We develop new methods for approximating conformal blocks as positive functions times polynomials, with applications to the numerical bootstrap. We argue that to obtain accurate bootstrap bounds, conformal block approximations should minimize a certain error norm related to the asymptotics of dispersive functionals. This error norm can be made small using interpolation nodes with an appropriate optimal density. The optimal density turns out to satisfy a kind of force-balance equation for charges in 1-dimension, which can be solved using standard techniques from large-$\textit{N}$ matrix models. We also describe how to use optimal density interpolation nodes to improve condition numbers inside the semidefinite program solver SDPB. Altogether, our new approximation scheme and improvements to condition numbers lead to more accurate bootstrap bounds with fewer computational resources. They were crucial in the recent bootstrap study of stress tensors in the 3d Ising CFT.

United States

Cyuan-Han Chang

Vasiliy Dommes

United Kingdom

Petr Kravchuk

David Poland

David Simmons-Duffin

Accurate bootstrap bounds from optimal interpolation

We perform a systematic classification of (2+1)d Gross-Neveu-Yukawa-like models built out of one or more 4-component Dirac fermions and $\textit{M}$ scalar fields, which preserve an O($\textit{M}$) symmetry rotating the scalars. We then identify the perturbative fixed points of these models in the 4 – $\textit{ϵ}$ expansion. Our classification highlights several targets for the conformal bootstrap and reveals a new fixed point with $\textit{M}$ = 3, which we call the “orthogonal Heisenberg” CFT.

Italy

Matthew Mitchell

Classifying GNY-like models

A thermal model describing hadron production in heavy-ion collisions in the few-GeV energy regime is combined with the concept of nucleon coalescence to make predictions for the production of $^{3}$H and $^{3}$He nuclei. A realistic parametrization of the freeze-out conditions is employed, which accurately reproduces the spectra of protons and pions. It also correctly predicts the deuteron yield, which agrees with experimental observations. However, the predicted yields of $^{3}$H and $^{3}$He are lower than the experimental results by approximately a factor of two. The model predictions for the spectra can be compared with future experimental data.

Poland

Zbigniew Drogosz

Wojciech Florkowski

Nikodem Witkowski

Radoslaw Ryblewski

$^{3}$H and $^{3}$He nuclei production in a combined thermal and coalescence framework for heavy-ion collisions in the few-GeV energy regime * Supported in part by the National Science Centre, Poland (NCN) (2022/47/B/ST2/01372 (W.F.) , 2018/30/E/ST2/00432 (R.R.))

We study five-point correlators of the $\textit{σ}$, $\textit{ϵ}$, and $\textit{ϵ}$ operators in the critical 3d Ising model. We consider the $\textit{σ}$ × $\textit{σ}$ and $\textit{σ}$ × $\textit{ϵ}$ operator product expansions (OPEs) and truncate them by including a finite set of exchanged operators. We approximate the truncated operators by the corresponding contributions in appropriate disconnected five-point correlators. We compute a number of OPE coefficients that were previously unknown and show that these are consistent with predictions obtained using the fuzzy sphere regularization of the critical 3d Ising model.

Montenegro

Valentina Prilepina

Petar Tadić

Mixed five-point correlators in the 3d Ising model

We study the operator product expansion (OPE) of identical scalars in a conformal four-point correlator as a Stieltjes moment problem, and use Riemann-Liouville type fractional differential operators to generate classical moments from the correlation function. We use crossing symmetry to derive leading and subleading relations between moments in ∆ and $\textit{J}$ ≡ $\textit{ℓ}$($\textit{ℓ}$ + $\textit{d}$ − 2) in the “heavy” limit of large external scaling dimension, and combine them with constraints from unitarity to derive two-sided bounds on moment sequences in ∆ and the covariance between ∆ and $\textit{J}$ . The moment sequences which saturate these bounds produce “saddle point” solutions to the crossing equations which we identify as particular limits of correlators in a generalized free field (GFF) theory. This motivates us to study perturbations of heavy GFF four-point correlators by way of saddle point analysis, and we show that saddles in the OPE arise from contributions of fixed-length operator families encoded by a decomposition into higher-spin conformal blocks. To apply our techniques, we consider holographic correlators of four identical single scalar fields perturbed by a bulk interaction, and use their first few moments to derive Gaussian weight-interpolating functions that predict the OPE coefficients of interacting double-twist operators in the heavy limit.

Gordon Rogelberg

Moments and saddles of heavy CFT correlators

We compute observables of the critical 3d Ising model to high precision by applying the numerical conformal bootstrap to mixed correlators of the leading scalar operators and , and the stress tensor . We obtain new precise determinations of scaling dimensions (∆ , ∆ ) = (0.518148806(24), 1.41262528(29)) as well as OPE coefficients involving , , and . We also describe several improvements made along the way to algorithms and software tools for the numerical bootstrap.

France

Rajeev Erramilli

Canada

Alexandre Homrich

Aike Liu

Bootstrapping the 3d Ising stress tensor

We perform a numerical bootstrap study of scalar operators in the critical 3d Gross-Neveu-Yukawa models, a family of conformal field theories containing N Majorana fermions in the fundamental representation of an O(N) global symmetry. We compute rigorous bounds on the scaling dimensions of the next-to-lowest parity-even and parity-odd singlet scalars at N = 2, 4, and 8. All of these dimensions have lower bounds greater than 3, implying that there are only two relevant singlet scalars and placing constraints on the RG flow structure of these theories.

Bounding irrelevant operators in the 3d Gross-Neveu-Yukawa CFTs

We present a new algorithm for the numerical evaluation of five-point conformal blocks in d-dimensions, greatly improving the efficiency of their computation. To do this we use an appropriate ansatz for the blocks as a series expansion in radial coordinates, derive a set of recursion relations for the unknown coefficients in the ansatz, and evaluate the series using a Padé approximant to accelerate its convergence. We then study the 〈σσϵσσ〉 correlator in the 3d critical Ising model by truncating the operator product expansion (OPE) and only including operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We approximate the contributions of the operators above the cutoff by the corresponding contributions in a suitable disconnected five-point correlator. Using this approach, we compute a number of OPE coefficients with greater accuracy than previous methods.

Improving the five-point bootstrap

Abstract Recently, lattice formulations of Abelian chiral gauge theory in two dimensions have been devised on the basis of the Abelian bosonization. A salient feature of these 2D lattice formulations is that the gauge invariance is exactly preserved for anomaly-free theories and thus is completely free from the question of the gauge mode decoupling. In the present paper, we propose yet another lattice formulation sharing this desired property. A particularly unique point in our formulation is that the vertex operator of the dual scalar field, which carries the vector charge of the fermion and the “magnetic charge” in the bosonization, is represented by a “hole” excised from the lattice; this is the excision method formulated recently by Abe et al. in a somewhat different context.

Japan

Okuto Morikawa

Soma Onoda

Hiroshi Suzuki

Yet Another Lattice Formulation of 2D (1) Chiral Gauge Theory via Bosonization David Tong: lectures on gauge theory

We study five-point correlation functions of scalar operators in $\textit{d}$-dimensional conformal field theories. We develop a new approach to computing the five-point conformal blocks for exchanged primary operators of arbitrary spin by introducing a generalization of radial coordinates, using an appropriate ansatz, and perturbatively solving two quadratic Casimir differential equations. We then study five-point correlators 〈$\textit{σσϵσσ}$〉 in the critical 3d Ising model. We truncate the operator product expansions (OPEs) in the correlator by including a finite number of primary operators with conformal dimension below a cutoff ∆ ⩽ ∆ . We then compute several OPE coefficients involving $\textit{ϵ}$ and two spinning operators by demanding that the truncated correlator approximately satisfies the crossing relation.