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 Kähler cone, in which moduli spaces of sheaves are independent of Kähler 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.

Notes on Certain (0,2) Correlation Functions

“…However, the techniques of localization can be applied if X admits a torus action and the bundle E is equivariant for that torus action. Such bundles are studied in [18,19] if X is a toric variety. We do not develop these techniques here since one of our goals in the present paper is to verify the claims of [1] and the gauge bundles appearing there are not of this type.…”

Compactification, Geometry and Duality: N = 2

“…This fixes the mirror map as follows. First we need to switch back to the dual language of periods defined in (25). We will find one period which we denote t 0 which is completely invariant under the monodromies M k .…”

“…As we move around the moduli space we will often encounter degenerations of the bundle data which can be interpreted easily in the algebraic picture by using the language of "sheaves". See [24][25][26] for example for more on this.…”

(0,2) Duality

“…This notion of mirror symmetry inti-mately involves a superpotential in both the original and dual descriptions. In related work, a description of equivariant sheaves and their relevence to (0, 2) mirror symmetry appears in [12], while an extension of the monomial divisor mirror map [13] to a class of (0, 2) theories appears in [14]. Note that unlike the (2, 2) case, we believe that (0, 2) mirror symmetry should map certain instanton sums on M to instanton sums on the mirror.…”