We investigate analyticity properties of correlation functions in conformal field theories (CFT) in the Wightman formulation. The goal is to determine domain of holomorphy of permuted Wightman functions. We focus on crossing property of three-point functions. The domain of holomorphy of a pair of three-point functions is determined by appealing to Jost's theorem and by adopting the technique of analytic completion. This program paves the way to address the issue of crossing for the four-point functions on a rigorous footing.
In the framework of quantum field theory (QFT) on noncommutative (NC) space-time with the symmetry group $O(1,1)\times SO(2)$, we prove that the Jost-Lehmann-Dyson representation, based on the causality condition taken in connection with this symmetry, leads to the mere impossibility of drawing any conclusion on the analyticity of the $2\to 2$-scattering amplitude in $\cosΘ$, $Θ$ being the scattering angle. Discussions on the possible ways of obtaining high-energy bounds analogous to the Froissart-Martin bound on the total cross-section are also presented.
Analyticity and crossing properties of four point function are investigated in conformal field theories in the frameworks of Wightman axioms. A Hermitian scalar conformal field, satisfying the Wightman axioms, is considered. The crucial role of microcausality in deriving analyticity domains is discussed and domains of analyticity are presented. A pair of permuted Wightman functions are envisaged. The crossing property is derived by appealing to the technique of analytic completion for the pair of permuted Wightman functions. The operator product expansion of a pair of scalar fields is studied …
It is shown that self-dual theories generalize to four dimensions both the conformal and analytic aspects of two-dimensional conformal field theories. In the harmonic space language there appear several ways to extend complex analyticity (natural in two dimensions) to quaternionic analyticity (natural in four dimensions). To be analytic, conformal transformations should be realized on $CP^3$, which appears as the coset of the complexified conformal group modulo its maximal parabolic subgroup. In this language one visualizes the twistor correspondence of Penrose and Ward and consistently formul…
Quaternionic formulation of D=4 conformal group and of its associated twistors and their relation to harmonic analyticity is presented. Generalization of $SL(2,\cal{C})$ to the D=4 conformal group SO(5,1) and its covering group $SL(2,\cal{Q})$ that generalizes the euclidean Lorentz group in $R^4$ [namely $SO(3,1)\approx SL(2,\cal{C})$ which allow us to obtain the projective twistor space $CP^3$] is shown. Quasi-conformal fields are introduced in D=4 and Fueter mappings are shown to map self-dual sector onto itself (and similarly for the anti-self-dual part). Differentiation of Fueter series an…
We argue that certain apparently consistent low-energy effective field theories described by local, Lorentz-invariant Lagrangians, secretly exhibit macroscopic non-locality and cannot be embedded in any UV theory whose S-matrix satisfies canonical analyticity constraints. The obstruction involves the signs of a set of leading irrelevant operators, which must be strictly positive to ensure UV analyticity. An IR manifestation of this restriction is that the "wrong" signs lead to superluminal fluctuations around non-trivial backgrounds, making it impossible to define local, causal evolution, and …
Conformal theory correlators are characterized by the spectrum and three- point functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an "imaginary part" which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields 1/J expansions with controlled errors. In large-N theories, the imaginary part is saturated by…
Causality places nontrivial constraints on QFT in Lorentzian signature, for example fixing the signs of certain terms in the low energy Lagrangian. In d-dimensional conformal field theory, we show how such constraints are encoded in crossing symmetry of Euclidean correlators, and derive analogous constraints directly from the conformal bootstrap (analytically). The bootstrap setup is a Lorentzian four-point function corresponding to propagation through a shockwave. Crossing symmetry fixes the signs of certain log terms that appear in the conformal block expansion, which constrains the interact…
We derive a simple relation between the Mellin amplitude for AdS/CFT correlation functions and the bulk S-Matrix in the flat spacetime limit, proving a conjecture of Penedones. As a consequence of the Operator Product Expansion, the Mellin amplitude for any unitary CFT must be a meromorphic function with simple poles on the real axis. This provides a powerful and suggestive handle on the locality vis-a-vis analyticity properties of the S-Matrix. We begin to explore analyticity by showing how the familiar poles and branch cuts of scattering amplitudes arise from the holographic description. For…
We explore the relation between positivity of the energy constraints in conformal field theories and causality in their dual gravity description. Our discussion involves CFTs with different central charges whose description, in the gravity side, requires the inclusion of quadratic curvature corrections. It is enough, indeed, to consider the Gauss-Bonnet term. We find that both sides of the AdS/CFT correspondence impose a restriction on the Gauss-Bonnet coupling. In the case of 6d supersymmetric CFTs, we show the full matching of these restrictions. We perform this computation in two ways. Firs…
Within the low-energy effective field theories of QED and gravity, the low-energy speed of light or that of gravitational waves can typically be mildly superluminal in curved spacetimes. Related to this, small scattering time advances relative to the curved background can emerge from known effective field theory coefficients for photons or gravitons. We clarify why these results are not in contradiction with causality, analyticity or Lorentz invariance, and highlight various subtleties that arise when dealing with superluminalities and time advances in the gravitational context. Consistent low…
In this paper we use the analyticity properties of the scattering amplitude in the context of the conformal mapping techniques. The Schwarz-Christoffel and Riemann-Schwarz functions are used to map the upper half -plane onto a triangle. We use the known asymptotic and threshold behaviors of the scattering amplitude to establish a connection between the values of the Regge trajectory functions and the angles of the triangle. This geometrical interpretation allows a link between values of the Regge trajectory functions and the generators of the invariance group of Moebius transformations associa…
In the framework of quantum field theory (QFT) on noncommutative (NC) space-time with $SO(1,1)\times SO(2)$ symmetry, which is the feature arising when one has only space-space noncommutativity ($θ_{0i}=0$), we prove that the Jost-Lehmann-Dyson representation, based on the causality condition usually taken in connection with this symmetry, leads to the mere impossibility of drawing any conclusion on the analyticity of the $2\to 2$-scattering amplitude in $\cosΘ$, $Θ$ being the scattering angle. A physical choice of the causality condition rescues the situation and as a result an analog of Lehm…
We consider the holographic prescription problem in a (Lorentzian) AdS background, deriving from first principles the explicit formulas that relate the field at infinity with the field in the bulk. In contrast with the previous studies of the "real-time" holography problem, our derivation uses purely classical arguments that involve causality, as in the usual treatment of the holographic prescription problem in Wick-rotated spaces of Euclidean signature. We show that there is a unique propagator that preserves causality and see that this provides a simple picture of the relationship between th…
The expression of causality depends on an underlying choice of chronology. Since a chronology is provided by any Lorentzian metric in relativistic theories, there are as many expressions of causality as there are non-conformally related metrics over spacetime. Although tempting, a definitive choice of a preferred metric to which one may refer to is not satisfying. It would indeed be in great conflict with the spirit of general covariance. Moreover, a theory which appear to be non causal with respect to (hereafter, w.r.t) this metric, may well be causal w.r.t another metric. In a theory involvi…
All conformal correlation functions of 3 scalar primary operators are constructed in an axiomatic way, relying only on conformal symmetry and causality. The construction makes use of the R-product and its analyticity properties in Minkowski momentum space. The R-product is completely determined by a system of partial differential equations (the Ward identities for conformal transformations) and a boundary condition (permutation symmetry), up to an OPE coefficient. Wightman functions and time-ordered products are then derived from the R-product using operator identities, and the Schwinger funct…
In this paper we fill some gaps in the arguments of our previous papers [hep-th/9412229,hep-th/9604044]. In particular, we give a proof that the L operators of Conformal Field Theory indeed satisfy the defining relations of the Yang-Baxter algebra. Among other results we present a derivation of the functional relations satisfied by ${\bf T}$ and ${\bf Q}$ operators and a proof of the basic analyticity assumptions for these operators used in [hep-th/9412229,hep-th/9604044].
By carefully analyzing the radial part of the wave-equation for a scalar field in AdS, we show that for a particular range of boundary conditions on the scalar field, the radial spectrum contains a bound state. Using the AdS/CFT correspondence, we interpret this peculiar phenomenon as being dual to an unstable double trace deformation of the boundary conformal field theory. We thus show how the bulk theory holographically detects whether a boundary perturbation is stable.
We show that 4--dimensional conformal field theory is most naturally formulated on Kulkarni 4--folds, i. e. real 4--folds endowed with an integrable quaternionic structure. This leads to a formalism that parallels very closely that of 2--dimensional conformal field theory on Riemann surfaces. In this framework, the notion of Fueter analyticity, the quaternionic analogue of complex analyticity, plays an essential role. Conformal fields appear as sections of appropriate either harmonic real or Fueter holomorphic quaternionic line bundles. In the free case, the field equations are statements of e…
In this paper, I study the conditions imposed on a normal charged fluid so that the causality and stability criteria hold for this fluid. I adopt the newly developed General Frame (GF) notion in the relativistic hydrodynamics framework which states that hydrodynamic frames have to be fixed after applying the stability and causality conditions. To my purpose, I take a charged conformal matter in the flat and $3+1$ dimension to analyze better these conditions. The causality condition is applied by looking to the asymptotic velocity of sound hydro modes at the large wave number limit and stabilit…
We study a simple version of the AdS/CFT (anti-de Sitter spacetime/Conformal Field Theory) correspondence, where operators have integer conformal dimensions. In this model, bulk causality follows from boundary analyticity, even in nontrivial black hole backgrounds that break the underlying conformal symmetry. This allows a natural set of quasi-local bulk observables to be constructed. Estimates of finite central charge corrections to semiclassical correlators are made. These corrections are used to determine the regime of validity of effective field theory in the bulk spacetime. The results ar…
We analyze properties of the Sp(2M) conformally invariant field equations in the recently proposed generalized $\half M(M+1)$-dimensional space-time $\M_M$ with matrix coordinates. It is shown that classical solutions of these field equations define a causal structure in $\M_M$ and admit a well-defined decomposition into positive and negative frequency solutions that allows consistent quantization in a positive definite Hilbert space. The effect of constraints on the localizability of fields in the generalized space-time is analyzed. Usual d-dimensional Minkowski space-time is identified with …
The mapping between bulk supergravity fields in anti-de Sitter space and operators in the dual boundary conformal field theory usually relies heavily on the available global symmetries. In the present work, we study a generalization of this mapping to time dependent situations, for the simple case of collapsing shock waves in two spacetime dimensions. The construction makes use of analyticity of the conformal field theory and the properties of the asymptotic bulk geometry to reconstruct the non-analytic bulk observables. Many of the features of this construction are expected to apply to higher…
We show that the causal properties of asymptotically flat spacetimes depend on their dimensionality: while the time-like future of any point in the past conformal infinity $\mathcal{I}^-$ contains the whole of the future conformal infinity $\mathcal{I}^+$ in $(2+1)$ and $(3+1)$ dimensional Schwarzschild spacetimes, this property (which we call the Penrose property) does not hold for $(d+1)$ dimensional Schwarzschild if $d>3$. We also show that the Penrose property holds for the Kerr solution in $(3+1)$ dimensions, and discuss the connection with scattering theory in the presence of positive ma…
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…
We study the two point correlation function of a local operator on an $n$-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. In analogy with the inverse Laplace transform, we define the Renyi transform of this correlation function, which is a function of one complex variable $w$, dual to the Renyi parameter $n$. Inspired by the inversion formula of Caron-Huot, we argue that if the Renyi transform $f(w)$ has bounded behavior at infinity in the complex $w$ plane, the discontinuity of the Renyi transform disc $f(w)$ provides the unique analyt…
Dynamical symmetries are of considerable importance in elucidating the complex behaviour of strongly interacting systems with many degrees of freedom. Paradigmatic examples are cooperative phenomena as they arise in phase transitions, where conformal invariance has led to enormous progress in equilibrium phase transitions, especially in two dimensions. Non-equilibrium phase transitions can arise in much larger portions of the parameter space than equilibrium phase transitions. The state of the art of recent attempts to generalise conformal invariance to a new generic symmetry, taking into acco…
Four-point correlation functions of hypermultiplet bilinear composites are analysed in N=2 superconformal field theory using the superconformal Ward identities and the analyticity properties of the composite operator superfields. It is shown that the complete amplitude is determined by a single arbitrary function of the two conformal cross-ratios of the space-time variables.
The analyticity of response functions and scattering amplitudes implies powerful relations between low-energy observables and the underlying short-distance dynamics. These 'IR/UV' relations are rooted in basic physical principles, such as causality and unitarity. In this paper, we seek similar connections in inflation, relating cosmological observations to the physics responsible for the accelerated expansion. We assume that the inflationary theory is Lorentz invariant at short distances, but allow for non-relativistic interactions and a non-trivial speed of propagation at low energies. Focusi…
A new approach is described to help improve the foundations of relativistic viscous fluid dynamics and its coupling to general relativity. Focusing on neutral conformal fluids constructed solely in terms of hydrodynamic variables, we derive the most general viscous energy-momentum tensor yielding equations of motion of second order in the derivatives, which is shown to provide a novel type of generalization of the relativistic Navier-Stokes equations for which causality holds. We show how this energy-momentum tensor may be derived from conformal kinetic theory. We rigorously prove existence, u…
Unitary, Lorentz-invariant quantum field theories in flat spacetime obey microcausality: commutators vanish at spacelike separation. For interacting theories in more than two dimensions, we show that this implies that the averaged null energy, $\int du T_{uu}$, must be positive. This non-local operator appears in the operator product expansion of local operators in the lightcone limit, and therefore contributes to $n$-point functions. We derive a sum rule that isolates this contribution and is manifestly positive. The argument also applies to certain higher spin operators other than the stress…
In a weakly coupled gravity theory in the anti-de Sitter space, local states in the bulk are linear superpositions of Ishibashi states for a crosscap in the dual conformal field theory. The superposition structure can be constrained either by the microscopic causality in the bulk gravity or the bootstrap condition in the boundary conformal field theory. We show, contrary to some expectation, that these two conditions are not compatible to each other in the weak gravity regime. We also present an evidence to show that bulk local states in three dimensions are not organized by the Virasoro symme…
We derive spectral sum rules in the shear channel for conformal field theories at finite temperature in general $d\geq 3$ dimensions. The sum rules result from the OPE of the stress tensor at high frequency as well as the hydrodynamic behaviour of the theory at low frequencies. The sum rule states that a weighted integral of the spectral density over frequencies is proportional to the energy density of the theory. We show that the proportionality constant can be written in terms the Hofman-Maldacena variables $t_2, t_4$ which determine the three point function of the stress tensor. For theorie…
We study the analytic properties of tree-level wavefunction coefficients in quasi-de Sitter space. We focus on theories which spontaneously break dS boost symmetries and can produce significant non-Gaussianities. The corresponding inflationary correlators are (approximately) scale invariant, but are not invariant under the full conformal group. We derive cutting rules and dispersion formulas for the late-time wavefunction coefficients by using factorization and analyticity properties of the dS bulk-to-bulk propagator. This gives a unitarity method which is valid at tree-level for general $n$-p…
We investigate the structure of conformal Regge trajectories for the maximally supersymmetric (2,0) theories in six dimensions. The different conformal multiplets in a single superconformal multiplet must all have similarly-shaped Regge trajectories. We show that these super-descendant trajectories interact in interesting ways, leading to new constraints on their shape. For the four-point function of the stress tensor multiplet supersymmetry also softens the Regge behavior in some channels, and consequently we observe that 'analyticity in spin' holds for all spins greater than -3. All the phys…
We study stress tensor correlation functions in four-dimensional conformal field theories with large $N$ and a sparse spectrum. Theories in this class are expected to have local holographic duals, so effective field theory in anti-de Sitter suggests that the stress tensor sector should exhibit universal, gravity-like behavior. At the linearized level, the hallmark of locality in the emergent geometry is that stress tensor three-point functions $\langle TTT\rangle$, normally specified by three constants, should approach a universal structure controlled by a single parameter as the gap to higher…
Exact expectation values of the fields e^{aϕ} in the Bullough-Dodd model are derived by adopting the ``reflection relations'' which involve the reflection S-matrix of the Liouville theory, as well as special analyticity assumption. Using this result we propose explicit expressions for expectation values of all primary operators in the c<1 minimal CFT perturbed by the operator Φ_{1,2} or Phi_{2,1}. Some results concerning the $Φ_{1,5}$ perturbed minimal models are also presented.
In this work, we present a holographic renormalization scheme for asymptotically anti-de Sitter spacetimes in which the dual renormalization scheme of the boundary field theory is dimensional regularization. This constitutes a new level of precision in the holographic dictionary and paves the way for the exact matching of scheme dependent quantities, such as holographic beta functions, with field theory computations. Furthermore, the renormalization procedure identifies a local source field which satisfies the equations of motion along renormalization group flows, resolving a long-standing puz…
We consider higher derivative corrections to the graviton three-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond the one present in the Einstein theory. We argue that these are constrained by causality. We devise a thought experiment involving a high energy scattering process which leads to causality violation if the graviton three-point vertex contains the additional structures. This violation cannot be fixed by adding conventional particles with spins $J \leq 2$. But, it can be fixed by adding an infinite tower of extra massive par…
We derive bounds on couplings in the standard model effective field theory (SMEFT) as a consequence of causality and the analytic structure of scattering amplitudes. In the SMEFT, there are 64 independent operators at mass dimension eight that are quartic in bosons (either Higgs or gauge fields) and that contain four derivatives and/or field strengths, including both CP-conserving and CP-violating operators. Using analytic dispersion relation arguments for two-to-two bosonic scattering amplitudes, we derive 27 independent bounds on the sign or magnitude of the couplings. We show that these bou…
The mutual and tripartite information between pairs and triples of disjoint regions in a quantum field theory are sensitive probes of the spread of correlations in an equilibrating system. We compute these quantities in strongly coupled two-dimensional conformal field theories with a gravity dual following the homogenous deposition of energy. The injected energy is modeled in anti-de Sitter space as an infalling shell, and the information shared by disjoint intervals is computed in terms of geodesic lengths in this background. For given widths and separation of the intervals, the mutual inform…
In both QCD and supersymmetric QCD (SQCD) with N_f flavors there are conformal windows where the theory is asymptotically free in the ultraviolet while the infrared physics is governed by a non-trivial fixed-point. In SQCD, the lower N_f boundary of the conformal window, below which the theory is confining is well understood thanks to duality. In QCD there is just a sufficient condition for confinement based on superconvergence. Studying the Banks-Zaks expansion and analyzing the conditions for the perturbative coupling to have a causal analyticity structure, it is shown that the infrared fixe…
We introduce a classical field theory based on a concept of extended causality that mimics the causality of a point-particle Classical Mechanics by imposing constraints that are equivalent to a particle initial position and velocity. It results on a description of discrete (pointwise) interactions in terms of localized particle-like fields. We find the propagators of these particle-like fields and discuss their physical meaning, properties and consequences. They are conformally invariant, singularity-free, and describing a manifestly covariant $(1+1)$-dimensional dynamics in a $(3+1)$ spacetim…
In conformally invariant quantum field theories one encounters besides the standard DHR superselection theory based on spacelike (Einstein-causal) commutation relations and their Haag duality another timelike (''Huygens'') based superselection structure. Whereas the DHR theory based on spacelike causality of observables confirmed the Lagrangian internal symmetry picture on the level of the physical principles of local quantum physics, the attempts to understand the timelike based superselection charges (associated with the center of the conformal covering group) in terms of timelike localized …
A physical characterization of Landau singularities is emphasized, which should trace the lower boundary N_f^* of the conformal window in QCD and supersymmetric QCD. A natural way to disentangle ``perturbative'' from ``non-perturbative'' contributions to amplitudes below N_f^* is suggested. Assuming an infrared fixed point persists in the perturbative part of the QCD coupling even below N_f^* leads to the condition γ(N_f^*)=1, where γis the critical exponent. Using the Banks-Zaks expansion, one gets 4<N_f^*<6. This result is incompatible with the existence of an analogue of Seiberg duality in …
