Euclidean Conformal Field Theory Research Articles

From celestial correlators to AdS, and back

We present a general relation between celestial correlation functions in d-dimensions and Witten diagrams in (d + 1)-dimensional Euclidean anti-de Sitter (EAdS) space, to all orders in perturbation theory. Contact diagram processes are proportional to contact Witten diagrams and particle exchanges can be recast as a continuum of particle exchanges in EAdS where the exchanged particles carrying unitary Principal Series representations of SO(d + 1, 1). One can then try to import familiar EAdS techniques to study the properties of celestial correlators. In this work we use this relation to infer the analytic structure of the spectral density in the conformal partial wave expansion of celestial correlators which, at least perturbatively, should be a meromorphic function of the spectral parameter. We also discuss non-perturbative constraints from unitarity in Euclidean Conformal Field Theory, which requires positivity of the spectral density. This extends similar relations recently uncovered between boundary correlation functions in de Sitter space and Witten diagrams in EAdS, suggesting that EAdS could play a central role in efforts towards holography for all lambdas.

Accelerating Cosmology from a Holographic Wormhole.

We consider cosmological models in which the cosmology is related via analytic continuation to a Euclidean asymptotically AdS planar wormhole geometry defined holographically via a pair of three-dimensional Euclidean conformal field theories (CFTs). We argue that these models can generically give rise to an accelerating phase for the cosmology due to the potential energy of scalar fields associated with relevant scalar operators in the CFT. We explain how cosmological observables are related to observables in the wormhole spacetime and argue that this leads to a novel perspective on naturalness puzzles in cosmology.

Conformal correlation functions in four dimensions from quaternionic Lauricella system

Correlation functions in four-dimensional Euclidean conformal field theories are expressed in terms of representations of the conformal group SL(2,H), H being the field of quaternions, on the configuration space of points. The representations are obtained in terms of a Lauricella system derived using quaternions. It generalizes the two-dimensional case, wherein the N-point correlation function is expressed in terms of solutions of Lauricella system on the configuration space of N points on the complex plane, furnishing representation of the conformal group SL(2,C).

Loops in dS/CFT

We consider the semi-classical expansion of the Bunch-Davies wavefunction with future boundary condition in position space for a real scalar field, conformally coupled to a classical de Sitter background in the expanding Poincaré patch with quartic selfinteraction. In the future boundary limit the wave function takes the form of the generating functional of a Euclidean conformal field theory for which we calculate the anomalous dimensions of the double trace deformations at one loop order using results obtained from Euclidean Anti de Sitter space. We find analytic expressions for some subleading twist operators and an algorithm to obtain expressions for general twist.

Analysis for Lorentzian conformal field theories through sine-square deformation

Abstract We reexamine two-dimensional Lorentzian conformal field theory using the formalism previously developed in a study of sine-square deformation of Euclidean conformal field theory. We construct three types of Virasoro algebra. One of them reproduces the result by Lüscher and Mack, while another type exhibits divergence in the central charge term. The third leads to a continuous spectrum and contains no closed time-like curve in the system.

Generalized Hawking-Page transitions

We construct holographic backgrounds that are dual by the AdS/CFT correspondence to Euclidean conformal field theories on products of spheres {S}^{d_1}times {S}^{d_2} , for conformal field theories whose dual may be approximated by classical Einstein gravity (typically these are large N strongly coupled theories). For d2 = 1 these backgrounds correspond to thermal field theories on {S}^{d_1} , and Hawking and Page found that there are several possible bulk solutions, with two different topologies, that compete with each other, leading to a phase transition as the relative size of the spheres is modified. By numerically solving the Einstein equations we find similar results also for d2> 1, with bulk solutions in which either one or the other sphere shrinks to zero smoothly at a minimal value of the radial coordinate, and with a first order phase transition (for d1 + d2< 9) between solutions of two different topologies as the relative radius changes. For a critical ratio of the radii there is a (sub-dominant) singular solution where both spheres shrink, and we analytically analyze the behavior near this radius. For d1 + d2< 9 the number of solutions grows to infinity as the critical ratio is approached.

Constraints on Holographic Multifield Inflation and Models Based on the Hamilton-Jacobi Formalism.

In holographic inflation, the 4D cosmological dynamics is postulated to be dual to the renormalization group flow of a 3D Euclidean conformal field theory with marginally relevant operators. The scalar potential of the 4D theory-in which inflation is realized-is highly constrained, with use of the Hamilton-Jacobi equations. In multifield holographic realizations of inflation, fields additional to the inflaton cannot display underdamped oscillations (that is, their wave functions contain no oscillatory phases independent of the momenta). We show that this result is exact, independent of the number of fields, the field space geometry, and the shape of the inflationary trajectory followed in multifield space. In the specific case where the multifield trajectory is a straight line or confined to a plane, it can be understood as the existence of an upper bound on the dynamical masses m of extra fields of the form m≤3H/2 up to slow roll corrections. This bound corresponds to the analytic continuation of the well-known Breitenlohner-Freedman bound found in anti-de Sitter spacetimes in the case when the masses are approximately constant. The absence of underdamped oscillations implies that a detection of "cosmological collider" oscillatory patterns in the non-Gaussian bispectrum would not only rule out single-field inflation, but also holographic inflation or any inflationary model based on the Hamilton-Jacobi equations. Hence, future observations have the potential to exclude, at once, an entire class of inflationary theories, regardless of the details involved in their model building.

{{SO(d,1)}}-Invariant Yang\u2013Baxter Operators and the dS/CFT Correspondence

We propose a model for the dS/CFT correspondence. The model is constructed in terms of a “Yang–Baxter operator” R for unitary representations of the de Sitter group {SO(d,1)}. This R-operator is shown to satisfy the Yang–Baxter equation, unitarity, as well as certain analyticity relations, including in particular a crossing symmetry. With the aid of this operator we construct: (a) a chiral (light-ray) conformal quantum field theory whose internal degrees of freedom transform under the given unitary representation of {SO(d,1)}. By analogy with the O(N) non-linear sigma model, this chiral CFT can be viewed as propagating in a de Sitter spacetime. (b) A (non-unitary) Euclidean conformal quantum field theory on {mathbb{R}^{d-1}}, where SO(d, 1) now acts by conformal transformations in (Euclidean) spacetime. These two theories can be viewed as dual to each other if we interpret {mathbb{R}^{d-1}} as conformal infinity of de Sitter spacetime. Our constructions use semi-local generator fields defined in terms of R and abstract methods from operator algebras.

Disorder in large-N theories

We consider Euclidean Conformal Field Theories perturbed by quenched disorder, namely by random fluctuations in their couplings. Such theories are relevant for second-order phase transitions in the presence of impurities or other forms of disorder. Theories with quenched disorder often flow to new fixed points of the renormalization group. We begin with disorder in free field theories. Imry and Ma showed that disordered free fields can only exist for d>4. For d>4 we show that disorder leads to new fixed points which are not scale-invariant. We then move on to large-N theories (vector models or gauge theories in the `t Hooft limit). We compute exactly the beta function for the disorder, and the correlation functions of the disordered theory. We generalize the results of Imry and Ma by showing that such disordered theories exist only when disorder couples to operators of dimension \Delta > d/4. Sometimes the disordered fixed points are not scale-invariant, and in other cases they have unconventional dependence on the disorder, including non-trivial effects due to irrelevant operators. Holography maps disorder in conformal theories to stochastic differential equations in a higher-dimensional space. We use this dictionary to reproduce our field theory results. We also study the leading 1/N corrections, both by field theory methods and by holography. These corrections are particularly important when disorder scales with the number of degrees of freedom.

Baby de Sitter black holes and dS3/CFT2

Unlike three-dimensional Einstein gravity, three-dimensional massive gravity admits asymptotically de Sitter space (dS) black hole solutions. These black holes present interesting features and provide us with toy models to study the dS/CFT correspondence. A remarkable property of these black holes is that they are always in thermal equilibrium with the cosmological horizon of the space that hosts them. This invites us to study the thermodynamics of these solutions within the context of dS/CFT. We study the asymptotic symmetry group of the theory and find that it indeed coincides with the local two-dimensional conformal algebra. The charge algebra associated to the asymptotic Killing vectors consists of two copies of the Virasoro algebra with non-vanishing central extension. We compute the mass and angular momentum of the dS black holes and verify that a naive application of Cardy's formula exactly reproduces the entropy of both the black hole and the cosmological horizon. By adapting the holographic renormalization techniques to the case of dS space, we define the boundary stress tensor of the dual Euclidean conformal field theory.

Lattice radial quantization: 3D Ising

Lattice radial quantization is introduced as a nonperturbative method intended to numerically solve Euclidean conformal field theories that can be realized as fixed points of known Lagrangians. As an example, we employ a lattice shaped as a cylinder with a 2D Icosahedral cross-section to discretize dilatations in the 3D Ising model. Using the integer spacing of the anomalous dimensions of the first two descendants (l=1,2), we obtain an estimate for η=0.034(10). We also observed small deviations from integer spacing for the 3rd descendant, which suggests that a further improvement of our radial lattice action will be required to guarantee conformal symmetry at the Wilson–Fisher fixed point in the continuum limit.

Shape Dynamics and AdS/CFT

I present a semi-classical correspondence between three-dimensional euclidean conformal field theory and the large-volume dynamics of gravity with positive cosmological constant in 3+1 dimensions. The correspondence is based on a theory of gravity called Shape Dynamics, whose solutions are a subset of those of General Relativity, namely those that can be foliated with constant mean extrinsic curvature. Shape Dynamics makes it possible to solve exactly for all the local degrees of freedom, while in GR this is possible only approximately, within a derivative expansion, due to the non-linearity of the scalar constraint.

Entanglement entropy of 3-d conformal gauge theories with many flavors

Three-dimensional conformal field theories (CFTs) of deconfined gauge fields coupled to gapless flavors of fermionic and bosonic matter describe quantum critical points of condensed matter systems in two spatial dimensions. An important characteristic of these CFTs is the finite part of the entanglement entropy across a circle. The negative of this quantity is equal to the finite part of the free energy of the Euclidean CFT on the three-sphere, and it has been proposed to satisfy the so called F-theorem, which states that it decreases under RG flow and is stationary at RG fixed points. We calculate the three-sphere free energy of non-supersymmetric gauge theory with a large number N F of bosonic and/or fermionic flavors to the first subleading order in 1/N F . We also calculate the exact free energies of the analogous chiral and non-chiral $ \mathcal{N} = 2 $ supersymmetric theories using localization, and find agreement with the 1/N F expansion. We analyze some RG flows of supersymmetric theories, providing further evidence for the F-theorem.

Holography at an extremal De Sitter horizon

Rotating maximal black holes in four-dimensional de Sitter space, for which the outer event horizon coincides with the cosmological horizon, have an infinite near-horizon region described by the rotating Nariai metric. We show that the asymptotic symmetry group at the spacelike future boundary of the near-horizon region contains a Virasoro algebra with a real, positive central charge. This is evidence that quantum gravity in a rotating Nariai background is dual to a two-dimensional Euclidean conformal field theory. These results are related to the Kerr/CFT correspondence for extremal black holes, but have two key differences: one of the black hole event horizons has been traded for the cosmological horizon, and the near-horizon geometry is a fiber over dS_2 rather than AdS_2.

Holographic framework for eternal inflation

In this paper we provide some circumstantial evidence for a holographic duality between bubble nucleation in an eternally inflating universe and a Euclidean conformal field theory (CFT). The holographic correspondence (which is different than Strominger's de Sitter (dS)/CFT duality) relates the decay of ($3+1$)-dimensional de Sitter space to a two-dimensional CFT. It is not associated with pure de Sitter space, but rather with Coleman-De Luccia bubble nucleation. Alternatively, it can be thought of as a holographic description of the open, infinite, Friedmann-Robertson-Walker (FRW) cosmology that results from such a bubble. The conjectured holographic representation is of a new type that combines holography with the Wheeler-DeWitt formalism to produce a Wheeler-DeWitt theory that lives on the spatial boundary of a $k=\ensuremath{-}1$ FRW cosmology. We also argue for a more ambitious interpretation of the Wheeler-DeWitt CFT as a holographic dual of the entire Landscape.

Relationship between five-dimensional black holes and de Sitter spaces

We study a close relationship between the topological anti-de Sitter (TAdS) black holes and topological de Sitter (TdS) spaces including the Schwarzschild–de Sitter (SdS) black hole in five dimensions. We show that all thermal properties of the TdS spaces can be found from those of the TAdS black holes by replacing k by −k. Also we find that all thermal information for the cosmological horizon of the SdS black hole is obtained from either the hyperbolic-AdS black hole or the Schwarzschild–TdS space by substituting m with −m. For this purpose we calculate thermal quantities of bulk (Euclidean) conformal field theory (ECFT) and moving domain wall by using the A(dS)/(E)CFT correspondences. Further, we compute logarithmic corrections to the Bekenstein–Hawking entropy, Cardy–Verlinde formula and Friedmann equation due to thermal fluctuations. It implies that in the thermal relation between the TdS spaces and TAdS black holes, the cosmological horizon plays the same role as the horizon of TAdS black holes. Finally we note that the dS/ECFT correspondence is valid for the TdS spaces in conjunction with the AdS/CFT correspondence for the TAdS black holes.

Codimension-two holography

A holographic interpretation for some specific Ricci-flat string backgrounds of the form A 6× C 4 is proposed. The conjecture is that there is a four-dimensional Euclidean conformal field theory (ECFT) defined on a codimension two submanifold of the manifold A 6 (where one of the two remaining holographic coordinates of A 6 is timelike, and the other one spacelike), with central charge proportional to the radius of curvature of the six-dimensional manifold, c∼ l 4.

Tall tales from de Sitter space II: Field theory dualities

We consider the evolution of massive scalar fields in (asymptotically) de Sitter spacetimes of arbitrary dimension. Through the proposed dS/CFT correspondence, our analysis points to the existence of new nonlocal dualities for the Euclidean conformal field theory. A massless conformally coupled scalar field provides an example where the analysis is easily explicitly extended to 'tall' background spacetimes.

Exploring de Sitter space and holography

We explore aspects of the physics of de Sitter (dS) space that are relevant to holography with a positive cosmological constant. First we display a non-local map that commutes with the de Sitter isometries, transforms the bulk-boundary propagator and solutions of free wave equations in de Sitter onto the same quantities in Euclidean anti-de Sitter (EAdS), and takes the two boundaries of dS to the single EAdS boundary via an anti-podal identification. Second we compute the action of scalar fields on dS as a functional of boundary data. Third, we display a family of solutions to 3d gravity with a positive cosmological constant in which the equal time sections are arbitrary genus Riemann surfaces, and compute the action of these spaces as a functional of boundary data from the Einstein gravity and Chern–Simons gravity points of view. These studies suggest that if de Sitter space is dual to a Euclidean conformal field theory (CFT), this theory should involve two disjoint, but possibly entangled factors. We argue that these CFTs would be of a novel form, with unusual hermiticity conditions relating left movers and right movers. After exploring these conditions in a toy model, we combine our observations to propose that a holographic dual description of de Sitter space would involve a pure entangled state in a product of two of our unconventional CFTs associated with the de Sitter boundaries. This state can be constructed to preserve the de Sitter symmetries and and its decomposition in a basis appropriate to anti-podal inertial observers would lead to the thermal properties of static patch.

Notes on de Sitter space and holography

We explore aspects of the physics of de Sitter (dS) space that are relevant to holography with a positive cosmological constant. First, we display a non-local map that commutes with the de Sitter isometries, transforms the bulk–boundary propagator and solutions of free wave equations in de Sitter onto the same quantities in Euclidean anti-de Sitter (EAdS) space, and takes the two boundaries of dS to the single EAdS boundary via an antipodal identification. Second, we compute the action of scalar fields on dS as a functional of boundary data. Third, we display a family of solutions to three-dimensional gravity with a positive cosmological constant in which the equal time sections are arbitrary genus Riemann surfaces, and compute the action of these spaces as a functional of boundary data. These studies suggest that if de Sitter space is dual to a Euclidean conformal field theory (CFT), this theory should involve two disjoint, but possibly entangled factors. We argue that these CFTs would be of a novel form, with unusual hermiticity conditions relating left movers and right movers. After exploring these conditions in a toy model, we combine our observations to propose that a holographic dual description of de Sitter space would involve a pure entangled state in a product of two of our unconventional CFTs associated with the de Sitter boundaries. This state can be constructed to preserve the de Sitter symmetries and its decomposition in a basis appropriate to antipodal inertial observers would lead to the thermal properties of a static patch. To conclude, we discuss the one-parameter family of de Sitter-invariant vacua for a massive free scalar field, and their thermodynamic properties. At the free field level, we find no obvious thermodynamic reason to favour one vacuum over the other.