## Monday, February 23, 2009 ... /////

### Dualities vs singularities

A decade ago, when we wrote Dualities vs Singularities (PDF), I obviously had to be afraid of Mathematica. Otherwise I would have easily drawn diagrams similar to the following:

Click to zoom in. It is kind of pretty. But what does it show? I must tell you a few words.

Take the 11-dimensional M-theory and compactify it on a rectangular k-torus with no C-field. The shape of the torus is described by "k" radii,

{exp(p1), exp(p2), ..., exp(pk)}
times the Planck length (or more precisely, the self-dual radius under a U-duality described later). It is useful to write the radii as exponentials because the logarithms, "p_i", behave linearly under the multiplication of the radii and their powers.

And when you perform U-dualities, there's a lot of multiplication of radii by powers of other (or the same) radii. The "p_i" coordinates naturally go from "-infinity" to "+infinity".

The question was which values of the vectors "p" describe a physical situation whose rough behavior is well understood and where we have a natural perturbative expansion. For M-theory on tori, there are only three such descriptions:
• M-theory: on a large torus in Planck units (11D SUGRA etc.): BLUE
• Type IIA string theory: weakly coupled on a large torus in string units: PURPLE
• Type IIB string theory: weakly coupled on a large torus in string units: BLACK
Various inequalities for linear functions of "p_i" describe which conditions have to be satisfied for you to have each of these descriptions. Permutations of "p_i" don't matter. For example, if the smallest "p_i" is (still) positive, then you are in the blue M-theoretical region.

But the most important operation we can use are dualities. An elementary U-duality changes three radii by some linear transformation and the remaining ones by another transformation.

If you interpret one of the three special circles as the 11th coordinate added to a type IIA theory to obtain M-theory, this elementary U-duality may be interpreted as a double T-duality in type IIA string theory reflecting the other two circles, permuting them, and returning you to another version of type IIA string theory - that you subsequently reinterpret as a copy of M-theory again.