Effective computations of the Atiyah-Bott formula

TL;DR

This paper implements the Atiyah-Bott residue formula to compute genus 0 Gromov-Witten invariants and contact curve counts, confirming Mirror Symmetry predictions and providing publicly available code for broader applications.

Contribution

It provides a practical implementation of the Atiyah-Bott formula for computing Gromov-Witten invariants and contact curves, enabling extensive calculations and validation of theoretical predictions.

Findings

Confirmed Mirror Symmetry predictions for certain invariants.

Successfully computed intersection numbers of rational curves on complete intersections.

Developed publicly available code for diverse enumerative geometry computations.

Abstract

We present an implementation of the Atiyah-Bott residue formula for $\overline{M}_{0,m}(\mathbb{P}^{n},d)$. We use this implementation to compute a large number of Gromov-Witten invariants of genus $0$, including intersection numbers of rational curves on general complete intersections. We also compute some numbers of rational contact curves satisfying suitable Schubert conditions. Our computations confirm known predictions made by Mirror Symmetry. The code we developed for these problems is publicly available and can also be used for other types of computations.