Communications in Number Theory and Physics

Volume 1 (2007)

Number 2

Invariants of algebraic curves and topological expansion

Pages: 347 – 452



B. Eynard (Service de Physique Théorique de Saclay, Gif-sur-Yvette, France)

N. Orantin (Service de Physique Théorique de Saclay, Gif-sur-Yvette, France)


For any arbitrary algebraic curve, we define an infinite sequence ofinvariants. We study their properties, in particular their variationunder a variation of the curve, and their modular properties. We alsostudy their limits when the curve becomes singular. In addition, wefind that they can be used to define a formal series, which satisfiesformally an Hirota equation, and we thus obtain a new way ofconstructing a $\tau$-function attached to an algebraic curve.

These invariants are constructed in order to coincide with thetopological expansion of a matrix formal integral, when thealgebraic curve is chosen as the large $N$ limit of the matrixmodel’s spectral curve. Surprisingly, we find that the sameinvariants also give the topological expansion of other models,in particular the matrix model with an external field, and theso-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 itto the Kontsevitch integral, and we give a new and extremely easyproof that Kontsevitch integral depends only on odd times, and thatit is a KdV $\tau$-function.

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