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Toric Fano varieties and Chern-Simons quivers

Cyril Closset, Stefano Cremonesi

TL;DR

The work builds a principled bridge from toric CY4 geometry to 3d N=2 Chern-Simons quiver gauge theories by grounding the construction in type IIA reductions of M-theory and a robust quiver/dictionary framework. It develops a comprehensive toolkit—GIT quotients, θ-stability, tilting collections, and brane-charge dictionaries—to translate toric data into quiver data and to verify that moduli spaces reproduce the corresponding IIA geometries. The authors furnish explicit CS quivers for a wide class of Y^{p, q}(B4) backgrounds without torsion flux, and extend the construction to include torsion G4, detailing wall-crossing, Seiberg dualities, and equal-rank cases that connect to prior proposals in the literature. Higgsing and partial resolutions provide a coherent picture of RG flows aligned with geometric transitions, validating the stringy origin of the CS quivers and their moduli spaces. The results offer a concrete path toward an inverse algorithm that maps toric data to CS quivers and open avenues for broader toric CY4 geometries and nontrivial torsion flux configurations, with implications for the holographic description of M2-branes in these backgrounds.

Abstract

In favourable cases the low energy dynamics of a stack of M2-branes at a toric Calabi-Yau fourfold singularity can be described by an N=2 supersymmetric Chern-Simons quiver theory, but there still does not exists an "inverse algorithm" going from the toric data of the CY_4 to the CS quiver. We make progress in that direction by deriving CS quiver theories for M2-branes probing cones over a large class of geometries Ypq(B_4), which are S^3/Z_p bundles over toric Fano varieties B_4. We rely on the type IIA understanding of CS quivers, giving a firm string theory footing to our CS theories. In particular we give a derivation of some previously conjectured CS quivers in the case B_4= CP^1*CP^1, as field theories dual to M-theory backgrounds with nontrivial torsion G_4 fluxes.

Toric Fano varieties and Chern-Simons quivers

TL;DR

The work builds a principled bridge from toric CY4 geometry to 3d N=2 Chern-Simons quiver gauge theories by grounding the construction in type IIA reductions of M-theory and a robust quiver/dictionary framework. It develops a comprehensive toolkit—GIT quotients, θ-stability, tilting collections, and brane-charge dictionaries—to translate toric data into quiver data and to verify that moduli spaces reproduce the corresponding IIA geometries. The authors furnish explicit CS quivers for a wide class of Y^{p, q}(B4) backgrounds without torsion flux, and extend the construction to include torsion G4, detailing wall-crossing, Seiberg dualities, and equal-rank cases that connect to prior proposals in the literature. Higgsing and partial resolutions provide a coherent picture of RG flows aligned with geometric transitions, validating the stringy origin of the CS quivers and their moduli spaces. The results offer a concrete path toward an inverse algorithm that maps toric data to CS quivers and open avenues for broader toric CY4 geometries and nontrivial torsion flux configurations, with implications for the holographic description of M2-branes in these backgrounds.

Abstract

In favourable cases the low energy dynamics of a stack of M2-branes at a toric Calabi-Yau fourfold singularity can be described by an N=2 supersymmetric Chern-Simons quiver theory, but there still does not exists an "inverse algorithm" going from the toric data of the CY_4 to the CS quiver. We make progress in that direction by deriving CS quiver theories for M2-branes probing cones over a large class of geometries Ypq(B_4), which are S^3/Z_p bundles over toric Fano varieties B_4. We rely on the type IIA understanding of CS quivers, giving a firm string theory footing to our CS theories. In particular we give a derivation of some previously conjectured CS quivers in the case B_4= CP^1*CP^1, as field theories dual to M-theory backgrounds with nontrivial torsion G_4 fluxes.

Paper Structure

This paper contains 67 sections, 356 equations, 26 figures, 1 table.

Table of Contents

  1. Introduction
  2. Toric quivers, $CY_3$ cones and fractional brane charges
  3. Toric quiver and crepant resolutions of $Y$
  4. Example: The $PdP_2$ quiver.
  5. GIT quotient and moduli space
  6. Quiver representations, $\theta$-stability and Kähler chambers
  7. Definition: $\theta$-stability.
  8. The $PdP_2$ example.
  9. Tilting collection of line bundles from toric quiver
  10. Example.
  11. 2d toric Fano varieties and brane charge dictionaries
  12. Example.
  13. Kähler moduli space and quiver locus
  14. Dictionaries, monodromies and Freed-Witten anomaly
  15. Reduction of M-theory on $\mathbb{R}^{1,2}\times CY_4$ to type IIA
  16. ...and 52 more sections

Figures (26)

  • Figure 1: Quiver, brane tiling and toric diagram for the complex cone over $PdP_2$. Black numbers above toric points are the multiplicities of these points (when they are larger than one), red numbers are the names of the corresponding perfect matchings. The strictly internal point conventionally has coordinates $(0,0)$.
  • Figure 2: The four possible triangulations $T_{\Gamma}^{(1)}, \cdots, T_{\Gamma}^{(4)}$ of the toric diagram of $C_{\mathbb{C}}(PdP_2)$.
  • Figure 3: Examples of pseudo-Beilinson quivers obtained from the toric quiver for $PdP_2$, for the perfect matchings $p_5$, $p_6$ and $p_7$ respectively.
  • Figure 4: Toric diagrams for the 16 two-dimensional toric Fano varieties. The labels over the external points give the height of the point in the corresponding 3d toric diagrams which we shall introduce later on.
  • Figure 5: The 3d toric diagram for the cone over $Y^{p,\,q_1,\,q_2}(dP_1)$, with $(q_1,q_2,p)=(2,1,5)$. Looking at the toric diagram from the top or the bottom, we see the two different triangulations of the 2d toric diagram of $C_\mathbb{C}(dP_1)$.
  • ...and 21 more figures