M ar 2 00 4 D - branes and complex curves in c = 1 string theory

We give a geometric interpretation for D-branes in the c = 1 string theory. The geometric description is provided by complex curves which arise in both CFT and matrix model formulations. On the CFT side the complex curve appears from the partition function on the disk with Neumann boundary conditions on the Liouville field (FZZ brane). In the matrix model formulation the curve is associated with the profile of the Fermi sea of free fermions. These two curves are not the same. The latter can be seen as a certain reduction of the former. In particular, it describes only (m, 1) ZZ branes, whereas the curve coming from the FZZ partition function encompasses all (m,n) branes. In fact, one can construct a set of reductions, one for each fixed n. But only the first one has a physical interpretation in the corresponding matrix model. Since in the linear dilaton background the singularities associated with the ZZ branes degenerate, we study the c = 1 matrix model perturbed by a tachyon potential where the degeneracy disappears. From the curve of the perturbed model we give a prediction how D-branes flow with the perturbation and derive the two-point bulk correlation function on the disk with the FZZ boundary conditions.