Question on Gromov-Witten invariants when $A=0$

Question

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.

Answer 1

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 $\int_M a_1\cup a_2 \cup a_3$).