A Practical Mini-Course on Applied Holography

TL;DR

This paper provides a practical, step-by-step mini-course on applied holography, focusing on analytic and numerical techniques for real-world applications in physics, with open-source code and detailed computations.

Contribution

It offers a pragmatic, accessible introduction to applied holography with concrete examples, open-source tools, and practical insights for researchers and students.

Findings

Step-by-step computational examples provided

Open-source Mathematica codes included

Practical tricks and warnings shared

Abstract

This is a collection of notes based on lectures given at IIT Madras in September 2019 and at IFT Madrid in November 2019. It is supposed to be a concise (and therefore not comprehensive) and pragmatic course on applied holography and especially the (basic) analytic and numerical techniques involved. The lectures are not focused on the large theoretical and fundamental background which can be found already in several places in the literature, but rather on concrete applications of Bottom-Up AdS-CFT to Hydrodynamics, QCD and Condensed Matter. The idea is to accompany the reader step by step through the various benchmark examples with a classmate attitude, providing details of the computations and open-source numerical codes in Mathematica, and sharing simple tricks and warnings collected during my research experience. At the end of this path, the reader will be in possess of all the fundamental skills and tools to learn by himself/herself more advanced techniques and to produce independent and novel research on the topic.

Full text

Applied holography

A practical mini-course

– SpringerBriefs in Physics –

Matteo Baggioli

Tra il dire e il fare c’é di mezzo il mare.

Italian Proverb

When I was asked by my friend and colleague Ayan Mukhopadhyay to give a series of lectures on Applied Holography, I wanted to create something that could be useful for students or in general researchers, who are willing to learn the fundamental techniques of the field and then apply them towards new directions. Being already approximately $20$ years since this tool has been first employed, there are already several, and I must say excellent, reviews and books available McGreevy:2016myw; McGreevy:2009xe; Nastase:2007kj; Ramallo:2013bua; Ammon:2015:GDF:2834415; Aharony:1999ti; CasalderreySolana:2011us; Zaffaroni_2000; Polchinski:2010hw; Natsuume:2014sfa; zaanen2015holographic; Hartnoll:2016apf; Hartnoll:2009sz. They cover in great details the history, the derivation and the fundamental features of the AdS-CFT correspondence with a concrete view towards its applications to Condensed Matter, Heavy Ion physics, Hydrodynamics and strongly coupled field theory in general. Re-organizing and re-writing their content in my own words would have been totally useless. At the same time, there are several reviews regarding advanced numerical methods Krikun:2018ufr; Berti:2009kk; Dias:2015nua; Jansen:2017oag; Chesler:2013lia; Andrade:2017jmt; Zhang:2016coy, with which I cannot (and do not want to) compete. I refer to them for the reader interested in digging into much more advanced problems which need those complex techniques.

I remember when, not long ago, I had to go through the various reviews by myself and what I felt at the time. The first thing that came to my mind is an Italian proverb saying ”Tra il dire e il fare c’é di mezzo il mare”. I am not sure how this should sound in proper English but Google suggests something like ”It’s easier said than done”. In simpler words, reading through the lectures and books of the experts, everything seemed so obvious and obtainable in one line of computations, but then tons of hidden subtleties, caveats and derivations, taken for granted, were appearing everywhere. In this mini-course, I plan to go over them again with you, as if I were your school-mate. I selected a small and reasonable collection of benchmark computations and techniques which, in my personal opinion and experience, represent the necessary package to be able to float in this research landscape and to produce novel work within it. I tried to be the more pedagogical possible and to share with the reader all the tricks, warnings and shortcuts I have accumulated in these past years.

Do not take me wrong! I definitely recognize the pedagogical value of getting stuck with a problem and scratching your head with it. These lectures have not to be taken as a shortcut to avoid those fundamental moments of the learning curve. On the contrary, they have to be used at a second step to compare and integrate the possible resolution methods and results. They are not a substitute car for your trip but rather a nice accessory which can make your trip more comfortable and entertaining like a good song driving on the highways. In this direction, I will also suggest exercises, which I will not solve, but which can be solved using the techniques I will present and whose results can be checked directly using the existing literature. I will try to be the most complete I can, and where it will not be possible I will indicate the concrete references where more details can be found.

The various examples I chose are just an excuse to learn the fundamental analytic and numerical techniques. As such, I will not focus too much on the physics of the problems and the interpretation of the results but rather on the practical steps of the problem-solving process. This is certainly a practice that I highly discourage during a creative process of research111Before jumping into whatever computations you have in mind, ask yourself why you are doing that! If the response is simply: ”Because I can”, you are definitely on the wrong path. but that, in this case, for pedagogical purposes, might be efficient. The different benchmarks are accompanied by available and open-source Mathematica Mathematica10 codes, which I used to obtain some of the results presented in the main text and most of the figures appearing in it222As an aside personal note, try to avoid becoming slaves of Mathematica. Mathematica is an amazing instrument which can speed up a lot of tedious computations but between Mathematica and you, you are the only one able to think (whatever the hyped machine-learning fans claim). When you realize (and it happened to me) that you use Mathematica to do $8*4$, then something is going wrong.. The codes can be found at https://members.ift.uam-csic.es/matteo.baggioli/lectures/.

The lectures are organized as follows:

1. Section 1.

In this section, I will briefly introduce and motivate the tool we will be using and its origin. This part, even if already explained in several previous works, is necessary to convince the reader that the dictionary and the framework we use is not a collection of God-given laws as those which Moses received at Mount Sinai. Despite the counter-intuitive and not obvious nature of the playground, there are strong motivations behind its origin. At the same time, I will make an effort to motivate why such new tools are necessary to make advancements in certain open questions and how we should use them. If you already know the topic you can skip this section completely; if you do not care about these details or you are in a hurry, I suggest to read only subsection LABEL:skipskip for a brief advertising slogan of the duality. 2. Section 2.

In this second part, I will go through the basic concepts of the duality and the dictionary. I will use the simplest example of a bulk scalar field to introduce the standard way of extracting physical information about the dual field theory from simple gravitational computations. This part can be a bit tedious and abstract but it is fundamental to understand what we will do later on, in the various applications. If you get bored along this section you can always jump ahead and come back to it at the moment you need it. Learning is not a linear process but rather an out-of-time order one. 3. Section 3.

This section is the first one containing practical examples and fundamental techniques. More specifically I will explain the following points:

• (i)

How to derive, with Mathematica, the Einstein equations for the background and for the perturbations (example: the shear mode).

• (i)

How to use the membrane paradigm and its power (example: the Kovtun-Starinets-Son (KSS) bound).

• (ii)

How to numerically obtain the Green function of a specific operator at finite frequency (example: the viscosity and elasticity of holographic massive gravity (HMG)).

• (iii)

How to obtain the low frequency expansion of a correlator via analytic methods in the bulk (example: the violation of KSS and the elastic modulus in HMG).

• (iv)

How to use near-horizon arguments to extract analytically the behaviour of certain physical quantities close to $T=0$ (example: the scaling of $\eta/s$ at zero temperature in HMG). 4. Section 4.

This section contains the second block of concepts. In particular:

• (i)

How to analytically study the stability of the solution by using the BF bound argument (example: the holographic superconductor).

• (ii)

How to numerically verify the instability of the background solution at finite temperature (example: the holographic superconductor).

• (iii)

How to numerically compute the Green function at finite frequency when the operators are mixing (example: the electric conductivity).

• (iv)

How to exploit certain properties of the bulk theory to derive the transport coefficients at zero frequency (example: the DC conductivities).

• (v)

How to extract and the quasinormal modes for a simple decoupled operator using numerical and perturbative techniques (example: the decoupled phonon mode in HMG). 5. Section 5.

In this last section we will investigate how the holographic duality geometrizes the RG flows between two different fixed points. In particular we will consider:

• (i)

The correspondence between the symmetries of the field theory and the isometries of the corresponding geometric gravitational picture (example: AdS and Lifshitz spacetimes).

• (i)

The construction of (zero temperature) holographic RG flows by using interpolating geometries (example: Lifshitz flows).

Once you master with confidence all the techniques mentioned above, you could easily publish the same papers I did, and even more! At the same time, you will be able to learn fast the opportune generalizations of those tools to attack more complicated problems you might be interested in. The rest is on you! This is your driving license, but it does not mean you know how to drive, it just means you are ready to get on your car and learn by yourself and accumulate experience and knowledge.

I truly hope you will find these notes as useful as their writing has been to me. It gave me the opportunity of thinking about and re-derive by myself all those points that when you read a paper they are sold as obvious, and they are not! Things are simple once you did them. Now it is your turn to do them!

It is very likely that you will find through the next pages a lot of typos and even incorrect statements. I could say I did it on purpose to force you to spot them; of course that is not true, but your duty to spot them is still valid. Think of yourself not as a simple reader but more as the referee of this manuscript. Every comment, correction, suggestion and discussion will be useful to me, to you and especially for everyone that will use this book to learn. I therefore encourage you to contact me in any of these cases to help me improve the content of this manual. I like to think about it as a living creature (like Science), evolving and getting better in time, and this is possible only through your feedback.

Tour guide

Along the way you will find several colored boxes:

The yellow boxes contain technical and independent computations and/or proofs complementary to the main text. If you not are interested in these details, you can just skip them completely.

The blue boxes contain tricks that I have learned during my years of research. These manoeuvres speed up computations and make them simpler.

The red boxes contain warnings about common mistakes or bad practices. They can save you in several situations.

The green boxes contain exercises for you. Most of them are unsolved in these notes but the results can be found in the existing literature which I will refer to. For some of them, I will provide directly Mathematica notebooks with the solutions and you are always welcome to contact me to discuss them.

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Acknowledgements

It is my pleasure to sincerely thank Ayan Mukhopadhyay and Indian Institute Of Technology Madras, Chennai to provide the spark that started all of this. You gave me the opportunity, the motivation and the courage to write this up and share all I have learned in these last years of research. I am grateful to all the people that contributed to my current understanding of the problem and to the knowledge I acquired in these last years. Supervisors, collaborators, colleagues, speakers, competitors, enemies, lecturers: you all have been of great help. I hope this could be of the same relevance for all the students interested in this topic. I am particularly grateful to my colleagues who wasted some of their time to read a preliminary version of this book. I finally acknowledge the generous support of the Spanish MINECO’s “Centro de Excelencia Severo Ochoa” Programme under grant SEV-2012-0249.