Scaling Self–Similar Formulation of the String Equations of the Hermitian One–Matrix Model

The string equation appearing in the double scaling limit of the Hermitian one–matrix model, which corresponds to a Galilean self–similar condition for the KdV hierarchy, is reformulated as a scaling self–similar condition for the Ur–KdV hierarchy. A non–scaling limit analysis of the one–matrix model has led to the complexified NLS hierarchy and a string equation. We show that this corresponds to the Galilean self–similarity condition for the AKNS hierarchy and also its equivalence to a scaling self–similar condition for the Heisenberg ferromagnet hierarchy. 0. The Hermitian one–matrix model has received much attention in recent years as a non–perturbative formulation of string theory. In [2] the double scaling limit for the even potential case was used to show that the specific heat is a solution of the Korteweg–de Vries (KdV) hierarchy that satisfies an additional constraint, the so called string equation. In [12] it was prove that this corresponds to invariance under Galilean transformations, see also [9]. The model is also relevant for topological gravity and for the Witten–Kontsevich intersection theory of the moduli space of complex curves [18]. In [3] it was performed a non–scaling limit analysis of the the Hermitian one– matrix model with general potential. Now, the specific heat is the second conserved density of the Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy and the string equation, as we shall show, corresponds to invariance under the Galilean transformations ∗Research supported by British Council’s Fleming award —postdoctoral MEC fellowship GB92 00411668, and postdoctoral EC Human Capital and Mobility individual fellowship ERB40001GT922134