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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.

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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.

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Thanks. I wonder if there is a good book to learn this sort of thing from? Preferably something which is tailored for mathematicians... –  Kevin H. Lin Jan 29 '10 at 22:12
Interesting! So are there empirically testable predictions coming from calculations of genus 0 GW invariants? –  Sheikraisinrollbank Sep 30 '10 at 11:36
Unfortunately I do not know a mathematically-oriented book which develops precisely the story I told above. This might be only my personal ignorance. –  Andy Neitzke Feb 26 '12 at 17:47
Regarding empirical predictions, there is a thriving industry that tries to start with a particular "model" -- some flavor of string theory, some Calabi-Yau manifold $X$ and some extra geometric ingredients -- and study the resulting physics in R^{3,1}, with the aim of comparing it to our actual real-world physics. That industry seems to be far from settling on a particular candidate model. If it did settle on one, you could then ask whether the effects associated to the Gromov-Witten invariants are big enough to be measured. My guess is that this would depend on the details of the model. –  Andy Neitzke Feb 26 '12 at 17:58

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