Friday, October 22, 2004

Nati Seiberg and Stokes' phenomenon

Nati Seiberg from the Institute for Advanced Study in Princeton gave a nice talk at Harvard about their work on the target space of minimal string theory.

He started his talk by expressing his happiness that he was able to visit the Great Boston during the extraordinary period of its history; this comment was clearly related to the Boston bloody socks.

Consider string theory with the following background: take the (p,q) minimal model, one of these simple classified rational conformal field theories, and add a CFT for Liouville theory with the right central charge, to get c=26 in total so that it cancels the bc ghosts.

One can define the boundary states for the FZZT branes - which is some hybrid between the D1-branes and D0-branes - namely some spacetime filling branes with a profile that suddenly goes to zero expo-exponentially. The wavefunction for this boundary state is known exactly, much like many other things. It has a parameter - one that roughly measures the point in space where the profile starts to decrease to zero rapidly.

The moduli space of these FZZT branes is interpreted as the exact target space. A funny feature is that the complexification of this target space seems to have branch cuts - the partition function depends on some sqrt(x) for a properly defined variable x. The complexification is therefore some degenerate genus 2 Riemann surface, roughly speaking. But you can also calculate the exact nonperturbative answer from the dual matrix model - and it turns out that it is an entire function of x without any branch cuts, even in the large N limit.

That sounds crazy: the semiclassical answer has some branch cuts that the exact answer does not see at all. The resolution of this apparent paradox is a mathematical effect discovered in the 19th century - called Stokes' phenomenon. It implies the following thing:

The procedure of analytical continuation of a result does not commute with the procedure of expanding the result, e.g. in the powers of g.

Well, that's a shocking observation - the person who first discovered it in the 19th century needed 30 years to realize how it works. We with Andy Neitzke were puzzled by this crazy behavior when we analyzed the quasinormal modes using the monodromy method.

The puzzling point was the asymptotic behavior of Bessel's functions. Take some linear combination f(x) of sqrt(x).J_n(x) and sqrt(x).J_{-n}(x) such that it can be approximated by exp(x) for Re(x) greater than zero. This specific linear combination can also be approximated by something like exp(-x) for Re(x) smaller than zero. You would think that if you combine these two approximations of f(x) that are valid in each half-plane, the function must be single-valued if x rotates around the origin. But it's not the case: f(x) is multiplied by a phase because it has really the form x^n times an entire function, and x^n picks a phase, at least for fractional values of n. This was puzzling for us - and the reason why it's possible is that the in each half-plane, we are neglecting an exponentially small part of f(x). Nevertheless, the analytical continuation of an exponentially small function can become very important, in fact it becomes exponentially large in the other half-plane - and it is this "innocently small" piece of f(x) that silently changes its phase, so that when you enter the region where the piece starts to dominate, the whole result has a different phase. This innocent small piece may be interpreted as an "instanton effect" in Seiberg's setup.

I am not explaining it too clearly, but if you think about it for a while, you will see that it works and there is nothing illogical about it.

Nati was expecting us that we would be advocating the higher genus moduli space - and we would be asking "where did the other sheet go?" Unfortunately for Nati, Andy Strominger (and partially I) were saying, from the beginning, that the other branch did not exist and you should only pick one sign of the partition function in each region. Nati finally confirmed that Andy's (and my) comments were correct, but they were not really correct because we were intelligent. ;-) Instead, they were correct because of an interesting reason (which was Stokes' phenomenon).

Seiberg's model is nice because all the observables are exactly calculable, and there is no room to hide, as he emphasized, so you can study exactly what's going on.

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