Double Quiver Yangians & Algebras Overview

• Double quiver Yangians/algebras are infinite-dimensional quantum algebras that generalize Yangians by incorporating data from quiver varieties and crystal melting models.
• They organize BPS state interactions in quiver gauge theories, using combinatorial and geometric insights from JK residue computations and one-loop determinants.
• Their rich structure featuring both raising/lowering operators and a coproduct framework facilitates applications in integrable systems, wall-crossing, and cohomological Hall algebras.

Double Quiver Yangians/Algebras are a class of infinite-dimensional quantum algebras that generalize both Yangians and their double (universal enveloping) extensions to the context of quivers and related supersymmetric gauge theories. These algebras encode the interaction of BPS states arising from quiver gauge theories, often via combinatorial and geometric data extracted from moduli spaces, such as quiver varieties or crystal melting models. The “double” aspect refers to the presence of both raising and lowering operators, as well as a compatible coproduct or tensor structure, allowing for rich representation theory and applications to geometric representation theory, integrable models, and BPS state counting.

1. Geometric and Combinatorial Foundations

Double Quiver Yangians/Algebras are rooted in the geometry of quiver varieties and their corresponding moduli spaces. The key objects are:

• Quiver Q: Oriented graph with vertices (nodes) and arrows, possibly equipped with a potential W and equivariant parameters attached to arrows.
• Quiver varieties: Hyper-Kähler quotients or moduli spaces of quiver representations, often used to construct geometric actions of quantum algebras via convolution techniques.
• Crystals/melting models: Combinatorial models arising from the Jeffrey-Kirwan (JK) residue formulas in supersymmetric gauge theory partition functions, where each “atom” in a crystal corresponds to a fixed point of a torus action on the moduli space (Bao et al., 6 Jan 2025, Bao, 21 Sep 2025).

The algebraic structure is built by organizing the data of the fixed points (from JK residue computations) and their interaction rules (melting rules, additive/removable atoms) into a Fock-space representation, where algebra generators act as creation, annihilation, and Cartan-like diagonal operations on crystal states.

2. Algebraic Structure and Current Presentations

Generators and Relations

Double Quiver Yangians are generated by families of currents { $$\widetilde{\psi}^{(a)}(z),\ \widetilde{e}^{(a)}(z),\ \widetilde{f}^{(a)}(z),\ \widetilde{\omega}^{(a)}(z)$$ } for each node $a$:

• $\widetilde{\psi}^{(a)}(z)$: Cartan currents (charge functions).
• $\widetilde{e}^{(a)}(z)$, $\widetilde{f}^{(a)}(z)$: Raising and lowering currents, acting by adding/removing atoms in crystals.
• $\widetilde{\omega}^{(a)}(z)$: Additional currents encoding “inadmissible” poles and extra data from the full one-loop determinants, crucial for theories with reduced supersymmetry.

The defining commutation relations are encoded using “bond factors” $\widetilde{\phi}^{a\Leftarrow b}(z)$ derived from the one-loop determinants:

$$\begin{align*} \widetilde{\psi}^{(a)}_{\pm}(z)\,\widetilde{e}^{(b)}(w) &\simeq \widetilde{\phi}^{a\Leftarrow b}(z-w+c\mp c/2)\,\widetilde{e}^{(b)}(w)\,\widetilde{\psi}^{(a)}_{\pm}(z) \ \widetilde{\psi}^{(a)}_{\pm}(z)\,\widetilde{f}^{(b)}(w) &\simeq \widetilde{\phi}^{a\Leftarrow b}(z-w-c\pm c/2)^{-1}\,\widetilde{f}^{(b)}(w)\,\widetilde{\psi}^{(a)}_{\pm}(z) \ \widetilde{\phi}^{a\Leftarrow b}(z-w-c)\widetilde{e}^{(a)}(z)\widetilde{f}^{(b)}(w) - (-1)^{|a||b|}\widetilde{f}^{(b)}(w)\widetilde{e}^{(a)}(z) &\simeq \delta_{ab}\left( \delta(z-w-c)\,\widetilde{\psi}^{(a)}_+(w+c/2) - \delta(z-w+c)\,\widetilde{\psi}^{(a)}_-(z+c/2) - \delta(z-w)\,\widetilde{\omega}^{(a)}(z) \right) \end{align*}$$

(Bao et al., 6 Jan 2025, Bao, 21 Sep 2025)

These relations generalize those of both the (single) quiver Yangian and the usual Drinfeld–Jimbo Yangian, incorporating additional poles and interaction data specific to the full BPS spectrum of the underlying gauge theory.

Representation Theory

Crystal melting models yield a canonical Fock space realization where:

• $\widetilde{e}^{(a)}(z)$ adds an “atom” (or BPS instanton) of color $a$ at spectral parameter $z$, provided the melting rule admits it.
• $\widetilde{f}^{(a)}(z)$ removes an “atom” correspondingly.
• The charge function $\widetilde{\Psi}_C^{(a)}(z)$ is

$$\widetilde{\Psi}_C^{(a)}(z) = {}^{\#}\widetilde{\psi}^{(a)}(z)\prod_{b\in Q_0}\prod_{\mathfrak{b}\in C} \widetilde{\phi}^{a\Leftarrow b}(z-\epsilon(\mathfrak{b}))$$

where the product runs over all atoms in crystal $C$ and equivariant/fugacity parameters $\epsilon(\mathfrak{b})$ (Bao et al., 6 Jan 2025).

In the case of “standard” (single) quiver Yangians, only admissible poles contribute, and the algebra $\mathsf{Y}$ is realized as a subalgebra where inadmissible contributions are projected out (e.g., by setting auxiliary parameters to zero).

3. Comparison with Single Quiver Yangians and BPS Algebras

A distinguishing feature of double quiver Yangians is the retention of all data from the JK residue computation, including “inadmissible” poles (via $\widetilde{\omega}^{(a)}(z)$), and the precise combinatorial structure of the one-loop factors. In conventional quiver Yangians, such as for toric Calabi–Yau threefolds with four supercharges, only admissible simple poles are kept, and the structure reduces to familiar bond factors ("half-algebras"). The double structure is essential for:

• Theories with two supercharges (e.g., (0,2) supersymmetry), where one-loop determinants yield nontrivial rational or elliptic functions in fugacities.
• BPS wall-crossing phenomena where the index changes discontinuously and requires the refined algebra.

Relationship between the two:

• In cyclic chambers with four supercharges, double quiver Yangians specialize to single Yangians upon restriction to admissible poles (Bao et al., 6 Jan 2025).
• The double construction provides an explicit algebraic upgrade necessary for capturing all features of the BPS spectrum in less supersymmetric or refined wall-crossing situations.

4. Connections to Geometric Representation Theory and Hall Algebras

There exists a deep geometric underpinning linking double quiver Yangians to cohomological Hall algebras (COHA) of quivers and their relation to Yangians as constructed via the stable envelopes and R-matrix formalism:

• The preprojective COHA of a quiver $Q$ is isomorphic (after extension of scalars) to the positive half of the Maulik–Okounkov Yangian (Schiffmann et al., 2023).
• The isomorphism intertwines convolution actions (via Hecke correspondences) on the (equivariant) cohomology of quiver varieties with the action of the Yangian constructed from stable envelopes, establishing a bridge between geometric and quantum algebraic approaches.
• Kac polynomials (counting absolutely indecomposable quiver representations) directly govern the graded dimensions of the corresponding Lie algebras—as conjectured by Okounkov and proven in (Schiffmann et al., 2023)—linking the combinatorics of quiver representations to the character formulae for double quiver Yangians.

Invariants (such as Donaldson–Thomas or motivic DT invariants) as captured by the refined character of the algebraic representation coincide with those computed via BPS Hilbert space enumeration or using moduli space techniques (Li, 2023).

5. Field Theoretic Realizations, Coproducts, and Vertex Algebra Structures

Quantized Coproducts and Module Categories

A geometric coproduct—interpreted as a family of convolution homomorphisms parameterized by a class $c$—is realized via the following operator on the convolution algebra (Nakajima, 2012):

$$\Delta_c(\bullet) = c^{-1} \ast \bullet \ast c = p_{14*}(p_{12}^*(c^{-1}) \cap p_{23}^*(\bullet) \cap p_{34}^*(c))$$

This operator induces a homomorphism compatible with the known Yangian coproduct on generators, geometrically implementing the tensor product of representations as a splitting of exact sequences in the derived category of perverse sheaves.

Quantum Vertex Algebra Formulation

For simply-laced cases, double Yangians can be given a “current” or “field” presentation, facilitating their identification with (quantum) vertex algebra structures (Kong et al., 20 Sep 2025):

• Generators $x_i^{\pm}(z)$ and $h_i(z)$ are organized into generating series and satisfy field-theoretic commutation relations.
• The universal vacuum module $\mathcal{V}_A(\ell)$ of level $\ell$ carries a natural $\hbar$-adic weak quantum vertex algebra structure, and there is an equivalence between the categories of restricted $\widehat{\mathcal{DY}(A)}$-modules and $\mathcal{V}_A(\ell)$-modules.
• This formalism enhances the functorial and categorical understanding of double quiver Yangians via geometric and field-theoretic tools.

6. Connections to Integrable Structures, Deformations, and Dualities

Double quiver Yangians/Algebras provide a natural algebraic home for features found in quantum integrable systems:

• Cohomological Hall algebra and Yangian realizations intertwine with the universal R-matrix formalism and stable envelopes, directly relating to solutions of the Yang–Baxter and quantum Knizhnik–Zamolodchikov (qKZ) equations (Zinn-Justin, 2015).
• The “double” aspect encodes the fusion and factorization structures familiar from conformal field theory and quantum group theory.

Toroidal and elliptic deformations (trigonometric/elliptic quiver algebras) extend the rational case and are used to organize higher-dimensional enumerative and gauge theories (Noshita et al., 2021, Galakhov et al., 2021). These deformations retain a Hopf-like coproduct and triality among quantum parameters and are represented on crystal models corresponding to higher cohomology theories or quantum K-theory.

Seiberg duality and related isomorphism phenomena manifest as explicit algebra isomorphisms—sometimes involving nontrivial operator transformations—between double quiver Yangians arising from different but physically dual quivers (Jejjala et al., 23 Sep 2025).

7. Applications and Outlook

Double quiver Yangians/algebras are central to several interwoven developments:

• BPS state counting in supersymmetric gauge theories, including the most general quiver gauge theories with arbitrary superpotentials (Bao et al., 6 Jan 2025, Li, 2023, Bao, 21 Sep 2025).
• Compactifications and dualities in string theory, with the associated wall-crossing and integrability phenomena.
• Explicit computation of flavored/motivic/graded BPS invariants, modular partition functions, and refined Donaldson–Thomas invariants.
• Structured equivalence between quantum algebraic and geometric categories (vertex algebras, quantum groups, and derived categories of sheaves/moduli spaces).
• Integrable systems and associated solution spaces of quantum difference/differential equations encoded via module representation theory.

Recent constructions demonstrate that double quiver Yangians and their representations are not only universal in organizing the algebraic structure underlying BPS spectra but also adaptable to new contexts, including ${\mathcal N}=2$ and ${\mathcal N}=0$ gauge theories, non-toric Calabi–Yau spaces, and deformation quantization frameworks (Bao et al., 6 Jan 2025, Kong et al., 20 Sep 2025, Bao, 21 Sep 2025).

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Key Formulas and Definitions (see (Bao et al., 6 Jan 2025, Schiffmann et al., 2023, Kong et al., 20 Sep 2025)):

• Charge function: $$\widetilde{\Psi}_C^{(a)}(z) = {}^{\#}\widetilde{\psi}^{(a)}(z) \prod_{b\in Q_0}\prod_{\mathfrak{b}\in C}\widetilde{\phi}^{a\Leftarrow b}(z-\epsilon(\mathfrak{b}))$$
• Raising operator: $$\widetilde{e}^{(a)}(z)|C\rangle = \sum_{\mathfrak{a}\in\mathrm{Add}(C)} \pm \delta(z-\epsilon(\mathfrak{a})) \left[ \lim_{x\to \epsilon(\mathfrak{a})} \zeta(x-\epsilon(\mathfrak{a})) \widetilde{\Psi}_C^{(a)}(x) \right]^{1/2}|C+\mathfrak{a}\rangle$$
• Bond factor: explicit product formula in terms of $\zeta$-functions and one-loop data, as fully detailed above.

These structures underpin the modern understanding and classification of quantum algebras attached to quiver gauge theories and the mathematical physics of supersymmetric BPS spectra.