Counting rational curves with path integrals

Question

Can someone give a laymans explanation of how one can count rational curves of degree $d$ using path integrals in QFT?

As explained by Atiyah here

Answer 1

Here goes my best attempt...

In my experience with QFT through Kock and Vainsencher's book (colloquially called "the green book" in my department), when you do an integral like that you are really doing some kind of integral in a moduli space of rational curves of degree $d$ (possibly with special property in which you're interested like having certain marked points or genus, etc.). However, sometimes when seeing these integral symbols, they're...well they didn't seem like honest-to-goodness integrals to me. Some of these integrals really just mean "add up the number of points of intersection of these things." (I think there is a way to make these statements formal, but I'm pretty sure they don't even have differentials in the integrals that I'm meaning.) So the idea when you're enumerating curves with a given property (e.g. the property of passing through a certain number of points), you can do a summation over a moduli space in some sense.

The moduli space in question is often (and certainly in the green book) some quotient of a space of maps from $\mathbb{P}^1$ to $\mathbb{P}^r$ by identifying two maps basically when their images are the same. (That is, you are counting curves by counting equivalence classes of maps which give those curves.) There are some nice theorems that the moduli spaces with which we are concerned for the Kontsevich formula turn out to be varieties with some desirable properties. You can find the theorems for yourself easily enough in the green book.

I didn't read all of the Atiyah article, but perhaps that answers your question to some extent. Atiyah is referencing the Kontsevich formula which comes up in the Kock and Vainsencher book. If you want another reference, Renzo Cavalieri taught a topics class at Colorado State University (my school) last year on moduli spaces of rational curves with marked points, which is foundation material for the moduli spaces of maps above. You can probably find some of his course notes on his website if you like.