Witten conjecture

In algebraic geometry, the Witten conjecture is a conjecture about intersection numbers of stable classes on the moduli space of curves, introduced by Witten (1991), and generalized in Witten (1993). Witten's original conjecture was proved by 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 M_g,n is the moduli stack of compact Riemann surfaces of genus g with n distinct marked points x₁,...,x_n, and M_g,n is its Deligne–Mumford compactification. There are n line bundles L_i on M_g,n, whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point x_i. The intersection index 〈τ_d1, ..., τ_dn〉 is the intersection index of Π c₁(L_i)^d_i on M_g,n where Σd_i = dimM_g,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

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