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D-branes on Orbifolds and Topology Change

Tomomi Muto

TL;DR

The paper investigates topology change in Calabi–Yau spaces probed by D-branes on the orbifolds $C^3/Z_n$ using toric geometry and gauged linear sigma models. It shows that for $n=11$ there are five Calabi–Yau phases connected by flop transitions, while non-geometric phases are projected out for $n=7,9,11$, and it explores the role of non-isolated singularities via explicit cases. The approach maps D-brane vacua to toric data and FI-parameter regions, providing explicit phase structures and triangulations that encode topology changes. A proposed correspondence between the a-vector data and area-$n$ triangulations suggests a general rule for which phases can occur, highlighting a D-brane–toric framework for systematic topology-changing transitions in string theory. The results contribute a concrete, computable picture of geometric transitions in brane probes and hint at a broader combinatorial classification of phases via area-$n$ lattice triangulations.

Abstract

We consider D-branes on an orbifold $C^3/Z_n$ and investigate the moduli space of the D-brane world-volume gauge theory by using toric geometry and gauged linear sigma models. For $n=11$, we find that there are five phases, which are topologically distinct and connected by flops to each other. We also verify that non-geometric phases are projected out for $n=7,9,11$ cases as expected. Resolutions of non-isolated singularities are also investigated.

D-branes on Orbifolds and Topology Change

TL;DR

The paper investigates topology change in Calabi–Yau spaces probed by D-branes on the orbifolds using toric geometry and gauged linear sigma models. It shows that for there are five Calabi–Yau phases connected by flop transitions, while non-geometric phases are projected out for , and it explores the role of non-isolated singularities via explicit cases. The approach maps D-brane vacua to toric data and FI-parameter regions, providing explicit phase structures and triangulations that encode topology changes. A proposed correspondence between the a-vector data and area- triangulations suggests a general rule for which phases can occur, highlighting a D-brane–toric framework for systematic topology-changing transitions in string theory. The results contribute a concrete, computable picture of geometric transitions in brane probes and hint at a broader combinatorial classification of phases via area- lattice triangulations.

Abstract

We consider D-branes on an orbifold and investigate the moduli space of the D-brane world-volume gauge theory by using toric geometry and gauged linear sigma models. For , we find that there are five phases, which are topologically distinct and connected by flops to each other. We also verify that non-geometric phases are projected out for cases as expected. Resolutions of non-isolated singularities are also investigated.

Paper Structure

This paper contains 11 sections, 55 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Topology change in toric geometry
  3. D-branes on ${\bf C}^3/{\bf Z}_n$
  4. Orbifold resolution and topology change
  5. D-branes on ${\bf C}^3/{\bf Z}_3$
  6. D-branes on ${\bf C}^3/{\bf Z}_5$
  7. D-branes on ${\bf C}^3/{\bf Z}_7$
  8. D-branes on ${\bf C}^3/{\bf Z}_9$
  9. D-branes on ${\bf C}^3/{\bf Z}_{11}$ and topology change
  10. D-branes on orbifolds with non-isolated singularities
  11. Discussion

Figures (12)

  • Figure 1: The toric diagrams which represent a flop.
  • Figure 2: The toric diagram for $n=3$, $\vec{a}=(2,2,-1)$.
  • Figure 3: (a)The quiver diagram for $n=5$, $\vec{a}=(3,3,-1)$, (b) the quiver diagram for $n=5$, $\vec{a}=(1,1,3)$. The two diagrams are related by a permutation of vertices.
  • Figure 4: The toric diagram for $n=5$, $\vec{a}=(3,3,-1)$.
  • Figure 5: (a) The quiver diagram for $n=7$, $\vec{a}=(4,4,-1)$, (b) the quiver diagram for $n=7$, $\vec{a}=(3,5,-1)$.
  • ...and 7 more figures