Computing the quantum cohomology of primitive classes

For smooth complete intersections in the projective spaces, we make use of the deformation invariance and $S_n$-symmetry of the Gromov-Witten invariants and results in classical invariant theory to study the symmetric reduction of the WDVV equation by the monodromy groups. We discuss to what extent the quantum cohomology involving primitive cohomology classes can be determined. In particular, we obtain a reconstruction theorem for cubic hypersurfaces and odd dimensional complete intersection of two quadrics. By the way, we obtain a complete description of the cohomology ring of the Fano variety of lines on smooth cubic hypersurfaces.