Witten's Conjecture

Witten’s conjecture is a recursive constraint for top intersections of $\psi$-classes on ${\overline{\mathcal{M}}}_{g,n}$. More precisely, if one considers the generating function

$$F_g=\sum _{n\ge 0}\frac{1}{n!}\left(\sum _{k_1,\dots ,k_n}{\int }_{{\overline{\mathcal{M}}}_{g,n}}{\psi }_1^{k_1}\cdots {\psi }_n^{k_n}\right)t_{k_1}\cdots t_{k_n},$$

setting $F={\sum }_g{\lambda }^{2g-2}{F}_g$, Witten’s conjecture is that $Z=\exp F$ is annihilated by a certain partial differential operator that also arises in studying the KdV equation in soliton theory. It can be restated as a specialization of the Virasoro conjecture for the case where $X$ is a point.

Multiple proofs of this conjecture now exist, due to Kontsevich, Okounkov-Pandharipande, and most recently Mirzakhani.