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Quasimap critical cohomology, Coulomb branches, and quantum groups

This paper establishes a systematic framework for the critical cohomology of quasimap moduli spaces to Nakajima quiver varieties by presenting them as global critical loci, thereby constructing compatible actions of shifted Yangians and quantized Coulomb branches and proposing a geometric morphism between their associated algebras.

Original authors: Tommaso Maria Botta, Spencer Tamagni

Published 2026-09-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Tommaso Maria Botta, Spencer Tamagni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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

In the vast landscape of modern mathematics, there is a field dedicated to understanding the shapes formed by solutions to complex equations. These shapes, often called moduli spaces, are not static; they are dynamic collections of all possible configurations a system can take. Imagine trying to map every possible way a flexible wire could be bent and twisted while holding its ends in place; the collection of all those shapes forms a single, intricate geometric object. For decades, mathematicians have sought to understand the hidden symmetries within these objects, much like a physicist looks for the fundamental forces that govern the universe. A central tool in this quest is the study of "quiver varieties," which are specific types of these shape spaces built from networks of points and arrows. These varieties are deeply connected to quantum physics, particularly to theories describing how particles interact in three dimensions. The challenge has long been to find a clear, unified way to describe the "critical cohomology" of these spaces—a sophisticated way of counting and measuring the holes and twists in these shapes that reveals their deepest algebraic secrets.

Two researchers, Tommaso Maria Botta and Spencer Tamagni, have recently taken a significant step forward in this endeavor. They focused on a specific type of shape space known as a "quasimap," which describes how a simple curve, like a circle, can be mapped onto a complex quiver variety. Their goal was to understand the symmetries acting on these maps. To do this, they developed two completely different but equivalent ways of looking at the same problem. The first approach treats the problem as a network of linear algebra equations, while the second views it through the lens of infinite-dimensional bundles and local patches. By proving that these two perspectives are mathematically identical, the authors were able to unlock a powerful new understanding of the symmetries at play. They demonstrated that the cohomology of these quasimap spaces acts as a stage for two major algebraic structures: a "shifted Yangian" and a "quantized Coulomb branch." These are complex algebraic systems that mathematicians use to describe quantum integrable systems, essentially the mathematical rules that govern how certain quantum systems evolve without chaos.

The core of their discovery lies in how they modeled these spaces. The researchers showed that the space of quasimaps can be viewed as the "critical locus" of a specific function. In simpler terms, if you imagine a landscape with hills and valleys, the critical locus is the set of points where the ground is perfectly flat—the peaks, the valleys, and the saddle points. The authors proved that the complex space of quasimaps is exactly this set of flat points for a function defined on a much larger, more manageable space. This insight allowed them to apply a powerful mathematical tool called "vanishing cycles," which essentially counts the topological features of these flat points. Using this method, they constructed a canonical "sheaf," which is a mathematical object that organizes data across the space, effectively creating a blueprint for the cohomology they wanted to study.

With this blueprint in hand, they were able to show that the space of quasimaps is acted upon by a "shifted Yangian." This is a vast algebraic structure that generalizes the concept of symmetry found in quantum mechanics. They proved that this algebra acts on the cohomology of the quasimap space in a very specific, canonical way. Simultaneously, they showed that the same space is acted upon by a "quantized Coulomb branch." In the language of physics, the Coulomb branch is a space that describes the possible vacuum states of a gauge theory, and its quantized version is an algebra that encodes the quantum properties of that theory. The authors constructed a natural action of this algebra on the same cohomology space, effectively showing that the same geometric object can be understood through two different, yet compatible, algebraic lenses.

Perhaps the most profound part of their work is the connection they established between these two seemingly different algebraic actions. They proposed a conjecture that the action of the shifted Yangian and the action of the Coulomb branch are not just compatible, but are essentially the same thing viewed through different mathematical windows. They proved this conjecture in a specific case where the base point of the map lies on a special "polarized" sub-variety, a condition that simplifies the geometry enough to allow for a direct comparison. In this scenario, they showed that the two actions commute and align perfectly. While the general case remains a conjecture, the authors reduced the problem to a statement about sheaves that could potentially be resolved using future developments in the field. Their work provides a rigorous geometric foundation for a long-standing idea in mathematical physics, suggesting that the algebraic structures governing quantum systems are deeply rooted in the geometry of these mapping spaces.

The researchers also clarified the relationship between their findings and previous work. They showed that their construction of the Coulomb branch via vanishing cycles is equivalent to the standard definition used by physicists and mathematicians, but their method offers a new, more direct route to understanding its action on quasimaps. They further demonstrated that the "Hall algebra," a structure built from the geometry of the space itself, maps naturally into the Coulomb branch algebra. This mapping is surjective, meaning the Coulomb branch is essentially a quotient of the Hall algebra, confirming a deep structural link between the combinatorics of the quiver and the physics of the Coulomb branch.

In essence, Botta and Tamagni have provided a new map for navigating the complex terrain of geometric representation theory. By showing that two different critical models of the same space yield the same cohomological results, they have unified two distinct approaches to studying quantum symmetries. Their work confirms that the "shifted Yangian" and the "Coulomb branch" are not just abstract algebraic curiosities but are intrinsic properties of the geometry of quasimaps. This unification offers a clearer path for future research, potentially allowing mathematicians to use the tools of one algebraic system to solve problems in the other. The paper stands as a testament to the power of viewing a single mathematical object from multiple angles, revealing that the underlying symmetries of the universe are often more interconnected than they first appear.

Technical Summary: Quasimap Critical Cohomology, Coulomb Branches, and Quantum Groups

Problem Statement
This paper investigates the geometric representation theory of the moduli space of based quasimaps, denoted $QM_\xi(\mathbb{P}^1, X)$ , from the projective line $\mathbb{P}^1$ to a Nakajima quiver variety $X$ . The central object of study is the Donaldson-Thomas (DT) cohomology of this space, defined as the derived global sections of the canonical DT sheaf $\phi_{QM_\xi(\mathbb{P}^1, X)}$ . This sheaf arises from the intrinsic $(-1)$ -shifted symplectic structure of the moduli space.

The paper addresses two primary challenges:

Explicit Realization: The intrinsic definition of the DT sheaf is non-constructive (relying on gluing local Darboux charts). The authors seek explicit, computable models for this sheaf and its cohomology.
Symmetry Actions: The paper aims to construct and relate actions of two distinct classes of algebras on this DT cohomology:
- Shifted Yangians: Arising from the theory of Cohomological Hall Algebras (CoHA) and Hecke modifications.
- Quantized Coulomb Branches: Defined by Braverman, Finkelberg, and Nakajima (BFN) as the equivariant Borel-Moore homology of a space of triples parametrizing Hecke modifications of bundles with sections.

A longstanding conjecture posits that the quantized Coulomb branch algebra acts on the equivariant cohomology of quasimap spaces. This paper seeks to refine and prove this conjecture within the framework of critical cohomology (vanishing cycles) rather than standard Borel-Moore homology.

Methodology
The authors employ Derived Algebraic Geometry (DAG) to exhibit $QM_\xi(\mathbb{P}^1, X)$ as a global derived critical locus in two distinct ways. These "critical models" allow the DT sheaf to be realized explicitly as a vanishing cycle functor applied to a constant sheaf on a smooth ambient space.

The Quiver Model (Part 1):
- Building on previous work [85], the authors present $QM_\xi(\mathbb{P}^1, X)$ as the critical locus of a potential function $W$ on a moduli space of representations of a specific framed quiver with potential $(Q_{QM}, W)$ .
- This model utilizes the Beilinson resolution to trade vector bundles for vector spaces and sections for linear maps.
- The authors utilize the theory of Cohomological Hall Algebras (CoHA) associated with the tripled quiver $\tilde{Q}$ and its cubic potential $\tilde{W}$ . They construct actions of the nilpotent and $\omega$ -nilpotent subalgebras of the CoHA via Hecke correspondences.
- They extend this to an action of the shifted Yangian $Y^T_\mu(\tilde{Q}, \tilde{W})$ by verifying the defining relations of the Yangian using intersection-theoretic arguments on the ambient quiver moduli space.
The Čech Model (Part 2):
- The authors lift the uniformization theorem for $Bun_G(\mathbb{P}^1)$ to the setting of quasimaps. They present $QM_\xi(\mathbb{P}^1, X)$ as the critical locus of a function $W_\xi$ on an infinite-rank (Tate) bundle over the stack of $G$ -bundles on $\mathbb{P}^1$ with a trivialization at infinity.
- This construction is compatible with the critical description of the BFN Coulomb branch. The authors define an alternative construction of the Coulomb branch via vanishing cycles on a space of triples $\tilde{R}$ , establishing an isomorphism between the critical cohomology of $\tilde{R}$ and the standard BFN Coulomb branch algebra.
- Using this compatibility, they construct a natural convolution algebra action of the quantized Coulomb branch on the DT cohomology of the quasimaps.
Compatibility and Comparison (Part 3):
- The paper constructs a canonical morphism $\theta$ from the $\omega$ -nilpotent Hall algebra (related to the Yangian) to the positive Coulomb branch algebra.
- The authors formulate Conjecture B, stating that the action of the shifted Yangian (via the Hall algebra) and the action of the Coulomb branch are compatible via this morphism $\theta$ .
- They prove this conjecture under the geometric assumption that the base point $\xi \in X$ is polarized (lying in a specific Lagrangian subvariety). In this case, dimensional reduction allows the critical cohomology to be identified with Borel-Moore homology, where the comparison becomes tractable.

Key Contributions and Results

Theorem A: Establishes that the intrinsic $(-1)$ -shifted symplectic structure on $QM_\xi(\mathbb{P}^1, X)$ agrees with the canonical structure on the derived critical locus of the quiver potential in the Quiver Model. This provides a canonical isomorphism of sheaves, realizing the DT sheaf as a vanishing cycle sheaf.
Theorem B: Proves the existence of a canonical action of the cohomological Hall algebra $H^T_{\tilde{Q}, \tilde{W}}$ on the DT cohomology. Furthermore, this action extends to an action of the shifted Yangian $Y^T_\mu(\tilde{Q}, \tilde{W})$ , where the shift $\mu$ is determined by the dimension and framing vectors of $X$ .
Theorem C: Provides an alternative definition of the BFN Coulomb branch via vanishing cycles on a space of triples $\tilde{R}$ . It establishes a dimensional reduction isomorphism between the critical cohomology of $\tilde{R}$ and the standard Borel-Moore homology of the BFN space $R$ , compatible with algebra structures and quantization.
Theorem D: Establishes the equivalence of the intrinsic symplectic structure with the critical structure in the Čech Model, providing a second global critical locus presentation for the DT sheaf.
Theorem E: Proves that the DT cohomology of $QM_\xi(\mathbb{P}^1, X)$ carries a natural action of the quantized, flavor-deformed BFN Coulomb branch algebra. This constitutes an affirmative answer to the conjecture regarding the action of the Coulomb branch on quasimap cohomology.
Theorem F: Constructs a canonical morphism from the $\omega$ -nilpotent Hall algebra to the Coulomb branch algebra, which extends to a surjective morphism from the shifted Yangian after inverting the equivariant parameters.
Theorem G: Proves Conjecture B (the compatibility of the Yangian and Coulomb actions) in the specific case where the evaluation point $\xi$ is polarized. The proof relies on dimensional reduction to Borel-Moore homology.

Significance and Claims
The paper claims to initiate the systematic study of the geometric representation theory of DT cohomology for quasimap spaces. Its significance lies in:

Unification: It unifies two seemingly distinct approaches to the representation theory of quiver varieties: the Hall algebra/Yangian approach (via Hecke modifications) and the Coulomb branch approach (via gauge theory/convolution algebras).
Explicit Models: It resolves the "gluing problem" of the DT sheaf by providing two explicit global critical locus models (Quiver and Čech), making the sheaf computationally accessible.
Proof of Conjecture: It provides a rigorous proof of the action of the quantized Coulomb branch on quasimap cohomology, refining previous conjectures by placing them in the context of critical cohomology.
Geometric Realization of Morphisms: It offers a purely geometric construction of the morphism between Hall algebras and Coulomb branches, contrasting with combinatorial approaches (e.g., shuffle algebras) found in related recent works.
Reduction to Known Cases: The paper notes that in the specific case where $X$ is the cotangent bundle to a partial flag variety, their results recover known theorems regarding the action of Yangians and Coulomb branches, but via a completely different proof mechanism involving derived critical loci.

The authors acknowledge that the full compatibility of the Yangian and Coulomb actions (Conjecture B) for general $\xi$ remains open, suggesting that a sheaf-theoretic approach involving "Joyce conjectures" or intrinsic stable envelopes for $(-1)$ -shifted symplectic schemes may be required for a complete resolution.

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