Time-dependent backgrounds of 2D string theory

We study possible backgrounds of 2D string theory using its equivalence with a system of fermions in upside-down harmonic potential. Each background corresponds to a certain profile of the Fermi sea, which can be considered as a deformation of the hyperbolic profile characterizing the linear dilaton background. Such a perturbation is generated by a set of commuting flows, which form a Toda lattice integrable structure. The flows are associated with all possible left- and right-moving tachyon states, which in the compactified theory have discrete spectrum. The simplest nontrivial background describes the sine-Liouville string theory. Our methods can be also applied to the study of 2D droplets of electrons in a strong magnetic field.