Line bundle cohomology formulae on Calabi-Yau threefolds

Cohomology is a universal tool in mathematics: from topology and geometry to algebra and number theory, cohomology is used to quantify the possible ways in which a problem fails to meet the naive expectations of the problem solver. As such, estimates of cohomology represent a vital key to problem solving. In areas of theoretical physics such as string theory, cohomology is linked to the low-energy quantum excitations of fields and strings that can be experimentally observed.

In practice, cohomology computations are notoriously difficult to carry out in general. However, it has been recently shown that line bundle cohomology dimensions on several classes of two-dimensional and three-dimensional complex manifolds, including compact toric surfaces, generalised del Pezzo surfaces, K3 surfaces and Calabi-Yau threefolds, are described by closed-form formulae. This new technique for the computation of cohomology uses a combination of algebro-geometric methods and exploratory machine learning methods. The goal of the project will be to go one step further and discover exact generating functions for line bundle cohomology on Calabi-Yau threefolds.

References:

[1] C. Brodie, A. Constantin, J. Gray, A. Lukas, F. Ruehle, Recent Developments in Line Bundle Cohomology and Applications to String Phenomenology, Nankai Symposium on Mathematical Dialogues 2021 Conference Proceedings, arXiv: 2112.12107.

[2] C. Brodie, A. Constantin, R. Deen, A. Lukas, Machine Learning Line Bundle Cohomology, Fortsch.Phys. 68 (2020) 1, 1900087, arXiv: 1906.08730.

[3] C. Brodie, A. Constantin, A. Lukas, Flops, Gromov-Witten invariants and symmetries of line bundle cohomology on Calabi-Yau three-folds, J.Geom.Phys. 171 (2022), arXiv: 2010.06597.

[4] C. Brodie, A. Constantin, Cohomology Chambers on Complex Surfaces and Elliptically Fibered Calabi-Yau Three-folds, arXiv: 2009.01275.