AdS/CFT at Loop Order

Abstract

Since the early days of AdS/CFT, rigorous obtention of tree-level holographic correlators from a given bulk theory is known through the process of holographic renormalization, however the picture where quantum corrections are also taken into account is currently lacking. It is the purpose of this thesis to fill this gap in the literature and propose a method of holographic renormalization valid to all orders in the bulk loop expansion, where it is found that the same prescription from classical order is valid provided one replaces the classical action by the effective action. This gives a first principle derivation of Witten-Feynman rules for diagrams in AdS analogous to those in flat space.

In addition to the usual IR divergences present in holography, at loop order there are also UV divergences coming from the short-distance singularities of the bulk propagators, and this led us to construct a novel AdS invariant regularization scheme which we denote as geodesic point-splitting. Its derivation from a regularized action in AdS and its connection with an IR regulator for the dual CFT is also discussed.

The quantum corrections to the correlators take the form of loop Witten diagrams in AdS, and we show they obey the conformal Ward identities to all loop orders by explicitly writing a general loop diagram in the expected CFT form. Direct computation of loop Witten diagrams is challenging, and we also make progress in this direction by providing new and exact results for the most basic yet essential loop vertices in AdS appearing in almost every theory. How these are constrained by AdS symmetry is also discussed.

As an example of our methods, we work out in detail the case of a scalar Φ⁴ theory in the bulk, obtaining the renormalized 2-point function of the dual operator to 2 loops and the 4-point function to 1 loop, for operators of arbitrary dimension ∆ > d/2 and bulk spacetime dimensions up to d + 1 = 7.

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Identifiers

ORCID for Ernesto Bianchi: orcid.org/0000-0002-8758-1105

ORCID for Kostas Skenderis: orcid.org/0000-0003-4509-5472

ORCID for Oscar Dias: orcid.org/0000-0003-4855-4750

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