This is a followup of my previous question Gromov-Witten and integrability. As I have learned from the answer (but guessed before), GW potentials of the point and $P^1$ (with different modifications) are, more or less, the only examples of the GW generation functions with established integrable properties. So what about higher genera curves? Are they really so complicated to establish integrability, at least for stable sector? What is the main problem with them?
Here's a sketch of my understanding of where the difficulty lies with higher genus curves. It got kind of long and vague, at parts, but hopefully it explains a few problems.
In Gromov-Witten theory, I'm aware of two or three general approaches to integrability currently. Certainly there's overlap among these approaches:
The quantumn cohomology of $X$ turns the cohomology of $X$ into a Frobenius manifold, and then Dubrovin has connections to various integrable hierarchies.
In some cases, the Gromov-Witten theory can be nicely expressed in terms of various Fock-spaces and the infinite wedges, and then you can get connections to the integrable hierarchies related to various infinite dimensional lie groups from the Kyoto school.
- Matrix Models, and in particular the work of Eynard and Orantin. This seems to have mostly implemented in terms of the Topological vertex, which is rephrasing things in terms of combinatorics.
For higher genus curves, approach 1 is completely a no-go: there are no positive degree maps from a sphere to a higher genus curve, so the quantum cohomology is just the usual cohomology.
The GW-theory of higher genus curves is computed by Okounkov and Pandharipande very much in the general method of 2. The GW/Hurwitz correspondence shows that what they call the "stationary sector" (descendents of point classes only -- not the identity or odd cohomology classes. Is this the same as the stable sector?) is equivalent to Hurwitz theory, which is completely computable in terms of the symmetric group. There are nice ways of computing these characters, and this is where the connections to the infinite dimensional lie algebras and come up. However, this computation becomes much more complicated as we increase the genus. I'd like to explain how it's an entirely different beast for genus 2 or bigger.
The GW/Hurwitz correspondence turns insertions of point classes into ramification data, and hence as far as counting goes, into multiplying elements in the symmetric group. The ramification that shows up is nice and is easy to write in terms of the infinite wedge (free fermion) and I think can be written in terms of matrix model type stuff as well. One particular nice bit is that any given insertion only produces permutations with bounded supports: even if we let the degree get big (so considering symmetric groups $S_d$ for $d$ large, we're only going to have a few nontrivial cycles in these permutations. Most points in $S_d$ will be fixed.
For genus 0, we're only multiplying these special elements in the symmetric group, and this is why everything is so beautiful here.
For higher genus, one way to keep everything in terms of just the symmetric group is degenerate the curve by pinching off $g$ cycles, so that we again have a genus zero curve, but now with $g$ pairs of points identified. At each of these identified pairs of points, we're going to need to insert inverse permtuations, and we're going to have to sum over all permutations in $S_d$ this way for each pair of points.
This is where things get ugly -- in higher genus, we have to consider these arbitrary permutations.
In genus one, things aren't all that bad: we only have two arbitrary permutations. So we can start with one of them, multiply in turn by the nice permutations that we know how to do, and then at end, instead of multiplying two arbitrary permutations, we only have to check that we have the same permutation, as the whole product has to be the identity. Essentially, we're taking the trace of some nice operator on the infinite wedge, and this why the quasi-modular forms show up here -- work of Bloch-Okounkov shows these are quasimodular forms.
Once we get to genus two though, we loose this as well. We really have to be able to multiply two arbitrary permutations of $S_d$. I haven't read Mironov-Morozov's stuff on this closely at all, but seemed to recall them having some type of non-integrablity results for the general multiplication of three permutations, but couldn't find exactly this statement. The start of section three of this paper might touch on this, though.
The representation theory viewpoint is probably better for integrable systems, but I think it's similarly more difficult.