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D-Branes on Vanishing del Pezzo Surfaces

Paul S. Aspinwall, Ilarion V. Melnikov

TL;DR

This work develops a derived-category framework for D-branes on vanishing del Pezzo surfaces, showing how marginal decays into fractional branes are encoded by quiver gauge theories whose arrows correspond to Ext^1 groups. Seiberg dualities are realized as tilting equivalences, with left and right tilts corresponding to dualities at a quiver node and mutations of exceptional collections providing a concrete geometric interpretation. A central insight is the necessity to avoid nonzero Ext^3 groups to obtain valid, tachyon-free quivers; tilting can remove such obstructions, though it may introduce new ones elsewhere. The approach yields a rigorous, scalable method to analyze D-branes in CY threefolds, and points to extensions to more general singularities and deeper connections to $A_\ abla$-algebras and superpotential data via higher structures.

Abstract

We analyze in detail the case of a marginally stable D-Brane on a collapsed del Pezzo surface in a Calabi-Yau threefold using the derived category of quiver representations and the idea of aligned gradings. We show how the derived category approach to D-branes provides a straight-forward and rigorous construction of quiver gauge theories associated to such singularities. Our method shows that a procedure involving exceptional collections used elsewhere in the literature is only valid if some tachyon-inducing Ext3 groups are zero. We then analyze in generality a large class of Seiberg dualities which arise from tilting equivalences. It follows that some (but not all) mutations of exceptional collections induce Seiberg duality in this context. The same tilting equivalence can also be used to remove unwanted Ext3 groups and convert an unphysical quiver into a physical one.

D-Branes on Vanishing del Pezzo Surfaces

TL;DR

This work develops a derived-category framework for D-branes on vanishing del Pezzo surfaces, showing how marginal decays into fractional branes are encoded by quiver gauge theories whose arrows correspond to Ext^1 groups. Seiberg dualities are realized as tilting equivalences, with left and right tilts corresponding to dualities at a quiver node and mutations of exceptional collections providing a concrete geometric interpretation. A central insight is the necessity to avoid nonzero Ext^3 groups to obtain valid, tachyon-free quivers; tilting can remove such obstructions, though it may introduce new ones elsewhere. The approach yields a rigorous, scalable method to analyze D-branes in CY threefolds, and points to extensions to more general singularities and deeper connections to -algebras and superpotential data via higher structures.

Abstract

We analyze in detail the case of a marginally stable D-Brane on a collapsed del Pezzo surface in a Calabi-Yau threefold using the derived category of quiver representations and the idea of aligned gradings. We show how the derived category approach to D-branes provides a straight-forward and rigorous construction of quiver gauge theories associated to such singularities. Our method shows that a procedure involving exceptional collections used elsewhere in the literature is only valid if some tachyon-inducing Ext3 groups are zero. We then analyze in generality a large class of Seiberg dualities which arise from tilting equivalences. It follows that some (but not all) mutations of exceptional collections induce Seiberg duality in this context. The same tilting equivalence can also be used to remove unwanted Ext3 groups and convert an unphysical quiver into a physical one.

Paper Structure

This paper contains 14 sections, 3 theorems, 47 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Marginal Decays and Quivers
  3. Gradings and Periods for $S = {\mathbb P}^1 \times {\mathbb P}^1$
  4. The Category of B-branes on del Pezzo Surfaces
  5. Quivers and Algebras
  6. Tilting
  7. Del Pezzo Surfaces
  8. Decay into Fractional Branes
  9. Tilting Duality
  10. A class of tilts
  11. Mutations
  12. Removing $\operatorname{Ext}^3$'s
  13. Some conjectures
  14. Conclusions

Key Result

Theorem 1

The derived categories $\mathbf{D}(\hbox{$A$--\bf mod})$ and $\mathbf{D}(\hbox{$B$--\bf mod})$ are equivalent if and only if there exists a tilting complex $T$ such that $B=\operatorname{End}(T)^{\textrm{op}}$.

Figures (1)

  • Figure 1: A sequence of $\Psi_L$ transforms equivalent to the identity for a $\mathrm{dP}_{1}$.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Conjecture 1