Musings

The worldsheet action of the noncritical string is $$S=\frac{1}{8\pi} \int (\partial \phi)^2 - 2\sqrt{2} R \phi + \mu e^{-\sqrt{2}\phi} - (\partial X)^2$$ a Liouville theory coupled to a (negative-signature) free boson. The spectrum of the string theory contains a single propagating field, a massless scalar which we perversely refer to as the “tachyon”. But the theory is not 1+1 dimensionally Poincaré invariant. The string coupling varies linearly with the “spatial” coordinate $\phi$ and the tachyon also has a nontrivial (exponential) profile.

The physics (such as it is) is a bit boring. We can send tachyon pulses in from the right. They scatter off the “wall” and we study what escapes back to infinity. The S-Matrix in such a situation is a little awkward to define. The “in” states are left-moving tachyons, whereas the “out” states are right-moving tachyons, so there’s a certain ambiguity in expressing the latter in a basis of the former. Nevertheless, we learned over a decade ago that one can define (at least perturbatively) a unitary S-matrix and, moreover that it can be computed from a large-N Hermitian matrix quantum mechanics, with action

One side of the Fermi sea filled: the 1+1 dimensional noncritical bosonic string
Both sides filled: the 1+1 dimensional noncritical Type 0B string

Diagonalizing $M$ introduces a Vandermonde determinant, which makes the eigenvalues of $M$ act like free Fermions moving in the potential $V$. The continuum limit is achieved by tuning $V$ so that, at the top of the Fermi sea, it looks like an inverted harmonic oscillator potential. The tachyon represents oscillations of the Fermi surface, though the relation between the eigenvalue coordinate depicted in the graph at left and the $\phi$ coordinate is highly nonlocal.

Since the Fermions are free, the scattering is rather trivial to compute; the nontrivial structure comes from converting to and from the bosonic description. All that looks fine perturbatively, but nonperturbatively the theory is clearly sick. The Fermions can tunnel through the barrier and disappear down the abyss. Clearly, as stated, the nonperturbative S-matrix is nonunitary.

We can fix this in a number of ways, without altering the perturbative physics. One possibility is simply to put in an infinite wall at $\lambda=0$. Then there’s no other side to tunnel to. But it turns out that the nonperturbative physics depends on the precise details of the shape and location of the wall and we have no a-priori way to determine these “nonperturbative parameters.”

Alternatively, one could fill up both sides of the well, but then one seems to have a theory with two aymptotic regions, rather than one. Perturbatively, they don’t talk to each other, but nonperturbatively, they do.

All this was known a decade ago. The new insight comes from reinterpreting this latter theory not as a theory with a single scalar and two asymptotic regions, but as a theory with two scalars and a single asymptotic region. The even fluctuations of the Fermi sea correspond to the tachyon as before. The odd fluctuations correspond to a new scalar, which we identify as the Ramond-Ramond scalar, $C$ of the Type 0B string.

The two Fermi surfaces need not have precisely the same height and the difference corresponds to an expectation-value for $C$. Clearly, the bosonic string corresponds to taking the expectation-value of $C$ to infinity. And we now understand the “fate” of the nonperturbative instability of this bosonic string. It decays to the Type 0B string!

More amusing yet, is that Zamolodchikov recently discovered a D0 brane of the noncritical string, imposing Dirichlet boundary conditions on $\phi$. This D0 brane turns out to correspond to putting a fermion at the top of the potential. The D0 brane is unstable (it carries no conserved charges, since in neither the bosonic or Type 0B theory is there a Ramond-Ramond gauge field under which it is charged). In the fermionic language, the decay of the D0 brane is described by the fermion rolling down the hill.

The original promise of these low-dimensional string theories was that, though they were too simple to be interesting in their own right, they were simple enough to understand completely. If we really could thoroughly understand their nonperturbative physics, we might learn something. It turned out that we could learn essentially nothing about their nonperturbative physics back then. So this promise went unrealized. Finally, though, it seems that we do have an nonperturbatively well-defined 1+1 dimensional string theory.

Perhaps we might yet learn something from it …