Now and then, a sense of accomplishment is welcome in my tiny existence. In the greater scheme of things it means little – as Kafka is reputed to have said to Brod, “there is hope, but not for us.” Still, it is nice, if only for a moment. If you are interested, you can find the reason here, or here.
Working on the confinement problem for Yang-Mills theories in lower dimensions, led me to the problem of the principal chiral model in one space and one time (1+1) dimensions. This is a model of an N by N unitary matrix field, of determinant one. Axel Cortes Cubero and I studied this problem at large N, when he was my Ph.D. student (he is now a postdoc at SISSA).
So what is large N? There is a special limit which some problems of high-energy or condensed-matter physics can be solved with. This is the large-N limit, where the number of species of particle is taken to infinity. Actually, there are a few of these large-N limits:
- The large-N limit of an isovector field with N species. This sometimes goes by the name “random-phase approximation”. Lots of interesting problems have been solved this way. The Feynman diagrams have a chain-like or linear structure.
- ‘t Hooft realized that for models of matrices (like the principal chiral model above) the Feynman diagrams became planar. The 1/N-expansion is an expansion in the genus of Feynman diagrams. The original motivation was an approximation scheme for QCD, 1/N=1/3 being approximately zero.
It turns out that it is much harder to calculate anything in ‘t Hooft’s large-N limit than in isovector case. Perhaps that is because the problems ‘t Hooft was considering are potentially much more interesting.
Since the late 90’s, much of the activity in supersymmetric quantum field theory concerns ‘tHooft’s limit. I don’t want to just wax poetically about this, but there has been a lot of progress in superconformal field theories and their deformations.
The principal chiral model is in only two spacetime dimensions, whereas the supersymmetric theories above are in three or four spacetime dimensions. Nonetheless, this model is interesting, because it shares features with QCD that the supersymmetric theories do not. Most important, it is asymptotically free and has massive particles in its spectrum. Furthermore, the solutions Axel and I obtained are complete, going beyond both perturbation theory and strong coupling approximations. These solutions are valid at all length scales.
But there was a difficult snag in all of this. Axel’s and my methods heavily use integrability and the “form-factor bootstrap”. It’s pretty easy to see how correlation functions (or renormalized propagators) look at large distances. Unfortunately, it was not obvious how they would behave at short distances. Moreover, there is a prediction for the short-distance behavior from the perturbative renormalization group; the two-point function behaves like logarithm of the separation squared. I was not at all confident that the form-factor bootstrap in Mr. ‘t Hooft’s limit would obey this prediction (and behave like the logarithm squared). There was no straightforward way to check it, and there were other things for me to do. I felt, however, that I was failing to put the ideas to the test.
Last July I asked an elephant in the Copenhagen Zoo to step on my head, should I be unable to solve the problem. She did not agree, because we could not shake on it (her trunk was too far from the fence). The entire summer, I worked only on this problem. It turned out that the mathematics is related to ideas well-known to statistical mechanicians (Levy flights) and mathematicians (non-integer powers of Laplacians).
In the end, the answer fit the prediction. It’s kind of magical that it worked, since the prediction was made using very different mathematics.
For a few more days, I will feel good about myself. Then I’ll get tired of my hubris. O Goddess Nemesis, mine eyes search for thine sword and shield amongst the constellations.