Exact vs. Semiclassical Target Space of the Minimal String

We study both the classical and the quantum target space of (p,q) minimal string theory, using the FZZT brane as a probe. By thinking of the target space as the moduli space of FZZT branes, parametrized by the boundary cosmological constant x, we see that classically it consists of a Riemann surface \CM_{p,q} which is a p-sheeted cover of the complex x plane. However, we show using the dual matrix model that the exact quantum FZZT observables exhibit Stokes' phenomenon and are entire functions of x. Along the way we clarify some points about the semiclassical limit of D-brane correlation functions. The upshot is that nonperturbative effects modify the target space drastically, changing it from \CM_{p,q} to the complex x plane. To illustrate these ideas, we study in detail the example of (p,q)=(2,1), which is dual to the Gaussian matrix model. Here we learn that the other sheets of the classical Riemann surface describe instantons in the effective theory on the brane. Finally, we discuss possible applications to black holes and the topological string.