TITLE: Negative Gromov-Witten invariants
QUESTION [13 upvotes]: I understand the heuristic reason why Gromov-Witten invariants can be rational; roughly it's because we're doing curve counts in some stacky sense, so each curve $C$ contributes $1/|\text{Aut}(C)|$ to the count rather than $1$.
However, I don't understand why or how Gromov-Witten invariants can be negative. What is the meaning of a negative GW invariant? What are some simple examples?

REPLY [4 votes]: The simplest example that I could find is: blow-up $P^3$ at one point and consider the following correlator $\langle pt, E^2, E^2\rangle_l=-1$ , where $l$ is the pull-back of a line from $P^3$. 
Proof: use the splitting axiom for the following 4 classes $(pt, E^2|E,E)_l$. To unravel this notation take a look at page 5 in http://www.mathematik.uni-kl.de/~gathmann/pub/blowup-9804043.pdf
The splitting axiom gives:
$\langle pt, E^2, E^2\rangle_l=\langle pt, E, E\rangle_{l-E'}\cdot\langle E^2, E^2, E\rangle_{E'}=1\cdot(-1)=-1$
I hope this is corect. 
Does anybody know an explicit example in dimension 2?