On tachyon condensation and open-closed duality in the c=1 string theory

We present an exact representation for decaying ZZ-branes within the dual matrix model formulation of c = 1 string theory. It is shown how to trade the insertion of decaying ZZ-branes for a shift of the closed string background. Our formalism allows us to demonstrate that the conjectured world-sheet mechanism behind the open-closed dualities (summing over disc insertions) is realized here in a clear way. On the way we need to clarify certain infrared issues — insertion of ZZ-branes creates solitonic superselection sectors