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An Overview of Crystals and Double Quiver Yangians

Jiakang Bao

Abstract

In this review, we summarize the recent progress on the crystal melting models and the quiver algebras regarding the BPS counting. We shall consider the constructions of crystals for generic quivers and discuss the so-called double quiver Yangians/algebras. This is an invited review for International Journal of Modern Physics A.

An Overview of Crystals and Double Quiver Yangians

Abstract

In this review, we summarize the recent progress on the crystal melting models and the quiver algebras regarding the BPS counting. We shall consider the constructions of crystals for generic quivers and discuss the so-called double quiver Yangians/algebras. This is an invited review for International Journal of Modern Physics A.

Paper Structure

This paper contains 18 sections, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Crystal Melting and Jeffrey-Kirwan Residues
  3. The Jeffrey-Kirwan residue formula
  4. The one-loop determinant
  5. The space M
  6. The Jeffrey-Kirwan residue
  7. Example
  8. Weights of the arrows
  9. Fermi multiplets and brane brick models
  10. Crystals
  11. No-overlap condition
  12. Molecules and cyclic chambers
  13. Example
  14. Modules of truncated Jacobi algebras
  15. Quiver Algebras
  16. ...and 3 more sections

Figures (1)

  • Figure 1: The crystal of the quiver \ref{['C3quiver']}. The right plot is equivalent to the left one but with a view from a different angle where the plane partition nature is more clear.