Lectures on the holographic duality of gauge fields and strings

TASI lectures on quantum matter (with a view toward holographic duality)

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.

Lectures on Holographic Superfluidity and Superconductivity

Four lectures on holography and the AdS/CFT correspondence applied to condensed matter systems. The first lecture introduces the concept of a quantum phase transition. The second lecture discusses linear response theory and Ward identities. The third lecture presents transport coefficients derived from AdS/CFT that should be applicable in the quantum critical region associated to a quantum phase transition. The fourth lecture builds in the physics of a superconducting or superfluid phase transition to the simple holographic model of the third lecture.

Lectures on Heterotic-Type I Duality

We present a review of heterotic-type I string duality. In particular, we discuss the effective field theory of six- and four-dimensional compactifications with N>1 supersymmetries. We then describe various duality tests by comparing gauge couplings, N=2 prepotentials, as well as higher-derivative F-terms. Based on invited lectures delivered at: 33rd Karpacz Winter School of Theoretical Physics ``Duality, Strings and Fields,'' Przesieka, Poland, 13 - 22 February 1997; Trieste Conference on Duality Symmetries in String Theory, Trieste, Italy, 1 - 4 April 1997; Cargese Summer School ``Strings, B…

TASI Lectures on the Higher Spin - CFT duality

In these lectures we give an overview of the duality between gravitational theories of massless higher spin fields in AdS and large N vector models. We first review the original higher spin/vector model duality conjectured by Klebanov and Polyakov, and then discuss its generalizations involving vector models coupled to Chern-Simons gauge fields. We proceed to review some aspects of the theory of massless higher spins, starting with the Fronsdal equations for free fields and moving on to the fully non-linear Vasiliev equations in four dimensions. We end by reviewing some recent tests of the hig…

Lectures on Holographic Space Time

Summary of three talks on the Holographic Space Time models of early universe cosmology, particle physics, and the asymptotically de Sitter final state of our universe.

TASI Lectures on Compatification and Duality

We describe the moduli spaces of theories with 32 or 16 supercharges, from several points of view. Included is a review of backgrounds with D-branes (including type I' vacua and F-theory), a discussion of holonomy of Riemannian metrics, and an introduction to the relevant portions of algebraic geometry. The case of K3 surfaces is treated in some detail.

预弦理论讲座：全息关联函数与解析自举

Pre-Strings Lectures on Holographic Correlators and Analytic Bootstrap

该讲义基于预弦理论2026年学校的五次讲座，介绍了用自举策略计算全息关联函数的方法，以4d $\boldsymbol{\textrm{N}=4}$超杨-米尔斯理论为例展开，还涉及相关扩展内容。

Lectures on holographic methods for condensed matter physics

These notes are loosely based on lectures given at the CERN Winter School on Supergravity, Strings and Gauge theories, February 2009 and at the IPM String School in Tehran, April 2009. I have focused on a few concrete topics and also on addressing questions that have arisen repeatedly. Background condensed matter physics material is included as motivation and easy reference for the high energy physics community. The discussion of holographic techniques progresses from equilibrium, to transport and to superconductivity.

Double Field Theory, T-Duality, and Courant Brackets

These lecture notes are based on three lectures, each ninety minutes long, given by the author at the "International School on Strings and Fundamental Physics" which took place in Garching/Munich from July 25 to August 6, 2010. These lectures, aimed at graduate students, require only a basic knowledge of string theory and give a simple introduction to double field theory. These notes were prepared by Marco Baumgartl and Nicolas Moeller.

Brief lectures on duality, integrability and deformations

We provide a pedagogical introduction to some aspects of integrability, dualities and deformations of physical systems in 0+1 and in 1+1 dimensions. In particular, we concentrate on the T-duality of point particles and strings as well as on the Ruijsenaars duality of finite many-body integrable models, we review the concept of the integrability and, in particular, of the Lax integrability and we analyze the basic examples of the Yang-Baxter deformations of non-linear sigma-models. The central mathematical structure which we describe in detail is the E-model which is the dynamical system exhibi…

Lectures on Celestial Amplitudes

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.

Dualities in Supersymmetric Field Theories

Lectures presented at the 33rd Karpacz Winter School ``Duality: Strings and Fields'' briefly introducing dualities in four-dimensional quantum field theory, and summarizing results found in supersymmetric field theories. The first lecture describes physical aspects of electric-magnetic (EM), strong-weak coupling (S), and infrared (IR) dualities. The second lecture focuses on results and conjectures concerning S-duality in N=2 supersymmetric gauge theories. The third lecture discusses IR-dualities and their relation to S-duality in N=1 supersymmetric field theories.

Space-time random tensor networks and holographic duality

In this paper we propose a space-time random tensor network approach for understanding holographic duality. Using tensor networks with random link projections, we define boundary theories with interesting holographic properties, such as the Renyi entropies satisfying the covariant Hubeny-Rangamani-Takayanagi formula, and operator correspondence with local reconstruction properties. We also investigate the unitarity of boundary theory in spacetime geometries with Lorenzian signature. Compared with the spatial random tensor networks, the space-time generalization does not require a particular ti…

特殊凯勒几何与电磁对偶旋转讲义

Lectures on Special Kahler Geometry and Electric--Magnetic Duality Rotations

本文系统回顾了偶数维时空中的电磁对偶旋转结构，重点讨论四维 N=2 超对称中辛协变性与特殊凯勒、超凯勒及四元数几何的关系，并以 Seiberg-Witten 模型为例说明模空间几何与精确对偶的联系。

关于AdS$_3$中弦的讲义：从世界面和AdS$_3$/CFT$_2$对偶性角度

Lecture notes on strings in AdS$_3$ from the worldsheet and the AdS$_3$/CFT$_2$ duality

讲义从世界面视角介绍AdS$_3$背景中NSNS flux支持的弦研究，涵盖最新成果，总结25年历史与近期发展，重点在世界面关联函数的计算。

On S-duality for holographic p-wave superconductors

We consider the generalization of the S-duality transformation previously investigated in the context of the FQHE and s-wave superconductivity to p-wave superconductivity in 2+1 dimensions in the framework of the AdS/CFT correspondence. The vector Cooper condensate transforms under the S-duality action to the pseudovector condensate at the dual side. The 3+1-dimensional Einstein-Yang-Mills theory, the holographic dual to p-wave superconductivity, is used to investigate the S-duality action via the AdS/CFT correspondence. It is shown that in order to implement the duality transformation, chemic…

Non-Abelian T-duality for nonrelativistic holographic duals

We find new type II backgrounds with non-relativistic symmetries via non-Abelian T-duality. First we consider geometries with Galilean symmetries in type IIA, which have been identified as non-relativistic generalizations of the ABJM background and massive IIA supergravities. We then consider the non-Abelian T-duality transformation on the backgrounds with Lifshitz symmetry constructed by Donos and Gauntlett. Using gauge/gravity duality we study aspects of the field theory dual to these backgrounds.

TASI Lectures on D-Branes

This is an introduction to the properties of D-branes, topological defects in string theory on which string endpoints can live. D-branes provide a simple description of various nonperturbative objects required by string duality, and give new insight into the quantum mechanics of black holes and the nature of spacetime at the shortest distances. The first two thirds of these lectures closely follow the earlier ITP lectures hep-th/9602052, written with S. Chaudhuri and C. Johnson. The final third includes more extensive applications to string duality.

Lectures on supersymmetric gauge theories and electric-magnetic duality

We review some of the recent work on the dynamics of four dimensional, supersymmetric gauge theories. The kinematics are largely determined by holomorphy and the dynamics are governed by duality. The results shed light on the phases of gauge theories. Some results and interpretations which have not been published before are also included.

Solitons, Black Holes and Duality in String Theory

These lectures are intended as an introduction to some of the basic aspects of string solitons, duality and black holes. We begin with a discussion of the role of classical solutions in duality, then focus on string/string duality and fundamental membranes. Finally, we examine the feature of compositeness of string solitons, and its implications for bound states and black hole thermodynamics. As these lectures are aimed primarily at those less familiar with this field, technical details are minimized.

Black Holes and the Holographic Principle

This lecture reviews the black hole information paradox and briefly appraises some proposed resolutions in view of developments in string theory. It goes on to give an elementary introduction to the holographic principle.

Lectures on the Plane-Wave String/Gauge Theory Duality

These lectures give an introduction to the novel duality relating type IIB string theory in a maximally supersymmetric plane-wave background to N=4, d=4, U(N) Super Yang-Mills theory in a particular large N and large R-charge limit due to Berenstein, Maldacena and Nastase. In the first part of these lectures the duality is derived from the AdS/CFT correspondence by taking a Penrose limit of the AdS_5 x S^5 geometry and studying the corresponding double-scaling limit on the gauge theory side. The resulting free plane-wave superstring is then quantized in light-cone gauge. On the gauge theory si…

Classical Geometry and Target Space Duality

This is the written version of lectures presented at Cargese 95. A new formulation for a ``restricted'' type of target space duality in classical two dimensional nonlinear sigma models is presented. The main idea is summarized by the analogy: euclidean geometry is to riemannian geometry as toroidal target space duality is to ``restricted'' target space duality. The target space is not required to possess symmetry. These lectures only discuss the local theory. The restricted target space duality problem is identified with an interesting problem in classical differential geometry.

Concepts in Gauge Theory Leading to Electric--Magnetic Duality

Gauge theory, which is the basis of all particle physics, is itself based on a few fundamental concepts, the consequences of which are often as beautiful as they are deep. In this short lecture course I shall try to give an introduction to these concepts, both from the physical and mathematical points of view. Then I shall show how these considerations lead to a nonabelian generalization of the well-known electric--magnetic duality in electromagnetism. I shall end by sketching some of the many consequences in quantum field theory that this duality engenders in particle physics. These are notes…

Holographic flavor in the gauge/gravity duality

In this paper we review some results on the generalization of the gauge/gravity duality to include fundamental matter by means of probe branes. We compute the meson spectrum of maximally supersymmetric gauge theories in several dimensions, both in the Coulomb and Higgs branch. We also study the addition of flavor and compute the meson spectrum for non-commutative theories. Additionally, we present a thorough search of supersymmetric embeddings of probe branes in the Klebanov-Witten model.

The Holographic Renormalization Group

In this lecture, we review the derivation of the holographic renormalization group given in hep-th/9912012. Some extra background material is included, and various applications are discussed.

全息复杂度的电磁对偶敏感性

Electromagnetic Duality Sensitivity of Holographic Complexity

本文在爱因斯坦-莫德麦克斯理论中研究电磁对偶对全息复杂度的敏感性，发现物质敏感泛函能区分具有相同几何的电磁构型，展示了广义全息复杂度的自由度。

Lectures on Spectrum Generating Symmetries and U-duality in Supergravity, Extremal Black Holes, Quantum Attractors and Harmonic Superspace

We review the underlying algebraic structures of supergravity theories with symmetric scalar manifolds in five and four dimensions, orbits of their extremal black hole solutions and the spectrum generating extensions of their U-duality groups. For 5D, N=2 Maxwell-Einstein supergravity theories (MESGT) defined by Euclidean Jordan algebras, J, the spectrum generating symmetry groups are the conformal groups Conf(J) of J which are isomorphic to their U-duality groups in four dimensions. Similarly, the spectrum generating symmetry groups of 4D, N=2 MESGTs are the quasiconformal groups QConf(J) ass…

Entanglement String and Spin Liquid with Holographic Duality

We show that the quantum entanglement can be transmuted to a force using the holographic duality. First, we prove that there is an open string in the spectrum of the holographic fermion coupled with scalar. The string ends at two fermions and its tension vanishes in the limit of zero scalar condensation. We associate such string with a dimer and identify the scalar condensation as the degree of the dimerization. Together with divergently large entanglement entropy, the model is expected to describe the Spin Liquid. As a consistency check, we show that there is a Mott transition as the dimeriza…

LonTI Lectures on Sine-Gordon and Thirring

This is the extended write-up of a series of lectures on the duality between the Sine-Gordon model and the Thirring model. Prepared for the London Theory Institute (LonTI) - Fall 2022: a PhD-level mini-course, with exercises and a guide to the literature.

Universal features of the holographic duality: conformal anomaly and brane gravity trapping from 5d AdS Black Hole

We calculate the holographic conformal anomaly and brane Newton potential when bulk is 5d AdS BH. It is shown that such anomaly is the same as in the case of pure AdS or (asymptotically) dS bulk spaces, i.e. it is (bulk) metric independent one. While Newton potential on the static brane in AdS BH is different from the one in pure AdS space, the gravity trapping still occurs for two branes system. This indicates to metric independence of gravity localization.

Holographic entanglement entropy in open-closed string duality

We study minimal co-dimension-2 surfaces in the asymptotically flat background of extremal 3-brane solutions in ten-dimensional type IIB supergravity. A conjectured open-closed string duality combined with the Ryu-Takayanagi prescription implies that the area of the surfaces we consider could be interpreted as the entanglement entropy of a dual (3+1)-dimensional large-N, strongly-coupled open string field theory on D3-branes. As the size of the surface is varied we observe a transition from a volume law to an area law in agreement with expectations from non-locality in an open string field the…

Holographic duality with a view toward many-body physics

These are notes based on a series of lectures given at the KITP workshop "Quantum Criticality and the AdS/CFT Correspondence" in July, 2009. The goal of the lectures was to introduce condensed matter physicists to the AdS/CFT correspondence. Discussion of string theory and of supersymmetry is avoided to the extent possible.

Holographic duality, supersymmetry, and Painleve equation

An example of the holographic correspondence between 2d, N=2 quantum field theories and classical 4d, N=2 supergravity theories is found. The constraints on the target space geometry of the 4d, N=2 non-linear sigma-models in N=2 supergravity background are interpreted as the renormalization flow equations in two dimensions. Our geometrical description of the renormalization flow is manifestly covariant under reparametrization of the 2d coupling constants. The proposed holography is described in terms of the (Weyl) anti-self-dual Einstein metrics, whose exact regular (Tod-Hitchin) solutions are…

Electric-Magnetic Duality and Effective Field Theories

Lectures given by C.G. in the Advanced School on Effective Theories (Almuñecar, Granada, 1995) on duality in N=2 supersymmetric Yang-Mills, and the coupling to gravity.

Lectures on holographic non-Fermi liquids and quantum phase transitions

In these lecture notes we review some recent attempts at searching for non-Fermi liquids and novel quantum phase transitions in holographic systems using gauge/gravity duality. We do this by studying the simplest finite density system arising from the duality, obtained by turning on a nonzero chemical potential for a U(1) global symmetry of a CFT, and described on the gravity side by a charged black hole. We address the following questions of such a finite density system: 1. Does the system have a Fermi surface? What are the properties of low energy excitations near the Fermi surface? 2. Does …

Improved Holographic QCD and the Quark-gluon Plasma

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.

Holographic duality and applications

In this thesis we review some results on the generalization of the gauge/gravity duality to new cases by using T-duality and by including fundamental matter, finding applications to condensed matter physics. First, we construct new supersymmetric solutions of type IIA/B and eleven-dimensional supergravity by using non-abelian T-duality. Second, we construct a type IIA supergravity solution with D6-brane sources, dual to an unquenched massive flavored version of the ABJM theory. Third, we study a probe D6-brane with worldvolume gauge fields in the ABJM background, obtaining the dual description…

Lectures on entanglement entropy in field theory and holography

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…

Lectures on Superstring and M Theory Dualities

These lectures begin by reviewing the evidence for S duality of the toroidally compactified heterotic string in 4d that was obtained in the period 1992--94. Next they review recently discovered dualities that relate all five of the 10d superstring theories and a quantum extension of 11d supergravity called M theory. The study of p-branes of various dimensions (some of which are D-branes) plays a central role. The final sections survey supersymmetric string vacua in 6d and some of the dual constructions by which they can be obtained. Special emphasis is given to a class of N=1 models that exhib…

Duality in String Theory

In this lecture we review some of the recent developments in string theory on an introductory and qualitative level. In particular we focus on S-T-U dualities of toroidally compactified ten-dimensional string theories and outline the connection to M-theory. Dualities among string vacua with less supersymmetries in six and four space-time dimensions is discussed and the concept of F-theory is briefly presented. (Lecture given by J. Louis at the Workshop on Gauge Theories, Applied Supersymmetry and Quantum Gravity, Imperial College, London, UK, July 5--10, 1996.)

Holographic relations for OPE blocks in excited states

We study the holographic duality between boundary OPE blocks and geodesic integrated bulk fields in quotients of AdS$_3$ dual to excited CFT states. The quotient geometries exhibit non-minimal geodesics between pairs of spacelike separated boundary points which modify the OPE block duality. We decompose OPE blocks into quotient invariant operators and propose a duality with bulk fields integrated over individual geodesics, minimal or non-minimal. We provide evidence for this relationship by studying the monodromy of asymptotic maps that implement the quotients.

Introduction to Gauge/Gravity Duality

These lectures are an introduction to gauge/gravity duality, presented at TASI 2010. The first three sections present the basics, focusing on $AdS_5 \times S^5$. The last section surveys a variety of ways to generate duals of reduced symmetry.

Notes on D-Branes

This is a series of remedial lectures on open and unoriented strings for the heterotic string generation. The particular focus is on the interesting features that arise under T-duality---D-branes and orientifolds. The final lecture discusses the application to string duality. There will be no puns. Lectures presented by J. P. at the ITP from Nov. 16 to Dec. 5, 1995. References updated through Jan. 25, 1996.

Understanding strongly coupling magnetism from holographic duality

The unusual magnetic materials are significant in both science and technology. However, because of the strongly correlated effects, it is difficult to understand their novel properties from theoretical aspects. Holographic duality offers a new approach to understanding such systems from gravity side. This paper will give a brief review of our recent works on the applications of holographic duality in understanding unusual magnetic materials. Some quantitative compare between holographic results and experimental data will be shown and some predictions from holographic duality models will be dis…

Lectures on Strings and Dualities

In this set of lectures I review recent developments in string theory emphasizing their non-perturbative aspects and their recently discovered duality symmetries. The goal of the lectures is to make the recent exciting developments in string theory accessible to those with no previous background in string theory who wish to join the research effort in this area. Topics covered include a brief review of string theory, its compactifications, solitons and D-branes, black hole entropy and web of string dualities. (Lectures presented at ICTP summer school, June 1996)

non-Abelian T-duality, G2-structure Rotation and Holographic Duals of N=1 Chern-Simons Theories

A new dynamic SU(3)-structure solution in type-IIA is found by T-dualising a deformation of the Maldacena-Nastase solution along an SU(2) isometry. It is argued that this is dual to a quiver gauge theory with multiple Chern-Simons levels. A clear way of defining Chern-Simons levels in terms of Page charges is presented, which is also used to define a Chern-Simons term for the G2-structure analogue of Klebanov-Strassler, providing evidence of a cascade in both the ranks and levels of the dual quiver.

TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effective Field Theory

I argue that the conventional field theoretic notion of vacuum state is not valid in quantum gravity. The arguments use gravitational effective field theory, as well as results from string theory, particularly the AdS/CFT correspondence. Different solutions of the same low energy gravitational field equations correspond to different quantum systems, rather than different states in the same system. I then introduce {\it holographic space-time} a quasi-local quantum mechanical construction based on the holographic principle. I argue that models of quantum gravity in asymptotically flat space-tim…

Free partition functions and an averaged holographic duality

We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with $U(1)^c \times U(1)^c$ symmetry and a composite boundary graviton. Additionally, for small central charge $c$, we obtain general constraints on the spectral gap of f…

Holographic Duality for three-dimensional Super-conformal Field Theories

This thesis is based on the publications arXiv:1106.4253, arXiv:1206.2920, arXiv:1210.2590. We construct warped "AdS_4 x S^2 x S^2 x Σ" type IIB supergravity solutions dual to the infrared fixed points of all 3d N=4 linear and circular quivers. We provide checks of the correspondence by evaluating the supergravity action in a large N limit and by matching it with the computation of the free energy obtained from matrix models. To complete previous work, we establish the holographic dictionary for the more general supergravity solutions corresponding to domain walls (non-compact Σ) and dual to 4…

Quantum Information in Holographic Duality

We give a pedagogical review of how concepts from quantum information theory build up the gravitational side of the AdS/CFT correspondence. The review is self-contained in that it only presupposes knowledge of quantum mechanics and general relativity; other tools--including holographic duality itself--are introduced in the text. We have aimed to give researchers interested in entering this field a working knowledge sufficient for initiating original projects. The review begins with the laws of black hole thermodynamics, which form the basis of this subject, then introduces the Ryu-Takayanagi p…

Holographic superconductors with Weyl Corrections via gauge/gravity duality

In this paper, we analytically compute the basic parameters of the p-wave holographic superconductors with Weyl geometrical corrections using the matching method. The explicit correspondence between the critical temperature $T_c$ and the dual charge density $ρ$ has been calculated as $T_c\proptoρ^{{1}{3}}$ and the dependence of the vacuum expectation value for the dual condensate operator $\cal{O}$ on the temperature has been found analytically in the form $<{\cal O}_{+}>\propto T_c^{{3}{2}}T^{Δ-{1}{2}}\sqrt{1-({T}{T_c})^3}$. The critical exponent ${1}{2}$ is an universal quantity according to…

Consistency Conditions for Holographic Duality

We show that if the beta functions of a field theory are given by the gradient of a certain potential on the space of couplings, a gravitational background in one more dimension can express the renormalization group (RG) flow of the theory. The field theory beta functions and the gradient flow constraint together reconstruct the second order spacetime equations of motion. The RG equation reduces to the conventional gravitational computation of the spacetime quasilocal stress tensor, and a c-theorem holds true as a consequence of the Raychaudhuri equation. Conversely, under certain conditions, …

Holographic State Complexity from Group Cohomology

Topological phases of matter are often described using auxiliary systems in one extra dimension. I review the one-dimensional cluster state--the simplest quantum state with Symmetry-Protected Topological (SPT) order--as a toy model of holographic duality, and inspect it for clues about defining holographic complexity. Group cohomology, which classifies SPT orders, is a viable candidate for a robust definition of complexity in gauge/gravity duality.