[Paper Review] Quiver algebras and their representations for arbitrary quivers

Wei Li|arXiv (Cornell University)|Mar 9, 2023

Algebraic structures and combinatorial models4 citations

TL;DR

This paper generalizes quiver Yangians—previously defined only for quivers from toric Calabi-Yau threefolds—to arbitrary quivers with potentials, constructing their representations via posets instead of 3D crystals. It extends the character computation to include refined BPS indices (motivic DT invariants), demonstrating consistency across BPS quivers of 4D $ mathcal{N}=2$ theories and knot-quiver correspondences, and establishes a link to affine Yangians of $ mathfrak{gl}_{m|n}$ and $D(2,1|α)$ via isomorphism to universal enveloping algebras of $ mathfrak{g}$-extended $ mathcal{W}_{1+∞}$ algebras.

ABSTRACT

The quiver Yangians were originally defined for the quiver and superpotential from string theory on general toric Calabi-Yau threefolds, and serve as BPS algebras of these systems. Their characters reproduce the unrefined BPS indices, which correspond to classical Donaldson-Thomas (DT) invariants. We generalize this construction in two directions. First, we show that this definition extends to arbitrary quivers with potentials. Second, we explain how to define the characters to incorporate the refined BPS indices, which correspond to motivic DT invariants. We focus on two main classes of quivers: the BPS quivers of 4D $N=2$ theories and the quivers from the knot-quiver correspondence. The entire construction allows for straightforward generalizations to trigonometric, elliptic, and generalized cohomologies.

Motivation & Objective

• To extend the definition of quiver Yangians beyond quivers from toric Calabi-Yau threefolds to arbitrary quivers with potentials.
• To construct representations of quiver Yangians for arbitrary quivers, replacing the 3D crystal structure with poset-based representations.
• To generalize the character computation of quiver Yangians to include refined BPS indices, corresponding to motivic Donaldson-Thomas invariants.
• To demonstrate that the refined vacuum characters of quiver Yangians reproduce known results for BPS quivers of 4D $ mathcal{N}=2$ theories and knot-quiver systems.
• To establish a connection between affine Yangians of certain Lie superalgebras and the universal enveloping algebra of $ mathfrak{g}$-extended $ mathcal{W}_{1+∞}$ algebras via the refined character structure.

Proposed method

• The quiver Yangian is constructed via a bootstrap procedure based on its action on poset representations derived from the quiver and its potential.
• The quadratic part of the algebra is defined directly from the quiver structure and potential, with relations determined by the action on the poset representations.
• Poset representations are built from partially ordered sets defined by the quiver's path algebra, serving as the state space for the algebra's action.
• The vacuum-like representation is constructed first, followed by non-vacuum representations using the vacuum as a building block.
• Characters are computed by enumerating states in the poset, with unrefined characters matching classical DT invariants and refined characters matching motivic DT invariants.
• Refined characters are defined via a consistent sign prescription for square roots in the wavefunction, ensuring algebraic consistency and reciprocity.

Experimental results

Research questions

• RQ1 Can quiver Yangians be consistently defined for arbitrary quivers with potentials, not just those from toric Calabi-Yau threefolds?
• RQ2 How can representations of quiver Yangians be constructed for arbitrary quivers when the 3D crystal structure is absent?
• RQ3 Can the character of the quiver Yangian be refined to reproduce motivic Donaldson-Thomas invariants, and how is this refinement implemented?
• RQ4 Does the refined vacuum character of the quiver Yangian for BPS quivers of 4D $ mathcal{N}=2$ theories agree with known results from other methods?
• RQ5 Is there a universal algebraic structure—specifically, a $ mathcal{W}_{1+∞}$ algebra—that underlies the refined quiver Yangian for certain Lie superalgebras?

Key findings

• The quiver Yangian construction is generalized to arbitrary quivers with potentials, with the quadratic part of the algebra manifestly depending on the quiver and potential.
• Representations of the quiver Yangian are constructed using posets derived from the quiver, replacing the 3D crystal structure used in the original toric CY 3 case.
• The unrefined characters of the quiver Yangian, computed by enumerating states in the poset, reproduce classical Donaldson-Thomas invariants.
• The refined characters are defined via a consistent sign choice for square roots in the wavefunction, allowing the recovery of motivic DT invariants.
• For BPS quivers of 4D $ mathcal{N}=2$ theories, the refined vacuum characters agree with results obtained using the same choice of subtorus, and are consistent across examples including Kronecker and McKay quivers.
• The refined vacuum characters reveal that the affine Yangian of $ mathfrak{g}$, with $ mathfrak{g} = mathfrak{gl}_{m|n}$ or $D(2,1|α)$, is isomorphic to the universal enveloping algebra of the $ mathfrak{g}$-extended $ mathcal{W}_{1+∞}$ algebra, providing a $ mathcal{W}$-algebra explanation for the plethystic exponential form of the BPS partition function.