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D-Branes, Pi-Stability and Theta-Stability

Paul S. Aspinwall

Abstract

We investigate some aspects of Pi-stability of D-branes on Calabi-Yau threefolds in cases where there is a point in moduli space where the grades nearly or completely align. We prove that an example of complete alignment is the case of a collapsed del Pezzo surface. It is shown that there is an open neighbourhood of such a point for which Pi-stability reduces to theta-stability of quiver representations. This should be contrasted to the case of the large radius limit where mu-stability of sheaves cannot be extended out over an open neighbourhood.

D-Branes, Pi-Stability and Theta-Stability

Abstract

We investigate some aspects of Pi-stability of D-branes on Calabi-Yau threefolds in cases where there is a point in moduli space where the grades nearly or completely align. We prove that an example of complete alignment is the case of a collapsed del Pezzo surface. It is shown that there is an open neighbourhood of such a point for which Pi-stability reduces to theta-stability of quiver representations. This should be contrasted to the case of the large radius limit where mu-stability of sheaves cannot be extended out over an open neighbourhood.

Paper Structure

This paper contains 5 sections, 5 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Quivers and del Pezzo Surfaces
  3. Points and Stability
  4. $\theta$-stability
  5. Complete alignment of the gradings

Key Result

Theorem 1

The bounded derived category of coherent sheaves on $S$, $\mathbf{D}(S)$, is equivalent to $\mathbf{D}(\hbox{$A$--\bf mod})$, the bounded derived category of left $A$-modules, where

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5