What are Gromov-Witten invariants in terms of physics? What do Gromov-Witten invariants (of say a Calabi-Yau 3-fold) represent, or what are they supposed to represent, in terms of string theory? When I compute GW invariants, am I actually computing some interactions between some particles, or what ...?
Please be gentle, and use only undergraduate-level physics words, if possible. (Perhaps this is too much to ask! Ok well, I'd rather get a response that involves fancier physics words than no response at all.)
I suppose this question is more physics than math -- I hope that's ok.
 A: Here is a very rough answer.
The Gromov-Witten invariants show up in a few a priori different 
contexts within string theory.  Let me focus on one particular place they show up that is 
directly related to conventional physics, as opposed to topological
quantum field theory.
Type IIA string theory is formulated on a spacetime "background" 
which is, in the simplest setup, just a Lorentzian 10-manifold.  The 
equations of motion of the theory require (at least in their leading 
approximation) that the metric on this 10-manifold should be Ricci-flat.
A popular thing to do is to take this 10-manifold of the form 
X x R^{3,1}, where X is a compact Calabi-Yau threefold.  
We can simplify matters by taking X to be very small --- 
smaller than the Compton wavelength of any of the particles we are able 
to create.  (Remember that in quantum mechanics particles have a 
wavelike character, with wavelength inversely related to their energy; 
since we only have limited energy available to us, we can't make 
particles with arbitrarily short wavelength.)  A little more precisely, 
let's take X such that the first nonzero eigenvalue of the Laplacian is 
larger than the energy scale we can access.
In this case we low-energy 
observers will not be able to detect X directly in any experiments.  To 
us, spacetime will appear to be R^{3,1}.
What will be the physics we see on this R^{3,1}?  We will see 
various different species of particle.  Each species of particle that 
we see corresponds to some zero-mode of the Laplacian of X.
In particular, there are particles corresponding to classes in H^{1,1}(X).
The genus 0 Gromov-Witten invariants are giving
information about the interactions between these particles.  (So if you want to calculate what will come out when you 
shoot two of these particles at each other, one of the inputs to that calculation
would be the genus 0 Gromov-Witten invariants.)  The higher genus Gromov-Witten 
invariants are giving information about interactions which involve these particles 
together with other particles related to the gravitational interaction.
