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Quasimaps to Nakajima varieties as critical loci

This paper demonstrates that the moduli space of quasimaps from $\mathbb{P}^1$ to a Nakajima quiver variety, with a fixed value at infinity, can be globally realized as the critical locus of an explicit function.

Original authors: Spencer Tamagni

Published 2026-08-27

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CC BY 4.0

Original authors: 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 landscape of modern mathematics, there is a rich field dedicated to understanding shapes that arise from symmetry and algebra. These are not the smooth curves of a coastline or the jagged peaks of a mountain range, but rather abstract spaces constructed by organizing vast collections of linear equations and vector spaces. One particularly important family of these shapes is known as Nakajima varieties. They serve as a bridge between pure algebra and geometry, appearing in contexts ranging from quantum physics to the study of how particles might interact. To navigate these spaces, mathematicians often look for ways to map one shape onto another, tracing paths that reveal the hidden structure of the destination. A specific type of path, called a quasimap, allows researchers to explore these varieties by looking at how they behave when stretched over a simple circle, much like wrapping a string around a sphere to see how it fits.

The question of how to describe the collection of all such paths has long been a challenge. While the destination shapes are well-understood, the space containing all possible paths leading to them is often incredibly complex and difficult to pin down with a single, clear definition. In a recent note, Spencer Tamagni from the Leinweber Institute for Theoretical Physics at the University of California, Berkeley, offers a fresh perspective on this problem. He demonstrates that the entire space of these paths can be described as the critical points of a specific function. In simpler terms, just as a ball rolling on a hilly landscape will eventually come to rest at the bottom of a valley or the top of a peak, the mathematical objects representing these paths are exactly those that sit at the "bottom" of a carefully constructed mathematical surface. This discovery provides a concrete, global way to visualize and work with these abstract spaces, turning a difficult geometric problem into a question of finding the lowest points of a function.

Tamagni's work focuses on a specific scenario where the destination is a Nakajima variety, and the paths start from a fixed point on a circle and end at a chosen location on the variety. He constructs a new mathematical object, which he calls a quiver moduli space, to represent these paths. A quiver is essentially a diagram made of dots and arrows, where the dots represent vector spaces and the arrows represent linear maps between them. By adding extra layers to this diagram and defining a specific function, or potential, on the data associated with it, Tamagni shows that the solutions to the equations defining the critical points of this function are exactly the same as the space of quasimaps. This means that instead of wrestling with the abstract definition of the paths directly, one can study the simpler, more tangible data of the quiver and its associated function.

The construction is remarkably precise. It involves taking the data that defines a point on the destination variety and treating it as a fixed background. Then, one introduces new variables that represent the "winding" or degree of the path. These variables are organized into a larger quiver diagram. Tamagni defines a function on this diagram that depends on the fixed background point. He proves that if you look for the points where the slope of this function is zero—where the function is flat in every direction—you find exactly the space of quasimaps. This result is not just a loose connection; it is a rigorous isomorphism, meaning the two mathematical descriptions are identical in every structural detail. The paper establishes that the space of paths is not just similar to the critical locus of this function, but is precisely that locus.

One of the most significant aspects of this finding is the nature of the symmetry involved in the construction. To define the space of paths, one must account for the fact that many different mathematical descriptions can represent the same physical path. This usually involves dividing out by a group of symmetries. In this case, the group is a mix of standard linear transformations and a more complex, non-reductive part. Tamagni shows that despite the complexity of this group, the critical locus description remains valid and explicit. He also notes that while it is possible to simplify the description by fixing a specific gauge, or choice of coordinates, doing so requires making arbitrary choices that depend on the specific point where the path ends. The beauty of his primary construction is that it avoids these arbitrary choices, offering a description that is canonical and works uniformly for any chosen endpoint.

The implications of this work extend beyond the immediate definition of the space. By presenting the space of quasimaps as a critical locus, the paper reveals that this space possesses a natural geometric structure known as a shifted symplectic structure. This is a sophisticated type of geometry that has become central to modern research in mathematical physics and representation theory. It suggests that the space of paths has an intrinsic "volume" and geometric properties that can be studied using tools from derived algebraic geometry. The paper does not merely list these properties but provides the explicit machinery to describe them, laying the groundwork for future investigations into how these spaces interact with other areas of mathematics.

To illustrate the power of this approach, the author examines a special case where the destination variety is the Hilbert scheme of points on a plane, a space that describes configurations of points in two dimensions. In this specific instance, the general construction simplifies to a known problem involving a different type of quiver, often called a handsaw quiver. The paper shows that the new method recovers the established description of this space, confirming its validity. More importantly, it demonstrates that the general method works for all Nakajima varieties, not just this special case. The ability to translate the complex geometry of quasimaps into the language of quivers and potentials opens up new avenues for calculation and understanding.

The paper concludes by emphasizing that this construction is a direct, explicit proof. The author builds maps in both directions between the space of paths and the critical locus of the function and shows that they are perfect inverses of each other. This means that every path corresponds to a unique critical point, and every critical point corresponds to a unique path. There is no ambiguity or approximation in this relationship. The work stands as a solid, canonical bridge between two different ways of thinking about these mathematical objects. By grounding the abstract concept of a quasimap in the concrete reality of a function's critical points, Tamagni provides a new lens through which to view the intricate geometry of Nakajima varieties, offering clarity to a field that often deals with high levels of abstraction.

Technical Summary: Quasimaps to Nakajima varieties as critical loci

Problem Statement
The paper addresses the problem of providing an explicit, canonical construction of the moduli space of quasimaps from the projective line $\mathbb{P}^1$ to a Nakajima quiver variety $X$ . Specifically, it focuses on the subspace of quasimaps $f: \mathbb{P}^1 \to X$ that send a distinguished point $\infty \in \mathbb{P}^1$ to a fixed point $p \in X$ . While the moduli space of quasimaps is well-defined, the author seeks a description of this space as the critical locus of an explicit potential function, analogous to the ADHM construction for instanton moduli spaces or the identification of handsaw quiver varieties with quasimaps to flag manifolds.

Methodology
The author employs a two-step construction involving the interplay between quasimap data and quiver representations, utilizing the Beilinson resolution of the diagonal on $\mathbb{P}^1$ and cohomological base change.

From Quasimaps to Quiver Representations:
Given a family of quasimaps over a base scheme $B$ , the author defines vector bundles $U_i$ and $F_i$ on $B$ via derived pushforwards of the universal bundles $V_i$ on $B \times \mathbb{P}^1$ :
- $U_i = R^1 p_* (V_i(-1))$
- $F_i = R^1 p_* (V_i(-2))$
  Using the coordinate $z$ on $\mathbb{P}^1$ (viewed as a section of $\mathcal{O}(1)$ ), the author constructs a morphism $\zeta_i: F_i \to U_i$ and identifies $V_i$ as the kernel of the map $\zeta_i - z\pi_i$ , where $\pi_i$ is the projection. The sections of the quasimap ( $B_2, B_3, I, J$ ) induce morphisms between these new bundles. The stability of the quasimap at the generic point ensures that these constructions yield well-defined vector bundles and morphisms satisfying specific commutative diagrams.
From Quiver Representations to Quasimaps:
Conversely, starting with a quiver representation in a specific enlarged space $R(v, w; d)$ , the author defines the bundle $V_i$ on $B \times \mathbb{P}^1$ as the kernel of a matrix involving the quiver data ( $B_1, I$ ) and the coordinate $z$ . The critical locus condition ( $dW_p = 0$ ) for a specific potential function $W_p$ ensures that the induced morphisms ( $B_2, B_3, I, J$ ) are well-defined on the sub-bundle $V_i$ and satisfy the moment map equations required for quasimaps.
The Potential Function:
The core of the construction is the explicit potential function $W_p$ defined on the space of representations of an enlarged quiver. This quiver includes the original vertices and edges, plus new vertices $U_i$ (representing the degree) and specific arrows connecting them to the original framing and vertex spaces. The function is given by:
$W_p = \text{tr}(B_1[B_2, B_3]) + \text{tr}(J_2(B_2I - I\xi_2)) + \text{tr}(J_3(B_3I - I\xi_3)) + \text{tr}(I\iota\alpha)$
Here, $(\xi_2, \xi_3, \iota, \jmath)$ are the fixed quiver data corresponding to the point $p$ .

Key Results
The main result is Theorem 2.1, which establishes an isomorphism of algebraic stacks:
$QM^p_d(X) \simeq MX(p, d)$
where:

$QM^p_d(X)$ is the moduli space of stable quasimaps of degree $d$ from $\mathbb{P}^1$ to $X$ with $f(\infty) = p$ .
$MX(p, d)$ is the critical locus of the potential $W_p$ on the quotient stack $[R(v, w; d)^{st} / (GL(U) \ltimes \text{Hom}(U, V))]$ .

The paper proves this by constructing explicit inverse morphisms $\Psi$ and $\Phi$ between the two spaces and verifying that they satisfy the necessary stability and commutativity conditions.

Specific Contributions and Examples

Explicit Critical Locus Description: The paper provides a concrete realization of quasimap spaces as critical loci, a feature that endows these spaces with a natural derived enhancement and a $(-1)$ -shifted symplectic structure (in the sense of [PTVV13]).
Hilbert Scheme Case: For the specific case $X = \text{Hilb}^n(\mathbb{C}^2)$ , the construction yields an explicit critical locus description of a Pandharipande-Thomas moduli space of stable pairs.
Recovery of Handsaw Varieties: In Section 3.3.1, the author demonstrates that specializing the construction to the cotangent bundle of partial flag varieties ( $X = T^*(GL_n/P)$ ) recovers the "handsaw quiver varieties" described by Nakajima [Nak11]. This is achieved by gauge-fixing the non-reductive part of the quotient group and solving specific equations in the critical locus to eliminate redundant arrows.

Significance and Claims
The author claims that this construction is "reasonably canonical" and presents the space of quasimaps as the critical locus of an explicit function. The primary significance lies in the ability to describe the derived structure of quasimap spaces explicitly, which is detailed in a companion paper [BT26].

The paper acknowledges that the quotient by the non-reductive group $GL(U) \ltimes \text{Hom}(U, V)$ is a somewhat nonstandard feature in the context of quiver representations. However, it notes that this can be traded for a quotient by $GL(U)$ alone at the cost of non-canonical choices (gauge fixing), as illustrated in the partial flag example. The work is presented as a short note establishing the isomorphism and the explicit construction, with further implications for geometric representation theory to be explored in future joint work.

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