Euclidean AdS Research Articles

Asymptotic boundary conditions and square integrability in the partition function of AdS gravity

There has been renewed interest in the path-integral computation of the partition function of AdS3 gravity, both in the metric and Chern-Simons formulations. The one-loop partition function around Euclidean AdS3 turns out to be given by the vacuum character of Virasoro group. This stems from the work of Brown and Henneaux (BH) who showed that, in AdS3 gravity with sensible asymptotic boundary conditions, an infinite group of (improper) diffeomorphisms arises which acts canonically on phase space as two independent Virasoro symmetries. The gauge group turns out to be composed of so-called “proper” diffeomorphisms which approach the identity at infinity fast enough. However, it is sometimes far from evident to identify where BH boundary conditions enter in the path integral, and much more difficult to see how the improper diffeomorphisms are left out of the gauge group. In particular, in the metric formulation, Giombi, Maloney and Yin obtained the one-loop partition function around thermal AdS3 resorting to the heat kernel method to compute the determinants coming from the path integral. Here we identify how BH boundary conditions follow naturally from the usual requirement of square-integrability of the metric perturbations. Also, and equally relevant, we clarify how the quotient by only proper diffeomorphisms is implemented, promoting the improper diffeomorphisms to symmetries in the path integral. Our strategy is general enough to apply to other approaches where square integrability is assumed. Finally, we show that square integrability implies that the asymptotic symmetries in higher dimensional AdS gravity are just isometries.

From bulk loops to boundary large-N expansion

We study the analytic structure of loop Witten diagrams in Euclidean AdS represented by their conformal partial wave expansions. We show that, as in flat space, amplitude’s singularities are associated with non-trivial cuts of the diagram and factorize into products of the coefficient functions for the subdiagrams resulting from these cuts. We consider an example of a one-loop four-point diagram in detail and then briefly discuss how the procedure can be extended to more general diagrams. Finally, we show that this analysis reproduces simple relations that follow from the large-N considerations on the boundary.

Uplifting AdS3/CFT2 to flat space holography

Four-dimensional (4D) flat Minkowski space admits a foliation by hyperbolicslices. Euclidean AdS3 slices fill the past and future lightcones of the origin, while dS3 slices fill the region outside the lightcone. The resulting link between 4D asymptotically flat quantum gravity and AdS3/CFT2 is explored in this paper. The 4D superrotations in the extended BMS4 group are found to act as the familiar conformal transformations on the 3D hyperbolic slices, mapping each slice to itself. The associated 4D superrotation charge is constructed in the covariant phase space formalism. The soft part gives the 2D stress tensor, which acts on the celestial sphere at the boundary of the hyperbolic slices, and is shown to be an uplift to 4D of the familiar 3D holographic AdS3 stress tensor. Finally, we find that 4D quantum gravity contains an unexpected second, conformally soft, dimension (2, 0) mode that is symplectically paired with the celestial stress tensor.

Chaos in three-dimensional higher spin gravity

We discuss SL(N, ℂ) Chern-Simons higher spin gauge theories in Euclidean AdS3. With appropriate boundary term, we derive the higher spin generalization of Schwarzian on-shell action. We investigate gravitationally dressed bi-locals, and we study the soft higher spin mode expansion to obtain soft mode eigenfunctions. We also derive the spin-s eigenfunction from Ward identity of mathcal{W} -algebra and a recursion relation. Using the on-shell action, we evaluate the contributions of the soft higher spin modes to the out-of-time-order correlators, and the corresponding Lyapunov exponent of SL(N) higher spin gravity is found to be frac{2pi }{beta}left(N - 1right) .

Massive (pesudo) scalars in AdS4, SO(4) invariant solutions and holography

We include a new 7-form ansatz in 11-dimensional supergravity over AdS_4 x S^7/Z_k when the internal space is considered as a U(1) bundle on CP^3. After a general analysis of the ansatz, we take a special form of it and obtain a scalar equation from which we focus on a few massive bulk modes that are SU(4) x U(1) R-singlet and break all supersymmetries. The mass term breaks the scale invariance and so the (perturbative) solutions we obtain are SO(4) invariant in Euclidean AdS_4 (or SO(3,1) in its dS_3 slicing). The corresponding bare operators are irrelevant in probe approximation; and to realize the AdS_4/CFT_3 correspondence, we need to swap the fundamental representations of $SO(8)$ for supercharges with those for scalars and fermions. In fact, we have domain-walls arising from (anti)M5-branes wrapping around S^3/Z_k of the internal space with parity breaking scheme. As a result, the duals may be in three-dimensional U(N) or O(N) Chern-Simon models with matters in fundamental representations. Accordingly, we present dual boundary operators and build instanton solutions in a truncated version of the boundary ABJM action; and, because of the unboundedness of bulk potential from below, it is thought that they lead to big crunch singularities in the bulk.

Space-time in the SYK model

We consider the question of identifying the bulk space-time of the SYK model. Focusing on the signature of emergent space-time of the (Euclidean) model, we explain the need for non-local (Radon-type) transformations on external legs of n-point Green’s functions. This results in a dual theory with Euclidean AdS signature with additional legfactors. We speculate that these factors incorporate the coupling of additional bulk states similar to the discrete states of 2d string theory.

Stochastic quantization and holographic Wilsonian renormalization group of free massive fermion

We examine a suggested relation between stochastic quantization and the holographic Wilsonian renormalization group in the massive fermion case on Euclidean AdS space. The original suggestion about the general relation between the two theories is posted in arXiv:1209.2242 . In the previous researches, it is already verified that scalar fields, U(1) gauge fields, and massless fermions are consistent with the relation. In this paper, we examine the relation in the massive fermion case. Contrary to the other case, in the massive fermion case, the action needs particular boundary terms to satisfy boundary conditions. We finally confirm that the proposed suggestion is also valid in the massive fermion case.

Instantons from geodesics in AdS moduli spaces

We investigate supergravity instantons in Euclidean AdS5 × S5/ℤk. These solutions are expected to be dual to instantons of mathcal{N} = 2 quiver gauge theories. On the supergravity side the (extremal) instanton solutions are neatly described by the (lightlike) geodesics on the AdS moduli space for which we find the explicit expression and compute the on-shell actions in terms of the quantised charges. The lightlike geodesics fall into two categories depending on the degree of nilpotency of the Noether charge matrix carried by the geodesic: for degree 2 the instantons preserve 8 supercharges and for degree 3 they are non-SUSY. We expect that these findings should apply to more general situations in the sense that there is a map between geodesics on moduli-spaces of Euclidean AdS vacua and instantons with holographic counterparts.

Minimal area surfaces in AdSn+1 and Wilson loops

The AdS/CFT correspondence relates the expectation value of Wilson loops in mathcal{N} = 4 SYM to the area of minimal surfaces in AdS5. In this paper we consider minimal area surfaces in generic Euclidean AdSn+1 using the Pohlmeyer reduction in a similar way as we did previously in Euclidean AdS3. As in that case, the main obstacle is to find the correct parameterization of the curve in terms of a conformal parameter. Once that is done, the boundary conditions for the Pohlmeyer fields are obtained in terms of conformal invariants of the curve. After solving the Pohlmeyer equations, the area can be expressed as a boundary integral involving a generalization of the conformal arc-length, curvature and torsion of the curve. Furthermore, one can introduce the λ-deformation symmetry of the contours by a simple change in the conformal invariants. This determines the λ-deformed contours in terms of the solution of a boundary linear problem. In fact the condition that all λ deformed contours are periodic can be used as an alternative to solving the Pohlmeyer equations and is equivalent to imposing the vanishing of an infinite set of conserved charges derived from integrability.

Wavelets and Lattice Field Theory.

When continuous fields are expanded in a wavelet basis, a D-dimensional continuum action becomes a (D+1)-dimensional lattice action on the naively discretized Poincare-patch coordinates of an Euclidean AdS(D+1). New possible criteria for acceptable actions open up.

Topologically twisted renormalization group flow and its holographic dual

Euclidean field theories admit more general deformations than usually discussed in quantum field theories because of mixing between rotational symmetry and internal symmetry (a.k.a topological twist). Such deformations may be relevant, and if the subsequent renormalization group flow leads to a non-trivial fixed point, it generically gives rise to a scale invariant Euclidean field theory without conformal invariance. Motivated by an ansatz studied in cosmological models some time ago, we develop a holographic dual description of such renormalization group flows in the context of AdS/CFT. We argue that the non-trivial fixed points require fine-tuning of the bulk theory in general, but remarkably we find that the $O(3)$ Yang-Mills theory coupled with the four-dimensional Einstein gravity in the minimal manner supports such a background with the Euclidean AdS metric.

Interacting fields in real-time AdS/CFT

We compute time-ordered 2- and 3-pt correlation functions of CFT scalar operators between generic in/out states. The calculation is holographically carried out by considering a non backreacting AdS scalar field with a λϕ3 self-interaction term on a combination of Euclidean and Lorentzian AdS sections following the Skenderis-van Rees prescription. We show that, although working in an essentially different set up, the final result for the 3-pt correlators agree with those of Rastelli et al. for Euclidean AdS. By analyzing the inner product between the in/out excited states in the large N approximation, we argue that a cubic bulk interaction deforms the excited states from coherent into squeezed. Finally, a diagrammatic interpretation of the results suggests some general properties for the n-point correlation functions between excited states.

Localization of supersymmetric field theories on non-compact hyperbolic three-manifolds

We study supersymmetric gauge theories with an R-symmetry, defined on non-compact, hyperbolic, Riemannian three-manifolds, focusing on the case of a supersymmetry-preserving quotient of Euclidean AdS3. We compute the exact partition function in these theories, using the method of localization, thus reducing the problem to the computation of one-loop determinants around a supersymmetric locus. We evaluate the one-loop determinants employing three different techniques: an index theorem, the method of pairing of eigenvalues, and the heat kernel method. Along the way, we discuss aspects of supersymmetry in manifolds with a conformal boundary, including supersymmetric actions and boundary conditions.

4D scattering amplitudes and asymptotic symmetries from 2D CFT

We reformulate the scattering amplitudes of 4D flat space gauge theory and gravity in the language of a 2D CFT on the celestial sphere. The resulting CFT structure exhibits an OPE constructed from 4D collinear singularities, as well as infinite-dimensional Kac-Moody and Virasoro algebras encoding the asymptotic symmetries of 4D flat space. We derive these results by recasting 4D dynamics in terms of a convenient foliation of flat space into 3D Euclidean AdS and Lorentzian dS geometries. Tree-level scattering amplitudes take the form of Witten diagrams for a continuum of (A)dS modes, which are in turn equivalent to CFT correlators via the (A)dS/CFT dictionary. The Ward identities for the 2D conserved currents are dual to 4D soft theorems, while the bulk-boundary propagators of massless (A)dS modes are superpositions of the leading and subleading Weinberg soft factors of gauge theory and gravity. In general, the massless (A)dS modes are 3D Chern-Simons gauge fields describing the soft, single helicity sectors of 4D gauge theory and gravity. Consistent with the topological nature of Chern-Simons theory, Aharonov-Bohm effects record the “tracks” of hard particles in the soft radiation, leading to a simple characterization of gauge and gravitational memories. Soft particle exchanges between hard processes define the Kac-Moody level and Virasoro central charge, which are thereby related to the 4D gauge coupling and gravitational strength in units of an infrared cutoff. Finally, we discuss a toy model for black hole horizons via a restriction to the Rindler region.

Wilson loops and Riemann theta functions II

In this paper we extend and simplify previous results regarding the computation of Euclidean Wilson loops in the context of the AdS/CFT correspondence, or, equivalently, the problem of finding minimal area surfaces in hyperbolic space (Euclidean AdS3). If the Wilson loop is given by a boundary curve X(s) we define, using the integrable properties of the system, a family of curves X(lambda,s) depending on a complex parameter lambda known as the spectral parameter. This family has remarkable properties. As a function of lambda, X(lambda,s) has cuts and therefore is appropriately defined on a hyperelliptic Riemann surface, namely it determines the spectral curve of the problem. Moreover, X(lambda,s) has an essential singularity at the origin lambda=0. The coefficients of the expansion of X(lambda,s) around lambda=0, when appropriately integrated along the curve give the area of the corresponding minimal area surface. Furthermore we show that the same construction allows the computation of certain surfaces with one or more boundaries corresponding to Wilson loop correlators. We extend the area formula for that case and give some concrete examples. As the main example we consider a surface ending on two concentric circles and show how the boundary circles can be deformed by introducing extra cuts in the spectral curve.

Higher spinAdSd+1/CFTdat one loop

Following arXiv:1308.2337, we carry out one loop tests of higher spin AdS$_{d+1}$/CFT$_d$ correspondences for $d\geq 2$. The Vasiliev theories in AdS$_{d+1}$, which contain each integer spin once, are related to the $U(N)$ singlet sector of the $d$-dimensional CFT of $N$ free complex scalar fields; the minimal theories containing each even spin once -- to the $O(N)$ singlet sector of the CFT of $N$ free real scalar fields. Using analytic continuation of higher spin zeta functions, which naturally regulate the spin sums, we calculate one loop vacuum energies in Euclidean AdS$_{d+1}$. In even $d$ we compare the result with the $O(N^0)$ correction to the $a$-coefficient of the Weyl anomaly; in odd $d$ -- with the $O(N^0)$ correction to the free energy $F$ on the $d$-dimensional sphere. For the theories of integer spins, the correction vanishes in agreement with the CFT of $N$ free complex scalars. For the minimal theories, the correction always equals the contribution of one real conformal scalar field in $d$ dimensions. As explained in arXiv:1308.2337, this result may agree with the $O(N)$ singlet sector of the theory of $N$ real scalar fields, provided the coupling constant in the higher spin theory is identified as $G_N\sim 1/(N-1)$. Our calculations in even $d$ are closely related to finding the regularized $a$-anomalies of conformal higher spin theories. In each even $d$ we identify two such theories with vanishing $a$-anomaly: a theory of all integer spins, and a theory of all even spins coupled to a complex conformal scalar. We also discuss an interacting UV fixed point in $d=5$ obtained from the free scalar theory via an irrelevant double-trace quartic interaction. This interacting large $N$ theory is dual to the Vasiliev theory in AdS$_6$ where the bulk scalar is quantized with the alternate boundary condition.

DS/CFT at uniform energy density and a de Sitter “bluewall”

We describe a class of spacetimes that are asymptotically de Sitter in the Poincare slicing. Assuming that a dS/CFT correspondence exists, we argue that these are gravity duals to a CFT on a circle leading to uniform energy-momentum density, and are equivalent to an analytic continuation of the Euclidean AdS black brane. These are solutions with a complex parameter which then gives a real energy-momentum density. We also discuss a related solution with the parameter continued to a real number, which we refer to as a de Sitter "bluewall". This spacetime has two asymptotic de Sitter universes and Cauchy horizons cloaking timelike singularities. We argue that the Cauchy horizons give rise to a blue-shift instability.

Minimal Area Surfaces, Euclidean Wilson loops and Riemann Theta Functions

The AdS/CFT correspondence relates Wilson loops in supersymmetric N = 4 gauge theory to minimal area surfaces in anti-de Sitter space. In this paper we consider the case of Euclidean flat Wilson loops which are related to minimal area surfaces in Euclidean AdS3 space. Using known mathematical results for such minimal area surfaces we describe an infinite parameter family of analytic solutions for closed Wilson loops. Further, we also consider examples of surfaces with two boundaries dual to correlators of two Wilson loops.

Jet physics from static charges in AdS space

Soft interactions with high-energy jets are explored in radial coordinates which exploit the approximately conformal behavior of perturbative gauge theories. In these coordinates, the jets, approximated by Wilson lines, become static charges in Euclidean AdS. The anomalous dimension of the corresponding Wilson line operator is then determined by the potential energy of the charges. To study these Wilson lines we introduce a ``conformal gauge'' which does not have kinetic mixing between radial and angular directions, and show that a number of properties of Wilson lines are reproduced through relatively simple calculations. For example, certain nonplanar graphs involving multiple Wilson lines automatically vanish. We also discuss the linear growth of the charges' imaginary potential energy with separation, and a relationship between Wilson line diagrams and Witten diagrams.

U(1) instantons on AdS4 and the uplift to exact supergravity solutions

We consider self-duality equation of U(1) gauge fields on Euclidean AdS_4 space, and find a simple finite action solution. With a suitable ansatz, we are able to embed this solution into the 10d supergravity background of AdS_4\times CP^3. Further, we show that the solution can be uplifted to an exact solution in 11d supergravity background of AdS_4\times SE_7.