Big quantum cohomology of even dimensional intersections of two quadrics

TL;DR

This paper investigates the genus zero Gromov-Witten invariants of even-dimensional intersections of two quadrics, revealing their structure, computability, and properties of the associated Frobenius manifold.

Contribution

It computes key invariants for these intersections, reconstructs all invariants from length-4 data, and analyzes the Frobenius manifold's semisimplicity.

Findings

All genus zero Gromov-Witten invariants can be reconstructed from length-4 invariants.

The generating function of invariants has a positive radius of convergence.

The Frobenius manifold is generically tame semisimple despite non-semisimple small quantum cohomology.

Abstract

For even dimensional smooth complete intersections, of dimension at least 4, of two quadric hypersurfaces in a projective space, we study the genus zero Gromov-Witten invariants by the monodromy group of its whole family. We compute the invariants of length 4 and show that, besides a special invariant, all genus zero Gromov-Witten invariants can be reconstructed from the invariants of length 4. In dimension 4, we compute the special invariant by solving a curve counting problem. We show that the generating function of genus zero Gromov-Witten invariants has a positive radius of convergence. We show that, although the small quantum cohomology is not semisimple, the associated Frobenius manifold is generically tame semisimple.