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# Hidden dimensions

### by Marianne Freiberger

*The shape of inner space*(co-authored by Steve Nadis) Yau describes how the strange geometrical spaces he discovered turned out to be just what theoretical physicists needed in their attempt to build a theory of everything.

*Plus*met up with Yau on his recent visit to London, to find out more.

### Curvature and gravity

*spacetime*, made up of the three spatial dimensions we're used to and an extra dimension for time. His revolutionary insight was that gravity wasn't some invisible force that propagated through spacetime, but a result of massive bodies distorting the very fabric of spacetime. A famous analogy is that of a bowling ball sitting on a trampoline, which creates a dip that a nearby marble will roll into. According to general relativity, massive objects like stars and planets warp spacetime in a similar way, and thus "attract" other bodies that pass nearby.

*curvature*of spacetime. This had been around since the 19th century, when the mathematician Carl Friedrich Gauss and after him his brilliant student Bernhard Riemann had come up with ways of measuring the curvature of an object from the "inside": they no longer needed to refer to a larger space the object might be sitting in. This intrinsic concept of curvature was just what Einstein needed.

### Gravity in a vacuum?

*topology*. Topology is blind to exact measurements and only captures the overall form of an object. A sphere and a deflated football, for example, are very different geometrically, but they are topologically the same because one can be transformed into the other without any tearing or cutting. Similarly, topology regards a doughnut and a coffee cup as equivalent, because one can be morphed into the other. What differentiates the doughnut from a sphere is the fact that it has a hole.

### Calabi's question

*manifolds*. These are objects that when viewed from up close, look like the ordinary "flat" space (called

*Euclidean space*) we are used to. Spheres and doughnuts, for example, locally look like the flat 2D plane. If you're small enough, you won't notice their curvature, or whether there's a hole in them. You can easily draw a map of a patch of the sphere or doughnut on a flat piece of paper. So these are both 2D manifolds, also called surfaces.

*compact*: you'd only need a finite number of 2D maps to cover them. This means that they are finite in extent. Given a doughnut or sphere, you can always find a box to fit it into, even if it has to be a very big box.

*Ricci curvature*at the point you're looking at. Since it's an average, Ricci curvature only captures one component of the full notion of curvature as defined by Riemann. This means that a manifold can have zero Ricci curvature at every point without being flat (or having zero Riemannian curvature) overall. In terms of physics, the component captured by Ricci curvature happens to be just the one that describes the curvature of spacetime that's induced by matter being present. So a space with zero Ricci curvature corresponds to a space with no matter — a vacuum in other words.

*vanishing first Chern class*.

*complex manifolds*: the maps that chart them preserve angles and the manifolds display a certain local symmetry. (The term

*complex*refers to the fact that locally the manifolds look similar not just to plain old Euclidean space, but to something called complex space. In two dimensions this is just the complex plane you might be familiar with if you've studied complex numbers.) Being Kähler makes a manifold accessible to powerful mathematical machinery and also endows it with a special kind of symmetry.

### Yau's answer

*Calabi-Yau manifolds*.

### Hiding dimensions

^{-30}cm. That's more than a quadrillion times smaller than an electron. But there are other reasons too. To be consistent with the understanding of physics at the time, the manifolds harbouring the hidden dimensions had to have zero Ricci curvature. What is more, string theory assumes a special kind of symmetry, called

*supersymmetry*, which makes special demands of the geometry of spacetime. These demands make Calabi-Yau manifolds (with their special Kähler symmetry) excellent candidates for string theory, although we still don't know whether they are the only possible solution to the dimensional conundrum.

### String future

*mirror symmetry*(rather misleadingly, since it's much more complicated that its name suggests).

### About the author

*Plus*.

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