Chiralization of Quiver Varieties

Abstract: Given a quiver Q with gauge dimension $\bf v$ and framing dimension $\bf w$, one can define the extended quiver variety $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$, which is a smooth family of deformations of the Nakajima quiver variety $\mathcal M(\mathbf v,\mathbf w)$. In this paper we discuss two vertex algebras which chiralize the geometry $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$. We construct a sheaf of $\hbar$-adic vertex superalgebras $\mathscr D^{\mathrm{ch}}_{\widetilde{\mathcal M}(\mathbf v,\mathbf w),\hbar}$ on $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$ which quantizes the jet bundle of $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$, and define a vertex algebra $\mathsf D^{\mathrm{ch}}(\widetilde{\mathcal M}(\mathbf v,\mathbf w))$ to be the $\hbar=1$ specialization of the $\mathbb C^{\times}$-finite part of the vector space of global sections $Γ(\widetilde{\mathcal M}(\mathbf v,\mathbf w), \mathscr D^{\mathrm{ch}}_{\widetilde{\mathcal M}(\mathbf v,\mathbf w),\hbar})$. We define another vertex superalgebra $\mathcal V(\mathbf v,\mathbf w)$ by BRST reduction of the tensor product of the $βγbc$-system and Heisenberg VOA associated to the quiver Q, and show that there exists a natural vertex superalgebra map from $\mathcal V(\mathbf v,\mathbf w)$ to $\mathsf D^{\mathrm{ch}}(\widetilde{\mathcal M}(\mathbf v,\mathbf w))$. Under certain technical assumptions, we prove that the negative degree BRST cohomologies of the tensor product of $βγbc$-systems and Heisenberg VOA associated to the quiver Q are zero, and under stronger assumptions, that the aforementioned vertex superalgebra map is injective. Physically, the vertex superalgebra $\mathcal V(\mathbf v,\mathbf w)$ is closely related to the boundary VOA of the H-twisted 3D $\mathcal N=4$ quiver gauge theory associated to the quiver Q with gauge and framing dimension vectors $\bf v$ and $\bf w$.

• The paper introduces a dual chiralization framework, merging sheaf-theoretic quantization with BRST reduction to construct vertex superalgebras for extended quiver varieties.
• It establishes vanishing of negative BRST cohomologies and injectivity of key morphisms under precise geometric criteria like jet-flatness and rational singularities.
• The work bridges geometric representation theory with 3D gauge theory, offering new tools for categorifying boundary VOAs and advancing quiver gauge models.

Chiralization of Quiver Varieties: An Expert Overview

Introduction and Motivation

"Chiralization of Quiver Varieties" (2606.23807) develops a rigorous and systematic framework for constructing vertex superalgebras that encode the geometric and representation-theoretic data of (extended) Nakajima quiver varieties. The work is situated at the intersection of algebraic geometry, geometric representation theory, and mathematical physics, with precise applications to the study of boundary VOAs of three-dimensional $\mathcal{N}=4$ quiver gauge theories.

The principal aim is to formalize and relate two approaches to chiralizing the geometry of these varieties:

1. Sheaf-theoretic Chiralization: Construction of a sheaf of $\hbar$-adic vertex superalgebras on the extended quiver variety, quantizing its jet bundle.
2. BRST (BRST/Hamiltonian) Reduction: Algebraic construction of a global vertex superalgebra via BRST reduction from free field systems associated to the quiver data.

The paper further develops structural criteria that ensure the vanishing of negative BRST cohomologies and injectivity of natural morphisms between these vertex superalgebras—results of essential importance for both mathematical structure and physical applications.

Construction of Chiralized Structures

Extended Quiver Varieties and Chiral Quantization

Given a quiver $Q$, with dimension vectors $\mathbf v$ (gauge) and $\mathbf w$ (framing), one studies the extended quiver variety $\widetilde{\mathcal M}(\mathbf v, \mathbf w)$, which can be regarded as a $\mathcal Z$-parametrized deformation of Nakajima's quiver variety. The crucial geometric objects are:

• The extended representation space $$\mathfrak{R} = T^*\mathrm{Rep}(Q, \mathbf{v}, \mathbf{w}) \times \mathcal{Z}$$,
• The extended moment map $\widetilde{\mu}: \mathfrak{R} \to \mathfrak{g}^*$.

Chiralization, in the sense of Arakawa et al., associates to this space a sheaf $$\mathscr{D}^{\mathrm{ch}}_{\widetilde{\mathcal M}(\mathbf v, \mathbf w), \hbar}$$ of $\hbar$0-adic vertex superalgebras, designed to quantize the jet bundle $\hbar$1. The $\hbar$2-finite part of its global sections, at $\hbar$3, defines the vertex algebra $\hbar$4.

BRST Reduction and Free-Field Realizations

A parallel, algebraic construction is given by the BRST (quantum Hamiltonian reduction) approach: Starting from the free-field vertex algebra generated by a tensor product of $\hbar$5-systems and Heisenberg VOAs for the quiver, one constructs a complex whose cohomology realizes the naive chiralization of $\hbar$6. Explicitly, one defines

$\hbar$7

where the BRST differential enforces equivariance via the quiver's moment map data.

A distinguished morphism from $\hbar$8 to $\hbar$9 is constructed via descent from global to local data.

Main Theorems: Vanishing and Embedding

The core results concern precise conditions under which:

• Negative BRST cohomology vanishes ($Q$0 for $Q$1),
• The natural morphism $Q$2 is injective,
• Analogous results hold for modules of the respective vertex algebras.

Geometric Criteria

These properties are proved under technical, yet explicit, conditions on the quiver data—namely, flatness and good singularity structure:

• Jet-flatness of the moment map (termed $Q$3): Ensures regular sequences defining the reduction at the level of jets, allowing for the cohomological vanishing.
• Rational and Reduced Singularities (termed $Q$4): When the extended moment map is flat, and its zero fiber is reduced, irreducible, and has rational singularities, the injectivity of the morphism follows.

The paper provides combinatorial and geometric criteria (dimension, stratification, slice arguments) to verify whether these properties hold for specific classes of quivers, including ADE (Dynkin), Jordan, and "totally negative" quivers.

Main Results

• Theorem: Under the above geometric hypotheses, the negative-degree BRST cohomologies of the global BRST reduction vanish.
• Theorem: Under stronger assumptions, the map $Q$5 is injective (and analogously for their modules).

These results provide chiral (vertex-algebraic) analogues to well-known statements about global section rings and their function-theoretic counterparts on quiver varieties.

Physical and Representation-Theoretic Context

From a physical viewpoint, the vertex algebra $Q$6 coincides, under the H-twist, with boundary VOAs of the corresponding three-dimensional $Q$7 quiver gauge theory. This links the geometric structure of the quiver variety with the (categorified) representation category of line defects in the 3D TQFT.

Mathematically, this framework builds a bridge between geometric representation theory (via quiver/GIT constructions and symplectic singularities), the theory of vertex (super)algebras (as organized by chiralization and BRST reduction), and the physical theory of boundary conditions and dualities in gauge theory.

Implications and Future Directions

The techniques and results of this work have several noteworthy consequences:

• They provide a blueprint for constructing quantizations of more general symplectic singularities and moduli spaces, particularly those arising in geometric representation theory and physics.
• The explicit criteria for vanishing and embedding facilitate the categorification of structures in gauge theory—enabling finer control over the representation categories relevant in both mathematics and physics.
• Methodologically, the synthesis of chiral, algebraic, and sheaf-theoretic constructions suggests possible extensions to derived and higher-categorical settings, potentially relevant to the geometric Langlands program and to logarithmic-geometric representation theory.

One can anticipate applications to the classification of "quasi-lisse" or "good" boundary vertex (super)algebras, extensions to quantizations over positive characteristic, or to the study of dualities (such as 3D mirror symmetry) via explicit chiral structures.

Conclusion

"Chiralization of Quiver Varieties" gives a comprehensive and technically robust account of how quiver geometries admit compatible, functorial chiral quantizations realized as vertex superalgebras—constructed via both geometric and BRST-theoretic means. The demonstration of vanishing and embedding criteria grounds the approach rigorously and opens several avenues for both mathematical investigation and applications in quantum field theory. The explicit treatment of jet-flatness and singularity conditions stands out for its utility in categorifying and quantizing the data underlying quiver varieties, facilitating advances across multiple related domains.