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Deformations of Toric Singularities and Fractional Branes

Agostino Butti

Abstract

Fractional branes added to a large stack of D3-branes at the singularity of a Calabi-Yau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann's rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional branes can be classified by looking at suitable weights associated with the external legs of the (p,q) web. We check the proposal on many examples and match in some interesting cases the moduli space of the gauge theory with the deformed geometry.

Deformations of Toric Singularities and Fractional Branes

Abstract

Fractional branes added to a large stack of D3-branes at the singularity of a Calabi-Yau cone modify the quiver gauge theory breaking conformal invariance and leading to different kinds of IR behaviors. For toric singularities admitting complex deformations we propose a simple method that allows to compute the anomaly free rank distributions in the gauge theory corresponding to the fractional deformation branes. This algorithm fits Altmann's rule of decomposition of the toric diagram into a Minkowski sum of polytopes. More generally we suggest how different IR behaviors triggered by fractional branes can be classified by looking at suitable weights associated with the external legs of the (p,q) web. We check the proposal on many examples and match in some interesting cases the moduli space of the gauge theory with the deformed geometry.

Paper Structure

This paper contains 12 sections, 1 theorem, 44 equations, 23 figures.

Table of Contents

  1. Introduction and overview
  2. Generalities about the gauge theory
  3. Distribution of charges in the dimer
  4. Classes of fractional branes and deformations in toric geometry
  5. Fractional branes and baryonic symmetries
  6. Matching deformations with fractional branes
  7. Examples and further observations
  8. Comments on the $\Psi$-map
  9. Computing the deformed geometry: a detailed example
  10. The deformed geometry
  11. The moduli space of the gauge theory
  12. Conclusions

Key Result

Theorem 7.1

If $L$ is a closed loop of chiral fields (oriented or not), then the $\Psi$-map for $L$ is: where $(n,m,c)$ are integer numbers and $(n,m)$ are the homotopy numbers of the loop $L$ on the torus $T^2$ of the periodic quiver or dimer.

Figures (23)

  • Figure 1: Dimer configuration and toric diagram for the Suspended Pinch Point.
  • Figure 2: Zig-zag paths for the Suspended Pinch Point and their correspondence with external legs of the (p,q) web.
  • Figure 3: Deformation of the cone over SPP: decomposition of the toric diagram in Minkowski summands and splitting of the web into subwebs in equilibrium.
  • Figure 4: Deformation of the cone over $dP_3$. a) One parameter branch. b) Two parameters branch.
  • Figure 5: The parameters $b_i$ for global charges.
  • ...and 18 more figures

Theorems & Definitions (1)

  • Theorem 7.1