We introduce two new graphical-level relations among possible contributions to the four-point correlation function and scattering amplitude in planar, maximally supersymmetric Yang-Mills theory. When combined with the rung rule, these prove powerful enough to fully determine both functions through ten loops. This then also yields the full five-point amplitude to eight loops and the parity-even part to nine loops. We derive these rules, illustrate their applications, compare their relative strengths for fixing coefficients, and survey some of the features of the previously unknown nine and ten loop expressions. Explicit formulae for amplitudes and correlators through ten loops are available at: http://goo.gl/JH0yEc.
We apply the numerical bootstrap program to chiral operators in four-dimensional ${\mathcal N}=2$ SCFTs. In the first part of this work we study four-point functions in which all fields have the same conformal dimension. We give special emphasis to bootstrapping a specific theory: the simplest Argyres-Douglas fixed point with no flavor symmetry. In the second part we generalize our setup and consider correlators of fields with unequal dimension. This is an example of a mixed correlator and allows us to probe new regions in the parameter space of ${\mathcal N}=2$ SCFTs. In particular, our resul…
We show that the scaling dimensions of lowest operators in conformal field theories (CFTs) can be isolated in small and closed regions from single correlator bootstrap. We find the conserved currents play crucial roles in bootstrapping the crossing equation. By imposing a mild gap between the scaling dimensions of the conserved current and its next operator, the scaling dimensions of lowest operators are forced to lie in small isolated regions, i.e., these CFTs can be almost fixed by few lowest operators in certain channels. For CFTs with extended supersymmetry, the single correlator crossing …
How much spectral information is needed to determine the correlation functions of a conformal theory? We study this question in the context of planar supersymmetric Yang-Mills theory, where integrability techniques accurately determine the single-trace spectrum at finite 't Hooft coupling. Corresponding OPE coefficients are constrained by dispersive sum rules, which implement crossing symmetry. Focusing on correlators of four stress-tensor multiplets, we construct combinations of sum rules which determine one-loop correlators, and we study a numerical bootstrap problem that nonperturbatively b…
We present the full form of a four-point correlation function of large BPS operators in planar N = 4 Super Yang-Mills to any loop order. We do this by following a bootstrap philosophy based on three simple axioms pertaining to (i) the space of functions arising at each loop order, (ii) the behaviour in the OPE in a double-trace dominated channel and (iii) the behaviour under a double null limit. We discuss how these bootstrap axioms are in turn strongly motivated by empirical observations up to nine loops unveiled through integrability methods in our previous work [9] on this simplest correlat…
We use large spin perturbation theory and the Lorentzian inversion formula to compute order-$\varepsilon$ corrections to mixed correlators in the $O(n)$ Wilson-Fisher CFT in $4 - \varepsilon$ dimensions. In particular, we find the scaling dimensions and averaged OPE coefficients appearing in all correlators involving $φ$ and $φ^2$, for $φ^2$ in both the singlet and symmetric traceless representations of $O(n)$. We extend some computations to the next order, and find order-$\varepsilon^2$ data for a number of quantities for the Ising case at $n = 1$. Along the way, we discuss several interestin…
We revisit the problem of bootstrapping CFT correlators of charged fields. After discussing in detail how bounds for uncharged fields can be recycled to the charged case, we introduce two sets of analytic functional bases for correlators on the line. The first, which we call "simple", is essentially a direct sum of analytic functionals for the uncharged case. We use it to establish very general bounds on the OPE density appearing in charged correlators. The second basis is dual to generalized free fields and we explain how it is related to a charged version of the Polyakov bootstrap. We apply …
We study two-point functions of single-trace half-BPS operators in the presence of a supersymmetric Wilson line in $\mathcal{N}=4$ SYM. We use inversion formula technology in order to reconstruct the CFT data starting from a single discontinuity of the correlator. In the planar strong coupling limit only a finite number of conformal blocks contributes to the discontinuity, which allows us to obtain elegant closed-form expressions for two-point functions of single-trace operators $\mathcal{O}_J$ of weight $J=2,3,4$. Our final result passes a number of non-trivial consistency checks: it has the …
We use the conformal bootstrap approach to explore $5D$ CFTs with $O(N)$ global symmetry, which contain $N$ scalars $ϕ_i$ transforming as $O(N)$ vector. Specifically, we study multiple four-point correlators of the leading $O(N)$ vector $ϕ_i$ and the $O(N)$ singlet $σ$. The crossing symmetry of the four-point functions and the unitarity condition provide nontrivial constraints on the scaling dimensions ($Δ_ϕ$, $Δ_σ$) of $ϕ_i$ and $σ$. With reasonable assumptions on the gaps between scaling dimensions of $ϕ_i$ ($σ$) and the next $O(N)$ vector (singlet) scalar, we are able to isolate the scaling…
We propose a method to analytically solve the bootstrap equation for two point functions in boundary CFT. We consider the analytic structure of the correlator in Lorentzian signature and in particular the discontinuity of bulk and boundary conformal blocks to extract CFT data. As an application, the correlator $\langle ϕϕ\rangle$ in $ϕ^4$ theory at the Wilson-Fisher fixed point is computed to order $ε^2$ in the $ε$ expansion.
New Cornerstone Laboratory, Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所新基石实验室）；School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（中国科学院大学杭州高等研究院基础物理与数学学院）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；Centre for Theoretical Physics and Astronomy, Department of Physics and Astronomy, Queen Mary University of London（伦敦大学玛丽女王学院物理与天文系理论物理与天文学中心）
We elaborate and extend the method of Wronskian differential equations for conformal blocks to compute four-point correlation functions on the plane for classes of primary fields in rational (and possibly more general) conformal field theories. This approach leads to universal differential equations for families of CFT's and provides a very simple re-derivation of the BPZ results for the degenerate fields $ϕ_{1,2}$ and $ϕ_{2,1}$ in the c < 1 minimal models. We apply this technique to compute correlators for the WZW models corresponding to the Deligne-Cvitanović exceptional series of Lie algebr…
We consider unitary CFTs with continuous global symmetries in $d>2$. We consider a state created by the lightest operator of large charge $Q \gg 1$ and analyze the correlator of two light charged operators in this state. We assume that the correlator admits a well-defined large $Q$ expansion and, relatedly, that the macroscopic (thermodynamic) limit of the correlator exists. We find that the crossing equations admit a consistent truncation, where only a finite number $N$ of Regge trajectories contribute to the correlator at leading nontrivial order. We classify all such truncated solutions to …
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 $σ$ and $\mathbb{Z}_2$-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 yie…
Near lightcone correlators are dominated by operators with the lowest twist. We consider the contributions of such leading lowest twist multi-stress tensor operators to a heavy-heavy-light-light correlator in a CFT of any even dimensionality with a large central charge. An infinite number of such operators contribute, but their sum is described by a simple ansatz. We show that the coefficients in this ansatz can be determined recursively, thereby providing an operational procedure to compute them. This is achieved by bootstrapping the corresponding near lightcone correlator: conformal data for…
The observation of primordial correlators by cosmological surveys is a very promising avenue to probe high energies and the perturbative regime of quantum gravity. Hence, it is imperative that we understand how these observables are shaped by the pillars of fundamental physics, namely unitarity, locality and symmetries. To this end, we study the three-point correlators of gravitons and scalar curvature perturbations around a quasi de Sitter spacetime. We identify a set of Bootstrap Rules that fully fix the form of these correlators in the asymptotic future, i.e. at the boundary, and make no re…
We use the superconformal bootstrap to derive exact relations between OPE coefficients in three-dimensional superconformal field theories with ${\cal N} \geq 4$ supersymmetry. These relations follow from a consistent truncation of the crossing symmetry equations that is associated with the cohomology of a certain supercharge. In ${\cal N} = 4$ SCFTs, the non-trivial cohomology classes are in one-to-one correspondence with certain half-BPS operators, provided that these operators are restricted to lie on a line. The relations we find are powerful enough to allow us to determine an infinite numb…
The recent emergence of the modern conformal bootstrap method for the study of conformal field theories (CFTs) has enabled the revisiting of old problems in classical critical phenomena described by three-dimensional CFTs. The study of such CFTs with $O(m)^n \rtimes S_n$ global symmetry, also known as MN models, is pursued in this work. Systems of mixed correlators involving scalar operators in two different representations of the global symmetry group are considered. Isolated allowed regions are found in parameter space for various values of $m$ and $n$. These "islands" can be separated into …
II. Institut für Theoretische Physik, Universität Hamburg（汉堡大学理论物理研究所）；Deutsches Elektronen-Synchrotron（德国电子同步加速器）；Independent Researcher（独立研究员）；Department of Mathematical Sciences, Durham University（杜ham大学数学科学系）
We investigate two aspects of conformal field theories. In the first part, we study the general 4-point correlator of identical scalars around the fully crossing symmetric point $u=v=1$, where $u,v$ are conformally invariant cross ratios. Since this point is fully crossing invariant, we can deduce some general properties of the 4-point correlators from crossing symmetry. In the second part, we discuss the conformal bootstrap with OPE truncations. As a generalization of Gliozzi's method, we propose to extract the low-lying CFT data by minimizing the "error" induced by an OPE truncation. The err…
We study families of one-dimensional CFTs relevant for describing gapped QFTs in AdS$_2$. Using the Polyakov bootstrap as our main tool, we explain how S-matrices emerge from the flat space limit of CFT correlators. In this limit we prove that the CFT OPE density matches that of a generalized free field, and that this implies unitarity of the S-matrix. We establish a CFT dispersion formula for the S-matrix, proving its analyticity except for singularities on the real axis which we characterize in terms of the CFT data. In particular positivity of the OPE establishes that any such S-matrix must…
We study the stress tensor four-point function for $\mathcal{N}=4$ SYM with gauge group $G=SU(N)$, $SO(2N+1)$, $SO(2N)$ or $USp(2N)$ at large $N$. When $G=SU(N)$, the theory is dual to type IIB string theory on $AdS_5\times S^5$ with complexified string coupling $τ_s$, while for the other cases it is dual to the orbifold theory on $AdS_5\times S^5/\mathbb{Z}_2$. In all cases we use the analytic bootstrap and constraints from localization to compute 1-loop and higher derivative tree level corrections to the leading supergravity approximation of the correlator. We give perturbative evidence that…
An important part of a CFT four-point function, the stress tensor sector, comprises the exchanges of the stress tensor and its composites. The OPE coefficients of these multi-stress tensor operators and consequently, the complete stress tensor sector of four-point functions in CFTs with a large central charge, can be determined by computing a heavy-heavy-light-light correlator. We show how one can make substantial progress in this direction by bootstrapping a certain ansatz for the stress tensor sector of the correlator, iteratively computing the OPE coefficients of multi-stress tensor operato…
Bootstrap equations for conformal correlators that mimic the early theory of conformal bootstrap are written down in frames of the AdS/CFT approach. The simplified version of these equations, that may be justified if Schwinger-Keldysh formalism is used in AdS/CFT instead of conventional Feynman-Witten diagrams technique, permits to calculate values of conformal dimensions in the $O(N)$ symmetric model with conformal or composite Hubbard-Stratonovich field.
We study the statistical Ising model of spins on the infinite lattice using a bootstrap method that combines spin-flip identities with positivity conditions, including reflection positivity and Griffiths inequalities, to derive rigorous two-sided bounds on spin correlators through semi-definite programming. For the 2D Ising model on the square lattice, the bootstrap bounds based on correlators supported in a 13-site diamond-shaped region determine the nearest-spin correlator to within a small window, which for a wide range of coupling and magnetic field is narrower than the precision attainabl…
We introduce simple group-theoretic techniques for classifying conformally-invariant tensor-structures. With them, we classify tensor structures of general n-point functions of non-conserved operators, and $n\geq 4$-point functions of general conserved currents, with or without permutation symmetries, and in any spacetime dimension d. (The case n=3 for conserved operators will appear in subsequent work.) Our techniques are useful for bootstrap applications. The rules we derive simultaneously count tensor structures for flat-space scattering amplitudes in d+1 dimensions.
A key insight of the bootstrap approach to cosmological correlations is the fact that all correlators of slow-roll inflation can be reduced to a unique building block---the four-point function of conformally coupled scalars, arising from the exchange of a massive scalar. Correlators corresponding to the exchange of particles with spin are then obtained by applying a spin-raising operator to the scalar-exchange solution. Similarly, the correlators of massless external fields can be derived by acting with a suitable weight-raising operator. In this paper, we present a systematic and highly strea…
We study local operator insertions on 1/2-BPS line defects in ABJM theory. Specifically, we consider a class of four-point correlators in the CFT$_1$ with SU$(1, 1|3)$ superconformal symmetry defined on the 1/2-BPS Wilson line. The relevant insertions belong to the short supermultiplet containing the displacement operator and correspond to fluctuations of the dual fundamental string in AdS$_4 \times \mathbb{C}\textrm{P}^3$ ending on the line at the boundary. We use superspace techniques to represent the displacement supermultiplet and we show that superconformal symmetry determines the four-po…
We extend the cosmological bootstrap to correlators involving massless particles with spin. In de Sitter space, these correlators are constrained both by symmetries and by locality. In particular, the de Sitter isometries become conformal symmetries on the future boundary of the spacetime, which are reflected in a set of Ward identities that the boundary correlators must satisfy. We solve these Ward identities by acting with weight-shifting operators on scalar seed solutions. Using this weight-shifting approach, we derive three- and four-point correlators of massless spin-1 and spin-2 fields w…
