Databases of quantum periods for Fano manifolds

Area of Science:

• Algebraic Geometry
• Theoretical Physics
• Computational Mathematics

Background:

• Fano manifolds are fundamental objects in algebraic geometry, analogous to atomic elements in chemistry.
• The classification of Fano manifolds has been a long-standing open problem since the 1930s.
• Mirror symmetry, a technique from theoretical physics, offers a new perspective for Fano manifold classification.

Purpose of the Study:

• To create comprehensive databases of regularized quantum periods for Fano manifolds.
• To facilitate the classification of Fano manifolds by providing computable data.
• To address the challenge of scattered and incomplete literature on Fano manifold properties.

Main Methods:

• Encoding Fano manifolds using sequences of integers representing regularized quantum periods.
• Utilizing mirror symmetry principles for computational analysis.
• Compiling and structuring data into accessible databases.

Main Results:

• Databases of regularized quantum periods for Fano manifolds up to dimension four have been established.
• Databases for dimensions one, two, and three are complete.
• The database for four-dimensional Fano manifolds is ongoing and will be updated.

Conclusions:

• The developed databases represent a significant step towards the complete classification of Fano manifolds.
• These resources will aid researchers by providing accessible and organized computational data.
• The integration of theoretical physics concepts has proven effective in advancing geometric problems.