Toda Lattice Hierarchy and Generalized String Equations

String equations of the p-th generalized Kontsevich model and the compactified c = 1 string theory are re-examined in the language of the Toda lattice hierarchy. As opposed to a hypothesis postulated in the literature, the generalized Kontsevich model at p = −1 does not coincide with the c = 1 string theory at self-dual radius. A broader family of solutions of the Toda lattice hierarchy including these models are constructed, and shown to satisfy generalized string equations. The status of a variety of c ≤ 1 string models is discussed in this new framework. E-mail: [email protected] 1