Lecture notes on holographic renormalization

By Kostas Skenderis. Classical and quantum gravity, 2002

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We review the formalism of holographic renormalization. We start by discussing mathematical results on asymptotically anti-de Sitter (AdS) spacetimes. We then outline the general method of holographic renormalization. The method is illustrated by working all details in a simple example: a massive scalar field on anti-de Sitter spacetime. The discussion includes the derivation of the on-shell renormalized action, holographic Ward identities, anomalies and renormalization group (RG) equations, and the computation of renormalized one-, two- and four-point functions. We then discuss the application of the method to holographic RG flows. We also show that the results of the near-boundary analysis of asymptotically AdS spacetimes can be analytically continued to apply to asymptotically de Sitter spacetimes. In particular, it is shown that the Brown–York stress energy tensor of de Sitter spacetime is equal, up to a dimension-dependent sign, to the Brown–York stress energy tensor of an associated AdS spacetime.

Subfields: Astronomy and Astrophysics, Atomic and Molecular Physics, and Optics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/96e41bd4588e284557bc0389973518e1ced6b637

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Citations in the corpus

Comments on holographic complexity (2016, background): “1) is expressed in terms of the dimensionless radial coordinate ρ = z(2)/L(2) — see [26, 27].”
Thermodynamics and instabilities of a strongly coupled anisotropic plasma (2011, background): “ravity action with respect to the boundary metric. The action suers from large-volume divergences, which can be regularized and subtracted by a procedure called holographic renormalization (see e.g. [107] and references therein). The total on-shell action takes the form Son-shell = Sbulk Sct ; (20) ”
Comparison between various notions of conserved charges in asymptotically AdS spacetimes (2005, methodology): “KBL approach [9, 10], the spinor definition [11, 12] based on an original idea by Witten [13], and the “counterterm subtraction method” by Henningson and Skenderis and by Balasubramanian and Kraus [14, 15, 16, 17, 18, 19 , 20, 21, 22, 23].”
Integrating out geometry: holographic Wilsonian RG and the membrane paradigm (2010, background): “with the corresponding canonical momenta along z-direction (as dened in ( 1; 2. Then Sct can be found to be (see e.g., [ 62 ])”
The holographic fluid dual to vacuum Einstein gravity (2011, background): “remains elusive and indeed the structure of the asymptotic solutions [1–3] and the divergences of the on-shell action suggest [4] that if there is a dual theory it is nonlocal, see also [5].”
The fuzzball proposal for black holes (2008, background): “Higher point functions can be obtained by solving the fluctuation equations around the original solution [31, 32, 33, 34] but this will not be reviewed here.”
Lectures on localization and matrix models in supersymmetric Chern–Simons-matter theories (2011, background): “Useful reviews, focused on AdS5, can be found in for example [91, 66].”
Chiral symmetry breaking as open string tachyon condensation (2007, methodology): “This is done by following the so-called holographic renormalization procedure (for a review, see [53]).”
Higher order singletons, partially massless fields and their boundary values in the ambient approach (2013, background): “g.[33]) where we focus on the particular case corresponding to the (higherorder) singleton and discuss some of its subtleties.”
Topologically Massive Gravity and the AdS/CFT Correspondence (2009, methodology): “For theories that admit asymptotically locally AdS solutions the most general asymptotic solution, which is sometimes called the “Fefferman-Graham” expansion, can always be found by solving algebraic equations, see [21] for a review.”
Holographic Superconductors (2008, background): “The counter terms we need to regulate it are standard (see for example [24]).”
Gauge-gravity duality and thermalization of a boost-invariant perfect fluid (2006, methodology): “In the above paper, a holographic renormalization procedure [2] using Fefferman-Graham coordinates [3] allowed to construct the gravity duals of a continuous 1-parametric set of 4-d stress-energy tensors of the gauge theory containing among others the duals of the free streaming and perfect fluid ca”
Fuzzballs with internal excitations (2007, methodology): “A systematic method for extracting vevs from asymptotically AdS ×X solutions (with X an arbitrary compact manifold) was only recently constructed, in [10], building on earlier work [24, 25, 26, 27, 28], see also the review [29].”
Non-relativistic holography (2008, methodology): “We will proceed using the standard principles of holographic renormalization [49]: we derive the general asymptotic solution to the field equations, parameterize the volume divergences of the onshell action, introduce local counterterms and then obtain the renormalized one point function by function”
Scalar 3-point functions in CFT: renormalisation, beta functions and anomalies (2015, positive): “The 1-point function in the presence of sources for the operator O dual to the bulk scalar Φ reads [48, 49] 〈O〉s = −(2∆− d)Φ(∆) +X[φ0], (D.”