Cohomology of Line Bundles: A Computational Algorithm

TL;DR

The paper addresses computing line bundle cohomology on $D$-dimensional toric varieties, a key step in determining massless spectra for string compactifications. It introduces a constructive Čech-cohomology framework that leverages the Stanley-Reisner ideal to recast cohomology as a counting problem over monomial rationoms, with a remnant cohomology structure to handle overlaps in SR generators. The authors propose a final conjectured algorithm, refine it with a remnant cohomology $\mathfrak{h}^i(\mathcal{Q})$, and validate it against known results and index theorems, later obtaining proofs in follow-up work. This approach enables efficient, manifold-independent computation of line bundle cohomology, facilitating applications to heterotic and F-theory compactifications and beyond, as summarized by the formula $\chi(X;L)=\int_X {\rm ch}(L)\, {\rm Td}(X)$ for consistency checks.

Abstract

We present an algorithm for computing line bundle valued cohomology classes over toric varieties. This is the basic starting point for computing massless modes in both heterotic and Type IIB/F-theory compactifications, where the manifolds of interest are complete intersections of hypersurfaces in toric varieties supporting additional vector bundles.

Figures (1)

• Figure 1: The region subdivision of the lattice. Note, that it is meant that we have the inclusions ${\cal R}_i\cup {\cal R}_j\subset {\cal R}_{ij}$ for $i<j$ and $\bigcup_i {\cal R}_i\cup \bigcup_{i<j} {\cal R}_{ij} \subset {\cal R}_{123}$, where ${\cal R}_{123}$ covers the entire lattice space.