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D-branes, Discrete Torsion and the McKay Correspondence

Paul S. Aspinwall, M. Ronen Plesser

TL;DR

The paper investigates D-branes on orbifolds with discrete torsion, proposing an α-twisted extension of the McKay correspondence that links D-brane data to projective representations and stringy homology. It develops a Γ-equivariant D-brane quiver framework and analyzes the brane moduli space, showing that for isolated singularities the moduli space recovers ${ m C}^n/ ext{Γ}$, while non-isolated cases produce wrapped-brane branches via induced representations. A stringy K-theory perspective is introduced, predicting torsion in stringy homology of order $p$, where $p$ is the order of α in $H^2( ext{Γ},U(1))$, and connecting this to a lattice $K_0$ generated by α-twisted irreducible projective representations; the Atiyah–Hirzebruch spectral sequence mediates between $K_0$ and even homology. Detailed examples with ${ m Z}_2 imes{ m Z}_2$ and a trihedral group illustrate how discrete torsion alters quivers, wrapped-brane content, and torsion phenomena, and the authors discuss why no exact S-duality exchanging strings and D1-branes is expected in this setup.

Abstract

We analyze the D-branes of a type IIB string theory on an orbifold singularity including the possibility of discrete torsion following the work of Douglas et al. First we prove some general results about the moduli space of a point associated to the "regular representation" of the orbifold group. This includes some analysis of the "wrapped branes" which necessarily appear when the orbifold singularity is not isolated. Next we analyze the stringy homology of the orbifold using the McKay correspondence and the relationship between K-theory and homology. We find that discrete torsion and torsion in this stringy homology are closely-related concepts but that they differ in general. Lastly we question to what extent the D-1 brane may be thought of as being dual to a string.

D-branes, Discrete Torsion and the McKay Correspondence

TL;DR

The paper investigates D-branes on orbifolds with discrete torsion, proposing an α-twisted extension of the McKay correspondence that links D-brane data to projective representations and stringy homology. It develops a Γ-equivariant D-brane quiver framework and analyzes the brane moduli space, showing that for isolated singularities the moduli space recovers , while non-isolated cases produce wrapped-brane branches via induced representations. A stringy K-theory perspective is introduced, predicting torsion in stringy homology of order , where is the order of α in , and connecting this to a lattice generated by α-twisted irreducible projective representations; the Atiyah–Hirzebruch spectral sequence mediates between and even homology. Detailed examples with and a trihedral group illustrate how discrete torsion alters quivers, wrapped-brane content, and torsion phenomena, and the authors discuss why no exact S-duality exchanging strings and D1-branes is expected in this setup.

Abstract

We analyze the D-branes of a type IIB string theory on an orbifold singularity including the possibility of discrete torsion following the work of Douglas et al. First we prove some general results about the moduli space of a point associated to the "regular representation" of the orbifold group. This includes some analysis of the "wrapped branes" which necessarily appear when the orbifold singularity is not isolated. Next we analyze the stringy homology of the orbifold using the McKay correspondence and the relationship between K-theory and homology. We find that discrete torsion and torsion in this stringy homology are closely-related concepts but that they differ in general. Lastly we question to what extent the D-1 brane may be thought of as being dual to a string.

Paper Structure

This paper contains 16 sections, 6 theorems, 45 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The $\Gamma$-Equivariant D-brane Quiver
  3. D-brane Moduli Space
  4. The regular representation
  5. Mobile wrapped branes
  6. The McKay Correspondence
  7. An attempt with homology
  8. K-theory
  9. Examples
  10. ${\mathbb Z}_2\times{\mathbb Z}_2$
  11. No discrete torsion
  12. With discrete torsion
  13. A trihedral group
  14. No discrete torsion
  15. With discrete torsion
  16. ...and 1 more sections

Key Result

Proposition 1

The space ${\Scr M}_R$ represents the space of allowed positions of the D-brane associated to $R$ in the target space ${\mathbb C}^n/\Gamma$.

Figures (5)

  • Figure 1: The quiver for ${\mathbb Z}_2\times{\mathbb Z}_2$ with no discrete torsion.
  • Figure 2: A blow-up of ${\mathbb C}^3/{\mathbb Z}_2\times{\mathbb Z}_2$.
  • Figure 3: The quiver for ${\mathbb Z}_2\times{\mathbb Z}_2$ with discrete torsion.
  • Figure 4: The quiver for ${\mathbb Z}_3\ltimes({\mathbb Z}_2\times{\mathbb Z}_2)$ with no discrete torsion.
  • Figure 5: The quiver for ${\mathbb Z}_3\ltimes({\mathbb Z}_2\times{\mathbb Z}_2)$ with discrete torsion.

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Conjecture 1
  • Proposition 2
  • Theorem 3
  • Theorem 4