TITLE: Gromov-Witten and integrability.
QUESTION [10 upvotes]: The generation function of the Gromow-Witten invariants (with descendants) of the point is known to be Kontsevich-Witten tau-function of KdV, partition functions of $P^1$ and equivariant $P^1$ are known to be tau-functions of extended Toda and 2-Toda respectively. Are there any other manifolds (except of orbifolds made of mentioned previously manifolds ) for which generation functions of GW invariants are identified with tau-functions of some integrable hierarchy?

REPLY [4 votes]: Let $X$ be a smooth projective variety. Actually, as long as the quantum cohomology of $X$ is semisimple, the partition function of the (descendent) GW-invariants of $X$ is always identified with a tau-function of the Dubrovin--Zhang integrable hierarchy associated to $X$. So for the case of $\mathbb{P}^N$, any $N$, there is a nice integrable hierarchy whose topological tau-function gives the partition function of GW-invariants of $\mathbb{P}^N$. The point is that for $N=0,1$, the hierarchy was known in the traditional literature of integrable systems; whereas for $N\geq 2$, the Dubrovin--Zhang hierarchy is new.