The gauge/gravity duality – or how dimensions may not be what they seem

Here’s another enlightening installment from Jon Shock, string theorist…

OK, so we were talking about how QCD, the strong force, is a particularly tricky beast when we’re trying to calculate measurable quantities. This was because the usual ‘perturbative’ methods we used to study QED (quantum electrodynamics) don’t work when your coupling constant (the expansion parameter in the series) is large.

I mentioned that there was this amazing duality between two very different theories from string theory which meant that we could actually learn something about the strong force.

Let me first explain a little bit about these theories. First of all I need to tell you something about the ingredients of string theory. (I will write this expecting that the reader has a very very rough idea what string theory is, though not many details are needed)

One of the lessons we learnt from the 90s was that string theory wasn’t just a theory of string. In fact there are other objects in this theory too. Whereas a string is a 2-dimensional object (filling one time and one space direction) it turns out that there are other objects of both higher and lower dimension, called D-branes (brane as in membrane). In fact we label these as Dp-branes where p is a number between 0 and 9 indicating how many spatial dimensions they fill.

There is a D0-brane which is very similar to a point particle, a D1-brane which is very similar to a string and then higher Dp-branes. These can be thought of as hypersurfaces living in a ten dimensional spacetime.

For instance, a D3-brane fills 3 spatial dimensions (just like the three dimensions we see around us). All of these objects live in a ten dimensional spacetime, where they can move, interact and oscillate in a variety of ways. One of the interesting things about a D-brane is that certain types of string can live on them. In string theory there are two types of string, open and closed. The open strings are just like a piece of elastic with two ends which can move about while a closed string is like a loop, without any ends.

A D-brane can have open strings living on it and the way this happens is that the ends of the strings live on the surface of the D-brane while the rest of the string can wander about in the rest of space time. The strings have tension and so naturally they like to contract to as small a size as possible. Sometimes you can stretch a string by having it live between two D-branes which are separated (think of two pieces of paper sitting parallel but some distance apart with a piece of elastic stretched in between them). A closed string can be stretched too, usually by winding it about a circle.

So, these D-brane can have open strings ending on them. Well, it turns out that by taking a stack of the D3-branes and putting them all on top of one another, one can calculate what the open strings living on it will look like (their masses and how they transform under Lorentz transformations). The lightest modes of the strings, in a particular limit, give you a theory which looks quite a bit like the strong force (it’s not exact by any means).

It sounds strange at first because the strong force is a theory of point particles while we are talking about strings here, but the limits we take are such that the strings become very very short and we can treat them like particles. So, this theory is a four dimensional theory (because it’s living on the surface of the D3-branes) and looks a bit like the strong force.

The other desciption of the D3-branes is in terms of closed strings. While open strings end on the D-brane, closed strings can interact with it. One of the modes of the closed string is the graviton, the force carrying particle for gravity. We find that the stack of D3-branes warps the space around them. In fact it can warp the space a lot and creates a black hole, but that’s not too important for the discussion. What is important is that it warps the space into some strange ten-dimensional spacetime – I won’t go into the details of this geometry, but it’s very well understood.

There are closed strings moving about in this geometry and we can formulate a description of these closed strings moving about and interacting with one another. In this picture we do not have the open strings, but just the closed strings moving in the geometry created by the stack of D-branes.

OK so far. The two pictures are an open string description and a closed string description of a stack of D3-branes. In the limits we are interested in one of them is a strongly coupled four dimensional theory while the other one is a weakly coupled theory of gravity in ten dimensions.

The remarkable fact is that these two descriptions are actually equivalent. Somehow all the useful information in one of the theories is encoded in the other theory, even though they look very different and have different numbers of dimensions.

There are many reasons for believing this, the first one coming from studying the symmetries of the two descriptions and finding that they are the same.

Well, although we have a dual description of a strong interaction in terms of a weakly coupled theory of gravity this is of no use unless we can actually calculate anything. The idea is as follows:

You should be able to ask a question about the strongly coupled theory, a question which you won’t be able to answer in that theory precisely because it is strongly coupled. You should then be able to translate that question into the language of the theory of gravity where you can ask the question relatively easily, because the theory is weakly coupled. You will then get an answer which can be translated back into the original theory which was the one you asked the question in in the first place.

In fact there’s a very precise relation between mathematical entities on both sides which allows us to perform calculations using this duality.

One problem with this duality is that the four dimensional theory doesn’t look very much like the strong force we see in nature. In fact it’s a type of theory called a conformal, or scale invariant theory. This means that however much you zoom in or zoom out of the system it will look the same, clearly very much unlike the real world (though there are systems in the real world which approximate this symmetry, in particular systems at their critical points – fluids and magnetic systems being the two normal examples).

It turns out that we can look at the exact mathematical relationship between the two theories and we are told how we have to change the ten-dimensional gravitational theory in order for its dual, four dimensional construction, to look more like the real world. Generally what this means for the gravitational side is that the geometry will be changed and more fields switched on in the ten-dimensional background. This makes the system a little harder to study and often we have to rely on numerical methods for performing the calculation which we may have been able to perform analytically in the original, simpler correspondence.

This is the aspect that I work most closely on. I take a ten dimensional geometry which is dual to a theory which looks (in some aspects) like the strong force, and try and calculate the phenomenolgy of the dual four dimensional theory. I might try and calculate what are the symmetries of the four dimensional theory, the spectrum of states of the theory, the phase structure if I turn the temperature up in the theory, amongst many other things.

So, in summary for this post, what I am doing is using a duality which tells me that a four dimensional theory, a bit like the real world, and a ten dimensional theory of gravity, are in some sense equivalent. I then perform calculations in the gravitational theory and am able to relate the results I get in the gravitational theory with results in the four dimensional theory.

It turns out that the results we get can be extremely close (in the approximations which we must make) to those of the real world. The spectrum of states may just be 10% or so away from the real world and given that the theory isn’t really QCD, this is pretty good.

In comparison with the big computer simulations which can be used to understand QCD, the correspondence, which is variously known as the gauge-gravity duality or the AdS/CFT correspondence, can in some cases give us a more intuitive understanding of what is happening as well as taking a fraction of the time and computer power to perform the calculations. It must be stated however that there are many questions we can’t answer using this correspondence, at the moment, but progress is being made all the time and the hope that we can discover the true gravity dual of QCD is alive and well.

This may sounds like some mathematical trickery but in fact the AdS/CFT correspondence may be telling us some truly remarkable things about the structure of space time. This is something I’d like to talk about in more detail at a later date.


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