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On the geometry of quiver gauge theories (Stacking exceptional collections)

Christopher P. Herzog, Robert L. Karp

TL;DR

This work advances the dictionary between derived-category methods and D-brane gauge theories by proving that a full strong exceptional collection on the exceptional divisor $S$ lifts to a full set of fractional branes on the total space $KS$, and by giving a complete strong exceptional collection for all $Y^{p,q}$ singularities with $p-q>2$. It offers two independent verifications of strongness: explicit cohomology calculations and toric Kawamata–Viehweg vanishing, reinforcing the stability of the associated quiver and helix. The authors also extend the framework to stacky (singular) bases and demonstrate the invariance of the quiver under helix shifts, tying together quiver representations, tilting theory, and toric geometry to illuminate D-brane gauge theories at Calabi–Yau singularities. The results provide concrete, computable tools for constructing quivers and checking tachyon absence in a broad class of geometries, with a particular emphasis on the $Y^{p,q}$ family and its toric-stacky refinements.

Abstract

In this paper we advance the program of using exceptional collections to understand the gauge theory description of a D-brane probing a Calabi-Yau singularity. To this end, we strengthen the connection between strong exceptional collections and fractional branes. To demonstrate our ideas, we derive a strong exceptional collection for every Y^{p,q} singularity, and also prove that this collection is simple.

On the geometry of quiver gauge theories (Stacking exceptional collections)

TL;DR

This work advances the dictionary between derived-category methods and D-brane gauge theories by proving that a full strong exceptional collection on the exceptional divisor lifts to a full set of fractional branes on the total space , and by giving a complete strong exceptional collection for all singularities with . It offers two independent verifications of strongness: explicit cohomology calculations and toric Kawamata–Viehweg vanishing, reinforcing the stability of the associated quiver and helix. The authors also extend the framework to stacky (singular) bases and demonstrate the invariance of the quiver under helix shifts, tying together quiver representations, tilting theory, and toric geometry to illuminate D-brane gauge theories at Calabi–Yau singularities. The results provide concrete, computable tools for constructing quivers and checking tachyon absence in a broad class of geometries, with a particular emphasis on the family and its toric-stacky refinements.

Abstract

In this paper we advance the program of using exceptional collections to understand the gauge theory description of a D-brane probing a Calabi-Yau singularity. To this end, we strengthen the connection between strong exceptional collections and fractional branes. To demonstrate our ideas, we derive a strong exceptional collection for every Y^{p,q} singularity, and also prove that this collection is simple.

Paper Structure

This paper contains 18 sections, 20 theorems, 88 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary ideas
  3. Physical motivation
  4. Quiver representations and path algebras
  5. $\mathbf{D}(S)$ versus $\mathbf{D}(K S)$
  6. The physical quiver
  7. The stacky version
  8. Vanishing results -- absence of tachyons
  9. Absence of Ext$^0$'s
  10. Absence of Ext$^{d+1}$'s
  11. Invariance of the quiver
  12. The geometry of the $Y^{p,q}$ spaces
  13. An exceptional collection on ${\mathbb F}_n$
  14. The exceptional collection
  15. The toric approach
  16. ...and 3 more sections

Key Result

Theorem 2.2

If $S$ has a full strongly exceptional collection $\{E_i\}_{i=1}^n$, then the derived category of coherent sheaves on $S$ is equivalent to the derived category of right $A$-modules $\mathbf{D}(\hbox{\rm mod-}A)$. The equivalence is given by the functor

Figures (1)

  • Figure 1: The toric fan for the $X_{p,q}$ space.

Theorems & Definitions (34)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3: 3.1 of Kawamata:Francia
  • Definition 2.4
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • Theorem 2.7: 4.5 of Kawamata:GMcKay
  • Lemma 3.1: Sect. 7.2.3 of Rudakov:Book
  • Lemma 3.2
  • ...and 24 more