Extended Quiver Varieties

• Extended quiver varieties are holomorphic symplectic moduli spaces that extend Nakajima’s construction by incorporating vertex multiplicities, central deformations, and marked structures.
• They employ Hamiltonian reduction, complex moment maps, and stability conditions to model singular local behavior and to realize non-symmetric Kac–Moody symmetries.
• These varieties have practical applications in classifying Painlevé equations, studying meromorphic connections, and constructing gauge-theoretic moduli on Kähler manifolds.

An extended quiver variety is a holomorphic symplectic or hyper-Kähler moduli space obtained by enlarging the standard Nakajima quiver-variety construction. In the literature, the term appears in several closely related senses: quiver varieties with multiplicities attached to vertices, deformation families $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$ obtained by allowing the complex moment map to take values in the center, marked quiver-like varieties produced by Legendre transforms, and bundle-valued globalizations on curves or Kähler manifolds. This suggests that “extended quiver variety” functions as an umbrella term for extensions of the Nakajima framework rather than a single universally fixed definition (Yamakawa, 2010, Coman et al., 22 Jun 2026, Rimanyi et al., 2021, Jeffrey et al., 2024).

1. Terminology and basic scope

The shared core of these constructions is a quotient of representation-theoretic data by a gauge group, controlled by real and/or complex moment maps and stability conditions. What varies is the form of the input data: vertex multiplicities, central deformation parameters, marked half-edges, or vector bundles over a base curve.

Usage	Defining feature	Source
Quiver variety with multiplicities	Positive integers $d_i$ attached to vertices	(Yamakawa, 2010)
Extended Nakajima quiver variety $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$	Stable quotient of $\mu_{\mathbb C}^{-1}(Z)$	(Coman et al., 22 Jun 2026)
Extended quiver-like variety	Marked quiver with partially Legendre-transformed half-edges	(Rimanyi et al., 2021)
Bundle-valued/globalized extension	Quotient of doubled-quiver bundle data on $X$	(Jeffrey et al., 2024)

All of these generalizations retain the structural role of doubled quivers, Hamiltonian reduction, and symplectic geometry. In several cases they also preserve the links to Kac–Moody theory, integrable systems, and gauge theory.

2. Quiver varieties with multiplicities

A foundational extension was introduced by Yamakawa. One starts with a finite quiver $Q=(I,\Omega,s,t)$ with no loops, a multiplicity vector

$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$

a nonzero dimension vector

$$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$

and parameters

$$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$

For complex vector spaces $V_i$ of dimension $d_i$0, one sets

$d_i$1

and forms the cotangent bundle of the corresponding representation space built from

$d_i$2

The acting group is

$d_i$3

This space carries a canonical holomorphic symplectic form

$d_i$4

where $d_i$5 is the double quiver and $d_i$6 on $d_i$7, $d_i$8 on $d_i$9. Its moment map

$\widetilde{\mathcal M}(\mathbf v,\mathbf w)$0

has components

$\widetilde{\mathcal M}(\mathbf v,\mathbf w)$1

Stability is defined by requiring that the only proper $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$2-submodules of $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$3 preserved by all $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$4 be zero. The quiver variety with multiplicities is then

$\widetilde{\mathcal M}(\mathbf v,\mathbf w)$5

When all multiplicities satisfy $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$6, the construction reduces to the stable locus of Nakajima’s quiver variety $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$7 with $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$8 (Yamakawa, 2010).

This multiplicity formalism is significant because it inserts truncated local algebra data directly at the vertices. The resulting spaces are still holomorphic symplectic quotients, but they model more singular local behavior than ordinary linear quiver representations.

3. Symplectic reduction, reflection functors, and Kac–Moody symmetry

In the multiplicity setting, the quotient inherits a unique holomorphic symplectic form whenever the group action is free on the stable locus: $\widetilde{\mathcal M}(\mathbf v,\mathbf w)$9 This places the construction squarely inside complex symplectic reduction, even though the coefficient rings $\mu_{\mathbb C}^{-1}(Z)$0 differ from the classical case (Yamakawa, 2010).

The same framework generalizes Nakajima’s reflection functors. If $\mu_{\mathbb C}^{-1}(Z)$1 denotes the adjacency matrix of the underlying graph and $\mu_{\mathbb C}^{-1}(Z)$2, then

$\mu_{\mathbb C}^{-1}(Z)$3

is a symmetrizable, possibly non-symmetric generalized Cartan matrix. One therefore obtains a Kac–Moody algebra $\mu_{\mathbb C}^{-1}(Z)$4 with simple roots $\mu_{\mathbb C}^{-1}(Z)$5. The corresponding simple reflections act on dimension vectors and parameters by

$\mu_{\mathbb C}^{-1}(Z)$6

and

$\mu_{\mathbb C}^{-1}(Z)$7

Whenever $\mu_{\mathbb C}^{-1}(Z)$8, Theorem 4.1 gives a natural symplectic isomorphism

$\mu_{\mathbb C}^{-1}(Z)$9

These maps satisfy $X$0, and the full Coxeter relations are stated conjecturally in the source (Yamakawa, 2010).

A key conceptual point is that multiplicities lead naturally to Weyl groups of symmetrizable non-symmetric Kac–Moody algebras, not only the symmetric Cartan data familiar from ordinary Nakajima varieties. This enlarges the symmetry class of quiver-variety constructions without abandoning their reflection-functor mechanism.

4. Meromorphic connections and the Painlevé correspondence

Yamakawa’s construction identifies certain moduli spaces of meromorphic connections with quiver varieties with multiplicities. For a star-shaped quiver of length $X$1, with $X$2, $X$3, $X$4, and $X$5, one considers systems on the rank $X$6 trivial bundle over $X$7: $X$8 After fixing coadjoint orbits $X$9 containing residue-free normal forms

$Q=(I,\Omega,s,t)$0

Proposition 6.1 gives a symplectic bijection

$Q=(I,\Omega,s,t)$1

via

$Q=(I,\Omega,s,t)$2

Moreover, stability of $Q=(I,\Omega,s,t)$3 is equivalent to irreducibility of the connection (Yamakawa, 2010).

The same paper derives a rank-$Q=(I,\Omega,s,t)$4 Painlevé classification from the dimension formula

$Q=(I,\Omega,s,t)$5

In rank $Q=(I,\Omega,s,t)$6,

$Q=(I,\Omega,s,t)$7

Up to permutation, the possible tuples are as follows.

$Q=(I,\Omega,s,t)$8	Kac–Moody type
$Q=(I,\Omega,s,t)$9	$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$0
$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$1	$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$2
$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$3	$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$4
$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$5	$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$6
$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$7	$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$8

A normalization or “shifting trick” shows that these extended Dynkin types are symplectomorphic to the untwisted affine types

$d=(d_i)_{i\in I}\in \mathbb Z_{>0}^I,$9

which are exactly Okamoto’s symmetry groups for $$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$0 through $$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$1. In this sense, the quiver-with-multiplicities formalism reproduces the Painlevé II–VI symmetry classification, with a list of Dynkin diagrams that is “slightly different from (but equivalent to) Okamoto’s” (Yamakawa, 2010).

5. Central deformations and chiralization of Nakajima varieties

A second major usage of “extended quiver variety” concerns central deformations of framed Nakajima varieties. For a finite quiver $$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$2 with gauge and framing dimension vectors

$$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$3

one defines

$$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$4

with cotangent coordinates $$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$5. The group

$$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$6

acts by change of basis on each $$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$7. The complex moment map is

$$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$8

and if

$$v=(v_i)_{i\in I}\in\mathbb Z_{\ge 0}^I\setminus\{0\},$$9

then the affine reductions are

$$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$0

while the stable quotients are

$$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$1

Under suitable numerical conditions on $$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$2, including $$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$3 and further Crawley–Boevey inequalities, both $$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$4 and $$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$5 are smooth, of complex dimension

$$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$6

Equivalently, $$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$7 is the family of all Nakajima varieties deformed by $$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$8 (Coman et al., 22 Jun 2026).

This deformation family admits a chiralization. Under flatness and rational singularity hypotheses, including property $$\lambda=(\lambda_i)_{i\in I}\in\bigoplus_{i\in I} z^{-d_i}\mathbb C[z]/\mathbb C[z], \qquad \lambda_i(z)=\sum_{k=1}^{d_i}\lambda_{i,k}z^{-k}.$$9, the extended shell $V_i$0 is reduced, irreducible, and has rational singularities, while all jet schemes $V_i$1 remain reduced and irreducible. These conditions imply vanishing of higher BRST cohomology in negative degree and injectivity of natural maps from classical functions on jets to global sections on $V_i$2. One then constructs a sheaf of $V_i$3-adic vertex superalgebras

$V_i$4

on $V_i$5, together with the global vertex algebra

$V_i$6

A second vertex superalgebra,

$V_i$7

is defined by BRST reduction of the tensor product of the $V_i$8-system and a Heisenberg VOA associated to $V_i$9. Under stronger assumptions, there is a canonical injective VOA-homomorphism

$d_i$00

Physically, $d_i$01 is closely related to the boundary VOA of the $d_i$02-twisted $d_i$03 quiver gauge theory with gauge and framing dimensions $d_i$04 (Coman et al., 22 Jun 2026).

6. Extended Dynkin singularities and hyper-Kähler cobordisms

A different but adjacent use of “extended” concerns quiver varieties over extended Dynkin quivers. Let $d_i$05 have underlying unoriented graph of type $d_i$06, $d_i$07, or $d_i$08, and let $d_i$09 be the unique minimal positive imaginary root spanning the radical of the symmetric form, so that

$d_i$10

For parameters $d_i$11 satisfying

$d_i$12

the unframed Nakajima variety

$d_i$13

is nonempty, and for generic $d_i$14 it is smooth of dimension four (Helle, 2022).

When $d_i$15, one has the affine GIT identification

$d_i$16

Its singularities are classified by the root-theoretic decomposition

$d_i$17

where $d_i$18 is the ADE root system of the finite part and $d_i$19. Each singular point $d_i$20 has a neighborhood analytically isomorphic to the Kleinian surface singularity

$d_i$21

with $d_i$22 the corresponding binary polyhedral group. Varying $d_i$23 yields hyper-Kähler resolutions

$d_i$24

and suitable parameter choices produce a connected $d_i$25-manifold with cylindrical ends modeled on

$d_i$26

This gives explicit hyper-Kähler bordisms between $d_i$27 and the disjoint union $d_i$28 (Helle, 2022).

These results do not define an “extended quiver variety” in the same sense as the central-deformation family $d_i$29, but they clarify how extended Dynkin input controls the singularity theory and asymptotic geometry of quiver-variety quotients.

7. Other extensions: Legendre-transformed quiver-like varieties and quiver bundles

Another generalization replaces Hamiltonian reduction by intersections of generalized Lagrangian subvarieties. For a finite quiver with vertices carrying Lie algebras $d_i$30, each edge $d_i$31 is assigned a symplectic variety $d_i$32 with moment maps to the adjacent Lie algebras. The quiver variety can then be realized as

$d_i$33

where $d_i$34 is the generalized Lagrangian determined by the edge potential $d_i$35. If some half-edges are marked and replaced by one-sided Legendre transforms, the resulting marked quiver $d_i$36 defines

$d_i$37

called an extended quiver-like variety. With no marks one recovers the ordinary Nakajima–Cherkis bow variety, while marking all half-edges gives a different but isomorphic presentation. For $d_i$38, special cases produce $d_i$39 and three related vector bundles over $d_i$40, together with super stable envelopes, super Tarasov–Varchenko weight functions, and geometric $d_i$41-matrices matching the Yangian $d_i$42-matrices of $d_i$43, $d_i$44, and $d_i$45 (Rimanyi et al., 2021).

A further extension globalizes the construction to bundle-valued data on a compact Kähler manifold $d_i$46, especially a Riemann surface. A Nakajima bundle representation of the doubled quiver $d_i$47 assigns to each vertex $d_i$48 a Hermitian vector bundle $d_i$49, a unitary connection $d_i$50, and a Higgs field $d_i$51, and to each arrow $d_i$52 a section

$d_i$53

and a cotangent section

$d_i$54

The real and complex moment maps are

$d_i$55

$d_i$56

Fixing central levels $d_i$57, the extended quiver variety is

$d_i$58

When $d_i$59 is a point, this reduces to the usual Nakajima quiver variety; when $d_i$60 is trivial, it recovers the moduli of Higgs bundles. The corresponding Hitchin–Kobayashi theorem identifies polystability with solvability of the moment-map equations, and the Zariski tangent space at a smooth point is computed by a hypercohomology complex, with expected dimension

$d_i$61

The one-vertex no-arrow case gives Hitchin’s equations, while a bundle-valued ADHM quiver yields a curve-valued version of the framed-instanton moduli space (Jeffrey et al., 2024).

Taken together, these constructions show that extended quiver varieties form a broad research program rather than a single object. Their common role is to preserve the quiver-variety mechanism—moment maps, stability, and quotient geometry—while enlarging the class of moduli spaces to include irregular connections, non-symmetric Kac–Moody symmetries, central deformation families, superalgebraic geometrizations, and bundle-valued gauge-theoretic moduli.