TITLE: Why would I want to know (equivariant) quantum cohomology?
QUESTION [12 upvotes]: Let's say that I have a variety I think is interesting, and based on some papers I don't fully understand, I can compute quite explicitly its equivariant quantum cohomology in terms of explicit formulae for multiplying by a degree 2 class.
Being something of a newcomer to quantum cohomology, I'm genuinely a bit unsure of how interesting a result this is, and have doubts about writing a paper whose content is "The quantum cohomology of variety X is blah."
Does it tell me anything particularly interesting?   Might it have cool implications in integrable systems or something like that?

REPLY [12 votes]: You might have already found an answer by now but still let me add my motivation for the
subject. The reason I am interested in quantum cohomology is that I am thinking
about it as the correct version of semi-infinite cohomology of the loop space of the variety
in question (this point of view by the way is sometimes very useful for guessing the answer).
Personally I find the definition of quantum cohomology rather ugly, but in geometric
representation theory this kind of phenomenon is sort of well-known: one way to do 
"semi-infinite geometry" which formally is not clear how to tackle, is by working with
the global curve ${\mathbb P}^1$ instead of the formal disc. For example, one can develop
rather non-trivial theory of "semi-infinite flag varieties" using spaces of quasi-maps,
which is defined using global  ${\mathbb P}^1$ as well. I sort of regard this as a similar
phenomenon to why quantum cohomology (which is supposed to be a local object) is defined
via a global curve.
A more "physical" explanation of the above phenomenon is this: in 2-dimensional conformal
field theory it happens very often that some genus 0 correlators actually coincide
with some local objects (like coefficients of the OPE and such things). One example of this
is the Kazhdan-Lusztig tensor product for representations of affine Lie algebras.
It is actually a local thing, but the Kazhdan-Lusztig definition goes through coinvariants
on global ${\mathbb P}^1$.
Let me conclude by saying that while the "global" approach is often useful, because
that's the only way to define things regorously, the price you have to pay for this is the
fact that calculations become rather difficult and mysterious. Quantum cohomology is probably a good example of this -- many answers there become much more understandable when you think in terms of loop spaces.