Euclidean Anti-de Sitter Research Articles

Cosmological correlators for Bogoliubov initial states

We consider late-time correlators in de Sitter (dS) space for initial states related to the Bunch-Davies vacuum by a Bogoliubov transformation. We propose to study such late-time correlators by reformulating them in the familiar language of Witten diagrams in Euclidean anti-de Sitter space (EAdS), showing that they can be perturbatively re-cast in terms of corresponding dS boundary correlators in the Bunch-Davies vacuum and in turn, Witten diagrams in EAdS. Unlike the standard relationship between late-time correlators in the Bunch-Davies vacuum and EAdS Witten diagrams, this involves points on the upper and lower sheet of the EAdS hyperboloid which account for antipodal singularities of the two-point functions. Such Bogoliubov states include an infinite one parameter family of de Sitter invariant vacua as a special case, where the late-time correlators are constrained by conformal Ward identities. In momentum space, it is well known that their late-time correlators exhibit singularities in collinear (“folded”) momentum configurations. We give a position space interpretation of such solutions to the conformal Ward identities, where in embedding space they can be generated from the solution without collinear singularities by application of the antipodal map. We also discuss the operator product expansion (OPE) limit of late-time correlators in a generic dS invariant vacuum. Many results are derived using the Mellin space representation of late-time correlators, which in this work we extend to accommodate generic dS invariant vacua.

The semi-classical saddles in three-dimensional gravity via holography and mini-superspace approach

We determine the complex geometries dual to the semi-classical saddles in three-dimensional gravity with positive or negative cosmological constant. We examine the semi-classical saddles in Liouville field theory and interpret them in terms of gravity theory. For this, we describe the gravity theory by Chern-Simons theory and classify the possible saddles based on the homotopy group argument. We further realize the semi-classical saddles using the mini-superspace model of quantum gravity and explicitly determine the integral contour. In the case of positive cosmological constant, we recovered the geometry used for no-boundary proposal of Hartle and Hawking. In the case of negative cosmological constant, the geometry can be identified with Euclidean anti-de Sitter space attached with imaginary radius spheres. The geometry should be unphysical and several arguments on this issue are provided. Partial results were already presented in our earlier letter, and more detailed derivations and explanations on the results are given along with additional results. In particular, we reproduce the classical Liouville action from the Chern-Simons formulation of dual gravity theory.

Spinors in (Anti-)de Sitter Space

We explore analytical aspects of correlators involving Dirac spinors in d + 1-dimensional de Sitter space. Adapting the formalism of Sleight and Taronna, we show how to relate processes involving fermions in the in-in formalism to equivalent Witten diagrams in (complexified) Euclidean anti-de Sitter space. We exemplify the method for a fermion-exchange diagram. We establish a positive spectral decomposition over the principal series of the Wightman function of two spinors, showing the consequences of unitarity.

Quartic locality of higher-spin gravity in de Sitter and Euclidean anti-de Sitter space

We consider higher-spin gravity in (A)dS4, defined as the holographic dual of a free O(N) or Sp(N) vector model. At the quartic level, this theory has been judged non-local at distances greater than the (A)dS radius, due to a mismatch of massless (twist=1) exchange-type terms in its boundary OPE behavior. We review the non-locality argument, and note that it relies on a double-lightcone limit, which requires a Lorentzian boundary. In the Euclidean OPE limit, we demonstrate the absence of massless exchange-type non-localities of any spin, by inspecting a known formula for the bulk exchange diagrams in Euclidean AdS, and constructing upper bounds in which the spin-dependence factorizes from the position-dependence. Our results suggest that higher-spin theory is local (at distances greater than the curvature radius) in spacetimes with Euclidean boundary signature. For Lorentzian bulk, this implies locality in de Sitter space, as opposed to anti-de Sitter.

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.

Breitenlohner-Freedman Bound on Hyperbolic Tilings.

We establish how the Breitenlohner-Freedman (BF) bound is realized on tilings of two-dimensional Euclidean Anti-de Sitter space. For the continuum, the BF bound states that on Anti-de Sitter spaces, fluctuation modes remain stable for small negative mass squared m^{2}. This follows from a real and positive total energy of the gravitational system. For finite cutoff ϵ, we solve the Klein-Gordon equation numerically on regular hyperbolic tilings. When ϵ→0, we find that the continuum BF bound is approached in a manner independent of the tiling. We confirm these results via simulations of a hyperbolic electric circuit. Moreover, we propose a novel circuit including active elements that allows us to further scan values of m^{2} above the BF bound.

Analytical evaluation of cosmological correlation functions

Using the Schwinger-Keldysh-formalism, reformulated in [1] as an effective field theory in Euclidean anti-de Sitter, we evaluate the one-loop cosmological four-point function of a conformally coupled interacting scalar field in de Sitter. Recasting the Witten cosmological correlator as flat space Feynman integrals, we evaluate the one-loop cosmological four-point functions in de Sitter space in terms of single-valued multiple polylogarithms. From it we derive anomalous dimensions and OPE coefficients of the dual conformal field theory at space-like, future infinity. In particular, we find an interesting degeneracy in the anomalous dimensions relating operators of neighboring spins.

From dS to AdS and back

We describe in more detail the general relation uncovered in our previous work between boundary correlators in de Sitter (dS) and in Euclidean anti-de Sitter (EAdS) space, at any order in perturbation theory. Assuming the Bunch-Davies vacuum at early times, any given diagram contributing to a boundary correlator in dS can be expressed as a linear combination of Witten diagrams for the corresponding process in EAdS, where the relative coefficients are fixed by consistent on-shell factorisation in dS. These coefficients are given by certain sinusoidal factors which account for the change in coefficient of the contact sub-diagrams from EAdS to dS, which we argue encode (perturbative) unitary time evolution in dS. dS boundary correlators with Bunch-Davies initial conditions thus perturbatively have the same singularity structure as their Euclidean AdS counterparts and the identities between them allow to directly import the wealth of techniques, results and understanding from AdS to dS. This includes the Conformal Partial Wave expansion and, by going from single-valued Witten diagrams in EAdS to Lorentzian AdS, the Froissart-Gribov inversion formula. We give a few (among the many possible) applications both at tree and loop level. Such identities between boundary correlators in dS and EAdS are made manifest by the Mellin-Barnes representation of boundary correlators, which we point out is a useful tool in its own right as the analogue of the Fourier transform for the dilatation group. The Mellin-Barnes representation in particular makes manifest factorisation and dispersion formulas for bulk-to-bulk propagators in (EA)dS, which imply Cutkosky cutting rules and dispersion formulas for boundary correlators in (EA)dS. Our results are completely general and in particular apply to any interaction of (integer) spinning fields.

Recursive construction of the operator product expansion in curved space

I derive a formula for the coupling-constant derivative of the coefficients of the operator product expansion (Wilson OPE coefficients) in an arbitrary curved space, as the natural extension of the quantum action principle. Expanding the coefficients themselves in powers of the coupling constants, this formula allows to compute them recursively to arbitrary order. As input, only the OPE coefficients in the free theory are needed, which are easily obtained using Wick’s theorem. I illustrate the method by computing the OPE of two scalars ϕ in hyperbolic space (Euclidean Anti-de Sitter space) up to terms vanishing faster than the square of their separation to first order in the quartic interaction gϕ4, as well as the OPE coefficient at second order in g.

Gauged Dirac operator on the q-deformed fuzzy Euclidean anti-de Sitter space using the pseudo-generalization of q-deformed Ginsparg–Wilson algebra

In this paper, q-deformed pseudo-fuzzy Dirac and chirality operators on the q-deformed pseudo-fuzzy EAdS2, using the pseudo-generalization of the quantum pseudo-fuzzy Ginsparg–Wilson algebra, have been constructed. Gauged q-deformed pseudo-fuzzy Dirac and chirality operators have also been constructed. In the limit case q → 1, it will be shown that these operators will become Dirac and chirality operators on the pseudo-fuzzy EAdS2.

Holography in de Sitter and anti-de Sitter spaces and Gel'fand Graev Radon transform

Bulk reconstruction formulas similar to HKLL are obtained for de Sitter and anti-de Sitter spaces as the inverse Gel'fand Graev Radon transform. While these generalize our previous result on the Euclidean anti-de Sitter space, their validity in here is restricted only to odd dimensions in both instances. The exact Wightman function for the de Sitter space is then derived. The GGR transform fixes the coefficient of the Wightman function. For the anti-de Sitter space it is shown that a reconstruction formula exists for the case of timelike boundary as well. The restriction on the domain of integration on the boundary is derived. As a special case, we point out that the formula is valid for the BTZ black hole as well.

Kac-Moody and Virasoro characters from the perturbative Chern-Simons path integral

We evaluate to one loop the functional integral that computes the partition functions of Chern-Simons theories based on compact groups, using the background field method and a covariant gauge fixing. We compare our computation with the results of other, less direct methods. We find that our method correctly computes the characters of irreducible representations of Kac-Moody algebras. To extend the computation to non-compact groups we need to perform an appropriate analytic continuation of the partition function of the compact group. Non-vacuum characters are found by inserting a Wilson loop in the functional integral. We then extend our method to Euclidean Anti-de Sitter pure gravity in three dimensions. The explicit computation unveils several interesting features and lessons. The most important among them is that the very definition of gravity in the first-order Chern-Simons formalism requires non-trivial analytic continuations of the gauge fields outside their original domains of definition.

Quantum \u03d54 theory in AdS4 and its CFT dual

We compute the two- and four-point holographic correlation functions up to the second order in the coupling constant for a scalar ϕ4 theory in four-dimensional Euclidean anti-de Sitter space. Analytic expressions for the anomalous dimensions of the leading twist operators are found at one loop, both for Neumann and Dirichlet boundary conditions.

Emergence of AdS geometry in the simulated tempering algorithm

In our previous work [1], we introduced to an arbitrary Markov chain Monte Carlo algorithm a distance between configurations. This measures the difficulty of transition from one configuration to the other, and enables us to investigate the relaxation of probability distribution from a geometrical point of view. In this paper, we investigate the global geometry of a stochastic system whose equilibrium distribution is highly multimodal with a large number of degenerate vacua. We show that, when the simulated tempering algorithm is implemented to such a system, the extended configuration space has an asymptotically Euclidean anti-de Sitter (AdS) geometry. We further show that this knowledge of geometry enables us to optimize the tempering parameter in a simple, geometrical way.

Loops in Anti-de Sitter Space.

We obtain analytic results for the four-point amplitude, at one loop, of an interacting scalar field theory in four-dimensional, Euclidean anti-de Sitter space without exerting any conformal field theory knowledge. For the two-point function, we provide analytic expressions up to two loops. In addition, we argue that the critical exponents of correlation functions near the conformal boundary of anti-de Sitter space provide the necessary data for the renormalization conditions, thus replacing the usual on-shell condition.

Action and observer dependence in Euclidean quantum gravity

Given a Lorentzian spacetime and a non-vanishing timelike vector field with level surfaces Σ, one can construct on a Euclidean metric (Hawking and Ellis 1973 The Large Scale Structure of Space-Time (Cambridge: Cambridge University Press)). Motivated by this, we consider a class of metrics with an arbitrary function Θ that interpolates between the Euclidean () and Lorentzian () regimes, separated by the codimension one hypersurface defined by . Since can not, in general, be obtained from by a diffeomorphism, its Euclidean regime is in general different from that obtained from Wick rotation . For example, if is the k = 0 Lorentzian de Sitter metric corresponding to , the Euclidean regime of is the k = 0 Euclidean anti-de Sitter space with .We analyze the curvature tensors associated with for arbitrary Lorentzian metrics and timelike geodesic fields , and show that they have interesting and remarkable mathematical structures: (i) Additional terms arise in the Euclidean regime of . (ii) For the simplest choice of a step-profile for Θ, the Ricci scalar Ric of reduces, in the Lorentzian regime , to the complete Einstein–Hilbert lagrangian with the correct Gibbons–Hawking–York boundary term; the latter arises as a delta-function of strength 2K supported on . (iii) In the Euclidean regime , Ric also has an extra term of the -foliation. We highlight similar foliation dependent terms in the full Riemann tensor.We present some explicit examples for FLRW spacetimes in standard foliation and spherically symmetric spacetimes in the Painleve–Gullstrand foliation. We briefly discuss implications of the results for Euclidean quantum gravity and quantum cosmology.

Bulk reconstruction in AdS and Gel\u2019fand-Graev-Radon transform

The bulk reconstruction formula for a Euclidean anti-de Sitter space is directly related to the inverse of the Gel’fand-Graev-Radon transform. Correlation functions of a conformal scalar field theory in the boundary are thereby related to correlation functions of a self-interacting scalar field theory in the bulk at different loop orders.

Flow equation for the large N scalar model and induced geometries

We study the proposal that a (d+1)-dimensional induced metric is constructed from a d-dimensional field theory using gradient flow. Applying the idea to the O(N) ϕ4 model and normalizing the flow field, we have shown in the large N limit that the induced metric is finite and universal in the sense that it does not depend on the details of the flow equation and the original field theory except for the renormalized mass, which is the only relevant quantity in this limit. We have found that the induced metric describes Euclidean anti-de Sitter (AdS) space in both ultraviolet (UV) and infrared (IR) limits of the flow direction, where the radius of the AdS is bigger in the IR than in the UV.

Holographic calculations of Euclidean Wilson loop correlator in Euclidean anti-de Sitter space

The correlation functions of two or more Euclidean Wilson loops of various shapes in Euclidean anti-de Sitter space are computed by considering the minimal area surfaces connecting the loops. The surfaces are parametrized by Riemann theta functions associated with genus three hyperelliptic Riemann surfaces. In the case of two loops, the distance $L$ by which they are separated can be adjusted by continuously varying a specific branch point of the auxiliary Riemann surface. When $L$ is much larger than the characteristic size of the loops, then the loops are approximately regarded as local operators and their correlator as the correlator of two local operators. Similarly, when a loop is very small compared to the size of another loop, the small loop is considered as a local operator corresponding to a light supergravity mode.

On memory in exponentially expanding spaces

We examine the degree to which fluctuating dynamics on exponentially expanding spaces remember initial conditions. In de Sitter space, the global late-time configuration of a free scalar field always contains information about early fluctuations. By contrast, fluctuations near the boundary of Euclidean Anti-de Sitter may or may not remember conditions in the center, with a transition at \Delta=d/2. We connect these results to literature about statistical mechanics on trees and make contact with the observation by Anninos and Denef that the configuration space of a massless dS field exhibits ultrametricity. We extend their analysis to massive fields, finding that preference for isosceles triangles persists as long as \Delta_- < d/4.