D conformal field theories. In doing so we derive new results for conformal blocks appearing in four-point functions of scalars and present an efficient method for their computation in arbitrary space-time dimension. Comparing the resulting bounds on operator dimensions and product-expansion coefficients in 3D to known results, we find that the 3D Ising model lies at a corner point on the boundary of the allowed parameter space. We also derive general upper bounds on the dimensions of higher spin operators, relevant in the context of theories with weakly broken higher spin symmetries.

Solving the 3D Ising Model with the Conformal Bootstrap

We conjecture a sharp bound on the rate of growth of chaos in thermal quantum systems with a large number of degrees of freedom. Chaos can be diagnosed using an out-of-time-order correlation function closely related to the commutator of operators separated in time. We conjecture that the influence of chaos on this correlator can develop no faster than exponentially, with Lyapunov exponent $\lambda_L \le 2 \pi k_B T/\hbar$. We give a precise mathematical argument, based on plausible physical assumptions, establishing this conjecture.

A bound on chaos

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.

The Conformal Bootstrap: Theory, Numerical Techniques, and Applications

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| ≪ |υ| < 1. We prove that every CFT with a scalar operator ϕ must contain infinite sequences of operators $ {{\mathcal{O}}_{{\tau, \ell }}} $
 with twist approaching τ → 2Δ
 ϕ
 + 2n for each integer n as l → ∞. We show how the rate of approach is controlled by the twist and OPE coefficient of the leading twist operator in the ϕ × ϕ 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 → ∞. We interpret these results as a statement about superhorizon locality in AdS for general CFTs.

/pdf/the-analytic-bootstrap-and-ads-superhorizon-locality-1hf8pvh89w.pdf

The analytic bootstrap and AdS superhorizon locality

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 Z_2 -even operator dimensions and their OPE coefficients. We observe that a sharp transition in the operator spectrum occurs at the 3d Ising dimension Δ_σ=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.

/pdf/solving-the-3d-ising-model-with-the-conformal-bootstrap-ii-c-2a4qmm0qwe.pdf

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

We introduce SDPB: an open-source, parallelized, arbitrary-precision semidefinite program solver, designed for the conformal bootstrap. SDPB significantly outperforms less specialized solvers and should enable many new computations. As an example application, we compute a new rigorous high-precision bound on operator dimensions in the 3d Ising CFT, Δ
 σ
 = 0.518151(6), Δ
 ϵ
 = 1.41264(6).

/pdf/a-semidefinite-program-solver-for-the-conformal-bootstrap-1te1pfvwnt.pdf

A Semidefinite Program Solver for the Conformal Bootstrap

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 Z2 global symmetry. For the leading Z2-odd operator σ and Z2-even operator ǫ, we obtain numerical constraints on the allowed dimensions (�σ,�ǫ) assuming that σ and ǫ are the only relevant scalars in the theory. These constraints yield a small closed region in (�σ,�ǫ) space compatible with the known values in the 3D Ising CFT.

/pdf/bootstrapping-mixed-correlators-in-the-3d-ising-model-10pe41fwvh.pdf

David Simmons-Duffin

Papers

Solving the 3D Ising Model with the Conformal Bootstrap

The analytic bootstrap and AdS superhorizon locality

Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents

A Semidefinite Program Solver for the Conformal Bootstrap

Bootstrapping mixed correlators in the 3D Ising model