For any arbitrary algebraic curve, we define an infinite sequence of invariants. We study their properties, in particular their variation under a variation of the curve, and their modular properties. We also study their limits when the curve becomes singular. In addition, we find that they can be used to define a formal series, which satisfies formally an Hirota equation, and we thus obtain a new way of constructing a τ -function attached to an algebraic curve.These invariants are constructed in order to coincide with the topological expansion of a matrix formal integral, when the algebraic curve is chosen as the large N limit of the matrix model's spectral curve. Surprisingly, we find that the same invariants also give the topological expansion of other models, in particular the matrix model with an external field, and the so-called double scaling limit of matrix models, i.e., the (p, q) minimal models of conformal field theory.As an example to illustrate the efficiency of our method, we apply it to the Kontsevitch integral, and we give a new and extremely easy proof that Kontsevitch integral depends only on odd times, and that it is a KdV τ -function.

We rewrite the loop equations of the hermitian matrix model, in a way which involves no derivative with respect to the potential, we compute all the correlation functions, to all orders in the topological 1/N 2 expansion, as residues on an hyperelliptical curve. Those residues, can be represented as Feynmann graphs of a cubic field theory on the curve.

SPhT-T05/046 ITEP/TH-??/05We present the diagrammatic technique for calculating the free energy of the Hermitian onematrix model to all orders of 1/N expansion in the case where the limiting eigenvalue distribution spans arbitrary (but fixed) number of disjoint intervals (curves).