I started trying to learn about Gromov-Witten invariants by reading the book "$J$-holomorphic curves and Symplectic Topology" and I have a doubt in an example the authors provide. It's example $7.1.3$ where we consider the homology class $A=0$.


So when we are focusing in the case where $k=3$ and we want to compute the invariant $GW^{M}\_{0,3}(a\_1,a\_2,a\_3)$ the authors claim that this will be equal to $\int\_Ma\_1\cup a\_2\cup a\_3$.


However I'm having some difficulty seeing why this result is true. So by definition we have that $GW^{M}\_{0,3}(a\_1,a\_2,a\_3)$ is equal to $f\cdot ev\_{J}$ where $f$ is a pseudo-cycle Poincaré dual to $\pi\_1^\*a\_1\cup \pi\_2^\*a\_2\cup \pi\_3^\*a\_3$ and $ev\_{J}$ is the evaluation pseudo-cycle. By definition of Poincaré-dual we have that for any $2n-$submanifold $X\subset M^3$, $\int\_{X}a=f\cdot X$. So i'm guessing the idea would be to take $X=M\times \{pt\}\times \{pt\}$? However how can I know that the pseudocycle $ev\_{J}$ can represent this submanifold ? Is it because $k=3$ it does not matter which points we choose?


Any insight is appreciated. I'm fairly new to this so if anyone knows of any reference for this or other examples I would appreciate it. Thanks in advance.



Question on Gromov-Witten invariants when $A=0$
In this case, the J-holomorphic curves are all constant, so the
evaluation pseudocycle is the tridiagonal $\{(x,x,x) : x\in
M\}$. You take cycles $A\_1,A\_2,A\_3$ Poincare dual to
$a\_1,a\_2,a\_3$ respectively and intersect the tridiagonal with
$A\_1\times A\_2 \times A\_3$. This gives you precisely
$\{(x,x,x) : x\in A\_1\cap A\_2\cap A\_3\}$, which is the same
as the triple intersection of $A\_1,A\_2,A\_3$ (i.e. the integral $\int\_M a\_1\cup a\_2 \cup a\_3$).


