CFTs in Euclidean signature satisfy well-accepted rules, such as the convergent Euclidean OPE. It is nowadays common to assume that CFT correlators exist and have various properties also in Lorentzian signature. Some of these properties may represent extra assumptions, and it is an open question if they hold for familiar statistical-physics CFTs such as the critical 3d Ising model. Here we consider Wightman 4-point functions of scalar primaries in Lorentzian signature. We derive a minimal set of their properties solely from the Euclidean unitary CFT axioms, without using extra assumptions. We establish all Wightman axioms (temperedness, spectral property, local commutativity, clustering), Lorentzian conformal invariance, and distributional convergence of the s-channel Lorentzian OPE. This is done constructively, by analytically continuing the 4-point functions using the s-channel OPE expansion in the radial cross-ratios $ρ, \barρ$. We prove a key fact that $|ρ|, |\barρ| < 1$ inside the forward tube, and set bounds on how fast $|ρ|, |\barρ|$ may tend to 1 when approaching the Minkowski space.
We also provide a guide to the axiomatic QFT literature for the modern CFT audience. We review the Wightman and Osterwalder-Schrader (OS) axioms for Lorentzian and Euclidean QFTs, and the celebrated OS theorem connecting them. We also review a classic result of Mack about the distributional OPE convergence. Some of the classic arguments turn out useful in our setup. Others fall short of our needs due to Lorentzian assumptions (Mack) or unverifiable Euclidean assumptions (OS theorem).
We solve Klein-Gordon equation for massless scalars on d+1 dimensional Minkowski (Euclidean) space in terms of the Cauchy data on the hypersurface t=0. By inserting the solution into the action of massless scalars in Minkowski (Euclidean) space we obtain the action of dual theory on the boundary t=0 which is exactly the holographic dual of conformally coupled scalars on d+1 dimensional (Euclidean anti) de Sitter space obtained in (A)dS/CFT correspondence. The observed equivalence of dual theories is explained using the one-to-one map between conformally coupled scalar fields on Minkowski (Eucl…
We review how modular categories, and commutative and non-commutative Frobenius algebras arise in rational conformal field theory. For Euclidean CFT we use an approach based on sewing of surfaces, and in the Minkowskian case we describe CFT by a net of operator algebras.
We formulate a prescription for computing Minkowski-space correlators from AdS/CFT correspondence. This prescription is shown to give the correct retarded propagators at zero temperature in four dimensions, as well as at finite temperature in the two-dimensional conformal field theory dual to the BTZ black hole. Using the prescription, we calculate the Chern-Simons diffusion constant of the finite-temperature N=4 supersymmetric Yang-Mills theory in the strong coupling limit. We explain why the quasinormal frequencies of the asymptotically AdS background correspond to the poles of the retarded …
We describe how conformal Minkowski, dS- and AdS-spaces can be united into a single submanifold [N] of RP^5. It is the set of generators of the null cone in M^{2,4}. Conformal transformations on the Mink-, dS- and AdS-spaces are induced by O(2,4) linear transformations on M^{2,4}. We also describe how Weyl transformations and conformal transformations can be resulted in on [N]. In such a picture we give a description of how the conformal Mink-, dS- and AdS-spaces as well as [N] are mapped from one to another by conformal maps. This implies that a CFT in one space can be translated into a CFT i…
Minkowski space can be sliced, outside the lightcone, in terms of Euclidean Anti-de Sitter and Lorentzian de Sitter slices. In this paper we investigate what happens when we apply holography to each slice separately. This yields a dual description living on two spheres, which can be interpreted as the boundary of the light cone. The infinite number of slices gives rise to a continuum family of operators on the two spheres for each separate bulk field. For a free field we explain how the Green's function and (trivial) S-matrix in Minkowski space can be reconstructed in terms of two-point functi…
We discuss scattering in a CFT via the conformal partial-wave analysis and the Regge limit. The focus of this paper is on understanding an OPE with Minkowski conformal blocks. Starting with a t-channel OPE, it leads to an expansion for an s-channel scattering amplitude in terms of t-channel exchanges. By contrasting with Euclidean conformal blocks we see a precise relationship between conformal blocks in the two limits without preforming an explicit analytic continuation. We discuss a generic feature for a CFT correlation function having singular $F^{(M)}(u,v)\sim {u}^{-δ}\,$, $δ>0$, in the li…
By appeal to Distribution Theory we discuss in rigorous fashion, without appealing to {\bf any conjecture} (as usually done by other authors), the boundary-bulk propagators for the scalar field, both in the non-massive and massive cases. These calculations, new in the literature as far as we know, are carried out in two instances: (i) when the boundary is a Euclidean space and (ii) when it is of Minkowskian nature. In this last case we compute also three propagators: Feynman's, Anti-Feynman's, and Wheeler's (half advanced plus half retarded). For an operator corresponding to scalar field we ob…
We show that any conformal field theory in d-dimensional Minkowski space, in a phase with spontaneously broken conformal symmetry and with the dilaton among its fields, can be rewritten in terms of the static gauge (d-1)-brane on AdS_(d+1) by means of an invertible change of variables. This nonlinear holographic transformation maps the Minkowski space coordinates onto the brane worldvolume ones and the dilaton onto the transverse AdS brane coordinate. One of the consequences of the existence of this map is that any (d-1)-brane worldvolume action on AdS_(d+1)\times X^m (with X^m standing for th…
Asymptotic symmetries of AdS$_4$ quantum gravity and gauge theory are derived by coupling the dual CFT$_3$ to Chern-Simons gauge theory and 3D gravity in a "probe" large-level limit. The infinite-dimensional symmetries are shown to arise when one is restricted to boundary subspaces with effectively two-dimensional geometry. A canonical example of such a restriction occurs within the 4D subregion described by a Wheeler-DeWitt wavefunctional of AdS$_4$ quantum gravity. An AdS$_4$ analog of Minkowski "super-rotation" asymptotic symmetry is probed by 3D Einstein gravity, yielding CFT$_2$ structure…
The AdS/CFT transformation relates two nonlinear realizations of (super)conformal groups: their realization in the appropriate field theories in Minkowski space with a Goldstone dilaton field and their realization as (super)isometry groups of AdS (super)spaces. It exists already at the classical level and maps the field variables and space-time coordinates of the given (super)conformal field theory in $d$-dimensional Minkowski space ${\cal M}_d$ on the variables of a scalar codimension one (super)brane in AdS$_{d+1}$ in a static gauge, the dilaton being mapped on the transverse AdS brane coord…
Starting from the averaged null energy condition (ANEC) in Minkowski we show that conformal symmetry implies the ANEC for a conformal field theory (CFT) in a de Sitter and anti-de Sitter background. A similar and novel bound is also obtained for a CFT in the Lorentzian cylinder. Using monotonicity of relative entropy, we rederive these results for dS and the cylinder. As a byproduct we obtain the vacuum modular Hamiltonian and entanglement entropy associated to null deformed regions of CFTs in (A)dS and the cylinder. A third derivation of the ANEC in dS is shown to follow from bulk causality i…
We present a discussion on recent progress in high energy diffraction from the perspective of AdS/CFT, through which a unified treatment for both perturbative and nonperturbative Pomeron emerges. By working with Unitary Irreducible Representation of Conformal group, a frame is provided in extending AdS/CFT to both forward and nearforward scattering. We present an analysis involving an exact solution to conformal blocks in Minkowski CFT and discuss possible applications. Phenomenological applications can range from forward scattering to DIS/DVCS/TMD at LHC energies and beyond.
We give an AdS/CFT interpretation to homogeneous Yang-Baxter deformations of the AdS_5 x S^5 superstring as noncommutative deformations of the dual gauge theory, going well beyond the canonical noncommutative case. These homogeneous Yang-Baxter deformations can be of so-called abelian or jordanian type. While abelian deformations have a clear interpretation in string theory and many already had well understood gauge theory duals, jordanian deformations appear novel on both counts. We discuss the symmetry structure of the deformed string from the uniformizing perspective of Drinfeld twists and …
Recently the anomalous dimension of twist two operators in N=4 SYM theory was computed by Gubser, Klebanov and Polyakov in the limit of large 't Hooft coupling using semi-classical rotating strings in AdS_5. Here we reproduce their results for large angular momentum by using the cusp anomaly of Wilson loops in Minkowski signature also computed within the AdS/CFT correspondence. In this case the anomalous dimension is related to an Euclidean worldsheet whose properties are completely determined by the symmetries of the problem. This gives support to the proposed identification of rotating strin…
Two kinds of realizations of symmetry on classical domains or the Euclidean version of AdS space are used to study AdS/CFT correspondence. Mass of free particles is defined as an AdS group invariant, the Klein-Gordon and Dirac equations for relativistic particles in the AdS space are set up as a mimic in the case of Minkowskian space. The bulk-boundary propagator on the AdS space is given by the Poisson kernel. Theorems on the Poisson kernel guarantee the existence and sole of the bulk-boundary propagator. The propagator is used to calculate correlators of the theories that live on the boundar…
We show that D=4 Minkowski space is an emergent concept related to a class of operators in extended Hilbert space with no positive-definite scalar product. We start with the idea of position-like and momentum-like operators (Plewa 2019 J. Phys. A: Math. Theor. 52 375401), introduced discussing a connection between quantum entanglement and geometry predicted by ER=EPR conjecture. We examine eigenequations of the simplest operators and identify D=4 Minkowski space as to be spanned by normalized eigenvectors corresponding to the zero eigenvalue. Both spacetime dimension and signature of the metri…
4D CFTs have a scale anomaly characterized by the coefficient $c$, which appears as the coefficient of logarithmic terms in momentum space correlation functions of the energy-momentum tensor. By studying the CFT contribution to 4-point graviton scattering amplitudes in Minkowski space we derive a sum rule for $c$ in terms of $TT\mathcal{O}$ OPE coefficients. The sum rule can be thought of as a version of the optical theorem, and its validity depends on the existence of the massless and forward limits of the $\langle TTTT \rangle$ correlation functions that contribute. The finiteness of these l…
We extend the method of calculation of propagators in maximally symmetric spaces (Minkowski, dS, AdS and their Euclidean versions) in terms of intrinsic geometric objects to the case of massive spin 3/2 field. We obtain the propagator for arbitrary space-time dimension and mass in terms of Heun's function, which is a generalization of the hypergeometric function appearing in the case of other spins. As an application of this result we calculate the conformal dimension of the dual operator in the recently proposed dS/CFT correspondence both for spin 3/2 and for spin 1/2. We find that, in agreem…
We consider supergravity in five-dimensional Anti-De Sitter space $AdS_{5}$ with minimal supersymmetry, encoded by a Sasaki-Einstein metric on a five-dimensional compact manifold $M$. Our main result reveals how the Sasaki-Einstein metric emerges from a canonical state in the dual CFT, defined by a superconformal gauge theory in four dimensional Minkowski space $\mathbb{R}^{3,1}$in the t'Hooft limit where the rank $N$ tends to infinity. We obtain explicit finite $N-$approximations to the Sasaki-Einstein metric, expressed in terms of a canonical (i.e. background free) BPS-state on the gauge the…
We consider a 3-brane of positive cosmological constant (de Sitter) in D-dimensional Minkowski space. We show that the Poincare algebra in the bulk yields a SO(4,2) algebra when restricted to the brane. In the limit of zero cosmological constant (flat brane), this algebra turns into the conformal algebra on the brane. We derive a correspondence principle for Minkowski space analogous to the AdS/CFT correspondence. We discuss explicitly the cases of scalar and gravitational fields. For a 3-brane of finite thickness in the transverse directions, we obtain a spectrum for vector gravitational pert…
Extending the seminal work of Bilal and Gervais, we construct a tachyon-free, modular invariant partition function for critical superstrings on four-dimensional Minkowski x two-dimensional black hole. This model may be thought of as an SL(2,R)/U(1) version of Gepner models and corresponds to a conifold point on the moduli space of Calabi-Yau compactifications. We directly deal with N=2, c=9 unitary superconformal characters. Modular invariance is achieved by requiring the string to have a momentum along an extra noncompact direction, in agreement with the picture of singular CFTs advocated by …
The most common quantization scheme in which to study a conformal field theory is radial quantization, wherein a Hilbert space of states is defined on a sphere, whose Hamiltonian when mapped to the plane is the dilatation generator and which boasts a state/operator correspondence. In this paper, we consider an alternative quantization scheme for 2d CFTs in which the plane is foliated by constant-angle slices, as opposed to concentric circles, whose Hamiltonian is the rotation generator. In this angular quantization, there is no state/operator correspondence but instead an "asymptotics/operator…
We introduce a D7-brane probe in AdS_5 x S^5 background in a way that the 4d part of the induced metric on D7-brane becomes 4d de-Sitter space (dS_4) inside AdS_5 instead of 4d Minkowski space. Although supersymmetry is completely broken, we obtain a static configuration and show the absence of dynamical tachyonic modes. Following holographic renormalization we renormalize the Dirac-Born-Infeld action of D7-brane and we completely fix the counter terms including finite contributions from the consistency under various limits. Through the AdS/CFT correspondence we study the chiral condensate and…
We show that the four-point functions in conformal field theory are defined as distributions on the boundary of the region of convergence of the conformal block expansion. The conformal block expansion converges in the sense of distributions on this boundary, i.e. it can be integrated term by term against appropriate test functions. This can be interpreted as a giving a new class of functionals that satisfy the swapping property when applied to the crossing equation, and we comment on the relation of our construction to other types of functionals. Our language is useful in all considerations i…
We give evidence that spacelike BPS Wilson loops do not exist in Minkowski spacetime. We show that spacelike Wilson loops in Minkowski spacetime cannot preserve any supersymmetries, in $d = 4$ $\mathcal N = 4$ super Yang-Mills theory, $d = 3$ $\mathcal N = 2$ super Chern-Simons-matter theory, and $d = 3$ $\mathcal N = 6$ Aharony-Bergman-Jafferis-Maldacena theory. We not only show this using infinite straight lines and circles as examples, but also we give proofs for general curves. We attribute this to the conflicts of reality conditions of the spinors. However, spacelike Wilson loops do exist…
The AdS/CFT correspondence relates Wilson loops in N=4 SYM theory to minimal area surfaces in AdS5xS5 space. If the Wilson loop is Euclidean and confined to a plane (t,x) then the dual surface is Euclidean and lives in Minkowski AdS3. In this paper we study such minimal area surfaces generalizing previous results obtained in the Euclidean case. Since the surfaces we consider have the topology of a disk, the holonomy of the flat current vanishes which is equivalent to the condition that a certain boundary Schroedinger equation has all its solutions anti-periodic. If the potential for that Schro…
We define and study asymptotic Killing and conformal Killing vectors in $d$-dimensional Minkowski, (A)dS, $\mathbb{R}\times S^{d-1}$ and ${\rm AdS}_2\times S^{d-2}$. We construct the associated quantum charges for an arbitrary CFT and show they satisfy a closed algebra that includes the BMS as a sub-algebra (i.e. supertranslations and superrotations) plus a novel transformation we call `superdilations'. We study representations of this algebra in the Hilbert space of the CFT, as well as the action of the finite transformations obtained by exponentiating the charges. In the context of the AdS/C…
We use the membrane paradigm to analyze the horizon dynamics of a uniformly boosted black brane in a (d+2)-dimensional asymptotically Anti-de-Sitter space-time and a Rindler acceleration horizon in (d+2)-dimensional Minkowski space-time. We show that in these cases the horizon dynamics is governed by the relativistic CFT hydrodynamics equations. The fluid velocity and temperature correspond to the normal to the horizon and to the surface gravity, respectively. The second law of thermodynamics for the fluid is mapped into the area increase theorem of General Relativity. The analysis is applicab…
Globally conformal invariant quantum field theories in a D-dimensional space-time (D even) have rational correlation functions and admit an infinite number of conserved (symmetric traceless) tensor currents. In a theory of a scalar field of dimension D-2 they were demonstrated to be generated by bilocal normal products of free massless scalar fields with an O(N), U(N), or Sp(2N) (global) gauge symmetry [BNRT]. Recently, conformal field theories "with higher spin symmetry" were considered for D=3 in [MZ] where a similar result was obtained (exploiting earlier study of CFT correlators). We sugge…
Recently established rationality of correlation functions in a globally conformal invariant quantum field theory satisfying Wightman axioms is used to construct a family of soluble models in 4-dimensional Minkowski space-time. We consider in detail a model of a neutral scalar field $ϕ$ of dimension 2. It depends on a positive real parameter c, an analogue of the Virasoro central charge, and admits for all (finite) c an infinite number of conserved symmetric tensor currents. The operator product algebra of $ϕ$ is shown to coincide with a simpler one, generated by a bilocal scalar field $V(x_1,x…
We discuss thick domain walls interpolating between spaces with naked singularities and give arguments based on the $AdS$/CFT correspondence why such singularities may be physically meaningful. Our examples include thick domain walls with Minkowski, de Sitter, and anti-de Sitter geometries on the four-dimensional slice. For flat domain walls we can solve the equivalent quantum mechanics problem exactly, which provides the spectrum of graviton states. In one of the examples we discuss, the continuum states have a mass gap. We compare the graviton spectra with expectations from the $AdS$/CFT cor…
We use the flat space limit of the AdS/CFT correspondence to derive a simple relation between the 2 to 2 scattering amplitude of massless string states in type IIB superstring theory on ten-dimensional Minkowski space and a scaling limit of the N=4 super Yang-Mills four point functions. We conjecture that this relation holds non-perturbatively and at arbitrarily high energy.
Using the Minkowski space AdS/CFT prescription we explicitly compute in the low-energy limit the two-point correlation function of the boundary stress-energy tensor in a large class of type IIB supergravity backgrounds with a regular translationally invariant horizon. The relevant set of supergravity backgrounds includes all geometries which can be interpreted via gauge theory/string theory correspondence as being holographically dual to finite temperature gauge theories in Minkowski space-times. The fluctuation-dissipation theorem relates this correlation function computation to the previousl…
We study non-critical superstrings propagating in $d \le 6$ dimensional Minkowski space or equivalently, superstrings propagating on the two-dimensional Euclidean black hole tensored with d-dimensional Minkowski space. We point out a subtlety in the construction of supersymmetric theories in these backgrounds, and explain how this does not allow a consistent geometric interpretation in terms of fields propagating on a cigar-like spacetime. We explain the global symmetries of the various theories by using their description as the near horizon geometry of wrapped NS5-brane configurations. In the…
We develop the idea of an effective conformal theory describing the low-lying spectrum of the dilatation operator in a CFT. Such an effective theory is useful when the spectrum contains a hierarchy in the dimension of operators, and a small parameter whose role is similar to that of 1/N in a large N gauge theory. These criteria insure that there is a regime where the dilatation operator is modified perturbatively. Global AdS is the natural framework for perturbations of the dilatation operator respecting conformal invariance, much as Minkowski space naturally describes Lorentz invariant pertur…
We study quantum field theory on the two-dimensional Noncommutative Minkoswki space with a Grosse-Wulkenhaar potential. We explicitly construct the retarded propagator and show that it is not a tempered distribution. This leads to problems when trying to define planar products of such distributions, as they appear in the Yang-Feldman series. At and above the self-dual point, these can no longer be defined, not even at different points. This shows that we do not deal with an ordinary ultraviolet divergence.
Minkowski Functionals are a powerful tool for analyzing large scale structure, in particular if the distribution of matter is highly non-Gaussian, as it is in models in which cosmic strings contribute to structure formation. Here we apply Minkowski functionals to 21cm maps which arise if structure is seeded by a scaling distribution of cosmic strings embeddded in background fluctuations, and then test for the statistical significance of the cosmic string signals using the Fisher combined probability test. We find that this method allows for detection of cosmic strings with $G μ> 5 \times 10^{-…
Quantum Transitions Between Minkowski and de Sitter Spacetimes
University of Colorado, Boulder（科罗拉多大学博尔德分校）；ICTP（国际理论物理中心）；Dipartimento di Fisica, Universita’ di Trieste（的里雅斯特大学物理系）；DAMTP, Centre for Mathematical Sciences（剑桥大学数学科学中心应用数学与理论物理系）
We discuss how a spin system, which is subject to quenched disorder, might exhibit multicritical behaviors at criticality if the distribution of the impurities is arbitrary. In order to provide realistic candidates for such multicritical behaviors, we discuss several generalizations of the standard randomly diluted Ising's universality class adopting the $ε$-expansion close to several upper critical dimensions. In the presentation, we spend a special effort in bridging between CFT and RG results and discuss in detail the computation of quantities, which are of prominent interest in the case of…
