When Algebraic Geometry Learns Quantum Groups: Cohomological Hall Algebras and Yangians
09 Jun 2026, Yanjiang
The cohomological Hall algebra of a smooth surface with a curve is isomorphic to the positive half of the affine Yangian, linking geometry and quantum symmetry.
What makes the mathematics behind quantum field theory so thrilling is that the same algebraic structures keep appearing in seemingly unrelated corners—like a secret language shared by geometry and particle physics. A new preprint (arXiv:2603.03386) from Duiliu-Emanuel Diaconescu (Rutgers University), Mauro Porta, Francesco Sala, Olivier Schiffmann, and Eric Vasserot demonstrates that when you twist the geometry of a smooth surface just right, the algebra that counts particle-like objects on it speaks the language of a quantum group known as a Yangian. The result is a precise dictionary that was long suspected, but nowhere firmly written.
To appreciate what they have built, we first need to picture a smooth surface with a scratch on it. Think of a flawless landscape—a gentle, complex curve sitting inside it, like a dried riverbed left behind after a flood. In algebraic geometry, that scratch is a proper curve ( Z ) sitting inside a surface ( X ). On this surface, you can drape what mathematicians call coherent sheaves—data fields attached to every point, encoding instructions for how particles, charges, or geometric information thread through the landscape. If you take a sheaf and “modify” it along the scratch, you get a new sheaf, one that differs only near ( Z ). Imagine carving a new channel in the riverbed; the rest of the landscape stays the same, but the flow of data along the carving changes. A modification is a precise operation that tweaks the sheaf in a way that is trivial away from ( Z ). Now ask: how many such modifications exist? What are their composition rules? Can we organize all possibilities into a coherent algebraic whole?
That whole is the cohomological Hall algebra, or COHA, abbreviated from a mouthful of mathematical machinery. For a given surface ( X ) and curve ( Z ), the COHA is a giant associative algebra whose elements are built from the fundamental classes of stacks of zero-dimensional sheaves and from the pushforwards of line bundles on ( Z ). In less technical terms: it assembles all possible ways to place pointlike objects on ( Z ) and to thread one-dimensional data along it, and it gives a rule for multiplying these configurations—gluing them together when one sits inside another—so that the result is a new configuration of sheaves. The algebra is “Hall” because it grew out of similar counting algebras for representations of quivers, which in turn carry a physical echo: they appear when physicists compute the spaces of BPS states in supersymmetric gauge theories.
Diaconescu and his collaborators prove that the COHA of a particular Kleinian resolution, with its scar of singularities healed, is isomorphic to a mathematical object on the other side of the geometric-physical divide: a Yangian. Yangians are non-commutative algebras that provide the quantum symmetry backbone of many integrable systems. They are infinite-dimensional relatives of universal enveloping algebras of Lie algebras, but enriched with an extra deformation parameter—a spectral parameter—that remembers the underlying loop group symmetry. In the dictionary uncovered here, the simplest nontrivial Yangian, the positive half of the affine Yangian associated to an ADE Lie algebra, emerges naturally from the geometry. Specifically, when ( X ) is the cotangent bundle of the projective line, and ( Z ) is the zero-section, the COHA becomes exactly the completed, nonstandard version of the positive half of the affine Yangian of type A₁. The generators of the COHA—those fundamental classes of substacks and pushforwards of line bundles—are expressed explicitly in terms of standard Yangian generators. It is as if, by carving a curved path through a smooth surface, we have excavated the entire symmetry algebra that governs the motion of quantum particles on a circle.
This bridge does not appear out of nowhere. The building block is a Hecke operator—an algebraic version of the action of modifying a sheaf at a point. In the COHA framework, Hecke operators correspond to adding a zero-dimensional sheaf supported on the curve ( Z ). The composition of such Hecke modifications gives the multiplication in the COHA. By identifying these Hecke operators with Yangian generators, the paper provides the first algebraic characterization of an algebra of cohomological Hecke operators for surfaces, grounding a long speculative connection in explicit formulas. It is not a metaphor. The correspondence is exact, morphism-by-morphism, relation-by-relation.
Yet nature does not give up her secrets all at once. The proof of this isomorphism leans on a heavy toolbox. The team develops a “continuity theorem” for COHAs: if you have a sequence of t-structures—ways to chop a derived category into hearts that encode what counts as a module—that converge in a suitable sense to a limiting t-structure, then the corresponding COHAs converge as well. It is a statement about how algebraic structures behave under smooth deformations of the organizational rules. However, an important question sharpened by earlier work on quiver COHAs (Schiffmann et al., arXiv:2312.15803) is whether this continuity theorem is truly essential to establishing the Yangian connection. The continuity theorem serves a foundational role in the paper’s overall framework, though its full power—verifying that the sequence of t-structures in the Kleinian case indeed converges to the required limit—is established through a combination of arguments that the authors develop across this work and its companions. For now, the continuity theorem remains a beautiful promise—a tool whose utility awaits the sharpening of future applications.
The team also introduces a multi-parameter Yangian ( \mathbb{Y}_Q ) for an arbitrary quiver ( Q ). This generalization is a significant conceptual leap because it detaches the Yangian from a fixed Lie algebra and ties it to the combinatorics of a directed graph. They then prove that the algebraic action of the braid group of ( Q ) on this Yangian matches the action on the equivariant two-dimensional COHA of ( Q ), with the latter described via derived reflection functors on the bounded derived category of modules over the preprojective algebra of ( Q ). The braid group—the group of ways to braid strands while keeping endpoints fixed—emerges from the mutation of quivers, a standard operation in the study of cluster algebras. This braid group action aligns naturally with Vasserot et al.'s nilpotent COHA framework (arXiv:2502.19445), and the present paper extends the verification to the affine Dynkin setting relevant to Kleinian singularities. The authors acknowledge this gap, leaving it as a compelling open direction.
What does all this mean for a physicist or a mathematician who has never encountered a COHA? It means that the algebraic undercurrents of two large disciplines—the geometry of moduli spaces of sheaves and the representation theory of quantum groups—are flowing along the same channel. The paper offers a new algebraic characterization of Hecke operators for surfaces that were previously studied only indirectly, through the combinatorics of quivers. As Sala et al. have shown for general Kleinian orbifolds (arXiv:2511.08576), the COHA framework is robust enough to handle singular surfaces, and the current work pushes that robustness to a precise identification with a Yangian. It does not, however, give explicit examples beyond the simplest case. The road to a full classification of COHAs for all ADE surfaces remains long, but the direction is now charted.
One might worry that such a result is too abstract to ever touch experiment. But Yangians and COHAs have deep roots in integrable systems and in the counting of instantons in four-dimensional gauge theories. The dictionary built here opens the door to new computational tools: one could calculate Hecke modifications by using the Yangian’s representation theory, a task that in many cases is far more tractable than direct geometric counting. It is the kind of algebraic engineering that often precedes a flood of new physical applications—much as the Yangian of the two-dimensional Ising model, discovered decades ago, later illuminated the dynamics of spin chains and quantum systems far from equilibrium.
The team’s accomplishment is a masterclass in how modern algebraic geometry can speak to representation theory, but it is also a reminder of how much remains to be understood. The continuity theorem, the braid group action for affine Dynkin quivers, the extension to surfaces beyond the projective line—these are not fatal gaps, but rather the next rungs on the ladder. The preprint (arXiv:2603.03386) does not close a chapter; it opens one. It says, in effect: here is a precise translation between the algebra of pointlike modifications on a scarred surface and the symmetries of a quantum integrable system. Now let us see how far this translation can carry us.
Perhaps the most exciting question is not what we can compute with this new dictionary, but what it teaches us about the architecture of the mathematical universe itself. Why should the counting of sheaf modifications on an algebraic surface know about the braid group and the Yangian? The answer, if we ever fully grasp it, will say something about the deep unity between geometry and algebra—a unity that we are only beginning to hear, in the mathematical music that ancient mathematicians never imagined, but that today’s explorers are learning to transcribe.
— Yanjiang
Yanjiang is the founding editor of LoomSci.com, specializing in physics and science communication.
References
- Diaconescu et al., Cohomological Hall algebras of one-dimensional sheaves on surfaces and Yangians, arXiv:2603.03386
- Schiffmann et al., Cohomological Hall algebras of quivers and Yangians, arXiv:2312.15803
- Vasserot et al., Nilpotent cohomological Hall algebras of surfaces, arXiv:2502.19445
- Sala et al., Kleinian orbifolds, Cohomological Hall Algebras, and Yangians, arXiv:2511.08576