Computational Tools for Cohomology of Toric Varieties

TL;DR

The paper addresses the efficient computation of line-bundle cohomology on toric varieties relevant to string compactifications, introducing a novel combinatorial algorithm that uses the Stanley-Reisner ideal and the GLSM data to determine $h^i(X; \mathcal{O}_X(D))$ without heavy spectral sequences. It extends the framework to equivariant cohomology for finite group actions, Koszul sequences for complete intersections, and monad constructions of vector bundles, including a detailed (2,2) model dual to a (0,2) model with a consistent chiral spectrum and equal total moduli. The approach is implemented in the cohomCalg package and demonstrated on explicit examples, including orbifold and duality settings, highlighting practical impact for heterotic model building and spectrum computations. The work also discusses limitations (e.g., scaling with Picard number and SR generators) and points to future directions for broader applicability and efficiency improvements.

Abstract

In this review, novel non-standard techniques for the computation of cohomology classes on toric varieties are summarized. After an introduction of the basic definitions and properties of toric geometry, we discuss a specific computational algorithm for the determination of the dimension of line-bundle valued cohomology groups on toric varieties. Applications to the computation of chiral massless matter spectra in string compactifications are discussed and, using the software package cohomCalg, its utility is highlighted on a new target space dual pair of (0,2) heterotic string models.