Calabi-Yau Manifolds and Moduli Stabilization

For better or worse, string theory dominates modern research in theoretical physics. Naively, you might expect a theory consisting of tiny strings to be pretty simple. But the subject has grown into a vast and exciting playground for new ideas.

String theory is popular not just because it might unify physics or quantize gravity. In fact, many unexpected offshoots have proved more successful than the original idea! From particle physics to superconductors, string theory is having a surprising indirect impact. It’s certainly useful, even if it doesn’t prove to be the ultimate description of reality.

But what of the original plan – to describe nature using strings? A key sticking point is the existence of extra dimensions. String theory needs 6 of these to work consistently. Another problem is supersymmetry. 10 dimensional string theory must have lots of this to work correctly. But 4 dimensional physics only has a little bit of supersymmetry at most!

It turns out that these problems can be solved in one step. By coiling up the extra dimensions into a Calabi-Yau manifold we can make the extra dimensions effectively invisible, while reducing the supersymmetry we end up with in our 4D world. So what is this Calabi-Yau manifold, I hear you ask!

Well, Calabi-Yau is just a technical term for the shape of the compact extra dimensions. Different shapes break different amounts of symmetries, leaving us with different theories. Calabi-Yau’s are just symmetrical enough to break the right amount of supersymmetry, giving us a sensible theory in the end!

Technically Calabi-Yau manifolds must have a metric which is Kaehler and Ricci flat. These properties provide the correct information about the shape of the curled up dimensions. So we must look for 6-real dimensional manifolds with these properties.

Generically, you don’t have to put a notion of distance on a space. When I go for a walk, I don’t always carry around a yardstick so I can measure how far I’ve gone! You can have a perfectly good manifold without giving it a metric, but you get extra information once you have defined what distance means.

As it happens, finding a metric which is Calabi-Yau is quite difficult. But due to the genius of Shing-Tung Yau, we know that you don’t need to do this! There’s an equivalent definition of a Calabi-Yau manifold which doesn’t depend on metrics at all. All you need to know is the topological information about the manifold – roughly speaking, how “holey” it is.

If you know something about differential geometry, this kind of equivalence might sound familiar. Yau’s theorem relating geometry and topology is like a (much) more complicated version of the classic Gauss-Bonnet theorem!

It’s a darn sight easier to discover Calabi-Yau’s when you know it’s only the topological data that matters. At first people thought there might only be a few, but now we know there’s a huge number of potential candidates! The problem then becomes choosing one which produces the physics of our universe.

While people have made progress on this, the going is tough. One reason is that nobody knows the metric on a compact Calabi-Yau. This isn’t so important for string calculations, but it makes a big difference when you need to consider branes. So people have come up with various workarounds, which give promising physical results. One such success story is provided by my colleague Zac Kenton, who recently wrote a paper on brane inflation with his PhD supervisor, Steve Thomas.

There’s one final complication that I should mention. If string theory is to be a fundamental theory, then the Calabi-Yau shape should be dynamic. More specifically it will squeeze and stretch over time, unless there’s some mechanism to keep it stable. From the perspective of the 4 large dimensions, this freedom is seen as free scalar fields. These so-called “moduli” fields are bad, because we don’t observe anything like them in nature!

To solve this problem, we must find a way of constraining the fluctuations of the Calabi-Yau. Put another way we have to stabilize the moduli fields, by giving them potential terms, so that their fluctuations are small and essentially negligible at low energies. Hence this is known as the problem of moduli stabilization.

One popular way to solve the conundrum is to turn on some supergravity fields at high energy. These so-called fluxes generate potential terms for the moduli, solving the stabilization problem. Initially this idea was unpopular because of a famous no-go theorem by Witten. But since the advent of the D-brane revolution, the concept is back in vogue!

So there you have it – a 5 minute snapshot of “real” string theory. Now it’s time to get back to my calculations, where string theory is more the background Muse, and certainly not the main protagonist!

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