AdS/CFT: a deep duality

For those you that like to dabble in the world of theoretical physics you may have heard this string of letters before; AdS/CFT. This stands for the mouthful; Anti-de-Sitter/Conformal Field Theory correspondence, unfortunately no way to make a catchier acronym out of that one. If that’s already making your mouse hover near the little red cross bear with me. The concepts behind why this theory is useful and what physicists are trying to do with it is not so difficult to grasp. Bonus being, if you do you’re thinking about some of the theories at the forefront of modern physics today.

The key idea is that AdS/CFT is an example of a duality between two types of theories. On the left hand side we have theories in Anti-de-Sitter space, this is a special type of geometry that physicists use to model a particular spacetime. There number of dimensions for this spacetime can be chosen freely but the details of what exact kind of geometry this represents need not concern us today. When working with spacetimes it is easy to add gravity into your models and physicists try to model the gravitational interactions here in terms of string theory (see How long is a piece of string). Incorporating string theory to AdS space attempts to create a big and bold theory of quantum gravity. The inner workings of such a theory are still very much in the dark but here the assumed pillars of a quantum gravity set-up have been erected. Take-home message; AdS sets up an arena for physicists where the only field (see What is a Field?) is the gravitational field and the constituents are our little friends the strings. It is a proposed theory of quantum gravity, still being explored.

On the right hand side we have what is known as a Conformal Field Theory. Conformal field theories are particular types of quantum field theories (see What is Quantum Field Theory?) which in turn are models describing the interactions of elementary particles. Elementary particles interact heavily with the other three out of four fields in the universe’s playbook; the electromagnetic, strong and weak field. On this side of the correspondence there is no gravity and as such the geometry of the arena is a flat spacetime. The way Maldacena, the father of AdS/CFT, set up the correspondence is that if you choose the number of dimensions in your AdS theory to be D (any integer) the number of dimensions in the CFT theory must be D-1. We’ll explain the consequences of this later. Take home message; CFT sets up an arena for physicists where they can work with the interactions between the very small constituents of the universe and use all the fields except gravity.


Ok so both halves have been defined what next? Well we choose a number for our choice of dimensions and play around which each side of the theory. We then begin to see remarkable similarities in the behaviours of the models of quantum gravity formulated in AdS space and those the conformal quantum field theories. Important characteristics of the models, such as emergent symmetries and levels of chaos mirror each other on both sides of the correspondence and the main breakthroughs of this are twofold;

Firstly, the fields at play in quantum field theories on CFT side of things (electromagnetic, strong and weak) are subject to what is known as ‘strong coupling’. When studying theories in theoretical physics there are a bunch of mathematical procedures we very often use however when using maths to study theories with strong couplings our calculations essentially blow up. This is because the strong coupling is representing in the mathematics as a larg pre-factor, call it q here for example. As we try to expand the mathematical terms in the theory we get terms with q, q^2, q^3… and because q is a large number already the whole thing blows up in our face! The fancy terminology is that the theory is not ‘mathematically tractable’. However, theories on the AdS side of things only contain the gravitational field, which is subject to ‘weak coupling’ The pre-factor here is very small and less than 1, call it g. So when we expand the terms and get g, g^2, g^3 we can essentially ignore the terms which contain g raised to high powers as they would be so much smaller than 1 (check it!). As such we can work with the first few terms alone and the maths makes us happy. Take home message; we can study examples of CFTs in areas in difficult like nuclear and condensed matter physics by translating the projects into mathematically solvable problems of string theories on AdS spaces.

Secondly, the string theory on an AdS space set up is, as we said, a proposed theory of quantum gravity. A theory of quantum gravity is the holy grail of the theoretical physics world and this proposed theory is of course only an attempt at a formulation. However due to the similarity in the behaviours of models on both sides of the duality, we can probe the proposed quantum gravity with a quantum field theory – a domain much easier for humans to manipulate. Since we know very little about the inner workings of quantum gravity by considering its quantum analogue without gravity (the CFT theory) we can significantly enhance our understanding.  Take home message; we explore the quantum gravity AdS set up with what we know about the corresponding CFT.

A final point worth touching on because the idea is pretty funky and Susskind is a wonderful man, is holography. You may have recalled that in the set up the AdS quantum gravity theory has D dimensions but the CFT quantum field theory had D-1 dimensions, one less. The duality between the two theories was proposed by Susskind and ‘T Hooft as being the same kind of duality seen between a real world object in three-dimensions and its hologram on a two-dimensional surface… The idea is that all of the information in a theory of quantum gravity can be encoded within a theory without gravity on a lower-dimensional space. The geometric visualisation is akin to a sphere (3D) holding the full theory and the information being encoded on its surface, the boundary of the sphere (2D). Susskind’s interpretation of the AdS/CFT correspondence has conjured up an image of us living within an analogous hologram, a four-dimensional space, an embodiment of a richer five-dimensional space somewhere in the universe… a post in more detail on this to come.


4 responses to “AdS/CFT: a deep duality

  1. This is interesting. I occassionally try to make myself aware of the recent advancement in core sciences but I was totally I unware of AdS/CFT. One question though – as you said CFT yield strong coupling giving us increasing power of a bigger number while AdS models weak coupling with q< 1, so how exactly do you transfer the result of one side to another? Pardon me if my language is incorrect, I just want to know how this duality is being established mathematically. I understand our ultimate aim to unravel the mysteries of quantum gravity.


    • I suppose its more that the conclusions, drawn from the mathematics on one side of the theory (e.g. the field theory side), can be said to then be representative of the system on the other side (e.g. the gravitational side). Apologies it’s quite hard to explain this qualitatively – if you like I could link you to an introductory paper?


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