Why is the torus important for compactification in string theory? (aka much ado about the torus)


The torus is special because it's so simple, and because it provides the most tractable example of Mirror Symmetry https://en.wikipedia.org/wiki/Mirror_symmetry_(string_theory), a generalization of T-duality (which relates Type IIB with Type IIA with one another).

Toric compactifications are rather special, they're a special case of an incredibly large number of possible compactifications. The condition for the preservation of SUSY in the compactified theory was worked out in http://www.sciencedirect.com/science/article/pii/0550321385906029, and found to be that the 6-dimensional compactified space be a so-called Calabi-Yau manifold. Other compactifications are of course possible, but they do not preserve SUSY.

Both the torus and the Calabi-Yau's are Ricci flat, i.e. $R_{ab} = 0$. The $S^5$ has positive curvature, and when the compactified manifold has curvature, this curvature sources the other fields and makes a simple solution hard to find. Simple examples of compactifications on positively curved spaces include the famous $AdS_5 \times S^5$ compactification of Type IIB string theory. Here, the positive curvature of the sphere is balanced by the negative curvature of the AdS space, and moreover, the curvature lengths of the two are equal, so the sphere is in no sense "small", and this is therefore not a proper compactification in the usual sense.

It is not true that two manifolds which are topologically related will preserve the same symmetries, and also note that $S^5$ is not topological to $T^5$ as you say. This can be seen by comparing the Betti numbers, for example.


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Updated on August 01, 2022


  • leastaction
    leastaction over 1 year

    Why is the torus important in string theory and supergravity?

    To be specific, why does one care about something like compactification of Type IIB or IIA supergravity on a torus $T^5$, as opposed to say $S^5$ for the internal space?

    I know that the toric compactification preserves maximal SUSY just like the $S^5$ compactification because it is possible in principle to deform $T^5$ to $S^5$. However, this doesn't seem like a rigorous enough statement.

    Is it true that if two manifolds are related by deformation (i.e. they have the same genus) then both will preserve the same number of super symmetries?

  • Surgical Commander
    Surgical Commander over 8 years
    I'm not sure what the deformation is that takes $S^5$ to $T^5$, it certainly isn't a topology-preserving one. The allowed types of deformations depend on what you want to do. The sphere is not "small" because it's length is the same as the AdS space. So, contrary to the usual Kaluza-Klein scenario, the sphere would be very obvious to an AdS observer.