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

$$F_g = \sum_{n\geq 0} \frac{1}{n!} \left( \sum_{k_1,\ldots,k_n} \i... ...cal{M}}_{g,n}} \psi_1^{k_1} \cdots \psi_n^{k_n} \right) t_{k_1}\cdots t_{k_n},$$

Setting $F= \sum_{g}\lambda^{2g-2}F_{g}$, Witten's conjecture is that $Z=\mathrm{exp} F$ 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.