We review the formalism of holographic renormalization. We start by discussing mathematical results on asymptotically anti-de Sitter 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, of holographic Ward identities, anomalies and 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.
We introduce the gauge/gravity, or AdS/CFT, correspondence with an eye towards its application to strongly coupled systems. We provide an overview of the duality, including the AdS/CFT dictionary and holographic renormalization. We then discuss simple correlation functions in the context of renormalization group flow geometries, holographic thermodynamics and an application to the phase diagram of QCD, real-time correlators and the shear viscosity of strongly coupled field theories, holographic superconductors, and the application of holographic fermionic response to strange metals. These lect…
These are notes from my lectures at TASI 2015. The goal is to provide context for the study of strongly-correlated quantum many-body systems using quantum field theory, and possibly string theory.
Lecture notes prepared for the 2021 SAGEX PhD School in Amplitudes hosted by the University of Copenhagen August 10th through 13th. Topics covered include: the manifestation of asymptotic symmetries via soft theorems, their organization into currents in a celestial CFT, aspects of the holographic dictionary, a literature guide, and accompanying exercises.
Notes on holographic renormalization as a tool for UV analysis and derivation of the two-point Ward identity encoding symmetry breaking. Goldstone theorem is reviewed, discussing in particular the modifications required by its extension to non-relativistic field theories. The Ward identity encoding both spontaneous and explicit breaking are derived by general principles. Holographic renormalization is presented in a quite operational fashion. The derivation of the Ward identity in holography is illustrated by an example.
The bulk locality in the constructive holographic renormalization group requires miraculous cancellations among various local renormalization group functions. The cancellation is not only from the properties of the spectrum but from more detailed aspects of operator product expansions in relation to conformal anomaly. It is remarkable that one-loop computation of the universal local renormalization group functions in the weakly coupled limit of the $\mathcal{N}=4$ super Yang-Mills theory fulfils the necessary condition for the cancellation in the strongly coupled limit in its $SL(2,\mathbf{Z})…
We review holographic renormalization for non-conformal branes using the Hamilton-Jacobi formalism. We provide the tools required for the computation of the holographic dictionary, while we present some marginally new technical results regarding holographic renormalization for $Dp$-branes, $p<5$, for Einstein-Maxwell theory coupled to a dilaton. The computations are based on a recursive algorithm by Papadimitriou, that solves the radial Hamilton-Jacobi equation in the asymptotic boundary of the bulk. This paper is a short version of the Master thesis of the author written in 2014 at the Instit…
We construct a three-dimensional geometry interpolating two different AdS spaces. From the dual quantum field theory viewpoint, it corresponds to a nontrivial renormalization group flow from a UV to another IR conformal field theory. On this geometry, we discuss the change of the central charge in a momentum-space and real-space renormalization group flow. The result shows in both renormalization prescriptions that the central charge monotonically decreases along the renormalization group flow.
We look into the $AdS$ black holes from two-dimensional gravity perspective. In this work, we extend the previous results of holographic renormalization group flows to dimensions two. By introducing a superpotential, we derive the flow equations in two-dimensional dilaton gravity. We find a quantity which monotonically decreases along flows and give some comments on holographic $c$-theorem. As examples, we show that recently studied supersymmetric $AdS$ black hole solutions generically dimensionally reduce to two-dimensional dilaton gravity, and obtain the flow equations for black hole solutio…
We study a mathematical relationship between holographic Wilsonian renormalization group and stochastic quantization framework. We extend the original proposal given in arXiv:1209.2242 to interacting theories. The original proposal suggests that fictitious time(or stochastic time) evolution of stochastic 2-point correlation function will be identical to the radial evolution of the double trace operator of certain classes of holographic models, which are free theories in AdS space. We study holographic gravity models with interactions in AdS space and establish a map between the holographic ren…
We extend the relation between stochstic quantization and Holographic Wilsonian Renormalization Group to the massive fermion case in Euclidean AdS space. The original suggestion about the explicit relation between the two theories is in arXiv:1209.2242. To give a fermion bulk action the Dirichlet boundary condition and the Neumann boundary condition, there are some additional terms to the bulk action. We have confirmed that the relation also hold for massive fermion case.
We investigate subregion holographic complexity in the context of renormalization group flow geometries. We use both the Poincaŕe slicing and the Janus ansatz as holographic duals to renormalization group flows in the boundary conformal field theory. In the former metric, subregion complexity is computed for a disc and a strip shaped entangling region. For the disc shaped region, consistent emergence of length scales for flow to the deep infra-red is established. For strip shaped regions, we find that complexity cannot locate holographic phase transitions in a sharp domain wall scenario. For s…
Holographic renormalization is a systematic procedure for regulating divergences in observables in asymptotically locally AdS spacetimes. For dual boundary field theories which are supersymmetric it is natural to ask whether this defines a supersymmetric renormalization scheme. Recent results in localization have brought this question into sharp focus: rigid supersymmetry on a curved boundary requires specific geometric structures, and general arguments imply that BPS observables, such as the partition function, are invariant under certain deformations of these structures. One can then ask if …
A heavy quark moving through a strongly coupled deconfined plasma has a holographic dual description as a string moving in a black brane geometry. We apply the holographic Wilsonian renormalization method to derive a holographic effective string action dual to the heavy quark. The effective action only depends on the geometry between the black brane horizon and a cutoff localized in the radial direction, corresponding to the IR of the dual theory. We derive RG flow equations for the coefficients in the effective action and show that the force acting on the heavy quark is independent of the pos…
We investigate the holographic renormalization of a charged black brane with or without a dilaton field, whose dual field theory describes a dense medium at finite temperature. In a dense medium, two different thermodynamic descriptions are possible due to an additional conserved charge. These two different thermodynamic ensembles are classified by the asymptotic boundary condition of the bulk gauge field. We show that in the holographic renormalization regularity of all bulk fields can reproduce consistent thermodynamic quantities and that the Bekenstein-Hawking entropy is nothing but the ren…
The Hamilton-Jacobi method in holography has produced important results both at a renormalization group (RG) fixed point and away from it. In this paper we use the Hamilton-Jacobi method to compute the holographic trace anomaly for four- and six-dimensional boundary conformal field theories (CFTs), assuming higher-derivative gravity and interactions of scalar fields in the bulk. The scalar field contributions to the anomaly appear in CFTs with exactly marginal operators. Moving away from the fixed point, we show that the Hamilton-Jacobi formalism provides a deep connection between the holograp…
We study the holographic renormalization group (RG) flow in the presence of higher-order curvature corrections to the $(d+1)$-dimensional Einstein-Hilbert (EH) action for an arbitrary interacting scalar matter field by using the superpotential approach. We find the critical points of the RG flow near the local minima and maxima of the potential and show the existence of the bounce solutions. In contrast to the EH gravity, regarding the values of couplings of the bulk theory, superpotential may have both upper and lower bounds. Moreover, the behavior of the RG flow controls by singular curves. …
In this work, we present a holographic renormalization scheme for asymptotically anti-de Sitter spacetimes in which the dual renormalization scheme of the boundary field theory is dimensional regularization. This constitutes a new level of precision in the holographic dictionary and paves the way for the exact matching of scheme dependent quantities, such as holographic beta functions, with field theory computations. Furthermore, the renormalization procedure identifies a local source field which satisfies the equations of motion along renormalization group flows, resolving a long-standing puz…
We review construction of the improved holographic models for QCD-like confining gauge theories and their applications to the physics of the Quark-gluon plasma. We also review recent progress in this area of research. The lecture notes start from the vacuum structure of these theories, then develop thermodynamic and hydrodynamic observables and end with more advanced topics such as the holographic QCD in the presence of external magnetic fields. This is a summary of the lectures presented at the 56th Cracow School of Theoretical Physics in Spring 2016 at Zakopane, Poland.
Euclidean field theories admit more general deformations than usually discussed in quantum field theories because of mixing between rotational symmetry and internal symmetry (a.k.a topological twist). Such deformations may be relevant, and if the subsequent renormalization group flow leads to a non-trivial fixed point, it generically gives rise to a scale invariant Euclidean field theory without conformal invariance. Motivated by an ansatz studied in cosmological models some time ago, we develop a holographic dual description of such renormalization group flows in the context of AdS/CFT. We ar…
These notes, based on lectures given at various schools over the last few years, aim to provide an introduction to entanglement entropies in quantum field theories, including holographic ones. We explore basic properties and simple examples of entanglement entropies, mostly in two dimensions, with an emphasis on physical rather than formal aspects of the subject. In the holographic case, the focus is on how the Ryu-Takayanagi formula geometrically realizes general features of field-theory entanglement, while revealing special properties of holographic theories. In order to make the notes somew…
Holographic model of massive scalar field with its self-interaction $λϕ^n$ in AdS space is able to give a logarithmic scale dependence to marginal multi trace deformation couplings on its dual conformal field theory, where $λ$ is the self-interaction coupling of the scalar field, $ϕ$ and $n$ is an integral number. In arXiv:1501.06664, the authors realize this feature by looking at bulk scalar solutions near AdS boundary imposing a specific boundary condition between the coefficients of non-normalizable and normalizable modes of the scalar field excitations. We study the same holographic model …
