Structural and Dynamical Aspects of the AdS/CFT Correspondence: a Rigorous Approach

Pedro Lauridsen Ribeiro

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Structural and Dynamical Aspects of the AdS/CFT Correspondence: a Rigorous Approach

Pedro Lauridsen Ribeiro

Mathematical Physics · Submitted 3 Dec 2007

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The authors' main claim The main body of the present work consists in extending Rehren's result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity's quantum theory.

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We elaborate a detailed study of certain aspects of (a version of) the AdS/CFT correspondence, conjectured by Maldacena and Witten, between quantum field theories in a gravitational background given by an asymptotically anti-de Sitter (AAdS) spacetime, and conformally covariant quantum field theories in the latter's conformal infinity (in the sense of Penrose), aspects such that: (a) are independent from (the pair of) specific models in Quantum Field Theory, and (b) susceptible to a recast in a mathematically rigorous mould. We adopt as a starting point the theorem demonstrated by Rehren in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality. The main body of the present work consists in extending Rehren's result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity's quantum theory. The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch, extended afterwards by Sommer in order to incorporate boundary conditions.

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s0	We elaborate a detailed study of certain aspects of (a version of) the AdS/CFT correspondence, conjectured by Maldacena and Witten, between quantum field theories in a gravitational background given by an asymptotically anti-de Sitter (AAdS) spacetime, and conformally covariant quantum field theories in the latter's conformal infinity (in the sense of Penrose), aspects such that: (a) are independent from (the pair of) specific models in Quantum Field Theory, and (b) susceptible to a recast in a mathematically rigorous mould.	contribution 0.86	0.57	0.09	asserted 0.72
s1	We adopt as a starting point the theorem demonstrated by Rehren in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality.	method 0.83	0.29	0.04	asserted 0.98
s2	The main body of the present work consists in extending Rehren's result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity's quantum theory.	contribution 1.00	0.89	0.19	asserted 0.82
s3	The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch, extended afterwards by Sommer in order to incorporate boundary conditions.	method 0.99	0.16	0.02	asserted 0.99

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Original abstract

We elaborate a detailed study of certain aspects of (a version of) the AdS/CFT correspondence, conjectured by Maldacena and Witten, between quantum field theories in a gravitational background given by an asymptotically anti-de Sitter (AAdS) spacetime, and conformally covariant quantum field theories in the latter's conformal infinity (in the sense of Penrose), aspects such that: (a) are independent from (the pair of) specific models in Quantum Field Theory, and (b) susceptible to a recast in a mathematically rigorous mould. We adopt as a starting point the theorem demonstrated by Rehren in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality. The main body of the present work consists in extending Rehren's result to a reasonably general class of d-dimensional AAdS spacetimes (d>3), scrutinizing how the properties of such an extension are weakened and/or modified as compared to AdS spacetime, and probing how non-trivial gravitational effects manifest themselves in the conformal infinity's quantum theory. The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch, extended afterwards by Sommer in order to incorporate boundary conditions.

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At the end of that section, we compare our notion of wedges to other definitions which have been made in the literature [ref] , and point out the similarities and differences.

Deformations of quantum field theories on spacetimes with Killing vector fields · Geometric setup · 0.98

Cites it as background 4 papers

In fact, different definitions are possible, and wedges on special manifolds have been studied by many authors in the literature [ref] .

Deformations of quantum field theories on spacetimes with Killing vector fields · Introduction · 0.87
We consider generic boundary regions $$ and $$ (using the subscript $t$ to indicate when they lie on the same time slice) and describe simple relational properties[ See also e.g. [ref] for earlier related discussions. ] of causal wedges etc. associated with them, including observations pertaining to our constructs in causally non-trivial spacetimes.

Global properties of causal wedges in asymptotically AdS spacetimes · General features of $A$ and $χ_A$ · 0.94
This is a maximally symmetric solution of the Einstein equations with negative cosmological constant which is the arena for the renown AdS/CFT correspondence — see for example the recent monograph [ref] —, which has also been investigated rather extensively from the viewpoint of algebraic quantum field theory [ref] .

On the algebraic quantization of a massive scalar field in anti-de-Sitter spacetime · Introduction · 0.98

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We adopt as a starting point the theorem demonstrated by Rehren [ref] in the context of Local Quantum Physics (also known as Algebraic Quantum Field Theory) in anti-de Sitter (AdS) spacetimes, called algebraic holography or Rehren duality.

Algebraic Holography · Introduction · 0.84

Uses the method or tool of 14 papers

This result is original [ref] , and the proof involves an interesting application of the method employed by Penrose [ref] to prove the existence of singularities as result of gravitational collapse, together with a compactness argument, to obtain a geometric maximum principle that guarantees the precise envelope (Theorem [ref], page ch1t9).

Algebraic holography in asymptotically simple, asymptotically AdS spacetimes · On the structure of this Thesis · 0.86
The language employed for the quantum theories relevant for our generalization of Rehren duality follows the functorial formulation of Local Quantum Physics due to Brunetti, Fredenhagen and Verch [ref] , extended afterwards by Sommer [ref] in order to incorporate boundary conditions.

The generally covariant locality principle -- A new paradigm for local quantum physics · Introduction · 0.85
Employing the techniques of Borchers and Yngvason [ref] (stated in Theorems [ref], [ref] and [ref], page ch4t1ff.), we shall show that such a condition can be understood as a prescription for the scaling behaviour of observables in a neighborhood of infinity (Proposition [ref], page ch4p1), which can be generalized to asymptotically anti-de Sitter spacetimes.

Modular Groups of Quantum Fields in Thermal States · On the structure of this Thesis · 0.94

Compares with 5 papers

The presentation of this Chapter has several improvements with respect to [ref] .

Algebraic holography in asymptotically simple, asymptotically AdS spacetimes · On the structure of this Thesis · 0.92
The equivalence in item (iv) of Proposition [ref] shows that, indeed, ([ref]) and ([ref]) act effectively as a condition on scaling limits around points in $ℐ$, thus revealing the relation between this set of hypotheses and the formulation of Bertola et al. [ref] , based in $k$-point Wightman functions in $AdS_d$.

AdS/CFT correspondence for n-point functions · Boundary values as a dynamical condition · 0.80
A proof of a result similar to Theorem [ref], employing a somewhat different strategy, was proposed by Gao and Wald [ref] .

Theorems on gravitational time delay and related issues · Causality (bulk to boundary) · 0.97

Cites as support 3 papers

There are, inclusively, evidences [ref] that, for nontrivial AAdS geometries, Rehren duality becomes physically incompatible with known tests of the Maldacena conjecture, and, as such, needs to be reformulated.

The Maldacena Conjecture and Rehren Duality · Introduction · 0.97
An example which motivates such a conjecture is the relation between the poles of retarded two-point functions of the dual quantum theory at the boundary, which determine the properties of return to equilibrium, and quasinormal modes of AAdS black holes [ref] .

Minkowski-space correlators in AdS/CFT correspondence: recipe and applications · Obstructions to the inversion of Rehren duality · 0.98
An example which motivates such a conjecture is the relation between the poles of retarded two-point functions of the dual quantum theory at the boundary, which determine the properties of return to equilibrium, and quasinormal modes of AAdS black holes [ref] .

Quasinormal Modes of AdS Black Holes and the Approach to Thermal Equilibrium · Obstructions to the inversion of Rehren duality · 0.98

Cites as background 6 papers

We conclude this Subsection by outlining, for completeness, the original form of Rehren duality: [ref] .

Algebraic Holography · Locally covariant quantum theories in AdS · 0.88
Within the integrated description of gauge theories and gravity proposed by string theory, two results suggest that this theory may provide a realization of the holographic principle: the microscopic calculation of Bekenstein entropy made by Strominger and Vafa [ref] , and, above all, the work of Maldacena [ref] , which, by means of a(n effective limit of a) string model on the background spacetime $AdS_5×S^5$, where the $AdS_5$ factor is the anti-de Sitter (AdS) spacetime in five dimensions, obtained the field multiplet of a supersymmetric, conformally invariant gauge theory with structural group $SU(N)$ in the $N→∞$ limit, living at the conformal boundary of $AdS_5$ (the $S^5$ factor becomes a global group of internal (R-)symmetries $SU(4)∼SO(6)$).

The Large N Limit of Superconformal Field Theories and Supergravity · Introduction · 0.86
The formulation of this correspondence in terms of $k$-point functions was obtained by Gubser, Klebanov and Polyakov [ref] in a particular case, and more in general by Witten [ref] .

Anti De Sitter Space And Holography · Introduction · 0.90

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