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Reduced quiver quantum toroidal algebras

Andrei Neguţ

TL;DR

The paper provides a concrete generators-and-relations description of reduced quiver quantum toroidal algebras acting on BPS crystal spaces for toric Calabi-Yau threefolds. Using a shuffle-algebra realization and the notion of shrubby quivers, it proves that for shrubby $Q$ the kernel of the shuffle map is generated by coefficients of face-associated series $e_F$, giving an explicit Serre-like presentation of the reduced positive (and negative) halves. In the $X=\mathbb{C}^3$ example, this yields a cubic relation together with two face-wheel relations and ties the reduced algebra to the localized K-theoretic Hall algebra via a wheel-vanishing subspace $\mathcal{S}^+$. The results thereby deliver a tractable, torsion-free presentation for reduced quiver quantum toroidal algebras and illuminate their connection to BPS algebras, brane tilings, and $K$-theoretic Hall algebras on toric Calabi-Yau geometries.

Abstract

We give a generators-and-relations description of the reduced versions of quiver quantum toroidal algebras, which act on the spaces of BPS states associated to (non-compact) toric Calabi-Yau threefolds X. As an application, we obtain a description of the K-theoretic Hall algebra of (the quiver with potential associated to) X, modulo torsion.

Reduced quiver quantum toroidal algebras

TL;DR

The paper provides a concrete generators-and-relations description of reduced quiver quantum toroidal algebras acting on BPS crystal spaces for toric Calabi-Yau threefolds. Using a shuffle-algebra realization and the notion of shrubby quivers, it proves that for shrubby the kernel of the shuffle map is generated by coefficients of face-associated series , giving an explicit Serre-like presentation of the reduced positive (and negative) halves. In the example, this yields a cubic relation together with two face-wheel relations and ties the reduced algebra to the localized K-theoretic Hall algebra via a wheel-vanishing subspace . The results thereby deliver a tractable, torsion-free presentation for reduced quiver quantum toroidal algebras and illuminate their connection to BPS algebras, brane tilings, and -theoretic Hall algebras on toric Calabi-Yau geometries.

Abstract

We give a generators-and-relations description of the reduced versions of quiver quantum toroidal algebras, which act on the spaces of BPS states associated to (non-compact) toric Calabi-Yau threefolds X. As an application, we obtain a description of the K-theoretic Hall algebra of (the quiver with potential associated to) X, modulo torsion.
Paper Structure (4 sections, 13 theorems, 106 equations, 7 figures)

This paper contains 4 sections, 13 theorems, 106 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Shuffle algebras in general
  3. Shuffle algebras for shrubby quivers
  4. Appendix: the joys of gardening

Key Result

Theorem 1.13

If $Q$ is shrubby (as in Definition def:consistent intro), then the coefficients of the series eqn:series intro generate $\emph{Ker } \widetilde{\Upsilon}^+$ as a two-sided ideal. In other words, we have Similar results hold for $\mathbf{U}^-$, by replacing $e$'s with $f$'s and reversing the order of the factors in the product on the second line of eqn:formula series intro. While the quotient eqn

Figures (7)

  • Figure 1: The quiver associated to $X = {\mathbb{C}}^3$. The above square is the usual representation of the flat torus, so the quiver has one vertex, three edges and two faces.
  • Figure 2: A broken wheel (the path in red) and its mirror image (the path in blue). The black arrow is the interface.
  • Figure 3: An addable vertex $i$ (in black) to a shrub $S$ (in red).
  • Figure 4: Two situations of non-addable vertices $i$ (in black) to a shrub $S$ (in red).
  • Figure 5: A rhombus. The blue/red bullets represent the centers of the blue/red faces, while the other two vertices of the rhombus are vertices of $Q$ (with an arrow between them).
  • ...and 2 more figures

Theorems & Definitions (43)

  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.8
  • Definition 1.10
  • Theorem 1.13
  • Remark 1.14
  • Remark 1.15
  • Corollary 1.18
  • proof
  • ...and 33 more