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We construct backreacted geometries dual to the supersymmetric mass deformation of the IKKT matrix model. They are Euclidean type IIB supergravity solutions given in terms of an electrostatic potential, having $SO(7)\times SO(3)$ isometry and 16 supersymmetries. Quantizing the fluxes, we find that the supergravity solutions are in one-to-one correspondence with fuzzy sphere vacua of the matrix model., Comment: 22 pages + appendices, 3 figures

We revisit the Berenstein-Maldacena-Nastase (BMN) conjecture relating M-theory on a PP-wave background and Matrix Quantum Mechanics (MQM) of $N\times N$ matrices. In particular, we study the BMN MQM at strong coupling and finite $N$ and derive an effective Hamiltonian that describes non-relativistic free particles in a harmonic trap. The energy spectrum predicted by this Hamiltonian matches the supergravity excitation spectrum around the PP-wave background, if we further assume the existence of bound states. Our derivation is based on the strong coupling expansion of the wavefunction and supersedes the naive path integral approach that can lead to incorrect results, as we demonstrate in a simple toy model. We conclude with open questions about various regimes of the theory when we vary the size of the matrices, the coupling and the temperature., Comment: 40 pages + appendices, 6 figures

We consider 3+1 dimensional Quantum Field Theories (QFTs) coupled to the dilaton and the graviton. We show that the graviton-dilaton scattering amplitude receives a universal contribution which is helicity flipping and is proportional to $\Delta c-\Delta a$ along any RG flow, where $\Delta c$ and $\Delta a$ are the differences of the UV and IR $c$- and $a$-trace anomalies respectively. This allows us to relate $\Delta c-\Delta a$ to spinning massive states in the spectrum of the QFT. We test our predictions in two simple examples: in the theory of a massive free scalar and in the theory of a massive Dirac fermion (a more complicated example is provided in a companion paper [1]). We discuss possible applications., Comment: 45 pages + appendices, minor updates

We study two-point functions of symmetric traceless local operators in the bulk of de Sitter spacetime. We derive the K\"all\'en-Lehmann spectral decomposition for any spin and show that unitarity implies its spectral densities are nonnegative. In addition, we recover the K\"all\'en-Lehmann decomposition in Minkowski space by taking the flat space limit. Using harmonic analysis and the Wick rotation to Euclidean Anti de Sitter, we derive an inversion formula to compute the spectral densities. Using the inversion formula, we relate the analytic structure of the spectral densities to the late-time boundary operator content. We apply our technical tools to study two-point functions of composite operators in free and weakly coupled theories. In the weakly coupled case, we show how the K\"all\'en-Lehmann decomposition is useful to find the anomalous dimensions of the late-time boundary operators. We also derive the K\"all\'en-Lehmann representation of two-point functions of spinning primary operators of a Conformal Field Theory on de Sitter., Comment: 63 pages + appendices, 10 figures

We study correlation functions of the bulk stress tensor and boundary operators in Quantum Field Theories (QFT) in Anti-de Sitter (AdS) space. In particular, we derive new sum rules from the two-point function of the stress tensor and its three-point function with two boundary operators. In AdS2, this leads to a bootstrap setup that involves the central charge of the UV limit of the bulk QFT and may allow us to follow a Renormalization Group (RG) flow non-perturbatively by continuously varying the AdS radius. Along the way, we establish the convergence properties of the newly discovered local block decomposition of the three-point function., Comment: 43+27 pages, 10 figures, a few typos removed, a couple of references added, matches published version

We perform Monte-Carlo measurements of two and three point functions of charged operators in the critical O(2) model in 3 dimensions. Our results are compatible with the predictions of the large charge superfluid effective field theory. To obtain reliable measurements for large values of the charge, we improved the Worm algorithm and devised a measurement scheme which mitigates the uncertainties due to lattice and finite size effects., Comment: 16 pages, 12 figures. v2: Improved finite size scaling. v3: Added comparison between Monte Carlo updates

We study the decomposition of the Hilbert space of quantum field theory in $(d+1)$ dimensional de Sitter spacetime into Unitary Irreducible Representations (UIRs) of its isometry group \SO$(1,d+1)$. Firstly, we consider multi-particle states in free theories starting from the tensor product of single-particle UIRs. Secondly, we study conformal multiplets of a bulk Conformal Field Theory with symmetry group \SO$(2,d+1)$. Our main tools are the Harish-Chandra characters and the numerical diagonalization of the (truncated) quadratic Casimir of \SO$(1,d+1)$. We introduce a continuous density that encodes the spectrum of irreducible representations contained in a reducible one of $\SO(1,d+1)$. Our results are complete for $d=1$ and $d=2$. In higher dimensions, we rederive and extend several results previously known in the literature. Our work provides the foundation for future nonperturbative bootstrap studies of Quantum Field Theory in de Sitter spacetime., Comment: 58 pages + appendices, 44 figures and 4 tables

We merge together recent developments in the S-matrix bootstrap program to develop a dual setup in 2 space-time dimensions incorporating scattering amplitudes of massive particles and matrix elements of local operators. In particular, the stress energy tensor allows us to input UV constraints on IR observables in terms of the central charge $c_{UV}$ of the UV Conformal Field Theory. We consider two applications: (1) We establish a rigorous lower bound on $c_{UV}$ of a class of $\mathbb{Z}_2$ symmetric scalar theories in the IR (including $\phi^4$); (2) We target Ising Field Theory by, first, minimizing $c_{UV}$ for different values of the magnetic field and, secondly, by determining the allowed range of cubic coupling and one-particle form-factor for fixed $c_{UV} = 1/2$ and magnetic field., Comment: 24 pages + appendices, 29 figures. v2: minor changes, file size reduced

We use the S-matrix bootstrap to carve out the space of unitary, analytic, crossing symmetric and supersymmetric graviton scattering amplitudes in nine, ten and eleven dimensions. We extend and improve the numerical methods of our previous work in ten dimensions. A key new tool employed here is unitarity in the celestial sphere. In all dimensions, we find that the minimal allowed value of the Wilson coefficient $\alpha$, controlling the leading correction to maximal supergravity, is very close but not equal to the minimal value realized in Superstring theory or M-theory. This small difference may be related to inelastic effects that are not well described by our numerical extremal amplitudes. Although $\alpha$ has a unique value in M-theory, we found no evidence of an upper bound on $\alpha$ in 11D., Comment: 17 pages, 18 figures

We study 2-to-2 scattering amplitudes of massless spin one particles in $d=4$ space-time dimensions, like real world photons. We define a set of non-perturbative observables (Wilson coefficients) which describe these amplitudes at low energies. We use full non-linear unitarity to construct various novel numerical bounds on these observables. For completeness, we also rederive some bounds using positivity only. We discover and explain why some of these Wilson coefficients cannot be bounded., Comment: 39 pages + appendices, 20 figures

We study gapped 4d quantum field theories (QFTs) obtained from a relevant deformation of a UV conformal field theory (CFT). For simplicity, we assume the existence of a $\mathbb{Z}_2$ symmetry and a single $\mathbb{Z}_2$-odd stable particle and no $\mathbb{Z}_2$-even particles at low energies. Using unitarity, crossing and the assumption of maximal analyticity we compute numerically a lower bound on the value of the $a$-anomaly of the UV CFT as a function of various non-perturbative parameters describing the two-to-two scattering amplitude of the particle., Comment: 41 pages + appendices, 20 figures. In v2 section 2 improved, minor updates, references added

The S-matrix Bootstrap originated on the idea that the S-matrix might be fully constrained by global symmetries, crossing, unitarity, and analyticity without relying on an underlying dynamical theory that may or may not be a quantum field theory. Recently this approach was revived from a somewhat different point of view. Using the same constraints, one numerically maps out the (infinite-dimensional) space of allowed S-matrices that should contain all consistent quantum field theories (and quantum theories of gravity). Moreover, in the best case scenario one finds special points in the space that can be identified with a certain quantum field theory of interest. In that case, the approach allows the numerical computation of the S-matrix without relying on the particular Lagrangian of the theory. In this white paper we summarize the state of the art and discuss the future of the topic., Comment: 9 pages, v2: added reference

The boundary correlation functions for a Quantum Field Theory (QFT) in an Anti-de Sitter (AdS) background can stay conformally covariant even if the bulk theory undergoes a renormalization group (RG) flow. Studying such correlation functions with the numerical conformal bootstrap leads to non-perturbative constraints that must hold along the entire flow. In this paper we carry out this analysis for the sine-Gordon RG flows in AdS$_2$, which start with a free (compact) scalar in the UV and end with well-known massive integrable theories that saturate many S-matrix bootstrap bounds. We numerically analyze the correlation functions of both breathers and kinks and provide a detailed comparison with perturbation theory near the UV fixed point. Our bounds are often saturated to one or two orders in perturbation theory, as well as in the flat-space limit, but not necessarily in between., Comment: 62 pages, 22 figures, 1 menhir

We study quantum field theory on a de Sitter spacetime dS$_{d+1}$ background. Our main tool is the Hilbert space decomposition in irreducible unitary representations of its isometry group $SO(d+1,1)$. As the first application of the Hilbert space formalism, we recover the K\"allen-Lehmann spectral decomposition of the scalar bulk two-point function. In the process, we exhibit a relation between poles in the corresponding spectral densities and the boundary CFT data. Moreover, we derive an inversion formula for the spectral density through analytical continuation from the sphere and use it to find the spectral decompisiton for a few examples. Next, we study the conformal partial wave decomposition of the four-point functions of boundary operators. These correlation functions are very similar to the ones of standard conformal field theory, but have different positivity properties that follow from unitarity in de Sitter. We conclude by proposing a non-perturbative conformal bootstrap approach to the study of these late-time four-point functions, and we illustrate our proposal with a concrete example for QFT in dS$_2$., Comment: 54 pages + appendices, 3 figures; v2 corrections, fixed typos, added references, JHEP version

Quantum Field Theories (QFTs) in Anti-de Sitter (AdS) spacetime are often strongly coupled when the radius of AdS is large, and few methods are available to study them. In this work, we develop a Hamiltonian truncation method to compute the energy spectrum of QFTs in two-dimensional AdS. The infinite volume of constant timeslices of AdS leads to divergences in the energy levels. We propose a simple prescription to obtain finite physical energies and test it with numerical diagonalization in several models: the free massive scalar field, $\phi^4$ theory, Lee-Yang and Ising field theory. Along the way, we discuss spontaneous symmetry breaking in AdS and derive a compact formula for perturbation theory in quantum mechanics at arbitrary order. Our results suggest that all conformal boundary conditions for a given theory are connected via bulk renormalization group flows in AdS., Comment: 63 pages + appendices, 44 figures

We study the phase diagram of 5-dimensional $SU(2)$ Yang-Mills theory on the lattice. We consider two extensions of the fundamental plaquette Wilson action in the search for the continuous phase transition suggested by the $4+\epsilon$ expansion. The extensions correspond to new terms in the action: i) a unit size plaquette in the adjoint representation or ii) a two-unit sided square plaquette in the fundamental representation. We use Monte Carlo to sample the first and second derivative of the entropy near the confinement phase transition, with lattices up to $12^{5}$. While we exclude the presence of a second order phase transition in the parameter space we sampled for model i), our data is not conclusive in some regions of the parameter space of model ii)., Comment: 18 pages, 12 figures. Added discussion on expected critical scaling and frustrated phase. Added references

We use the S-matrix bootstrap to carve out the space of unitary, crossing symmetric and supersymmetric graviton scattering amplitudes in ten dimensions. We focus on the leading Wilson coefficient $\alpha$ controlling the leading correction to maximal supergravity. The negative region $\alpha<0$ is excluded by a simple dual argument based on linearized unitarity (the desert). A whole semi-infinite region $\alpha \gtrsim 0.14$ is allowed by the primal bootstrap (the garden). A finite intermediate region is excluded by non-perturbative unitarity (the swamp). Remarkably, string theory seems to cover all (or at least almost all) the garden from very large positive $\alpha$ -- at weak coupling -- to the swamp boundary -- at strong coupling., Comment: 3 pages + appendices, 9 figures. v2 with improved data, 1 figure added, 1 appendix added. The submission contains the Mathematica notebook used for the fit analysis and a .txt file with the data

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., Comment: 52+64 pages, 11 figures, v2: errors corrected, multiple typos fixed, plots updated

We use the numerical S-matrix bootstrap method to obtain bounds on the two leading Wilson coefficients of the chiral lagrangian controlling the low-energy dynamics of massless pions thus providing a proof of concept that the numerical S-matrix bootstrap can be used to derive non-perturbative bounds on EFTs in more than two spacetime dimensions., Comment: 12 pages + appendices

We use Mellin space dispersion relations together with Polyakov conditions to derive a family of sum rules for Conformal Field Theories (CFTs). The defining property of these sum rules is suppression of the contribution of the double twist operators. Firstly, we apply these sum rules to the Wilson-Fisher model in $d=4-\epsilon$ dimensions. We re-derive many of the known results to order $\epsilon^4$ and we make new predictions. No assumption of analyticity down to spin $0$ was made. Secondly, we study holographic CFTs. We use dispersive sum rules to obtain tree-level and one-loop anomalous dimensions. Finally, we briefly discuss the contribution of heavy operators to the sum rules in UV complete holographic theories., Comment: 36p, 5 fig

We define form factors and scattering amplitudes in Conformal Field Theory as the coefficient of the singularity of the Fourier transform of time-ordered correlation functions, as $p^2 \to 0$. In particular, we study a form factor $F(s,t,u)$ obtained from a four-point function of identical scalar primary operators. We show that $F$ is crossing symmetric, analytic and it has a partial wave expansion. We illustrate our findings in the 3d Ising model, perturbative fixed points and holographic CFTs., Comment: 70 pages, 15 figures

We argue that nonperturbative CFT correlation functions admit a Mellin amplitude representation. Perturbative Mellin representation readily follows. We discuss the main properties of nonperturbative CFT Mellin amplitudes: subtractions, analyticity, unitarity, Polyakov conditions and polynomial boundedness at infinity. Mellin amplitudes are particularly simple for large N CFTs and 2D rational CFTs. We discuss these examples to illustrate our general discussion. We consider subtracted dispersion relations for Mellin amplitudes and use them to derive bootstrap bounds on CFTs. We combine crossing, dispersion relations and Polyakov conditions to write down a set of extremal functionals that act on the OPE data. We check these functionals using the known 3d Ising model OPE data and other known bootstrap constraints. We then apply them to holographic theories., Comment: The erroneous claims regarding the consistency of certain AdS EFTs are corrected

We propose a new non-perturbative method for studying UV complete unitary quantum field theories (QFTs) with a mass gap in general number of spacetime dimensions. The method relies on unitarity formulated as positive semi-definiteness of the matrix of inner products between asymptotic states (in and out) and states created by the action of local operators on the vacuum. The corresponding matrix elements involve scattering amplitudes, form factors and spectral densities of local operators. We test this method in two-dimensional QFTs by setting up a linear optimization problem that gives a lower bound on the central charge of the UV CFT associated to a QFT with a given mass spectrum of stable particles (and couplings between them). Some of our numerical bounds are saturated by known form factors in integrable theories like the sine-Gordon, E8 and O(N) models., Comment: 57 pages, 12 figures. In v3 typos corrected, references added

We bootstrap the S-matrix of massless particles in unitary, relativistic two dimensional quantum field theories. We find that the low energy expansion of such S-matrices is strongly constrained by the existence of a UV completion. In the context of flux tube physics, this allows us to constrain several terms in the S-matrix low energy expansion or -- equivalently -- on Wilson coefficients of several irrelevant operators showing up in the flux tube effective action. These bounds have direct implications for other physical quantities; for instance, they allow us to further bound the ground state energy as well as the level splitting of degenerate energy levels of large flux tubes. We find that the S-matrices living at the boundary of the allowed space exhibit an intricate pattern of resonances with one sharper resonance whose quantum numbers, mass and width are precisely those of the world-sheet axion proposed in [1,2]. The general method proposed here should be extendable to massless S-matrices in higher dimensions and should lead to new quantitative bounds on irrelevant operators in theories of Goldstones and also in gauge and gravity theories., Comment: 7 + 10 pages, 13 figures; v2: Axionic String Ansatz appendix added. New SP bound added. Typos fixed, references added

We explore the space of consistent three-particle couplings in $\mathbb Z_2$-symmetric two-dimensional QFTs using two first-principles approaches. Our first approach relies solely on unitarity, analyticity and crossing symmetry of the two-to-two scattering amplitudes and extends the techniques of [arXiv:1607.06110] to a multi-amplitude setup. Our second approach is based on placing QFTs in AdS to get upper bounds on couplings with the numerical conformal bootstrap, and is a multi-correlator version of [arXiv:1607.06109]. The space of allowed couplings that we carve out is rich in features, some of which we can link to amplitudes in integrable theories with a $\mathbb Z_2$ symmetry, e.g., the three-state Potts and tricritical Ising field theories. Along a specific line our maximal coupling agrees with that of a new exact S-matrix that corresponds to an elliptic deformation of the supersymmetric Sine-Gordon model which preserves unitarity and solves the Yang-Baxter equation., Comment: 36 pages + 12 appendices, 29 figures

We set up a scattering experiment of matter against an impurity which separates two generic one-dimensional critical quantum systems. We compute the flux of reflected and transmitted energy, thus defining a precise measure of the transparency of the interface between the related two-dimensional conformal field theories. If the largest symmetry algebra is Virasoro, we find that the reflection and transmission coefficients are independent of the details of the initial state, and are fixed in terms of the central charges and of the two-point function of the displacement operator. The situation is more elaborate when extended symmetries are present. Positivity of the total energy flux at infinity imposes bounds on the coefficient of the two-point function of the displacement operator, which controls the free-energy cost of a small deformation of the interface. Finally, we study out-of-equilibrium steady states of a critical system connecting two reservoirs at different temperatures. In the absence of extended symmetries, our result implies that the energy flux across an impurity is proportional to the difference of the squared temperatures and controlled by the reflection coefficient., Comment: 32+10 pages, 14 figures, a discussion on non-equilibrium steady states added

We consider the S-matrix bootstrap of four dimensional scattering amplitudes with $O(3)$ symmetry and no bound-states. We explore the allowed space of scattering lengths which parametrize the interaction strength at threshold of the various scattering channels. Next we consider an application of this formalism to pion physics. A signature of pions is that they are derivatively coupled leading to (chiral) zeros in their scattering amplitudes. In this work we explore the multi-dimensional space of chiral zeros positions, scattering length values and resonance mass values. Interestingly, we encounter lakes, peninsulas and kinks depending on which sections of this intricate multi-dimensional space we consider. We discuss the remarkable location where QCD seems to lie in these plots, based on various experimental and theoretical expectations., Comment: 6 pages, 7 figures

How can a renormalization group fixed point be scale invariant without being conformal? Polchinski (1988) showed that this may happen if the theory contains a virial current -- a non-conserved vector operator of dimension exactly $(d-1)$, whose divergence expresses the trace of the stress tensor. We point out that this scenario can be probed via lattice Monte Carlo simulations, using the critical 3d Ising model as an example. Our results put a lower bound $\Delta_V>5.0$ on the scaling dimension of the lowest virial current candidate $V$, well above 2 expected for the true virial current. This implies that the critical 3d Ising model has no virial current, providing a structural explanation for the conformal invariance of the model., Comment: v2: 27pp; many explanations and some corrections added; Monte Carlo measurement reinterpreted as a lower bound; conclusions unchanged; v3: misprint in Eq. (2.5) corrected; v4: Eqs (1.4,1.5) and appendix B added, version to appear in JHEP; v5: Ref. [4] to 3d SAW conformal invariance added

We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 3+1 dimensions due to crossing symmetry, analyticity and unitarity. We extremize cubic couplings, quartic couplings and scattering lengths relevant for the elastic scattering amplitude of two identical scalar particles. In the cases where our results can be compared with the older S-matrix literature they are in excellent agreement. We also extremize a cubic coupling in 2+1 dimensions which we can directly compare to a universal bound for a QFT in AdS. This paper generalizes our previous 1+1 dimensional results of arXiv:1607.06109 and arXiv:1607.06110., Comment: 25+18 pages, 15 figures

We consider the Regge limit of the CFT correlation functions $\langle {\cal J} {\cal J} {\cal O}{\cal O}\rangle$ and $\langle TT {\cal O}{\cal O}\rangle$, where ${\cal J}$ is a vector current, $T$ is the stress tensor and ${\cal O}$ is some scalar operator. These correlation functions are related by a type of Fourier transform to the AdS phase shift of the dual 2-to-2 scattering process. AdS unitarity was conjectured some time ago to be positivity of the imaginary part of this bulk phase shift. This condition was recently proved using purely CFT arguments. For large $N$ CFTs we further expand on these ideas, by considering the phase shift in the Regge limit, which is dominated by the leading Regge pole with spin $j(\nu)$, where $\nu$ is a spectral parameter. We compute the phase shift as a function of the bulk impact parameter, and then use AdS unitarity to impose bounds on the analytically continued OPE coefficients $C_{{\cal J}{\cal J}j(\nu)}$ and $C_{TTj(\nu)}$ that describe the coupling to the leading Regge trajectory of the current ${\cal J}$ and stress tensor $T$. AdS unitarity implies that the OPE coefficients associated to non-minimal couplings of the bulk theory vanish at the intercept value $\nu=0$, for any CFT. Focusing on the case of large gap theories, this result can be used to show that the physical OPE coefficients $C_{{\cal J}{\cal J}T}$ and $C_{TTT}$, associated to non-minimal bulk couplings, scale with the gap $\Delta_g$ as $\Delta_g^{-2}$ or $\Delta_g^{-4}$. Also, looking directly at the unitarity condition imposed at the OPE coefficients $C_{{\cal J}{\cal J}T}$ and $C_{TTT}$ results precisely in the known conformal collider bounds, giving a new CFT derivation of these bounds. We finish with remarks on finite $N$ theories and show directly in the CFT that the spin function $j(\nu)$ is convex, extending this property to the continuation to complex spin., Comment: 47 pages, 3 figures, v2: JHEP version

We study correlation functions of a conserved spin-1 current $J_\mu$ in three dimensional Conformal Field Theories (CFTs). We investigate the constraints imposed by permutation symmetry and current conservation on the form of three point functions $\langle J_\mu J_\nu \mathcal O_{\Delta,\ell}\rangle $ and the four point function $\langle J_\mu J_\nu J_\rho J_\sigma \rangle $ and identify the minimal set of independent crossing symmetry conditions. We obtain a recurrence relation for conformal blocks for generic spin-1 operators in three dimensions. In the process, we improve several technical points, facilitating the use of recurrence relations. By applying the machinery of the numerical conformal bootstrap we obtain universal bounds on the dimensions of certain light operators as well as the central charge. Highlights of our results include numerical evidence for the conformal collider bound and new constraints on the parameters of the critical $O(2)$ model. The results obtained in this work apply to any unitary, parity-preserving three dimensional CFT., Comment: 67 pages, 18 figures

We numerically construct asymptotically $AdS_4$ solutions to Einstein-Maxwell-dilaton theory. These have a dipolar electrostatic potential turned on at the conformal boundary $S^2\times \mathbb{R}_t$. We find two classes of geometries: $AdS$ soliton solutions that encode the full backreaction of the electric field on the $AdS$ geometry without a horizon, and neutral black holes that are "polarised" by the dipolar potential. For a certain range of the electric field $\mathcal{E}$, we find two distinct branches of the $AdS$ soliton that exist for the same value of $\mathcal{E}$. For the black hole, we find either two or four branches depending on the value of the electric field and horizon temperature. These branches meet at critical values of the electric field and impose a maximum value of $\mathcal{E}$ that should be reflected in the dual field theory. For both the soliton and black hole geometries, we study boundary data such as the stress tensor. For the black hole, we also consider horizon observables such as the entropy. At finite temperature, we consider the Gibbs free energy for both phases and determine the phase transition between them. We find that the $AdS$ soliton dominates at low temperature for an electric field up to the maximum value. Using the gauge/gravity duality, we propose that these solutions are dual to deformed ABJM theory and compute the corresponding weak coupling phase diagram.

We present a retrieval algorithm based on generalized projections for ultrashort pulse characterization using dispersion scan (d-scan). The new algorithm is tested on several simulated cases and in two different experimental cases in the few-cycle regime. The proposed algorithm is much faster and leads to a drastic reduction of retrieval times, but performs less robust in the retrieval of noisy d-scan traces compared to the standard algorithm.

We introduce the AdS/CFT correspondence as a natural extension of QFT in a fixed AdS background. We start by reviewing some general concepts of CFT, including the embedding space formalism. We then consider QFT in a fixed AdS background and show that one can define boundary operators that enjoy very similar properties as in a CFT, except for the lack of a stress tensor. Including a dynamical metric in AdS generates a boundary stress tensor and completes the CFT axioms. We also discuss some applications of the bulk geometric intuition to strongly coupled QFT. Finally, we end with a review of the main properties of Mellin amplitudes for CFT correlation functions and their uses in the context of AdS/CFT., Comment: 63 pages, 7 figures

We propose a strategy to study massive Quantum Field Theory (QFT) using conformal bootstrap methods. The idea is to consider QFT in hyperbolic space and study correlation functions of its boundary operators. We show that these are solutions of the crossing equations in one lower dimension. By sending the curvature radius of the background hyperbolic space to infinity we expect to recover flat-space physics. We explain that this regime corresponds to large scaling dimensions of the boundary operators, and discuss how to obtain the flat-space scattering amplitudes from the corresponding limit of the boundary correlators. We implement this strategy to obtain universal bounds on the strength of cubic couplings in 2D flat-space QFTs using 1D conformal bootstrap techniques. Our numerical results match precisely the analytic bounds obtained in our companion paper using S-matrix bootstrap techniques., Comment: 25+21 pages, 14 figures; v2: minor corrections

We consider constraints on the S-matrix of any gapped, Lorentz invariant quantum field theory in 1 + 1 dimensions due to crossing symmetry and unitarity. In this way we establish rigorous bounds on the cubic couplings of a given theory with a fixed mass spectrum. In special cases we identify interesting integrable theories saturating these bounds. Our analytic bounds match precisely with numerical bounds obtained in a companion paper where we consider massive QFT in an AdS box and study boundary correlators using the technology of the conformal bootstrap., Comment: 25+6 pages, 15 figures; v2: minor corrections

This paper develops a method to compute any bosonic conformal block as a series expansion in the optimal radial coordinate introduced by Hogervorst and Rychkov. The method reduces to the known result when the external operators are all the same scalar operator, but it allows to compute conformal blocks for external operators with spin. Moreover, we explain how to write closed form recursion relations for the coefficients of the expansions. We study three examples of four point functions in detail: one vector and three scalars; two vectors and two scalars; two spin 2 tensors and two scalars. Finally, for the case of two external vectors, we also provide a more efficient way to generate the series expansion using the analytic structure of the blocks as a function of the scaling dimension of the exchanged operator., Comment: 42 pages, 17 figures, 7 Mathematica files, v2: minor changes in the text, typos corrected

In this paper we derive the projectors to all irreducible SO(d) representations (traceless mixed-symmetry tensors) that appear in the partial wave decomposition of a conformal correlator of four stress-tensors in d dimensions. These projectors are given in a closed form for arbitrary length $l_1$ of the first row of the Young diagram. The appearance of Gegenbauer polynomials leads directly to recursion relations in $l_1$ for seed conformal blocks. Further results include a differential operator that generates the projectors to traceless mixed-symmetry tensors and the general normalization constant of the shadow operator., Comment: 49 pages, 1 Mathematica notebook, many figures, v2: add references

We consider solutions in Einstein-Maxwell theory with a negative cosmological constant that asymptote to global $AdS_{4}$ with conformal boundary $S^{2}\times\mathbb{R}_{t}$. At the sphere at infinity we turn on a space-dependent electrostatic potential, which does not destroy the asymptotic $AdS$ behaviour. For simplicity we focus on the case of a dipolar electrostatic potential. We find two new geometries: (i) an $AdS$ soliton that includes the full backreaction of the electric field on the $AdS$ geometry; (ii) a polarised neutral black hole that is deformed by the electric field, accumulating opposite charges in each hemisphere. For both geometries we study boundary data such as the charge density and the stress tensor. For the black hole we also study the horizon charge density and area, and further verify a Smarr formula. Then we consider this system at finite temperature and compute the Gibbs free energy for both $AdS$ soliton and black hole phases. The corresponding phase diagram generalizes the Hawking-Page phase transition. The $AdS$ soliton dominates the low temperature phase and the black hole the high temperature phase, with a critical temperature that decreases as the external electric field increases. Finally, we consider the simple case of a free charged scalar field on $S^{2}\times\mathbb{R}_{t}$ with conformal coupling. For a field in the $SU(N)$ adjoint representation we compare the phase diagram with the above gravitational system.

In the context of conformal field theories in general space-time dimension, we find all the possible singularities of the conformal blocks as functions of the scaling dimension $\Delta$ of the exchanged operator. In particular, we argue, using representation theory of parabolic Verma modules, that in odd spacetime dimension the singularities are only simple poles. We discuss how to use this information to write recursion relations that determine the conformal blocks. We first recover the recursion relation introduced in 1307.6856 for conformal blocks of external scalar operators. We then generalize this recursion relation for the conformal blocks associated to the four point function of three scalar and one vector operator. Finally we specialize to the case in which the vector operator is a conserved current., Comment: 55 pages, 12 figures; v2 Typos corrected, conclusions changed, reference added

We perform Monte-Carlo simulations of the three-dimensional Ising model at the critical temperature and zero magnetic field. We simulate the system in a ball with free boundary conditions on the two dimensional spherical boundary. Our results for one and two point functions in this geometry are consistent with the predictions from the conjectured conformal symmetry of the critical Ising model., Comment: 9 figures; references added in v2

We construct the black hole geometry dual to the deconfined phase of the BMN matrix model at strong 't Hooft coupling. We approach this solution from the limit of large temperature where it is approximately that of the non-extremal D0-brane geometry with a spherical $S^8$ horizon. This geometry preserves the $SO(9)$ symmetry of the matrix model trivial vacuum. As the temperature decreases the horizon becomes deformed and breaks the $SO(9)$ to the $SO(6)\times SO(3)$ symmetry of the matrix model. When the black hole free energy crosses zero the system undergoes a phase transition to the confined phase described by a Lin-Maldacena geometry. We determine this critical temperature, whose computation is also within reach of Monte Carlo simulations of the matrix model., Comment: 28+7 pages, 9 figures

We introduce Mellin amplitudes for correlation functions of $k$ scalar operators and one operator with spin in conformal field theories (CFT) in general dimension. We show that Mellin amplitudes for scalar operators have simple poles with residues that factorize in terms of lower point Mellin amplitudes, similarly to what happens for scattering amplitudes in flat space. Finally, we study the flat space limit of Anti-de Sitter (AdS) space, in the context of the AdS/CFT correspondence, and generalize a formula relating CFT Mellin amplitudes to scattering amplitudes of the bulk theory, including particles with spin., Comment: 33+33 pages, 6 figures

We develop the embedding formalism to describe symmetric traceless tensors in Anti-de Sitter space. We use this formalism to construct the bulk-to-bulk propagator of massive spin J fields and check that it has the expected short distance and massless limits. We also and a split representation for the bulk-to-bulk propagator, by writing it as an integral over the boundary of the product of two bulk-to-boundary propagators. We exemplify the use of this representation with the computation of the conformal partial wave decomposition of Witten diagrams. In particular, we determine the Mellin amplitude associated to AdS graviton exchange between minimally coupled scalars of general dimension, including the regular part of the amplitude., Comment: 48 pages, 6 figures