Gromov-Witten theory of complete intersection

These notes were taken during talk given by Dimitri Zvonkine.

We have projective space $\mathbb{CP}^{m+r}$ with $r$ polynomial equations,
$X=\{f_1=f_2=\cdots=f_r=0\}$ and we want smooth complete intersection.
Let $(df_i)$ be linearly independent at every point of $X$ and it follows that $X$ is smooth with $dimension=m$.

Theorem (Lefschetz hyperplane Theorem):
$H^*(X)=H^{*}_{simple}\bigoplus H^*_{primitive}$

Next part (I am not sure) stated about monodromy. I don’t know anything about it.

$\mathcal{U}=\{smooth, complete ~intersections\}$ with polynomials $f_1, f_2, \cdots f_r$

Fundamental group of $\mathcal{U}$, $\pi_1(\mathcal{U}, X)$ acts on $H^*_{primitive}(X)$

$G=$ Zariski closure of image $Im\pi_1\subset GL(H^*_{primitive}).$

Theorem [Delign]:
$G=O (H^*_{primitive})$if $m$ is even and
$G=S_p (H^*_{primitive})$ if $m$ is odd.
($Sp$ denotes symplectic group)

Main Theorem of the talk: One can compute all Gromov-Witten invariants for all complete intersections.
Rest of the talk discussed this algorithm to compute the GW invariants.