Large N expansion of matrix models and tau functions of algebraic curves

by Eynard, B. (Saclay)

Tuesday 5 Dec 2006, 14:00 → 15:00 Europe/Zurich

TH Auditorium (CERN)

For every algebraic curve E, we define an infinite sequence of free energies F_g(E), as residues on the curve. We show that the F_g(E) are symplectic invariants, and that the formal series ln(τ_N(E)) = ∑_g N^2-2g F_g(E), obeys Hirota equations, and thus is a KP tau function. Moreover, we show that for various matrix models, if the algebraic curve is taken as the large N limit of the loop equations (classical spectral curve), then the matrix integral topological expansion is indeed given by F_g(E): ln(Matrix integral) = ∑_g N^2-2g F_g(E). In particular, this gives a new and very easy derivation of some known properties of the Kontsevitch integral (the fact that it depends only on odd times and that it is a KdV tau function).