Databases of quantum periods for Fano manifolds

TL;DR

This paper introduces comprehensive databases of regularized quantum periods for Fano manifolds up to four dimensions, facilitating classification and further research in algebraic geometry and mirror symmetry.

Contribution

It provides the first complete databases of quantum periods for Fano manifolds in dimensions one to three, with ongoing updates for four-dimensional cases.

Findings

Databases for dimensions 1-3 are complete.

The dimension 4 database is being expanded with new Fano manifolds.

These databases aid in the classification and study of Fano manifolds.

Abstract

Fano manifolds are basic building blocks in geometry - they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of this as building a Periodic Table for shapes. A recent breakthrough in Fano classification involves a technique from theoretical physics called Mirror Symmetry. From this perspective, a Fano manifold is encoded by a sequence of integers: the coefficients of a power series called the regularized quantum period. Progress to date has been hindered by the fact that quantum periods require specialist expertise to compute, and descriptions of known Fano manifolds and their regularized quantum periods are incomplete and scattered in the literature. We describe databases of regularized quantum periods for Fano manifolds in dimensions up to four. The databases in dimensions one, two, and three are complete; the database in dimension four will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed.