Exact vs. Semiclassical Target Space of the Minimal String

We study both the classical and the quantum target space of (p,q) minimalstring theory, using the FZZT brane as a probe. By thinking of the target spaceas the moduli space of FZZT branes, parametrized by the boundary cosmologicalconstant 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 thedual matrix model that the exact quantum FZZT observables exhibit Stokes'phenomenon and are entire functions of x. Along the way we clarify some pointsabout the semiclassical limit of D-brane correlation functions. The upshot isthat nonperturbative effects modify the target space drastically, changing itfrom \CM_{p,q} to the complex x plane. To illustrate these ideas, we study indetail 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 describeinstantons in the effective theory on the brane. Finally, we discuss possibleapplications to black holes and the topological string.Comment: 49 pages, 4 figure