Hi, I am not really an expert in this field, but I think (equivariant) quantum cohomology rings have a very nice combinatorics and show up at many places. I few month ago I was just finishing a paper (joint with Christian Korff) which connects the somehow completely understood quantum cohomology of the Grassmannian with the Verlinde algebra (which for me is the fusion algebra of certain tilting modules for a quantum group at a root of unity).

Have a look at arXiv:0909.2347!

There are indeed lots of connections with integrable systems. In our work we look somehow at a very easy situation: take the affine Dynkin diagram for affine sl(n) (that means a circle with n points!) Then consider the integrable system where you can place particles at this n places. Either "bosonic" that means however many you want or "fermionic" which means at most one at each place. Now there is the operation of moving a particle to the next place. This defines you linear maps on the space of all particle confugurations fixing the number of total particles. Then we define Schur functors where the variables are these (non-commuting) operators.

Now the quantum cohomology of the Grassmannian Gr(k,n+k) has a basis which we can identify with certain partitions or with fermionic particle configurations on an (n+k)-circle using k particles.

Whereas the fusion algebra at level k has a basis which can again be identified with certain partitions or with bosonic particle configurations on an n-circle.

The funny thing now is that the multiplication in either of the rings is just given by
a*b=Schur poly to a viewed as an operator applied to b.

This desription of quantum cohomology was found by Postnikov a few years ago, but he did not connect it to this integrable system. The whole thing reproves and makes explicit an old result due to Witten, Gepner, Vafa and Intrilligator that the fusion ring of gl(n) at level k is isomorphic to the quantum cohomology of the Grassmannian Gr(k,n+k) when we specialise q to 1.

So already in this really boring Baby-example something interesting shows up, so I guess one really should study all sort of equivariant cohomology rings!

Idiscovered the connections I mentioned above---other people have told me about them! – GS Jan 8 '10 at 19:49