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A bstract We study the conformal bootstrap for systems of correlators involving nonidentical 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 ℤ 2 global symmetry. For the leading ℤ 2 -odd operator σ and ℤ 2 -even operator ϵ , we obtain numerical constraints on the allowed dimensions (Δ σ , Δ ϵ ) assuming that σ and ϵ are the only relevant scalars in the theory. These constraints yield a small closed region in (Δ σ , Δ ϵ ) space compatible with the known values in the 3D Ising CFT.

A bstract We make precise determinations of the leading scaling dimensions and operator product expansion (OPE) coefficients in the 3d Ising, O (2), and O (3) models from the conformal bootstrap with mixed correlators. We improve on previous studies by scanning over possible relative values of the leading OPE coefficients, which incorporates the physical information that there is only a single operator at a given scaling dimension. The scaling dimensions and OPE coefficients obtained for the 3d Ising model, (Δ σ , Δ ϵ , λ σσϵ , λ ϵϵϵ ) = (0 . 5181489(10) , 1 . 412625(10) , 1 . 0518537(41) , 1 . 532435(19) , give the most precise determinations of these quantities to date.

A bstract We consider 5-point functions in conformal field theories in d > 2 dimensions. Using weight-shifting operators, we derive recursion relations which allow for the computation of arbitrary conformal blocks appearing in 5-point functions of scalar operators, reducing them to a linear combination of blocks with scalars exchanged. We additionally derive recursion relations for the conformal blocks which appear when one of the external operators in the 5-point function has spin 1 or 2. Our results allow us to formulate positivity constraints using 5-point functions which describe the expectation value of the energy operator in bilocal states created by two scalars.

A bstract Infrared fixed points of gauge theories provide intriguing targets for the modern conformal bootstrap program. In this work we provide some preliminary evidence that a family of gauged fermionic CFTs saturate bootstrap bounds and can potentially be solved with the conformal bootstrap. We start by considering the bootstrap for SO( N ) vector 4-point functions in general dimension D . In the large N limit, upper bounds on the scaling dimensions of the lowest SO( N ) singlet and traceless symmetric scalars interpolate between two solutions at ∆ = D/ 2 − 1 and ∆ = D − 1 via generalized free field theory. In 3D the critical O ( N ) vector models are known to saturate the bootstrap bounds and correspond to the kinks approaching ∆ = 1 / 2 at large N . We show that the bootstrap bounds also admit another infinite family of kinks T D , which at large N approach solutions containing free fermion bilinears at ∆ = D − 1 from below. The kinks T D appear in general dimensions with a D -dependent critical N * below which the kink disappears. We also study relations between the bounds obtained from the bootstrap with SO( N ) vectors, SU( N ) fundamentals, and SU( N ) × SU( N ) bi-fundamentals. We provide a proof for the coincidence between bootstrap bounds with different global symmetries. We show evidence that the proper symmetries of the underlying theories of T D are subgroups of SO( N ), and we speculate that the kinks T D relate to the fixed points of gauge theories coupled to fermions.

We derive general bounds on operator dimensions, central charges, and OPE coefficients in 4D conformal and superconformal field theories. In any CFT containing a scalar primary ϕ of dimension d we show that crossing symmetry of implies a completely general lower bound on the central charge c ≥ f c ( d ). Similarly, in CFTs containing a complex scalar charged under global symmetries, we bound a combination of symmetry current two-point function coefficients τ IJ and flavor charges. We extend these bounds to superconformal theories by deriving the superconformal block expansions for four-point functions of a chiral superfield Φ and its conjugate. In this case we derive bounds on the OPE coefficients of scalar operators appearing in the Φ × Φ † OPE, and show that there is an upper bound on the dimension of Φ † Φ when dim Φ is close to 1. We also present even more stringent bounds on c and τ IJ . In supersymmetric gauge theories believed to flow to superconformal fixed points one can use anomaly matching to explicitly check whether these bounds are satisfied.

A bstract We study the spectrum and OPE coefficients of the three-dimensional critical O(2) model, using four-point functions of the leading scalars with charges 0, 1, and 2 ( s , ϕ , and t ). We obtain numerical predictions for low-twist OPE data in several charge sectors using the extremal functional method. We compare the results to analytical estimates using the Lorentzian inversion formula and a small amount of numerical input. We find agreement between the analytic and numerical predictions. We also give evidence that certain scalar operators lie on double-twist Regge trajectories and obtain estimates for the leading Regge intercepts of the O(2) model.

A bstract 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 J 2 ≡ ℓ ( ℓ + 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 J 2 . 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.

A bstract We develop the analytic bootstrap in several directions. First, we discuss the appearance of nonperturbative effects in the Lorentzian inversion formula, which are exponentially suppressed at large spin but important at finite spin. We show that these effects are important for precision applications of the analytic bootstrap in the context of the 3d Ising and O(2) models. In the former they allow us to reproduce the spin-2 stress tensor with error at the 10 −5 level while in the latter requiring that we reproduce the stress tensor allows us to predict the coupling to the leading charge-2 operator. We also extend perturbative calculations in the lightcone bootstrap to fermion 4-point functions in 3d, predicting the leading and subleading asymptotic behavior for the double-twist operators built out of two fermions.

A bstract 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.

A bstract We develop new tools for isolating CFTs using the numerical bootstrap. A “cutting surface” algorithm for scanning OPE coefficients makes it possible to find islands in high-dimensional spaces. Together with recent progress in large-scale semidefinite programming, this enables bootstrap studies of much larger systems of correlation functions than was previously practical. We apply these methods to correlation functions of charge-0, 1, and 2 scalars in the 3d O(2) model, computing new precise values for scaling dimensions and OPE coefficients in this theory. Our new determinations of scaling dimensions are consistent with and improve upon existing Monte Carlo simulations, sharpening the existing decades-old 8 σ discrepancy between theory and experiment.