Localization in Gromov—Witten Theory of Toric Varieties in a Computer Algebra System

The Atiyah–Bott localization formula is a powerful tool for calculating the degree of equivariant classes of the moduli space of rational stable maps \(\overline{M}_{0,m}(X,\beta )\) , where X denotes a smooth toric variety, m is a non-negative integer, and \(\beta \) is an effective 1-cycle. Implementation of the formula entails intricate computational challenges, involving graph theory, colorings, partitions, and other discrete objects. Furthermore, the computed solution is a large summation of rational numbers, underscoring the imperative nature of computational efficiency. This formula has been applied in very specific cases for computing Gromov–Witten invariants, addressing enumerative problems, and determining the small quantum ring of X, among other applications. A comprehensive implementation as a Julia package has been recently presented by the author. We show the features of the package with a particular emphasis to the noteworthy contribution of the package Oscar.jl. Finally, we delve into the fundamental prerequisites for extending the implementation to encompass algebraic GKM manifolds.