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Critical CoHAs, vertex coalgebras and Deformed Drinfeld coproducts

This paper constructs a vertex coproduct on the Kontsevich--Soibelman cohomological Hall algebra of a quiver with potential to form a vertex bialgebra, which extends the algebra via bosonisation, recovers Drinfeld's deformed coproduct for ADE quivers, and provides a new proof of the cohomological integrality theorem for symmetric quivers with trivial potential.

Original authors: Shivang Jindal, Sarunas Kaubrys, Alexei Latyntsev

Published 2026-03-24

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CC0 1.0

Original authors: Shivang Jindal, Sarunas Kaubrys, Alexei Latyntsev

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a master architect trying to understand the blueprints of a massive, invisible city. This city isn't made of brick and mortar, but of mathematical shapes called "quivers" (which look like networks of dots and arrows) and "potentials" (which are like rules or energy fields governing how these shapes interact).

The paper you're asking about is a new set of instructions for how to navigate this city. It connects three very different worlds of mathematics: Geometry (shapes), Algebra (equations), and Physics (how particles interact).

Here is the story of the paper, broken down into simple concepts and analogies.

1. The City of Shapes (Quivers and Moduli Stacks)

Think of a Quiver as a subway map. The stations are dots (vertices), and the tracks are arrows. A Representation is like assigning a specific number of passengers to each station and deciding how they move along the tracks.

Now, imagine you want to study all possible subway maps at once. You can't just look at one; you need a "map of maps." In math, this giant collection is called a Moduli Stack. It's a chaotic, high-dimensional space where every point represents a different configuration of your subway system.

The authors are studying the "cohomology" of this space. Think of cohomology as a way to count the holes, loops, and twists in this giant city. It's like taking a census of the city's shape.

2. The Two Ways to Talk: The "Hall" and the "Vertex"

The paper introduces two different languages for describing this city.

The Hall Language (The Product): This is like a construction crew. If you have two small subway systems, the "Hall Algebra" tells you how to smash them together to build a bigger one. It's about combining things.
The Vertex Language (The Coproduct): This is like a splitting machine. If you have a big subway system, the "Vertex Coproduct" tells you how it can naturally split into two smaller systems. It's about dividing things.

The Big Discovery: The authors found a way to make these two languages talk to each other perfectly. They showed that the rules for building (Hall) and the rules for splitting (Vertex) are compatible. When you combine them, you get a Vertex Bialgebra.

Analogy: Imagine a Lego set.

The Hall part is the instruction manual on how to snap two bricks together.
The Vertex part is the instruction manual on how to pull a complex structure apart into its original bricks.
The Bialgebra is the realization that these two manuals are actually two sides of the same coin. If you know how to build, you automatically know how to split, and vice versa.

3. The "Magic Twist" (The Potential and Vanishing Cycles)

Usually, these subway maps are messy. But the authors focus on a special case where there is a "Potential" (a rule $W$ ). When this rule is applied, the city collapses into a smaller, more stable version called the Critical Cohomological Hall Algebra (CoHA).

Think of the Potential as a gravity well. It pulls all the chaotic, floating shapes down to the ground, leaving only the most stable, "critical" shapes behind. The authors built their new "splitting machine" (Vertex Coproduct) specifically for these stable shapes.

4. The Connection to Quantum Physics (Yangians)

Here is the most exciting part. The authors discovered that for a specific type of city (called ADE quivers, which relate to famous shapes in geometry), their new "splitting machine" produces the exact same results as a famous tool in quantum physics called the Yangian.

The Yangian is a mathematical object used to describe how particles (like electrons) interact in quantum mechanics. It has a "Drinfeld Coproduct," which is a specific way of splitting particles.
The Breakthrough: The authors proved that their geometric "splitting machine" (the Joyce-Liu coproduct) is identical to the physics "splitting machine" (Drinfeld's coproduct).

Analogy: Imagine a physicist and a geometer are both trying to describe how a drop of water splits into two smaller drops.

The Physicist uses complex equations about surface tension and quantum forces.
The Geometer uses shapes and topological maps.
This paper says: "Hey, your equations and my maps are actually describing the exact same thing!"

5. "Bosonisation" (Adding the Cartan Part)

In physics, to get a complete description of a system, you often need to add a "Cartan part" (think of it as adding a central control tower to your subway map).

The authors used a technique called Bosonisation (named after the physicist Bose) to attach this control tower to their geometric city. They showed that by doing this, the entire structure becomes a perfect match for the quantum Yangian. It's like taking a rough sketch of a city and adding the power grid and water lines to make it a fully functional metropolis.

6. Why Does This Matter?

New Proofs: Their method provides a fresh, geometric way to prove old theorems about counting shapes (integrality theorems). It's like proving a theorem about triangles using a ruler instead of a protractor.
Unifying Fields: It bridges the gap between pure geometry (studying shapes) and theoretical physics (studying particles). It suggests that the "rules" of the universe might be written in the language of these geometric shapes.
New Tools: They created a new "dictionary" (the Vertex Coproduct) that allows mathematicians to translate problems from one difficult area of math into another, easier area.

Summary

In simple terms, this paper builds a universal translator between the geometry of shapes and the algebra of quantum physics. It shows that when you look at a specific type of mathematical network (a quiver) and apply a "splitting" rule, you don't just get a geometric result; you get the exact same result that physicists use to describe the behavior of subatomic particles.

It's a beautiful example of how the deep structures of mathematics are interconnected, revealing that the "blueprints" of the universe are written in a single, elegant language.

1. Problem Statement

The paper addresses a gap in the geometric representation theory of quivers with potential. While Dominic Joyce has previously constructed vertex algebra structures on the homology of moduli spaces of objects in abelian categories, and similar structures exist for K-theory (Liu), the cohomological setting for critical Cohomological Hall Algebras (CoHAs) remained underdeveloped.

Specifically, the authors aim to:

Construct a vertex coproduct on the critical CoHA of a quiver with potential ( $Q, W$ ).
Establish the compatibility between the existing CoHA product and this new coproduct, forming a vertex bialgebra.
Relate this geometric construction to the Drinfeld coproduct on Yangians (quantum groups) for ADE quivers.
Provide a unified framework (via bosonisation) that extends CoHAs to include a "Cartan part," recovering the full structure of quantum groups.

2. Methodology

The authors employ a synthesis of algebraic geometry, derived categories, and vertex algebra theory.

A. Geometric Setup

Moduli Stacks: They work with the moduli stack $M_Q$ of representations of a quiver $Q$ with potential $W$ .
Critical Cohomology: The vector space of interest is the critical cohomology (vanishing cycles cohomology) of this stack:
$A_{Q,W} = \bigoplus_{d \in \mathbb{N}^{Q_0}} H^*(M_{Q,d}, \phi_{\text{Tr}W_d} \mathbb{Q}_{M_{Q,d}}[\dim M_{Q,d}])$
where $\phi$ denotes the vanishing cycles functor.
Torus Action: They introduce a torus $T$ action (grading) to handle equivariant cohomology, allowing for the definition of characteristic classes and the use of equivariant localization techniques.

B. Construction of the Vertex Coproduct

The core innovation is the construction of a Joyce–Liu vertex coproduct $\Delta(z)$ . This is defined via a geometric correspondence involving the direct sum of representations:

Direct Sum Map: A map $\oplus: M_Q \times M_Q \to M_Q$ .
Pullback: Using the pullback $\oplus^*$ on cohomology.
Translation/Scaling: An action of $\mathbb{G}_m$ (scaling automorphisms) induces a translation map $act^*$ .
Joyce-Borcherds Twist: The crucial step is twisting the result by a series $\Psi(\text{Ext}, -z)$ derived from the Ext complex (a perfect complex on the product of moduli stacks). This series acts as a "meromorphic" factor depending on a complex parameter $z$ .

The formula is roughly:
$\Delta(z)(\alpha) = \Psi(\text{Ext}, -z) \cdot act^*_1 (\oplus^* \alpha)$

C. Comparison with Localised Coproducts

The authors compare their vertex coproduct with the localised coproduct defined by Davison. Davison's approach involves inverting Euler classes of tangent complexes to define a coproduct on a localized ring. The paper proves that the Joyce–Liu vertex coproduct is essentially the "un-inverted" version of Davison's coproduct, where the inversion is replaced by a Laurent series expansion in $z^{-1}$ .

D. Bosonisation

To recover the full quantum group structure (including the Cartan subalgebra), the authors apply a vertex analogue of Majid–Radford bosonisation. They extend the CoHA by tensoring it with the tautological cohomology ring of the moduli stack (which acts as the Cartan part). This requires the CoHA to be a vertex bialgebra in a braided monoidal category.

3. Key Contributions and Results

Theorem A: Vertex Coalgebra Structure

For any quiver with potential $W$ satisfying a relative Künneth property, the critical CoHA $A_{Q,W}$ admits a coassociative vertex coalgebra structure.

The coproduct $\Delta(z)$ is braided colocal (a vertex analogue of cocommutativity), meaning $\Delta(z) = \sigma \cdot S(z) \Delta(-z) \cdot act^*$ , where $S(z)$ involves the R-matrix.

Theorem C: Vertex Bialgebra Structure

The CoHA product (twisted by a sign $\psi$ ) and the Joyce–Liu vertex coproduct are compatible. Together, they form a vertex bialgebra inside a category of modules over the tautological cohomology ring.

The compatibility condition involves a spectral R-matrix $R(z)$ , which braids the tensor factors before multiplication:
$\Delta(b \cdot b', z) = \Delta(b, z) \cdot R(z) \cdot \Delta(b', z)$
This establishes the CoHA as an object in a "meromorphic" braided monoidal category.

Theorem F: Recovery of Drinfeld's Coproduct

For ADE quivers (associated with simple Lie algebras $\mathfrak{g}$ ), the authors prove a major isomorphism:

The extended CoHA (after bosonisation) is isomorphic to the positive Borel part of the Yangian $Y_\hbar(\mathfrak{g})_{\geq 0}$ .
Under this isomorphism, the Joyce–Liu vertex coproduct corresponds exactly to Drinfeld's deformed coproduct on the Yangian.
This provides a geometric derivation of the Drinfeld coproduct, which was previously defined algebraically.

Theorem G & H: Cohomological Integrality and Dotsenko-Mozgovoy

In the case of symmetric quivers with trivial potential ( $W=0$ ):

The dual of the CoHA is shown to be a universal chiral envelope of a vertex Lie algebra.
This yields a new proof of the cohomological integrality theorem (originally by Efimov), showing that the CoHA is isomorphic to a symmetric algebra.
The authors prove that their constructed vertex algebra is isomorphic to the explicit construction by Dotsenko and Mozgovoy [DM25], providing a generators-and-relations description.

4. Significance

Geometric Realization of Quantum Groups: The paper bridges the gap between the geometric world of CoHAs (moduli of quiver representations) and the algebraic world of Quantum Groups (Yangians). It shows that the Drinfeld coproduct is not just an algebraic artifact but arises naturally from the geometry of vanishing cycles and Ext complexes.
Unification of Coproducts: It unifies three distinct approaches to coproducts in this field:
- The localised coproduct (Davison).
- The vertex coproduct (Joyce/Liu).
- The Drinfeld coproduct (Yangians).
  The paper demonstrates they are different facets of the same underlying structure, related by localization and series expansion.
Bosonisation Framework: By introducing vertex bosonisation, the authors provide a systematic method to "add a Cartan part" to CoHAs, which is essential for recovering the full quantum group structure rather than just the positive part.
New Proofs: The work offers a novel, geometric proof of cohomological integrality for symmetric quivers, independent of previous methods.
Physical Interpretation: The structure is motivated by and linked to 3d/4d holomorphic-topological Quantum Field Theories (QFTs), where the vertex coproduct corresponds to the collision of line operators, offering a physical interpretation of the algebraic structures.

In summary, this paper establishes a rigorous geometric foundation for vertex algebras on critical CoHAs, successfully linking them to the Drinfeld coproducts of Yangians and providing a powerful new tool for studying the representation theory of quivers and quantum groups.