Observables in the Kontsevich Model
Kontsevich introduced a hermitian random matrix model to compute the generating function of intersection numbers of the moduli space of (punctured) Riemann surfaces. He showed that this generating function is also a τ-function for the Korteveg-de Vries (KdV) hierarchy of differential equations. This model is fundamentally different from the usual double scaling limit of random matrix models known to yield analogous τ-functions. Our aim is to clarify the notion of “observables” in both pictures, as related to KdV time evolutions. As a result we prove two conjectures by Kontsevich and Witten about the form of these observables, which involve polynomial matrix averages.
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