Lovelock theory and the AdS/CFT correspondence

Lovelock theory is the natural extension of general relativity to higher dimensions. It can be also thought of as a toy model for ghost-free higher curvature corrections in gravitational theories. It admits a family of AdS vacua, which provides an appealing arena to explore different holographic aspects in a broader setup within the context of the AdS/CFT correspondence. We will elaborate on these features and review previous work concerning the constraints that Lovelock theory entails on the CFT parameters when imposing conditions like unitarity, positivity of the energy or causality.     

Applications of the AdS/CFT correspondence

In this writeup of a talk delivered at the Lepton Photon Conference 2011, applications of the AdS/CFT correspondence in diverse areas of physics are reviewed.     

Supersymmetry in the AdS/CFT correspondence

We study how local symmetry transformations of (p,q) anti de Sitter supergravities in three dimensions act on fields on the two-dimensional boundary. The boundary transformation laws are shown to be the same as those of two-dimensional (p,q) conformal supergravities for p,q≤2. Weyl and super Weyl transformations are generated from three-dimensional general coordinate and super transformations.     

Aspects of holography in the AdS/CFT correspondence

We study quantities which are important for the realization of the holographic principle in the AdS/CFT correspondence: boundaries, geodesics and the propagators of scalar fields. They should play a role in the holographic setup in the BMN limit as well. We review the backgrounds and their relations that are relevant in the AdS/CFT correspondence and in its BMN limit. We describe the realization of the holographic principle in the AdS/CFT correspondence and summarize proposals of how to translate it in the BMN limit. We analyze the boundaries, geodesics and propagators of scalar fields in detail and observe how they behave in the limiting process from AdS₅×S⁵ to the 10‐dimensional plane wave, which is the spacetime in the BMN limit.     

On the origin of supergravity boundary terms in the AdS /CFT correspondence

The standard formulation of the AdS/CFT correspondence is incomplete since it requires adding to a supergravity action some a priori unknown boundary terms. We suggest a modification of the correspondence principle based on the Hamiltonian formulation of the supergravity action, which does not require any boundary terms. Then all the boundary terms of the standard formulation naturally appear by passing from the Hamiltonian version to the Lagrangian one. As examples the graviton part of the supergravity action on the product of AdS_d+1 with a compact Einstein manifold ϵ and fermions on AdS_d+1, are considered. We also discuss conformal transformations of gravity fields on the boundary of AdS and show that they are induced by the isometrics of AdS.     

Higher impurity AdS/CFT correspondence in the near-BMN limit

The pp-wave/BMN limit of the AdS/CFT correspondence has exposed the Maldacena conjecture to a new regimen of direct tests. In one line of pursuit, finite-radius curvature corrections to the Penrose limit (which appear in inverse powers of the string angular momentum J) have been found to induce a complicated system of interaction perturbations to string theory on the pp-wave; these have been successfully matched to corresponding corrections to the BMN dimensions of N=4 super-Yang–Mills (SYM) operators to two loops in the 't Hooft coupling λ. This result is tempered by a well-established breakdown in the correspondence at three loops. Notwithstanding the third-order mismatch, we proceed with this line of investigation by subjecting the string and gauge theories to new and significantly more rigorous tests. Specifically, we extend our earlier results at O(1/J) in the curvature expansion to include string states and SYM operators with three worldsheet or R-charge impurities. In accordance with the two-impurity problem, we find a perfect and intricate agreement between both sides of the correspondence to two-loop order in λ and, once again, the string and gauge theory predictions fail to agree at third order.     

Parity asymmetric boost invariant plasma in AdS/CFT correspondence

We consider a simple extension to the previously found gravity solution corresponding to a boost invariant Bjorken plasma, by allowing components that are asymmetric under parity flipping of the spacetime rapidity. Besides the question whether this may have a realization in collisions of different species of projectiles, such as lead-gold collision, our new time-dependent gravity background can serve as a test ground for the recently proposed second order conformal viscous hydrodynamics. We find that non-trivial parity-asymmetric effects start to appear at second order in late time expansion, and we map the corresponding energy–momentum tensor to the second order conformal hydrodynamics to find certain second order transport coefficients. Our results are in agreement with the previous results in literature, giving one more corroborative evidence for the validity of the framework.     

Note on the massive Rarita-Schwinger field in the AdS/CFT correspondence

We consider a massive Rarita-Schwinger field on the Anti-de Sitter space and solve the corresponding equations of motion. We show that appropriate boundary terms calculated on-shell give two-point correlation functions for spin-3/2 fields of the conformal field theory on the boundary. The relation between Rarita-Schwinger field masses and conformal dimensions of corresponding operators is established.     

Conformal symmetry of relativistic and nonrelativistic systems and AdS/CFT correspondence

The nonlinear realization of conformal so(2,d) symmetry for relativistic systems and the dynamical conformal so(2,1) symmetry of nonrelativistic systems are investigated in the context of AdS/CFT correspondence. We show that the massless particle in d-dimensional Minkowski space can be treated as the system confined to the border of the AdS_d+1 of infinite radius, while various nonrelativistic systems may be canonically related to a relativistic (massless, massive, or tachyon) particle on the AdS₂ × S^d−1. The list of nonrelativistic systems “unified” by such a correspondence comprises the conformal mechanics model, the planar charge-vortex and three-dimensional charge-monopole systems, the particle in a planar gravitational field of a point massive source, and the conformal model associated with the charged particle propagating near the horizon of the extreme Reissner-Nordström black hole.     

Extremal static AdS black hole/CFT correspondence in gauged supergravities

A recently proposed holographic duality allows the Bekenstein–Hawking entropy of extremal rotating black holes to be calculated microscopically, by applying the Cardy formula to the two-dimensional chiral CFTs associated with certain reparameterisations of azimuthal angular coordinates in the solutions. The central charges are proportional to the angular momenta of the black hole, and so the method degenerates in the case of static (non-rotating) black holes. We show that the method can be extended to encompass such charged static extremal AdS black holes by using consistent Kaluza–Klein sphere reduction ansatze to lift them to exact solutions in the low-energy limits of string theory or M-theory, where the electric charges become reinterpreted as angular momenta associated with internal rotations in the reduction sphere. We illustrate the procedure for the examples of extremal charged static AdS black holes in four, five, six and seven dimensions.     

Unraveling AdS/CFT

In this review, I survey the conjectured correspondence between a string theory in ten dimensions and certain supersymmetric gauge theories in four. This duality has recently garnered considerable attention from scientists studying the hot matter produced in heavy-ion collisions. An important and immediate question is to what extent one can hope to describe the dynamics of the quark–gluon plasma in a supersymmetric conformal field theory. Here I explain recent applications of the AdS/CFT correspondence to the strongly interacting matter produced at the Relativistic Heavy Ion Collider. Progress in characterizing the medium with these techniques will be discussed, as well as limitations inherent to the method.     

Brick walls and AdS/CFT

We discuss the relationship between the bulk-boundary correspondence in Rehren’s algebraic holography (and in other ‘fixed-background’, QFT-based, approaches to holography) and in mainstream string-theoretic ‘Maldacena AdS/CFT’. Especially, we contrast the understanding of black-hole entropy from the point of view of QFT in curved spacetime—in the framework of ’t Hooft’s ‘brick wall’ model—with the understanding based on Maldacena AdS/CFT. We show that the brick-wall modification of a Klein–Gordon field in the Hartle–Hawking–Israel state on $1+2$ dimensional Schwarzschild AdS has a well-defined boundary limit with the same temperature and entropy as the brick-wall-modified bulk theory. One of our main purposes is to point out a close connection, for general AdS/CFT situations, between the puzzle raised by Arnsdorf and Smolin regarding the relationship between Rehren’s algebraic holography and mainstream AdS/CFT and the puzzle embodied in the ‘complementarity principle’ proposed by Mukohyama and Israel in their work on the brick-wall approach to black hole entropy. Working on the assumption that similar results will hold for bulk QFT other than the Klein–Gordon field and for Schwarzschild AdS in other dimensions, and recalling the first author’s proposed resolution to the Mukohyama–Israel puzzle based on his ‘matter–gravity entanglement hypothesis’, we argue that, in Maldacena AdS/CFT, the algebra of the boundary CFT is isomorphic only to a proper subalgebra of the bulk algebra, albeit (at non-zero temperature) the (GNS) Hilbert spaces of bulk and boundary theories are still the ‘same’—the total bulk state being pure, while the boundary state is mixed (thermal). We also argue from the finiteness of its boundary (and hence, on our assumptions, also bulk) entropy at finite temperature, that the Rehren dual of the Maldacena boundary CFT cannot itself be a QFT and must, instead, presumably be something like a string theory.