In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper Witten (1991), and generalized in Witten (1993). Witten's original conjecture was proved by Maxim Kontsevich in the paper Kontsevich (1992). In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Edward Witten in the paper Witten (1991), and generalized in Witten (1993). Witten's original conjecture was proved by Maxim Kontsevich in the paper Kontsevich (1992). Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the KdV hierarchy. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy. Suppose that Mg,n is the moduli stack of compact Riemann surfaces of genus g with n distinct marked points x1,...,xn, and Mg,n is its Deligne–Mumford compactification. There are n line bundles Li on Mg,n, whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point xi. The intersection index 〈τd1, ..., τdn〉 is the intersection index of Π c1(Li)di on Mg,n where Σdi = dimMg,n = 3g – 3 + n, and 0 if no such g exists, where c1 is the first Chern class of a line bundle. Witten's generating function