Matrix models, integrable structures and T-duality of type 0 string theory

Instanton matrix models (IMM) for two-dimensional string theories are obtained from the matrix quantum mechanics (MQM) of the T-dual theory. In this paper we study the connection between the IMM and MQM, which amounts to understand T-duality from the viewpoint of matrix models. We show that type 0A and type 0B matrix models perturbed by purely closed string momentum modes (or purely winding modes) have the integrable structure of Toda hierarchies, extending the well-known results for c=1 string. In particular, we show that type 0A (0B) MQM perturbed by momentum modes has the same integrable structure as type 0B (0A) MQM perturbed by winding modes, which is a nontrivial check of the T-duality between the matrix models. The MQM deformed by NSNS winding modes are used to study type 0 string in 2D black holes. We also find an intriguing connection between the IMM and the MQM via tachyon condensation. The array of alternating D-instantons and anti-D-instantons separated at the critical distance plays a key role in this picture. We discuss its implications on sD-branes in two-dimensional string theories.