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We present a generalisation of the embedding space formalism to conformal field theories (CFTs) on non-trivial states and curved backgrounds, based on the ambient metric of Fefferman and Graham. The ambient metric is a Lorentzian Ricci-flat metric in $d+2$ dimensions and replaces the Minkowski metric of the embedding space. It is canonically associated with a $d$-dimensional conformal manifold, which is the physical spacetime where the CFT${}_d$ lives. We propose a construction of CFT${}_d$ correlators in non-trivial states and on curved backgrounds using appropriate geometric invariants of the ambient space as building blocks. As a test of the formalism we apply it to thermal 2-point functions and find exact agreement with a holographic computation and expectations based on thermal operator product expansions (OPEs).

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A bstract
We compute thermal 2-point correlation functions in the black brane AdS 5 background dual to 4d CFT’s at finite temperature for operators of large scaling dimension. We find a formula that matches the expected structure of the OPE. It exhibits an exponentiation property, whose origin we explain. We also compute the first correction to the two-point function due to graviton emission, which encodes the proper time from the event horizon to the black hole singularity.

A bstract
The eikonal phase which determines the Regge limit of the gravitational scat- tering amplitude of a light particle off a heavy one in Minkowski spacetimes admits an expansion in the ratio of the Schwarzschild radius of the heavy particle to the impact parameter. Such an eikonal phase in AdS spacetimes of any dimensionality has been com- puted to all orders and reduces to the corresponding Minkowski result when both the impact parameter and the Schwarzschild radius are much smaller than the AdS radius. The leading term in the AdS eikonal phase can be reproduced in the dual CFT by a single stress tensor conformal block, but the subleading term is a result of an infinite sum of the double stress tensor contributions. We provide a closed form expression for the OPE coef- ficients of the leading twist double stress tensors in four spacetime dimensions and perform the sum to compute the corresponding lightcone behavior of a heavy-heavy-light-light CFT correlator. The resulting compact expression passes a few nontrivial independent checks. In particular, it agrees with the subleading eikonal phase at large impact parameter.

A bstract
We study the OPE coefficients c Δ, J for heavy-light scalar four-point functions, which can be obtained holographically from the two-point function of a light scalar of some non-integer conformal dimension Δ L in an AdS black hole. We verify that the OPE coefficient c d ,0 = 0 for pure gravity black holes, consistent with the tracelessness of the holographic energy-momentum tensor. We then study the OPE coefficients from black holes involving matter fields. We first consider general charged AdS black holes and we give some explicit low-lying examples of the OPE coefficients. We also obtain the recursion formula for the lowest-twist OPE coefficients with at most two current operators. For integer Δ L , although the OPE coefficients are not fully determined, we set up a framework to read off the coefficients γ Δ, J of the log( z $$ \overline{z} $$ z ¯ ) terms that are associated with the anomalous dimensions of the exchange operators and obtain a general formula for γ Δ, J . We then consider charged AdS black holes in gauged supergravity STU models in D = 5 and D = 7, and their higher-dimensional generalizations. The scalar fields in the STU models are conformally massless, dual to light operators with Δ L = d − 2. We derive the linear perturbation of such a scalar in the STU charged AdS black holes and obtain the explicit OPE coefficient c d −2,0 . Finally, we analyse the asymptotic properties of scalar hairy AdS black holes and show how c d ,0 can be nonzero with exchanging scalar operators in these backgrounds.

A bstract
We probe the conformal block structure of a scalar four-point function in d ≥ 2 conformal field theories by including higher-order derivative terms in a bulk gravitational action. We consider a heavy-light four-point function as the boundary correlator at large central charge. Such a four-point function can be computed, on the gravity side, as a two-point function of the light operator in a black hole geometry created by the heavy operator. We consider analytically solving the corresponding scalar field equation in a near-boundary expansion and find that the multi-stress tensor conformal blocks are insensitive to the horizon boundary condition. The main result of this paper is that the lowest-twist operator product expansion (OPE) coefficients of the multi-stress tensor conformal blocks are universal: they are fixed by the dimension of the light operators and the ratio between the dimension of the heavy operator and the central charge C T . Neither supersymmetry nor unitary is assumed. Higher-twist coefficients, on the other hand, generally are not protected. A recursion relation allows us to efficiently compute universal lowest-twist coefficients. The universality result hints at the potential existence of a higher-dimensional Virasoro-like symmetry near the lightcone. While we largely focus on the planar black hole limit in this paper, we include some preliminary analysis of the spherical black hole case in an appendix.

A bstract
We study conformal higher spin (CHS) fields on constant curvature backgrounds. By employing parent formulation technique in combination with tractor description of GJMS operators we find a manifestly factorized form of the CHS wave operators for symmetric fields of arbitrary integer spin s and gauge invariance of arbitrary order t ≤ s . In the case of the usual Fradkin-Tseytlin fields t = 1 this gives a systematic derivation of the factorization formulas known in the literature while for t > 1 the explicit formulas were not known. We also relate the gauge invariance of the CHS fields to the partially-fixed gauge invariance of the factors and show that the factors can be identified with (partially gauge-fixed) wave operators for (partially)-massless or special massive fields. As a byproduct, we establish a detailed relationship with the tractor approach and, in particular, derive the tractor form of the CHS equations and gauge symmetries.

A bstract
We initiate an approach to constraining conformal field theory (CFT) data at finite temperature using methods inspired by the conformal bootstrap for vacuum correlation functions. We focus on thermal one- and two-point functions of local operators on the plane. The KMS condition for thermal two-point functions is cast as a crossing equation. By studying the analyticity properties of thermal two-point functions, we derive a “thermal inversion formula” whose output is the set of thermal one-point functions for all operators appearing in a given OPE. This involves identifying a kinematic regime which is the analog of the Regge regime for four-point functions. We demonstrate the effectiveness of the inversion formula by recovering the spectrum and thermal one-point functions in mean field theory, and computing thermal one-point functions for all higher-spin currents in the critical O ( N ) model at leading order in 1 /N . Furthermore, we develop a systematic perturbation theory for thermal data in the large spin, low-twist spectrum of any CFT. We explain how the inversion formula and KMS condition may be combined to algorithmically constrain CFTs at finite temperature. Throughout, we draw analogies to the bootstrap for vacuum four-point functions. Finally, we discuss future directions for the thermal conformal bootstrap program, emphasizing applications to various types of CFTs, including those with holographic duals.

We develop the embedding formalism for conformal field theories, aimed at
doing computations with symmetric traceless operators of arbitrary spin. We use
an index-free notation where tensors are encoded by polynomials in auxiliary
polarization vectors. The efficiency of the formalism is demonstrated by
computing the tensor structures allowed in n-point conformal correlation
functions of tensors operators. Constraints due to tensor conservation also
take a simple form in this formalism. Finally, we obtain a perfect match
between the number of independent tensor structures of conformal correlators in
d dimensions and the number of independent structures in scattering amplitudes
of spinning particles in (d+1)-dimensional Minkowski space.

For conformal field theories in arbitrary dimensions, we introduce a method
to derive the conformal blocks corresponding to the exchange of a traceless
symmetric tensor appearing in four point functions of operators with spin.
Using the embedding space formalism, we show that one can express all such
conformal blocks in terms of simple differential operators acting on the basic
scalar conformal blocks. This method gives all conformal blocks for conformal
field theories in three dimensions. We demonstrate how this formalism can be
applied in a few simple examples.

We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the case of pure gravity up to six dimensions and of gravity coupled to scalars. The method can also be viewed as providing a holographic reconstruction of the bulk spacetime metric and of bulk fields on this spacetime, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. To continue the holographic reconstruction of the bulk fields one needs new CFT data: the expectation value of the dual operator. In particular, in order to obtain the bulk metric one needs to know the expectation value of stress-energy tensor of the boundary theory. We provide completely explicit formulae for the holographic stress-energy tensors up to six dimensions. We show that both the gravitational and matter conformal anomalies of the boundary theory are correctly reproduced. We also obtain the conformal transformation properties of the boundary stress-energy tensors. Comment: 27 pages,v2: typos corrected, two references added, to appear in CMP

In this paper we relate the Fefferman–Graham ambientmetric construction for conformal manifolds to the approach toconformal geometry via the canonical Cartan connection. We show thatfrom any ambient metric that satisfies a weakening of the usualnormalisation condition, one can construct the conformal standardtractor bundle and the normal standard tractor connection, which areequivalent to the Cartan bundle and the Cartan connection. This resultis applied to obtain a procedure to get tractor formulae for allconformal invariants that can be obtained from the ambient metricconstruction. We also get information on ambient metrics whichare Ricci flat to higher order than guaranteed by the results ofFefferman–Graham.

Motivated by the observed scale invariance of high-energy inelastic electron-proton scattering, we study the constraints on a Compton amplitude that follow if it is invariant under the full group of conformal coordinate transformations. Although the conformal group contains operations that interchange spacelike and timelike coordinate intervals, a large class of manifestly causal amplitudes can be constructed. We find that the strict application of conformal symmetry reduces the number of independent covariants from four to two in the case of a spin-zero target. The two covariants of the conformal Compton amplitude remain independent when it is restricted to forward scattering. Hence, invariance under the full conformal group yields no constraints on the structure functions of inelastic electron scattering other than that already provided by simple dilation invariance.

- A M Polyakov

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Massive Higher Spins from BRST and Tractors

- M Grigoriev
- A Waldron

M. Grigoriev and A. Waldron, Massive Higher Spins from
BRST and Tractors, Nucl. Phys. B 853, 291 (2011),
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A Calculus for Higher Spin Interactions

- E Joung
- M Taronna
- A Waldron

E. Joung, M. Taronna, and A. Waldron, A Calculus for
Higher Spin Interactions, JHEP 07, 186, arXiv:1305.5809
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Higher Spin Extension of Fefferman-Graham Construction

- X Bekaert
- M Grigoriev
- E D Skvortsov

X. Bekaert, M. Grigoriev, and E. D. Skvortsov, Higher
Spin Extension of Fefferman-Graham Construction, Universe 4, 17 (2018), arXiv:1710.11463 [hep-th].

An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity

- S Curry
- A R Gover

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Boundary calculus for conformally compact manifolds

- A R Gover
- A Waldron

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119 (2014), arXiv:1104.2991 [math.DG].

Poincare-Einstein Holography for Forms via Conformal Geometry in the Bulk

- A Rod Gover
- E Latini
- A Waldron

A. Rod Gover, E. Latini, and A. Waldron, Poincare-Einstein Holography for Forms via Conformal Geometry
in the Bulk, (2012), arXiv:1205.3489 [math.DG].

- C R Graham
- C Guillarmou
- P Stefanov
- G Uhlmann

C. R. Graham, C. Guillarmou, P. Stefanov, and
G. Uhlmann, X-ray Transform and Boundary Rigidity for Asymptotically Hyperbolic Manifolds, Annales
de l'Institut Fourier 69, 2857 (2019), arXiv:1709.05053
[math.DG].

Conformal field theories at nonzero temperature: Operator product expansions, Monte Carlo, and holography

- E Katz
- S Sachdev
- E S Sørensen
- W Witczak-Krempa

E. Katz, S. Sachdev, E. S. Sørensen, and W. Witczak-Krempa, Conformal field theories at nonzero temperature: Operator product expansions, Monte Carlo,
and holography, Phys. Rev. B 90, 245109 (2014),
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Constraining Quantum Critical Dynamics: (2+1)D Ising Model and Beyond

- W Witczak-Krempa

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