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Sheaves on Toric Varieties for Physics

A. Knutson, E. Sharpe

TL;DR

The paper develops an inherently toric framework for equivariant sheaves on toric varieties, using Klyachko filtrations (and Kaneyama’s complementary view) to encode bundles and reflexive/torsion-free sheaves, enabling explicit computation of Chern classes and cohomology. It then constructs moduli spaces of equivariant sheaves as GIT quotients, studies stability with respect to the Kaehler cone, and analyzes substructure within the Kaehler cone with applications to F-theory and mirror symmetry. The work connects these mathematical tools to heterotic (0,2) models, restriction stability to Calabi–Yau hypersurfaces, and potential (0,2) mirror symmetry maps that preserve equivariant data. It also outlines how non-equivariant moduli may be probed via equivariant strata (BB stratifications) and Beilinson/ADHM perspectives on bundles, while proposing ambitious future directions in (0,2) mirror symmetry and related physics. The results provide a detailed, calculable bridge between toric geometry, bundle moduli, and string-theoretic compactifications.

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.

Sheaves on Toric Varieties for Physics

TL;DR

The paper develops an inherently toric framework for equivariant sheaves on toric varieties, using Klyachko filtrations (and Kaneyama’s complementary view) to encode bundles and reflexive/torsion-free sheaves, enabling explicit computation of Chern classes and cohomology. It then constructs moduli spaces of equivariant sheaves as GIT quotients, studies stability with respect to the Kaehler cone, and analyzes substructure within the Kaehler cone with applications to F-theory and mirror symmetry. The work connects these mathematical tools to heterotic (0,2) models, restriction stability to Calabi–Yau hypersurfaces, and potential (0,2) mirror symmetry maps that preserve equivariant data. It also outlines how non-equivariant moduli may be probed via equivariant strata (BB stratifications) and Beilinson/ADHM perspectives on bundles, while proposing ambitious future directions in (0,2) mirror symmetry and related physics. The results provide a detailed, calculable bridge between toric geometry, bundle moduli, and string-theoretic compactifications.

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.

Paper Structure

This paper contains 43 sections, 175 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A rapid review of heterotic compactifications
  3. General remarks on bundles on toric varieties
  4. Equivariant sheaves
  5. Equivariant vector bundles
  6. Kaneyama's approach
  7. Applications
  8. Chern classes
  9. Sheaf cohomology groups
  10. Alternate gauge groups
  11. Equivariant torsion-free sheaves
  12. Basic principles
  13. Torsion-free sheaves on smooth toric varieties
  14. Global Ext
  15. An example on ${\bf P}_{1,1,2}^{2}$
  16. ...and 28 more sections

Figures (4)

  • Figure 1: A fan describing the weighted projective space ${\bf P}_{1,1,2}^{2}$ as a toric variety.
  • Figure 2: Schematically illustrated is a Kähler cone of a generic elliptic K3 with section. The outer boundaries of the Kähler cone are shown in bold lines, and a few of the chamber boundaries are shown as dotted lines.
  • Figure 3: A fan describing the Hirzebruch surface ${\bf F}_{n}$ as a toric variety.
  • Figure 4: Schematically illustrated is a stratum of a moduli space of sheaves of fixed Chern classes on a toric variety. Equivariant sheaves are located in the shaded region at the bottom, and the region described by a Distler-Kachru model is contained in the cylindrical region.