Sunday, March 04, 2007

Supercriticality vs negative curvature: a new duality

My review of Chris Herzog's interesting talk on Friday wouldn't go terribly beyond my comments about Subir Sachdev's talk.

Instead, let me focus on a special seminar by Eva Silverstein whose content I find kind of exciting, almost certainly true, and deep. It is mostly included in the paper by
Don't misspell the names. It is not McCartney but McGreevy and it is the same guy as John who is not a Lennon, while Starr is not Ringo and Silverstein is not Harrison. OK.

Let me start a bit differently. String theory as a complete theory of spacetime and other things predicts what happens when various things become very short. Our usual notions of "classical geometry" break down and they are replaced by all kinds of new phenomena that you may call "generalized or quantum geometry" but you don't have to. For the local purposes here, let us consider the compactification of string theory on a manifold M whose average Ricci scalar curvature is either
  • positive
  • zero
  • negative
and ask what happens when the size of M is very small.

Juan and positive curvature

In the case of positive curvature, there are no non-contractible curves on M in the most interesting examples. You may imagine that M is something like a sphere. More generally, it can be a base of a conical Calabi-Yau manifold such as the conifold. The compactifications with small higher-dimensional spherical hidden dimensions are described by a weakly-coupled dual CFT according to Maldacena's holographic duality. You may see that in some sense we're dividing the empire into three pieces that may be labeled e.g. "Maldacena", "Yau", and "Silverstein". ;-)




Vanishing curvature: tori

If the hidden dimensions have a vanishing curvature, you may very well imagine that they're toroidal. In that case, the most important insight is T-duality. The radius "R" is equivalent to the radius "1/R" in string units. Mirror symmetry is a special example of T-duality applied fiber-wise, as Strominger-Yau-Zaslow taught us. The Calabi-Yau manifolds where mirror symmetry is a key insight are Ricci-flat: they fit into the same intermediate group according to the average curvature.

Negative curvature: new analysis

What about the negative curvature? The simplest example of a negatively curved compact manifold is a genus "h" Riemann surface where "h" is greater than one. For example, the surface of eyeglasses without the glass is a genus two Riemann surface because there are two holes in it.

A genus "h" Riemann surface has a 2h-dimensional first homology group: there are "2h" independent one-cycles in it that define a symplectic form on the homology via the oriented intersection number. Moreover, all such Riemann surfaces may be thought of as quotients of the Lobachevsky hyperbolic plane - or the Poincare disk, if you wish - by a discrete group.

This covering space is the Euclidean counterpart of the anti de Sitter space and it shares a particular property with it: namely that the volume of a sufficiently large spherically symmetric region of it is proportional to the surface (times the curvature radius times a numerical constant). It's the property that makes holography in anti de Sitter space kind of less shocking than you could a priori think.

We can steal an idea from the "vanishing curvature" section. T-duality in that case creates circular dimensions whose dual momenta are obtained from the winding numbers of strings in the original, T-dual picture. In the case of genus "h" Riemann surfaces, there are many more ways how strings may wind around the surface.

Homology vs homotopy

The homology is the only description of the charges that is conserved once you include interactions. But for a free string, you may actually try to describe its winding in terms of the fundamental group i.e. the first homotopy group. If the characteristic size of the winding number is "W", you will find out that the element of the fundamental group is described by "W" letters or so, each of which can be one of "2h" different choices. So the number of different types of windings goes like "h^W" if you allow me to be very schematic. The authors are actually very precise about the numerical coefficients.

At any rate, it is an exponentially large number of winding states. The moral example of the torus tells you that every one-cycle leads to a new dimension. In the case of genus "h" Riemann surfaces, we seem to create many more dimensions in the dual picture than what we have in the original picture because there are many more 1-cycles. The greater "h" is, the more dimensions the dual theory seems to have.

Comparing additional central charges

You can easily see that for large values of "h", the corresponding dual theory that geometrizes the winding numbers around the diverse cycles of the Riemann surface is a supercritical theory - a theory whose spacetime dimension exceeds the critical dimension. It has a larger central charge than the critical string theory. A large central charge means, via Cardy's formula, a larger coefficient describing the exponential increase of the number of states.

You can see that in the supercritical descriptions, this larger increase simply arises from the non-zero oscillators associated with many new dimensions of spacetime. On the other hand, in the original Riemann surface picture, this additional factor in the degeneracy arises from the "zero modes", something that classifies the windings.

Volumes in hyperbolic spaces

There is a different way to think about the exponentially growing degeneracy of the winding states - a way that implies that the qualitative picture is valid not only for two-dimensional Riemann surfaces but for rather generic higher-dimensional negatively curved manifolds, too. Take the quotient of a hyperbolic plane or space. If the winding number goes like "W", you may see that the volume of the region of the hyperbolic plane where you can get with contours (winding strings) of proper length "W" grows exponentially with "W".

Because the density of copies of your starting point - copies induced by the discrete group identification - is constant in this hyperbolic space, you can see that the number of different ways how the periodic conditions may be satisfied is growing exponentially with "W", too.

You may want to count the number of dragons in Escher's picture of the Poincare disk below whose distance from a particular dragon is "W". Because the dragons (or fish) are essentially organized into trees, the result will clearly go as "exp(C.W)" for large "W".



Once again, the authors are able to check that both sides of the duality give you the same prediction for the numerical factor "C" in front of "W" in the exponent!

Supercritical dynamics

Because the dual theory of the negatively curved manifolds is supercritical, there are all kinds of phenomenological questions that you must ask - whether these extra dimensions may be made large and whether there are nice cosmological solutions with a time-dependent dilaton and tachyon that make the picture physical.

But even if they didn't exist, I think that it would be fair to say that they have shown that at least some supercritical string theories are parts of what we called "string theory" as long as you allow a natural dual description for some allowed (but non-static) compactifications on negatively curved manifolds. The tools used to derive the new description are as straightforward a generalization of T-duality as you can get.

String-string duality as a similar kind of a relation

In fact, I think that the string-string duality between K3 manifolds and heterotic strings might be reinterpreted as a generalized T-duality along these lines, too. One would have to generalize their construction a bit: the relevant windings on the K3 side involves 2-cycles. So interpret K3 as some fake S1-fibration over some fake 3-manifold, and treat is as type IIA on this 3-manifold (with a highly nontrivial dilaton profile). The two-cycles become type IIA strings of some kind, and a classification of their possible windings could give you the chiral worldsheet degrees of freedom you need for the heterotic string in the bosonic picture.

Some topological details don't yet work and I'm too busy with other things but the overall picture could work and lead to a semi-satisfactory proof of the K3-heterotic duality.

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