Sheaves on Toric Varieties for Physics

TL;DR

This paper develops a toric geometric framework for equivariant sheaves on toric varieties, applying it to heterotic and F-theory compactifications, and explores implications for moduli spaces and mirror symmetry.

Contribution

It provides an explicit toric description of equivariant sheaves and applies this to string compactifications, linking mathematical structures to physical models.

Findings

Equivariant sheaves can be described using toric data.

Stability of sheaves on Calabi-Yau hypersurfaces can be analyzed.

Moduli spaces of sheaves are substructure-dependent on the Kahler cone.

Abstract

In this paper we give an inherently toric description of a special class of sheaves (known as equivariant sheaves) over toric varieties, due in part to A. A. Klyachko. We apply this technology to heterotic compactifications, in particular to the (0,2) models of Distler, Kachru, and also discuss how knowledge of equivariant sheaves can be used to reconstruct information about an entire moduli space of sheaves. Many results relevant to heterotic compactifications previously known only to mathematicians are collected here -- for example, results concerning whether the restriction of a stable sheaf to a Calabi-Yau hypersurface remains stable are stated. We also describe substructure in the Kahler cone, in which moduli spaces of sheaves are independent of Kahler class only within any one subcone. We study F theory compactifications in light of this fact, and also discuss how it can be seen in the context of equivariant sheaves on toric varieties. Finally we briefly speculate on the application of these results to (0,2) mirror symmetry.