Thursday, April 20, 2006

Integrability: giant magnons

Diego Hofman and Juan Maldacena - these two physicists should not be confused with Diego Maradona - study the
  • excitations of N=4 gauge theory in d=4 in the planar limit.

Recall that according to the gauge-gravity holographic correspondence, the strong coupling limit describes type IIB string theory on the product space "AdS5 x S5". A few years ago, Berenstein, Maldacena, and Nastase have shown that the gauge theory is not equivalent to pure supergravity but the full string theory; they identified the strings with the long traces. This research direction has been transformed into the studies of integrability and spin chains (these are the discretized strings) and we have talked about this topic at various places, for example here.

This spin chain itself carries excitations and the most important ones are called magnons: it's an excitation that reverts the direction of a single spin (or the "magnetic moment" if you wish) in the spin chain and propagates as a wave along the chain. In the planar limit, i.e. up to the leading terms in the "1/N" expansion, physics should simplify. Many people have believed for some time that a full exact solution of string theory in this limit should exist. This task is equivalent to a full understanding of the worldsheet of a string propagating in the "AdS5 x S5" background for the simplest choice of its topology.

In the variables mentioned above, the question is reduced to the spectrum, the dispersion relations, and the S-matrix of the magnons. Effectively, one needs to study the S-matrix for various polarizations and encounters a "256 x 256" matrix. Its form was recently fixed by Niklas Beisert, up to an overall normalization. Moreover, one month ago, Romuald Janik of Poland has shown how the crossing symmetry emerges from the formulae for the S-matrix.

Hofman and Maldacena confirm the results but add something extremely interesting: the adjective "giant". In analogy with giant gravitons, you may suspect that there will be a new picture that replaces the original point-like magnon excitations by something big.




Graffiti inside the LLM disk

Indeed, there is one. The relevant picture is the LLM disk that represents the anti de Sitter space multiplied by the sphere with some extra drawings in it. Which drawings? Well, it may sound shocking but the magnons become straight lines stretched between two points on the boundary of the disk. Magnon is a bilocal object.

If the points are close to each other, this line looks like a point and you get the usual "small" excitations that propagate along the boundary of the disk, also known as the circle. ;-) If the points are far away, the magnon becomes "giant". How do things become giant? Much like in the case of giant gravitons or giant gluons, they become giant if they carry a large momentum. It turns out that the angle that separates the two points on the circle that we have just connected is nothing else than the momentum "p" of the magnon. This stringy, geometric description of the kinematical variables trivially explains why the S-matrix is periodic not just in one but in two different variables: "theta" and "p".

The scattering amplitudes for the magnons turn out to be governed by simple geometric quantities that you can extract from triangles inside the disks, drawn using lines that represent the individual magnons. This insight is a striking example of the ability of string theory to convert boring formulae into a geometric picture that is actually fully equivalent to its mathematical form. Many important phenomena in particle physics are interpreted as geometrical effects in which the hidden, compact dimensions and their number and other, more detailed properties dictated by string theory play a crucial role.

It has been established that various physical phenomena in field theories are equivalent to phenomena in string theory where some extra dimensions are added. Because our intuition what happens in the field theory context is often limited, the geometrical picture emerging from string theory is very useful and tells us what should we really expect. This paragraph is meant as an explanation that the extra dimensions and their features derived from string theory are completely natural and, to some extent, inevitable concepts for physics. I will describe the story of the conifold, Dijkgraaf-Vafa, gaugino condensation, and related topics when I have more time.

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