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On the C^n/Z_m fractional branes

Robert L. Karp

TL;DR

The paper develops a geometric framework for fractional branes at ${ t C}^n/{ t Z}_m$ singularities using derived-category methods and FM functors, connecting large-radius and conifold-type monodromies to brane transformations. By focusing on ${ t C}^3/{ t Z}_5$, it constructs a length-${5}$ orbit of fractional branes via the ${ t Z}_5$ monodromy, computes their Ext-quiver, and demonstrates a Seiberg duality linking this orbit to the McKay-collected branes. The approach relies on toric and GLSM techniques for the geometry, Horn uniformization for discriminants, and stack-theoretic resolutions (Kawamata) to extend to partial resolutions, yielding a coherent picture that ties monodromies, quivers, and the McKay correspondence. The results illuminate how different moduli-space patches yield equivalent brane categories and provide a practical method to derive quivers and superpotentials without relying on mirror symmetry, with potential applications to broader ${ t C}^n/{ t Z}_m$-type singularities and their quivers.

Abstract

We construct several geometric representatives for the C^n/Z_m fractional branes on either a partially or the completely resolved orbifold. In the process we use large radius and conifold-type monodromies, and provide a strong consistency check. In particular, for C^3/Z_5 we give three different sets of geometric representatives. We also find the explicit Seiberg-duality, in the Berenstein-Douglas sense, which connects our fractional branes to the ones given by the McKay correspondence.

On the C^n/Z_m fractional branes

TL;DR

The paper develops a geometric framework for fractional branes at singularities using derived-category methods and FM functors, connecting large-radius and conifold-type monodromies to brane transformations. By focusing on , it constructs a length- orbit of fractional branes via the monodromy, computes their Ext-quiver, and demonstrates a Seiberg duality linking this orbit to the McKay-collected branes. The approach relies on toric and GLSM techniques for the geometry, Horn uniformization for discriminants, and stack-theoretic resolutions (Kawamata) to extend to partial resolutions, yielding a coherent picture that ties monodromies, quivers, and the McKay correspondence. The results illuminate how different moduli-space patches yield equivalent brane categories and provide a practical method to derive quivers and superpotentials without relying on mirror symmetry, with potential applications to broader -type singularities and their quivers.

Abstract

We construct several geometric representatives for the C^n/Z_m fractional branes on either a partially or the completely resolved orbifold. In the process we use large radius and conifold-type monodromies, and provide a strong consistency check. In particular, for C^3/Z_5 we give three different sets of geometric representatives. We also find the explicit Seiberg-duality, in the Berenstein-Douglas sense, which connects our fractional branes to the ones given by the McKay correspondence.

Paper Structure

This paper contains 23 sections, 11 theorems, 136 equations, 6 figures.

Table of Contents

  1. Introduction
  2. ${\mathbb C}^3/{\mathbb Z}_5$ geometries
  3. The toric geometry of ${\mathbb C}^3/{\mathbb Z}_5$
  4. The ${\mathbb C}^3/{\mathbb Z}_5$ moduli space
  5. The discriminant
  6. Intersections
  7. ${\mathbb C}^3/{\mathbb Z}_5$ monodromies
  8. Fourier-Mukai functors
  9. Monodromies in general
  10. ${\mathbb C}^3/{\mathbb Z}_5$ monodromies
  11. Quantum symmetries
  12. The ${\mathbb C}^3/{\mathbb Z}_5$ fractional branes
  13. Generating fractional branes
  14. Computing $\mathsf{M}_{{\mathbb Z}_5}(j_*\mathcal{O}_{D_5})$
  15. Computing $(\mathsf{M}_{{\mathbb Z}_5})^2(j_*\mathcal{O}_{D_5})$
  16. ...and 8 more sections

Key Result

Theorem 3.1

Let $X_1$ and $X_2$ be smooth projective varieties. Suppose that $\mathsf{F}\!: \mathrm{D}(X_1)\to\mathrm{D}(X_2)$ is an equivalence of triangulated categories. Then there exists an object $\mathcal{K}\in \mathrm{D}(X_1\!\times \! X_2)$, unique up to isomorphism, such that the functors $\mathsf{F}$

Figures (6)

  • Figure 1: The toric fan for the resolution of the ${\mathbb C}^3/{\mathbb Z}_5$ singularity.
  • Figure 2: The four triangulations of the ${\mathbb C}^3/{\mathbb Z}_5$ model.
  • Figure 3: The phase structure of the ${\mathbb C}^3/{\mathbb Z}_5$ model.
  • Figure 4: The moduli space of the ${\mathbb C}^3/{\mathbb Z}_5$ model.
  • Figure 5: The two links: ${\bf L}({\mathcal{L}}_1,P_0)={\mathcal{L}}_1 \cap S^3_{\!\epsilon}$ and ${\bf L} (\Delta_0,P_0)=\Delta_0 \cap S^3_{\!\epsilon}$.
  • ...and 1 more figures

Theorems & Definitions (21)

  • Theorem 3.1
  • Proposition 3.2
  • Proposition 3.3
  • Conjecture 3.4
  • Definition 3.5
  • Theorem 3.6
  • Proposition 3.7
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 11 more