Witten's conjecture states that the partition function Z = exp F is a τ-function for the KdV hierarchy, in other words it satisfies a certain series of partial differential equations corresponding to the basis
Kontsevich used a combinatorial description of the moduli spaces in terms of ribbon graphs to show that Here the sum on the right is over the set Gg,n of ribbon graphs X of compact Riemann surfaces of genus g with n marked points.
By Feynman diagram techniques, this implies that F(t0,...) is an asymptotic expansion of as Λ lends to infinity, where Λ and Χ are positive definite N by N hermitian matrices, and ti is given by and the probability measure μ on the positive definite hermitian matrices is given by where cΛ is a normalizing constant.
From this he deduced that exp F is a τ-function for the KdV hierarchy, thus proving Witten's conjecture.
The Witten conjecture is a special case of a more general relation between integrable systems of Hamiltonian PDEs and the geometry of certain families of 2D topological field theories (axiomatized in the form of the so-called cohomological field theories by Kontsevich and Manin), which was explored and studied systematically by B. Dubrovin and Y. Zhang, A. Givental, C. Teleman and others.