Big quantum cohomology of Fano complete intersections

Abstract

For smooth complete intersections in the projective spaces, we use the deformation invariance of Gromov-Witten invariants and results in classical invariant theory to study the symmetric reduction of the WDVV equation by the monodromy groups. We propose a double root recursion conjecture for genus 0 invariants of non-exceptional Fano complete intersections other than the cubic hypersurfaces and the odd dimensional intersection of two quadrics. Based on it we develop an algorithm to compute the genus 0 invariants of any length and with any number of primitive insertions. The effectiveness of our algorithm is the main conjecture of this paper. We prove the conjecture at the first non-trivial order, which amounts to the computation of genus zero Gromov-Witten invariants with exactly 4 primitive insertions. We show a reconstruction theorem for the odd dimensional intersection of two quadrics. There are some 4-point and 8-point genus 0 Gromov-Witten invariants for cubic hypersurfaces that cannot be computed by the monodromy group method. We compute them by studying the cohomology ring structure of the Fano variety of lines and by Zinger's reduced genus 1 Gromov-Witten invariants. Then we get a reconstruction theorem for cubic hypersurfaces. At the end of the paper, we give numerical examples and some conjectural closed formulae.