Geometry and Integrability in Mathematical Physics GIMP'06
May 15 - 19, 2006, Moscow, Russia
Bertrand Eynard
SPhT CEA Saclay, France
Large N expansion of matrix models, and algebraic geometry
Given an arbitrary algebraic curve E, we construct an infinite sequence of symplectic invariants F_g(E). Those invariants have many interesting properties.
We show some explicit examples of applications, in particular:
- When we specialize the curve E to the large N limit of the Schwinger-Dyson equation of a (1 or 2) matrix model, the F_g(E) coincide with the topological expansion of the matrix model free energy.
- When the curve E is the classical limit of a (p,q) conformal minimal model, the F_g(E) give the KP tau function.
- When the curve E is the the large N limit of the Schwinger Dyson equation coming of the Kontsevitch integral, the F_g(E) give the Kontsevitch integral. In this formalism, the properties that the Kontsevitch integral depends only on odd moments, or the fact that it coincides with the KdV tau function are straightforward. This approach also allows to study the generalized Kontsevitch integral in a very easy way.
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