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?


Short answer: essentially, the point and $P^1$ are the only spaces where the GW generating function is a tau-function. However, you mentioned two variations on this spaces: equivariant orbifold versions. And there are other variations that go a bit further -- twisted and relative invariants, and, a little wilder, Landau-Ginzburg theory. But in all cases, everything seems to be a reduction of either the KP hierarchy or the 2-Today hierarchy.

Twisted Invariants

Let $L_i$ be some line bundles on $X$, and let $B$ be the total space of $\oplus L_i$. The GW theory of $B$ doesn't make sense since it's not compact, and so neither will $\overline{\mathcal{M}}_{g,n}(B, \beta)$, but we can take a $\mathbb{C}^*$ on $B$ that fixes $X$ and just acts on the fibers, and then use localization on $\overline{\mathcal{M}}_{g,n}(B, \beta)$.

The fixed point locus will be $\overline{\mathcal{M}}_{g,n}(X, \beta)$, but the invariants will be different because of extra terms coming from the euler class of the normal bundle. We call and so it will make sense to integrate, and we define these to be $GW(B)$, and call them twisted invariants of $X$.

A Grothendieck-Riemann-Roch calculation begun by Mumford and extended and expanded by Faber-Pandharipande and Coates-Givental will reduce $GW(B)$ to $GW(X)$, but in a messy way that would for instance change the invariants, and certainly change around the integrable hierarchy.

So, for instance, in the case of a point, we'll just get integrals over $\overline{\mathcal{M}}_{g,n}$, the normal bundle will essentially be copies of the Hodge bundle, and so we'll get $\lambda$ classes appearing in our integrals along with the usual $\psi$ classes: these are called Hodge integrals.

In the case that there is just one line bundle (so $B=\mathbb{C}$, the integral will be linear in the $\lambda$ classes, and will be the integral appearing in the ELSV formula. Kazarian shows that the resulting GW theory satisfies the KP hierarchy.

The case with three line bundles ($B=\mathbb{C}^3$) is the case covered by the topological vertex. As far as I know, there is no known integrable hierarchy satisfying the whole thing. But Zhou has shown that certain specializations lead to KP and 2-Toda type equations.

Similarly, we can twist the GW theory of $\mathbb{P}^1$ and hope to get integrable hierarchies here. Brini has made some progress here for the direct sum of two line bundles, so that $B$ is a three-fold. In particular, for the resolved conifold (the total space of $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, he gets connections with the Ablowitz-Ladik hierarchy, apparently some reduction of 2-Toda.

Relative Invariants

Another variation we can play is to use relative invariants, working relative to a divisor.

In dimension zero this doesn't work, but we can work relative to 0 and $\infty$ on $\mathbb{P}^1$. Okounkov and Pandharipande have shown that this flavor also satisfies some 2-Toda type hierarchy.

In the paper of Zhou mentioned above, he also sets up certain relative generating functions for toric varieties that satisfy KP and 2-Toda hierarchies.


In another direction, if we move slightly away from GW theory we can get more interesting examples. Fan, Jarvis and Ruan have recently finished rigorously constructing a Landau-Ginzburg A-model -- essentially, a Gromov-Witten theory for hypersurface singularities.

These theories have a central charge that acts as a dimension. The central charge 0 case has an ADE classification. Witten conjectured that the theory of these should satisfy the corresponding Kac-Wakimoto / Drinfeld-Sokolov reduction of the KP hierarchy. Note that for $A_1$, this is the Gromov-Witten theory of a point, and the usual KdV hierarchy.

In the $A_r$ case, the analytic machinery of FJR is not needed to define the theory, and it goes under the name $r$-spin curves -- essentially, we're doing integrals over the moduli space of orbifold curves with an chosen $r$-th root of the canonical bundle. Witten's conjecture was proven here by Faber, Shadrin and Zvonkine. FJRW have recently proven the D and E cases.

  • $\begingroup$ Thanks for one more perfect answer. Probably now I should refine my question in the following way: why no integrable structure is known for other manifolds like $P^N$ or higher genera curves? Is there any conceptual problem with it or this is just not so interesting question? $\endgroup$ – Sasha Sep 17 '10 at 10:01

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.

  • $\begingroup$ Thank you for the answer. Is a bilinear Hirota-type formulation of the Dubrovin--Zhang integrable hierarchy known? $\endgroup$ – Sasha Dec 5 '16 at 21:52
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    $\begingroup$ Existence of a Hirota-type formulation is still an open question for the Dubrovin--Zhang integrable hierarchy. Tau function of the Dubrovin--Zhang hierarchy comes from tau-symmetry. Dubrovin--Zhang's construction uses loop equation and quasi-triviality transformation in the principal hierarchy. Their construction keeps the Hamiltonian structure and tau symmetry property. However, one needs to check ``polynomiality" of the Hamiltonian structure which was later proved by Buryak--Posthuma--Shadrin. $\endgroup$ – Di Yang Dec 13 '16 at 21:10
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    $\begingroup$ Then due to tau symmetry + the Hamiltonian structure, one can show existence of tau function of an arbitrary solution. The tau function of the solution is uniquely determined up to an exponential of a linear function in the couplings. This linear function can be further fixed by the string equation. Now take the solution to be the so-called topological solution, then from the above way one obtains the ``partition function" (as the tau function). $\endgroup$ – Di Yang Dec 13 '16 at 21:12

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