  In this paper we will study the ground states of the toy model of 2D closed
string tachyon effective action. We will firstly construct the classical
solutions of the tachyon effective action that do not induce backreaction on
metric and dilaton. Then we will study the quantum mechanics of the zero mode
of the tachyon field. We will find family of vacuum states labelled with single
parameter. We will also perform the quantum mechanical analysis of the tachyon
effective action when we take into account dynamics of nonzero modes. We will
calculate the vacuum expectation values of components of the stress energy
tensor and dilaton source and we will argue that there is not any backreaction
on metric and dilaton.
