Dualities (String Theory)

Sometimes different mathematical theories describe the same physics. We call this situation a duality. In many cases, calculations which are very complicated in one theory become much easier in the other.

Usefully, string theory is awash with dualities. These variously offer us new perspectives on reality, improve our ability to compute hard sums and unite disparate areas of physics. Much of modern research focuses on using these dualities to better understand a broad spectrum of topics.

T-duality is the simplest to appreciate. Remember that string theory requires six extra dimensions tightly curled up in space. Naively, one would think that these dimensions could be arbitrarily big or arbitrarily small, with different physics holding in each case. However something strange happens when you make these dimensions very small. Of paramount importance is a tiny quantity known as the Planck length, which we denote by a.

How does the radius of a circular dimension affect the physics of string theory? We can appreciate this work with a thought experiment.

Set up a circular extra dimension the size of the Planck length. Start contracting the circle and measure the resulting physics. Your readings will vary depending on the size of the dimension. Now repeat the experiment, but with a crucial difference; instead of contracting your circle, expand it.

Observing the physics again, you realise that it’s exactly the same as for a contracting dimension! There is a duality between the two scenarios. Mathematically it can be proven that extra dimensions with radii r and a/r produce the same physics: they are identical theories.

An extension of T-duality produces mirror symmetry. In many string theory models, the extra dimensions form a six dimensional shape called a Calabi-Yau manifold. Sadly there are millions of different Calabi-Yau surfaces, each with a slightly different structure. The properties of the Calabi-Yau manifolds affect the expected four-dimensional physics. So we must pin down the correct possibility for the physics we observe.

This requires a lot of calculation. And maths is hard in six dimensions, as you might guess! But here’s where mirror symmetry comes in. In the late 1980s it became clear that Calabi-Yau shapes come in pairs. For any given pair, both lead to the same physics. We have a duality! Physicists could chop and change between mirror pairs, making computations more tractable.

Our third duality is more fundamental: it underpins the success of M-theory. We’ll refer to it as S-duality. All quantum field theories contain a coupling constant, which determines the strength of interactions between particles. String theory is no exception. The value of the coupling constant vastly affects the behaviour predicted.

During the First Superstring Revolution physicists realised that there were five different brands of string theory. At first it seemed like they were all completely separate. But the discovery of various S-dualities sparked a paradigm shift. These dualities related the different flavours of string theory through a framework called M-theory.

More precisely physicists paired up the different types of string model, like so. Take two distinct string theories, A and B. They each have an adjustable coupling constant. If A has a large coupling constant and B a small one, then they predict exactly the same physics. The end result was that the many different string theories were united under a single banner.

Finally we come to the hottest guy in town. The AdS-CFT correspondence is a conjectured duality which has been around for barely a decade. Subtle yet powerful, it has profound implications for string theory as a tool in research. It’s such an important idea that it requires a full explanation.