Sheaves on Toric Varieties for Physics

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.

arXiv comment83 pages, LaTeX, 4 figures, must run LaTeX 3 times, numerous minor cosmetic upgrades

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Cited by in corpus (20)

• D-Brane Stability and Monodromy
• Discrete Torsion
• Notes on Certain (0,2) Correlation Functions
• Fixed point loci of moduli spaces of sheaves on toric varieties
• Quantization of Fayet-Iliopoulos Parameters in Supergravity
• Exact results for $\mathcal{N}=2$ supersymmetric gauge theories on compact toric manifolds and equivariant Donaldson invariants
• Compactification, Geometry and Duality: N=2
• Exceptional collections and D-branes probing toric singularities
• Boundary Fermions, Coherent Sheaves and D-branes on Calabi-Yau manifolds
• Vector bundles, linear representations, and spectral problems
• Gauge theories on compact toric manifolds
• Equivariant Sheaves
• Non-compact Mirror Bundles and (0,2) Liouville Theories
• Recurrence relation for instanton partition function in SU(N) gauge theory
• Toric vector bundles, branched covers of fans, and the resolution property
• Representations of posets: Linear versus Unitary
• Soft noncommutative schemes via toric geometry and morphisms from an Azumaya scheme with a fundamental module thereto -- (Dynamical, complex algebraic) D-branes on a soft noncommutative space
• Positive formulae for K-types of SL_3(R)-irreps and a Blattner formula for smooth K-orbit closures
• Global aspects of (0,2) moduli space: toric varieties and tangent bundles
• Equivariant Chow cohomology of toric varieties