We use analytic conformal bootstrap methods to determine the anomalous dimensions and OPE coefficients for large spin operators in general conformal field theories in four dimensions containing a scalar operator of conformal dimension $Δ_ϕ$. It is known that such theories will contain an infinite sequence of large spin operators with twists approaching $2Δ_ϕ+2n$ for each integer $n$. By considering the case where such operators are separated by a twist gap from other operators at large spin, we analytically determine the $n$, $Δ_ϕ$ dependence of the anomalous dimensions. We find that for all $n$, the anomalous dimensions are negative for $Δ_ϕ$ satisfying the unitarity bound. We further compute the first subleading correction at large spin and show that it becomes universal for large twist. In the limit when $n$ is large, we find exact agreement with the AdS/CFT prediction corresponding to the Eikonal limit of a 2-2 scattering with dominant graviton exchange.
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.
We review the recent developments in the study of conformal field theories in generic space time dimensions using the methods of the conformal bootstrap, in its analytic aspect. These techniques are based solely on symmetries, in particular in the analytic structure and in the associativity of the operator product expansion. We focus on two applications of the analytic conformal bootstrap: the study of the $ε$ expansion of the Wilson Fisher model via the introduction of a dispersion relation and the large $N$ expansion of maximally supersymmetric Super Yang Mills theory in four dimensions.
We study skeleton expansion of $ϕ^3$ theory in $6+ε$ dimensions as well as its global symmetry generalizations. We use it to compute the four point function of the scalar field $ϕ$ up to $ε^2$. We also do a large spin bootstrap approach to the same model up to order $ε^2$ and check that both results agree.
This review aims to offer a pedagogical introduction to the analytic conformal bootstrap program via a journey through selected topics. We review analytic methods which include the large spin perturbation theory, Mellin space methods and the Lorentzian inversion formula. These techniques are applied to a variety of topics ranging from large-N theories, to the epsilon expansion and holographic superconformal correlators, and are demonstrated in a large number of explicit examples.
We review unitarity and crossing constraints on scattering amplitudes for particles with spin in four dimensional quantum field theories. As an application we study two to two scattering of neutral spin 1/2 fermions in detail. Assuming Mandelstam analyticity of its scattering amplitude, we use the numerical S-matrix bootstrap method to estimate various non-perturbative bounds on quartic and cubic (Yukawa) couplings.
We investigate the connection between spacetime wormholes and ensemble averaging in the context of higher spin AdS$_3$/CFT$_2$. Using techniques from modular bootstrap combined with some holographic inputs, we evaluate the partition function of a Euclidean wormhole in AdS$_3$ higher spin gravity. The fixed spin sectors of the dual CFT$_2$ exhibit features that starkly go beyond conventional random matrix ensembles: power-law ramps in the spectral form factor and potentials with a double-well/crest underlying the level statistics.
The analytic conformal bootstrap is an array of techniques to characterize, constrain, and solve strongly interacting quantum field theories using symmetries, causality, unitarity, and other general principles. In the last decade, bolstered by the development of new Lorentzian methods, it has been used to solve conformal field theories at large spin; to place bounds on energy distributions, event shapes, operator product coefficients, and other observables; and to understand aspects of quantum gravity in anti-de Sitter space. We review these advances and highlight several promising areas for f…
University of Science and Technology of China（中国科学技术大学）；Interdisciplinary Center for Theoretical Study, University of Science and Technology of China（中国科学技术大学理论物理交叉中心）；Peng Huanwu Center for Fundamental Theory（彭桓武基础理论研究中心）
In this work we discuss an analytic bootstrap approach [1,2] in the context of spinning 4D conformal blocks [3,4]. As an example we study the simplest spinning case, the scalar-fermion correlator $\langleϕψϕ\barψ\rangle$. We find that to every pair of primary scalar $ϕ$ and fermion $ψ$ correspond two infinite towers of fermionic large spin primary operators. We compute their twists and products of OPE coefficients using both s-t and u-t bootstrap equations to the leading and sub-leading orders. We find that the leading order is represented by the scalar-fermion generalized free theory and the …
Conformal field theories play a central role in theoretical physics with many applications ranging from condensed matter to string theory. The conformal bootstrap studies conformal field theories using mathematical consistency conditions and has seen great progress over the last decade. In this thesis we present an implementation of analytic bootstrap methods for perturbative conformal field theories in dimensions greater than two, which we achieve by combining large spin perturbation theory with the Lorentzian inversion formula. In the presence of a small expansion parameter, not necessarily …
We revisit analytical methods for constraining the nonperturbative $S$-matrix of unitary, relativistic, gapped theories in $d \geq 3$ spacetime dimensions. We assume extended analyticity of the two-to-two scattering amplitude and use it together with elastic unitarity to develop two natural expansions of the amplitude. One is the threshold (non-relativistic) expansion and the other is the large spin expansion. The two are related by the Froissart-Gribov inversion formula. When combined with crossing and a local bound on the discontinuity of the amplitude, this allows us to constrain scattering…
We apply the analytic conformal bootstrap method to study weakly coupled conformal gauge theories in four dimensions. We employ twist conformal blocks to find the most general form of the one-loop four-point correlation function of identical scalar operators, without any reference to Feynman calculations. The method relies only on symmetries of the model. In particular, it does not require introducing any regularisation and it is free from the redundancies usually associated with the Feynman approach. By supplementing the general solution with known data for a small number of operators, we rec…
We study analytically the constraints of the conformal bootstrap on the low-lying spectrum of operators in field theories with global conformal symmetry in one and two spacetime dimensions. We introduce a new class of linear functionals acting on the conformal bootstrap equation. In 1D, we use the new basis to construct extremal functionals leading to the optimal upper bound on the gap above identity in the OPE of two identical primary operators of integer or half-integer scaling dimension. We also prove an upper bound on the twist gap in 2D theories with global conformal symmetry. When the ex…
We apply the method of the large spin bootstrap to analyse fermionic conformal field theories with weakly broken higher spin symmetry. Through the study of correlators of composite operators, we find the anomalous dimensions and OPE coefficients in the Gross-Neveu model in $d=2+\varepsilon$ dimensions and the Gross-Neveu-Yukawa model in $d=4-\varepsilon$ dimensions, based only on crossing symmetry. Furthermore a non-trivial solution in the $d=2+\varepsilon$ expansion is found for a fermionic theory in which the fundamental field is not part of the spectrum. The results are perturbative in $\va…
Attempts to resolve the longstanding Fermion mass hierarchy (FMH) problem in frames of the AdS/CFT correspondence demand the knowledge of bulk fermion masses. The approach of the "old" conformal bootstrap in AdS context permitted to calculate bulk masses of scalar fields [1], [2]. In the present paper this approach is extended to physically more interesting spin 1/2 bulk fields. Calculation of spinor-scalar vertices is performed in physical AdS space, and unexpectedly simple expressions for spinor-scalar bubbles (2-point one-loop self-energy Witten diagrams) are obtained. The "double-trace fro…
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| << |v| < 1. We prove that every CFT with a scalar operator ϕmust contain infinite sequences of operators O_{τ,l} with twist approaching τ-> 2Δ_ϕ+ 2n for each integer n as l -> infinity. We show how the rate of approach is controlled by the twist and OPE coefficient of the leading twist operator in the ϕx ϕOPE, and we discuss SCFTs and the 3d Ising Model as examples. Additionally, we show that the OPE coefficients of ot…
We develop crossing symmetric dispersion relations for describing 2-2 scattering of identical external particles carrying spin. This enables us to import techniques from Geometric Function Theory and study two sided bounds on low energy Wilson coefficients. We consider scattering of photons, gravitons in weakly coupled effective field theories. We provide general expressions for the locality/null constraints. Consideration of the positivity of the absorptive part leads to an interesting connection with the recently conjectured weak low spin dominance. We also construct the crossing symmetric a…
Under the assumption that degenerate fields exist, diagonal CFTs such as Liouville theory can be solved analytically using the conformal bootstrap method. Here we generalize this approach to non-diagonal CFTs, i.e. CFTs whose primary fields have nonzero conformal spins. Assuming generic values of the central charge, we find that the non-diagonal sector of the spectrum must be parametrized by two integer numbers. We then derive and solve the equations that determine how three- and four-point structure constants depend on these numbers. In order to test these results, we numerically check crossi…
We compute the contribution of the vacuum Virasoro representation to the genus-two partition function of an arbitrary CFT with central charge $c>1$. This is the perturbative pure gravity partition function in three dimensions. We employ a sewing construction, in which the partition function is expressed as a sum of sphere four-point functions of Virasoro vacuum descendants. For this purpose, we develop techniques to efficiently compute correlation functions of holomorphic operators, which by crossing symmetry are determined exactly by a finite number of OPE coefficients; this is an analytic im…
Recently, boundary conformal field theories (BCFTs) have attracted much attention in the context of quantum gravity. This is because a BCFT can be dual to gravity coupled to a heat bath CFT, known as the island model. On this background, it would be interesting to explore the duality between the boundary and the braneworld. However, this seems to be a challenging problem. The reason is because although there has been much study of rational BCFTs, there has been comparatively little study of irrational BCFTs, and irrational BCFTs are expected to be the boundary duals of the braneworlds. For thi…
We study the space of 3d ${\cal N} = 6$ SCFTs by combining numerical bootstrap techniques with exact results derived using supersymmetric localization. First we derive the superconformal block decomposition of the four-point function of the stress tensor multiplet superconformal primary. We then use supersymmetric localization results for the ${\cal N} = 6$ $U(N)_k\times U(N+M)_{-k}$ Chern-Simons-matter theories to determine two protected OPE coefficients for many values of $N,M,k$. These two exact inputs are combined with the numerical bootstrap to compute precise rigorous islands for a wide …
We use the amplitude soft bootstrap method to explore the space of effective field theories (EFT) of massless vectors and scalars. It is known that demanding vanishing soft limits fixes uniquely a special class of EFTs: non-linear sigma model, scalar Galileon and Born-Infeld theories. Based on the amplitudes analysis, we conjecture no-go theorems for higher-derivative vector theories and theories with coupled vectors and scalars. We then allow for more general soft theorems where the non-trivial part of the soft limit of the (n+1)-pt amplitude is equal to a linear combination of n-pt amplitude…
Higher-point functions of scalar operators are a rich observable in CFTs, as they contain OPE data involving multiple spinning operators. We derive the lightcone blocks for five- and six-point functions in the snowflake channel and use them to bootstrap these correlators in the lightcone limit. As a result we determine the large spin expansion of OPE coefficients involving two or three spinning operators. We verify our results by comparing to the block decomposition of higher-point functions in generalized free theory and in theories with a cubic coupling.
We study logarithmic conformal field theory (LogCFT) in four dimensions using conformal bootstrap techniques in the large spin limit. We focus on the constraints imposed by conformal symmetry on the four point function of certain logarithmic scalar operators and compute the leading correction to the anomalous dimension of double trace operators in the large spin limit. There exist certain holographic duals to such LogCFTs, which involve higher derivative equations of motion. The anomalous dimension is related to the binding energy of a state where two scalars rotate around each other with a la…
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 introduce analytic functionals which act on the crossing equation for CFTs in arbitrary spacetime dimension. The functionals fully probe the constraints of crossing symmetry on the first sheet, and are in particular sensitive to the OPE, (double) lightcone and Regge limits. Compatibility with the crossing equation imposes constraints on the functional kernels which we study in detail. We then introduce two simple classes of functionals. The first class has a simple action on generalized free fields and their deformations and can be used to bootstrap AdS contact interactions in general dimen…
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 combine the large-$c$ ST modular bootstrap equations with the Cardy formula for the asymptotic growth of the density of states to prove that any $2d$ unitary, compact, conformal field theory (CFT) with no higher spin conserved currents leads to conflicting inequalities whenever the entire spectrum of non-trivial primaries lies above the BTZ threshold. As a consequence, the holographic dual of $3d$ pure gravity, if it exists, cannot be a $2d$ CFT. Consistent solutions of ST bootstrap equations require additional primaries lying below the BTZ threshold. The lowest non-trivial primary necessar…
We develop an algebraic approach to the analytic bootstrap in CFTs. By acting with the Casimir operator on the crossing equation we map the problem of doing large spin sums to any desired order to the problem of solving a set of recursion relations. We compute corrections to the anomalous dimension of large spin operators due to the exchange of a primary and its descendants in the crossed channel and show that this leads to a Borel-summable expansion. We analyse higher order corrections to the microscopic CFT data in the direct channel and its matching to infinite towers of operators in the cr…
We solve crossing equations analytically in the deep Euclidean regime. Large scaling dimension $Δ$ tails of the weighted spectral density of primary operators of given spin in one channel are matched to the Euclidean OPE data in the other channel. Subleading $1\over Δ$ tails are systematically captured by including more operators in the Euclidean OPE in the dual channel. We use dispersion relations for conformal partial waves in the complex $Δ$ plane, the Lorentzian inversion formula and complex tauberian theorems to derive this result. We check our formulas in a few examples (for CFTs and sca…
We clarify the relationships between different approaches to the conformal bootstrap. A central role is played by the so-called extremal functionals. They are linear functionals acting on the crossing equation which are directly responsible for the optimal bounds of the numerical bootstrap. We explain in detail that the extremal functionals probe the Regge limit. We construct two complete sets of extremal functionals for the crossing equation specialized to $z=\bar{z}$, associated to the generalized free boson and fermion theories. These functionals lead to non-perturbative sum rules on the CF…
We compute, using the method of large spin perturbation theory, the anomalous dimensions and OPE coefficients of all leading twist operators in the critical $ O(N) $ model, to fourth order in the $ ε$-expansion. This is done fully within a bootstrap framework, and generalizes a recent result for the CFT-data of the Wilson-Fisher model. The anomalous dimensions we obtain for the $ O(N) $ singlet operators agree with the literature values, obtained by diagrammatic techniques, while the anomalous dimensions for operators in other representations, as well as all OPE coefficients, are new. From the…
We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to th…
The modern conformal bootstrap program often employs the method of linear functionals to derive the numerical or analytical bounds on the CFT data. These functionals must have a crucial "swapping" property, allowing to swap infinite summation with the action of the functional in the conformal bootstrap sum rule. Swapping is easy to justify for the popular functionals involving finite sums of derivatives. However, it is far from obvious for "cut-touching" functionals, involving integration over regions where conformal block decomposition does not converge uniformly. Functionals of this type wer…
Using modular bootstrap we show the lightest primary fields of a unitary compact two dimensional conformal field theory(with $c, \bar{c}>1$) has a conformal weight $h_1\le \frac{c}{12}+\mathcal{O}(1)$.This implies that the upper bound on the dimension of the lightest primary fields depends on their spin. In particular if the set of lightest primary fields includes extremal or near extremal states whose spin to dimension ratio $\frac{j}Δ\approx 1$, the corresponding dimension is $Δ\le \frac{c}{12}+\mathcal{O}(1)$. From AdS/CFT correspondence, we obtain an upper bound on the spectrum of black ho…
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 o…
We propose the relaxation bootstrap method for the numerical solution of multi-matrix models in the large $N$ limit, developing and improving the recent proposal of H.Lin. It gives rigorous inequalities on the single trace moments of the matrices up to a given "cutoff" order (length) of the moments. The method combines usual loop equations on the moments and the positivity constraint on the correlation matrix of the moments. We have a rigorous proof of applicability of this method in the case of the one-matrix model where the condition of positivity of the saddle point solution appears to be e…
We study conformal partial waves (CPWs) in Mellin space with totally symmetric external operators of arbitrary integer spin. The exchanged spin is arbitrary, and includes mixed symmetry and (partially)-conserved representations. In a basis of CPWs recently introduced in arXiv:1702.08619, we find a remarkable factorisation of the external spin dependence in their Mellin representation. This property allows a relatively straightforward study of inversion formulae to extract OPE data from the Mellin representation of spinning 4pt correlators and in particular, to extract closed-form expressions f…
We revisit the critical two-dimensional Ashkin-Teller model, i.e. the $\mathbb{Z}_2$ orbifold of the compactified free boson CFT at $c=1$. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at $c=1$. We show that blocks with a degenerate channel field should be computed by…
We study half-BPS line defects in $\mathcal{N}=2$ superconformal theories using the bootstrap approach. We concentrate on local excitations constrained to the defect, which means the system is a $1d$ defect CFT with $\mathfrak{osp}(4^*|2)$ symmetry. In order to study correlation functions we construct a suitable superspace, and then use the Casimir approach to calculate a collection of new superconformal blocks. Special emphasis is given to the displacement operator, which controls deformations orthogonal to the defect and is always present in a defect CFT. After setting up the crossing equati…
Every conformal field theory (CFT) above two dimensions contains an infinite set of Regge trajectories of local operators which, at large spin, asymptote to "double-twist" composites with vanishing anomalous dimension. In two dimensions, due to the existence of local conformal symmetry, this and other central results of the conformal bootstrap do not apply. We incorporate exact stress tensor dynamics into the CFT$_2$ analytic bootstrap, and extract several implications for AdS$_3$ quantum gravity. Our main tool is the Virasoro fusion kernel, which we newly analyze and interpret in the bootstra…
We consider conformal field theories with slightly broken higher spin symmetry in arbitrary spacetime dimensions. We analyze the crossing equation in the double light-cone limit and solve for the anomalous dimensions of higher spin currents $γ_s$ with large spin $s$. The result depends on the symmetries and the spectrum of the unperturbed conformal field theory. We reproduce all known results and make further predictions. In particular we make a prediction for the anomalous dimensions of higher spin currents in the 3d Ising model.
Using the full conformal bootstrap method an analytic expression is given in d-dimensions for the anomalous dimension of the fermion at O(1/N^3) in a large N expansion of the Nambu--Jona-Lasinio model with SU(2) x SU(2) continuous chiral symmetry.
We apply bootstrap techniques in order to constrain the CFT data of the $(A_1,A_2)$ Argyres-Douglas theory, which is arguably the simplest of the Argyres-Douglas models. We study the four-point function of its single Coulomb branch chiral ring generator and put numerical bounds on the low-lying spectrum of the theory. Of particular interest is an infinite family of semi-short multiplets labeled by the spin $\ell$. Although the conformal dimensions of these multiplets are protected, their three-point functions are not. Using the numerical bootstrap we impose rigorous upper and lower bounds on t…
The axiomatic Wightman formulation for nonderivative conformal field theory is adopted to derive conformal bootstrap equation for the four point function. The equivalence between PCT theorem and {\it weak local commutativity}, due to Jost, play a very crucial role in axiomatic field theory. The theorem is suitably adopted for conformal field theory to derive the desired equations in CFT. We demonstrate that the two Wightman functions are analytic continuation of each other.
We employ a hybrid approach in determining the anomalous dimension and OPE coefficient of higher spin operators in the Wilson-Fisher theory. First we do a large spin analysis for CFT data where we use results obtained from the usual and the Mellin Bootstrap and also from Feynman diagram literature. This gives new predictions at $O(ε^4)$ and $O(ε^5)$ for anomalous dimensions and OPE coefficients, and also provides a cross-check for the results from Mellin Bootstrap. These higher orders get contributions from all higher spin operators in the crossed channel. We also use the Bootstrap in Mellin s…
We use analytic bootstrap techniques for a CFT with an interface or a boundary. Exploiting the analytic structure of the bulk and boundary conformal blocks we extract the CFT data. We further constrain the CFT data by applying the equation of motion to the boundary operator expansion. The method presented in this paper is general, and it is illustrated in the context of perturbative Wilson-Fisher theories. In particular, we find constraints on the OPE coefficients for the conformal interface CFT in $4 - ε$ dimensions (upto order $\mathcal{O}(ε^2)$) with $ϕ^4$-interactions in the bulk. We also …
Finding a method to combine the numerical bootstrap with the analytic lightcone bootstrap is an important goal to advance the conformal bootstrap program. We propose a hybrid bootstrap method to do just that. The numerical and analytic bootstrap approaches are sensitive to different regions of the spectrum and complement each other. When they are effectively combined, the hybrid bootstrap enjoys the best of both worlds and the prediction for the actual CFT can be significantly improved. In this work, we discuss the general strategy to perform such a hybrid bootstrap, and we make a partial impl…
We derive new constraints on the spectrum of two-dimensional conformal field theories with central charge $c>1.$ Employing the pillow representation of the four point correlator of identical scalars with dimension $Δ_{\mathcal{O}}$ and positivity of the coefficients of its expansion in the elliptic nome we place central charge dependent bounds on the dimension of the first excited Virasoro primary the scalar couples to, in the form $Δ_1<f(c,Δ_{\mathcal{O}}).$ We give an analytic expression for $f(c,Δ_{\mathcal{O}})$ and write down transcendental equations that significantly improve the analyti…
We initiate a study of the bootstrap programme for field theories with BMS symmetry. Specifically, we look at two-dimensional field theories with BMS3 symmetry and, using highest weight representations, we construct the BMS bootstrap equation by formulating the notion of crossing symmetry in the four-point functions of these field theories. In the limit of large central charges, we find analytic expressions for the BMS blocks that are the basic ingredients for the solution of the bootstrap equation. This constitutes, to the best of our knowledge, the first example of the formulation and signif…
