We show that AdS amplitudes are CFT correlators to all orders in the loop expansion by showing that they obey the conformal Ward identities. In particular, we provide explicit formulas for the constants and functions of cross-ratios that determine the CFT correlators in terms of bulk data.

Chile

Máximo Bañados

United Kingdom

Ernesto Bianchi

Iván Muñoz

Kostas Skenderis

AdS amplitudes as CFT correlators

We discuss the flat space limit of AdS using the momentum space representation of CFT correlators. The flat space limit involves sending the AdS radius and the dimensions of operators dual to massive fields to infinity while also scaling appropriately the sources of the dual operators. In this limit, d-dimensional CFT correlators become (d + 1)-dimensional scattering amplitudes. We exemplify our discussion with the computation of the flat-space limit of the CFT 3-point function of a conserved current, a non-conserved charged vector operator and its conjugate. The flat-space limit should yield the scattering amplitude of an Abelian gauge field with two massive vector fields. This scattering amplitude computes the electromagnetic form factors of the electromagnetic current in a spin-1 state, and these form factors encode the electromagnetic properties of the massive vector field (charge, magnetic moment and quadruple moment). In terms of the CFT, the flat-space limit amounts to zooming in the infrared region of the triple-K integrals that determine the 3-point function, while also scaling to infinity the order of (some of) the Bessel functions that feature in the triple-K integrals. In this limit the triple-K integral becomes proportional to the energy-preserving delta function, and the flat space limit correctly yields the corresponding flat space scattering amplitude in complete detail.

Italy

Raffaele Marotta

India

Mritunjay Verma

Flat space spinning massive amplitudes from momentum space CFT

We present a new formalism to solve the kinematical constraints due to Weyl invariance for CFTs in curved backgrounds and/or non-trivial states, and we apply it to thermal CFTs and to CFTs on squashed spheres. The ambient space formalism is based on constructing a class of geometric objects that are Weyl covariant and identifying them as natural building blocks of correlation functions. We construct (scalar) n-point functions and we illustrate the formalism with a detailed computation of 2-point functions. We compare our results for thermal 2-point functions with results that follow from thermal OPEs and holographic computations, finding exact agreement. In our holographic computation we also obtain the OPE coefficient of the leading double-twist contribution, and we discuss how the double-twist coefficients may be computed from the multi-energy-momentum contributions, given knowledge of the analytic structure of the correlator. The 2-point function for the CFT on squashed spheres is a new result. We also discuss the relation of our work to flat holography.

Enrico Parisini

Benjamin Withers

The ambient space formalism

We formulate a renormalisation procedure for IR divergences of tree-level in-in late-time de Sitter (dS) correlators. These divergences are due to the infinite volume of spacetime and are analogous to the divergences that appear in AdS dealt with by holographic renormalisation. Regulating the theory using dimensional regularisation, we show that one can remove all infinities by adding local counterterms at the future boundary of dS in the Schwinger-Keldysh path integral. The counterterms amount to renormalising the late-time bulk field. We frame the discussion in terms of bulk scalar fields in dS d+1, using the computation of tree-level correlators involving massless and conformal scalars for illustration. The relation to AdS via analytic continuation is discussed, and we show that different versions of the analytic continuation appearing in the literature are equivalent to each other. In AdS, one needs to add counterterms that are related to conformal anomalies, and also to renormalise the source part of the bulk field. The analytic continuation to dS projects out the traditional AdS counterterms, and links the renormalisation of the sources to the renormalisation of the late-time bulk field. We use these results to establish holographic formulae that relate tree-level dS d+1 in-in correlators to CFT correlators at up to four points, and we provide two proofs: one using the connection between the dS wavefunction and the partition function of the dual CFT, and a second by direct evaluation of the in-in correlators using the Schwinger-Keldysh formalism. The renormalisation of the bulk IR divergences is mapped by these formulae to UV renormalisation of the dual CFT via local counterterms, providing structural support for a possible duality. We also recast the regulated holographic formulae in terms of the AdS amplitudes of shadow fields, but show that this relation breaks down when renormalisation is required.

Poland

Adam Bzowski

Paul McFadden

Renormalisation of IR divergences and holography in de Sitter

We analyse the 3-point CFT correlators involving non-conserved spinning operators in momentum space. We derive a general expression for the conformal Ward identities defining the 3-point functions involving two generic spin s non-conserved operators and a spin 1 conserved current. We give explicit expressions for the 3-point function when the two non-conserved operators have spins 1 and 2 and generic conformal dimensions. We also systematically analyse the divergences appearing in these 3-point functions when the conformal dimensions of the two non-conserved operators coincide.

Momentum space CFT correlators of non-conserved spinning operators

We present a generalization of the embedding space formalism to conformal field theories (CFTs) on nontrivial states and curved backgrounds, based on the ambient metric of Fefferman and Graham. The ambient metric is a Lorentzian Ricci-flat metric in <math><mrow><mi>d</mi><mo>+</mo><mn>2</mn></mrow></math> dimensions and replaces the Minkowski metric of the embedding space. It is canonically associated with a <math><mi>d</mi></math>-dimensional conformal manifold, which is the physical spacetime where the <math><mrow><msub><mrow><mi>CFT</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math> lives. We propose a construction of <math><mrow><msub><mrow><mi>CFT</mi></mrow><mrow><mi>d</mi></mrow></msub></mrow></math> <math><mi>n</mi></math>-point functions in nontrivial states and on curved backgrounds using appropriate geometric invariants of the ambient space as building blocks. This captures the contributions of nonvanishing one-point functions of multi-stress-energy tensors, at least in holographic CFTs. We apply the formalism to two-point functions of thermal CFT, finding exact agreement with a holographic computation and expectations based on thermal operator product expansions (OPEs), and to CFTs on squashed spheres where no prior results are known and existing methods are difficult to apply, demonstrating the utility of the method.

Embedding formalism for CFTs in general states on curved backgrounds

We develop a systematic renormalization procedure for QFT in anti-de Sitter spacetime. UV infinities are regulated using a geodesic point-splitting method, which respects AdS isometries, while IR infinities are regulated by cutting off the radial direction (as in holographic renormalization). The renormalized theory is defined by introducing <math><mrow><mi>Z</mi></mrow></math> factors for all parameters in the Lagrangian and the boundary conditions of bulk fields (sources of dual operators), and a boundary counterterm action, <math><msub><mi>S</mi><mrow><mi>ct</mi></mrow></msub></math>, such that the limit of removing the UV and IR regulators exists. The results are in general scheme dependent (mirroring the analogous result in flat space) and require renormalization conditions. These may be provided by the dual CFT (or by string theory in AdS). Our analysis amounts also to a first principles derivation of the Feynman rules regarding Witten diagrams. The presence and treatment of IR divergences is essential for correctly accounting for anomalous dimensions of dual operators. We apply the method to scalar <math><mrow><msup><mrow><mi>Φ</mi></mrow><mrow><mn>4</mn></mrow></msup></mrow></math> theory and obtain the renormalized two-point function of the dual operator to two loops, and the renormalized four-point function to one-loop order, for operators of any dimension <math><mi>Δ</mi></math> and bulk spacetime dimension up to <math><mrow><mi>d</mi><mo>+</mo><mn>1</mn><mo>=</mo><mn>7</mn></mrow></math>.

Bulk renormalization and the <math><mrow><mi>AdS</mi><mo>/</mo><mi>CFT</mi></mrow></math> correspondence

We present a comprehensive discussion of tree-level holographic 4-point functions of scalar operators in momentum space. We show that each individual Witten diagram satisfies the conformal Ward identities on its own and is thus a valid conformal correlator. When the β = ∆ − d/2 are half-integral, with ∆ the dimensions of the operators and d the spacetime dimension, the Witten diagrams can be evaluated in closed form and we present explicit formulae for the case d = 3 and ∆ = 2, 3. These correlators require renormalization, which we carry out explicitly, and lead to new conformal anomalies and beta functions. Correlators of operators of different dimension may be linked via weight-shifting operators, which allow new correlators to be generated from given ‘seed’ correlators. We present a new derivation of weight-shifting operators in momentum space and uncover several subtleties associated with their use: such operators map exchange diagrams to a linear combination of exchange and contact diagrams, and special care must be taken when renormalization is required.

A handbook of holographic 4-point functions

We discuss the definition of conserved quantities in asymptotically locally de Sitter spacetimes. One may define an analog of holographic charges at future and past infinity and at other Cauchy surfaces <math><msub><mi>I</mi><mi>t</mi></msub></math> as integrals over the intersection of timelike surfaces <math><mi>C</mi></math> and the Cauchy surface <math><msub><mi>I</mi><mi>t</mi></msub></math>. In general, the charges <math><msup><mi>Q</mi><mi>t</mi></msup></math> defined on the Cauchy surface <math><msub><mi>I</mi><mi>t</mi></msub></math> depend on <math><mi>C</mi></math>, but if gravitational flux is absent the charges are independent of <math><mi>C</mi></math>. On the other hand, if there is a net gravitational flux entering or leaving the spacetime region bounded by <math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math>, <math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> and two Cauchy surfaces then <math><mi>Δ</mi><msup><mi>Q</mi><mi>t</mi></msup><mo>(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>,</mo><msub><mi>C</mi><mn>2</mn></msub><mo>)</mo><mo>=</mo><msup><mi>Q</mi><mi>t</mi></msup><mo>(</mo><msub><mi>C</mi><mn>1</mn></msub><mo>)</mo><mo>−</mo><msup><mi>Q</mi><mi>t</mi></msup><mo>(</mo><msub><mi>C</mi><mn>2</mn></msub><mo>)</mo></math> changes by the same amount.