Enumerative geometry of the mirror quintic

We evaluate the enumerative invariants of low degree on the mirror quintic threefold. Mirror symmetry burst upon the mathematical scene with the famous compu-tation made by Candelas, de la Ossa, Green, and Parkes [4] which in modern language proposed to count the number of rational curves of fixed degree on the quintic threefold using a technique from physics. In current language the “counts” are evaluations of Gromov–Witten invariants or Gopakumar–Vafa invariants, and the technique for counting these invariants has been expanded and extended in numerous ways. From the physics point of view, the compu-tation was made on a closely related algebraic variety – the mirror quintic. In a recent physics paper [10], a new technique was proposed for explicitly evaluating the Gromov–Witten or Gopakumar–Vafa invariants of the mirror quintic itself, not just the quintic. Such explicit evaluations seem rather daunt-ing, since the answer will be a function of 101 variables. One aspect of [10] is to use two variables only and arrive at a more reasonable count. Upon request of the authors of [10], the present authors worked out the enumerative geometry of the mirror quintic. We have made a conjecture about the Mori cone which comes with a plausibility argument rather than a proof. However, independent of the truth of that conjecture, we are able to explicitly evaluate the Gromov–Witten or Gopakumar–Vafa invariants of low degree for the mirror quintic, involving all 101 variables. It gives us great pleasure to dedicate this paper to our mentor and friend Herb Clemens. Herb’s work on rational curves on Calabi–Yau threefolds [5, 6, 7] gave an inspiration and a foundation to much of our own work, including this paper.