Nakajima quiver bundles

Lisa Jeffrey; Matthew Koban; Steven Rayan

arxiv: 2412.05457 · v1 · submitted 2024-12-06 · 🧮 math.AG · math-ph· math.DG· math.MP· math.RT· math.SG

Nakajima quiver bundles

Lisa Jeffrey , Matthew Koban , Steven Rayan This is my paper

Pith reviewed 2026-05-23 07:31 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.DGmath.MPmath.RTmath.SG

keywords Nakajima quiver representationsquiver bundlesHitchin-Kobayashi correspondencemoment mapsmoduli spacesgauge theoryalgebraic curvesRiemann surfaces

0 comments

The pith

Nakajima bundle representations assign vector bundles to quiver nodes on curves and use moment-map equations to build generalized moduli spaces.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines Nakajima bundle representations by assigning complex vector bundles to nodes of a doubled quiver and sections with connections to the edges, primarily on algebraic curves or Riemann surfaces. These objects satisfy gauge-theoretic equations modeled on the ADHM equations, which permits construction of generalized quiver varieties through moment-map reduction. The work proves a Hitchin-Kobayashi correspondence that identifies the representations with stable quiver bundles, examines deformation theory and torus actions on the resulting moduli spaces, and supplies examples that recover both known and new moduli spaces.

Core claim

Nakajima bundle representations are defined via assignments of vector bundles to quiver nodes together with sections and connections on associated twisted bundles. They admit gauge-theoretic characterizations via equations analogous to the ADHM equations, enabling the construction of generalized quiver varieties by moment-map reduction. A Hitchin-Kobayashi correspondence is established between such representations and stable quiver bundles.

What carries the argument

The Nakajima bundle representation, which augments ordinary quiver representations by replacing vector spaces with vector bundles and adding connections on a base curve, supplies the data that enters the moment-map equations and yields the moduli varieties.

If this is right

Generalized quiver varieties arise as quotients by moment-map reduction, recovering the original Nakajima varieties when the bundles are trivial.
The deformation theory of the representations determines the tangent spaces to the moduli spaces of stable quiver bundles.
Natural torus actions on the moduli varieties are induced by the quiver data and can be used to study fixed loci.
Concrete examples produce both previously known moduli spaces and new ones as special cases of the construction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same moment-map formalism could be tested on families of curves to see whether the resulting total spaces remain hyperkähler in appropriate cases.
The correspondence supplies a potential dictionary between stability parameters for quiver bundles and the parameters entering the ADHM-type equations.
Examples that recover new moduli spaces indicate that the framework can generate previously unstudied compactifications or resolutions in algebraic geometry.

Load-bearing premise

The constructions and correspondences assume that the base space is an algebraic curve or Riemann surface so that the stability conditions and moment maps are well-defined and the reduction procedure works as stated.

What would settle it

An explicit computation for a concrete quiver and curve in which the zero set of the moment maps fails to coincide with the moduli space of stable quiver bundles would show the claimed correspondence does not hold.

read the original abstract

We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold $X$, such a representation involves an assignment of a complex vector bundle on $X$ to each node of the doubled quiver; to the edges, we assign sections of, and connections on, associated twisted bundles. We for the most part restrict attention in our development to algebraic curves or Riemann surfaces. Our construction simultaneously generalizes ordinary Nakajima quiver representations on the one hand and quiver bundles on the other hand. These representations admit gauge-theoretic characterizations, analogous to the ADHM equations in the original work of Nakajima, allowing for the construction of these generalized quiver varieties using a reduction procedure with moment maps. We study the deformation theory of Nakajima bundle representations, prove a Hitchin-Kobayashi correspondence between such representations and stable quiver bundles, examine the natural torus action on the resulting moduli varieties, and comment on scenarios where the variety is hyperk\"ahler. Finally, we produce concrete examples that recover known and new moduli spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper defines Nakajima bundle representations on curves and proves a Hitchin-Kobayashi correspondence that merges quiver representations with quiver bundles.

read the letter

The main takeaway is that this paper defines Nakajima bundle representations on an algebraic curve or Riemann surface and proves a Hitchin-Kobayashi correspondence between those representations and stable quiver bundles. The construction assigns vector bundles to nodes of the doubled quiver and sections plus connections to edges, then uses moment-map reduction in the style of the original ADHM equations to build the moduli spaces. They also work out the deformation theory, look at the torus action, note hyperkahler cases, and give examples that recover both known and new moduli spaces. That package of definition, correspondence, and examples is the actual advance. It sits squarely in the overlap of quiver varieties and moduli of bundles on curves. The restriction to curves is stated clearly from the start, so the results stay within a setting where gauge theory and stability behave as expected. No circularity or hidden fitting shows up in the setup. The abstract presents the theorems as direct consequences of standard techniques applied to the new objects. This is aimed at people already working on Nakajima varieties, quiver bundles, or Hitchin systems on curves. A reader in that niche can use the new objects and the examples for further calculations or comparisons. The work has enough concrete content and supporting structure to justify sending it to a referee rather than desk rejecting it. I would recommend peer review.

Referee Report

2 major / 2 minor

Summary. The paper introduces the notion of a Nakajima bundle representation on a variety or manifold X (primarily restricted to algebraic curves or Riemann surfaces). Given a labelled quiver, it assigns complex vector bundles to nodes of the doubled quiver and sections of twisted bundles together with connections to the edges. This simultaneously generalizes ordinary Nakajima quiver representations and quiver bundles. The representations admit gauge-theoretic characterizations analogous to the ADHM equations, enabling construction of generalized quiver varieties via moment-map reduction. The paper studies the deformation theory of these representations, proves a Hitchin-Kobayashi correspondence between Nakajima bundle representations and stable quiver bundles, examines the natural torus action on the resulting moduli varieties, comments on hyperkähler scenarios, and produces concrete examples recovering known and new moduli spaces.

Significance. If the stated proofs of the deformation theory, moment-map reduction, and Hitchin-Kobayashi correspondence hold within the stated regime of curves, the work would provide a coherent gauge-theoretic framework unifying quiver representations with bundle moduli spaces. The explicit construction of examples and discussion of hyperkähler structures would add concrete value to the literature on quiver varieties and moduli problems in algebraic geometry.

major comments (2)

[Abstract / §1] The abstract and introduction assert that the moment-map reduction and generalized quiver varieties are constructed via gauge-theoretic characterizations analogous to ADHM, but the restriction to algebraic curves is stated only as 'for the most part'; a precise statement of the hypotheses under which the moment-map equations and stability notions are proven (including any dependence on the genus or degree of the curve) is needed in §1 or §2 to make the scope of the central claims unambiguous.
[§4 (presumed location of the correspondence)] The Hitchin-Kobayashi correspondence is claimed between Nakajima bundle representations and stable quiver bundles, but without an explicit reference to the stability parameter or the precise definition of stability used (e.g., in terms of the moment-map level set), it is unclear whether the correspondence is parameter-free or requires additional choices; this affects the load-bearing claim that the construction recovers known moduli spaces.

minor comments (2)

[§2] Notation for the doubled quiver and the twisting bundles associated to edges should be introduced with a single diagram or table early in the paper to aid readability.
[§6] The discussion of the torus action and hyperkähler cases would benefit from a brief comparison table listing which known moduli spaces are recovered in the examples section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying points where the scope and definitions require greater precision. We address each major comment below and will incorporate the necessary clarifications in a revised version.

read point-by-point responses

Referee: [Abstract / §1] The abstract and introduction assert that the moment-map reduction and generalized quiver varieties are constructed via gauge-theoretic characterizations analogous to ADHM, but the restriction to algebraic curves is stated only as 'for the most part'; a precise statement of the hypotheses under which the moment-map equations and stability notions are proven (including any dependence on the genus or degree of the curve) is needed in §1 or §2 to make the scope of the central claims unambiguous.

Authors: We agree that the current phrasing leaves the precise regime of the main theorems ambiguous. The moment-map equations, the associated stability notions, the deformation theory, and the Hitchin-Kobayashi correspondence are all established for smooth projective curves over ℂ (equivalently, compact Riemann surfaces), with no further restrictions on genus or on the degrees of the bundles beyond the standard topological data required to define the bundles and the twisting line bundles. The general definitions of Nakajima bundle representations are given for arbitrary varieties or manifolds, but the gauge-theoretic reduction and correspondence theorems are proved only in the curve case. We will add an explicit statement of these hypotheses at the end of §1 and again at the beginning of §2, replacing the phrase 'for the most part' with a clear delineation of which results hold on curves and which are stated more generally. revision: yes
Referee: [§4 (presumed location of the correspondence)] The Hitchin-Kobayashi correspondence is claimed between Nakajima bundle representations and stable quiver bundles, but without an explicit reference to the stability parameter or the precise definition of stability used (e.g., in terms of the moment-map level set), it is unclear whether the correspondence is parameter-free or requires additional choices; this affects the load-bearing claim that the construction recovers known moduli spaces.

Authors: The stability condition appearing in the Hitchin-Kobayashi correspondence is the one induced by the zero level set of the moment map (i.e., the equations that define a Nakajima bundle representation). This is the parameter-free case in the usual sense of quiver-variety constructions; no additional stability parameter is introduced beyond the choice of the moment-map level (which is fixed at zero). The correspondence therefore directly identifies the gauge-theoretic objects with the stable quiver bundles in the sense of the moment-map quotient. We acknowledge that an explicit cross-reference from the statement of the correspondence to the definition of stability via the moment-map equations is missing and will insert it in §4, together with a short remark clarifying how the resulting moduli spaces recover the classical Nakajima quiver varieties when X is a point. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper introduces the new notion of Nakajima bundle representation by explicit definition on a labelled quiver and variety/manifold X (restricted to curves or Riemann surfaces), then applies standard gauge-theoretic tools (moment-map reduction analogous to ADHM, deformation theory, Hitchin-Kobayashi correspondence) to construct moduli spaces and prove properties. These steps rely on established bundle theory and symplectic reduction rather than redefining inputs in terms of outputs, fitting parameters to data, or load-bearing self-citations. No equation or theorem reduces by construction to its own inputs; the central claims are independent theorems within the stated regime.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper introduces a new mathematical object (Nakajima bundle representation) via definition rather than deriving it from prior results; no numerical free parameters are mentioned, and the axioms invoked are standard facts about vector bundles and connections on curves.

axioms (1)

standard math Standard properties of holomorphic vector bundles and connections on algebraic curves or Riemann surfaces
Invoked when assigning bundles to nodes and sections/connections to edges of the doubled quiver.

invented entities (1)

Nakajima bundle representation no independent evidence
purpose: To generalize ordinary Nakajima quiver representations and quiver bundles to the setting of vector bundles on a base manifold X
New notion introduced in the paper; no independent falsifiable evidence outside the definition is provided in the abstract.

pith-pipeline@v0.9.0 · 5716 in / 1426 out tokens · 38094 ms · 2026-05-23T07:31:45.545461+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We for the most part restrict attention in our development to algebraic curves or Riemann surfaces... These representations admit gauge-theoretic characterizations, analogous to the ADHM equations... prove a Hitchin-Kobayashi correspondence between such representations and stable quiver bundles.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel (J-cost uniqueness) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the real and complex moment maps μ_R, μ_C ... μ_R(A,φ,x,y)_i : ... = τ ... μ_C(A,φ,x,y)_i : ... = τ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

Hitchin- K obayashi correspondence, quivers, and vortices

Luis \' A lvarez C\' o nsul and Oscar Garc\' a-Prada. Hitchin- K obayashi correspondence, quivers, and vortices. Comm. Math. Phys. , 238(1-2):1--33, 2003

work page 2003
[2]

M. F. Atiyah, N. J. Hitchin, V. G. Drinfel'd, and Y. I. Manin. Construction of instantons. Phys. Lett. A , 65(3):185--187, 1978

work page 1978
[3]

Moduli stacks of quiver bundles with applications to H iggs bundles, ar X iv:2407.11958, 2024

Mahmud Azam and Steven Rayan. Moduli stacks of quiver bundles with applications to H iggs bundles, ar X iv:2407.11958, 2024

work page arXiv 2024
[4]

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

Gwyn Bellamy, Alastair Craw, Steven Rayan, Travis Schedler, and Hartmut Weiss. All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces. J. Algebraic Geom. , 33(4):757--793, 2024

work page 2024
[5]

Towards a mathematical definition of C oulomb branches of 3-dimensional =4 gauge theories, II

Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. Towards a mathematical definition of C oulomb branches of 3-dimensional =4 gauge theories, II . Adv. Theor. Math. Phys. , 22(5):1071--1147, 2018

work page 2018
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Bradlow, Oscar Garc\'ia-Prada, and Peter B

Steven B. Bradlow, Oscar Garc\'ia-Prada, and Peter B. Gothen. Moduli spaces of holomorphic triples over compact R iemann surfaces. Math. Ann. , 328(1-2):299--351, 2004

work page 2004
[7]

Quantizations of conical symplectic resolutions II : category O and symplectic duality

Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster. Quantizations of conical symplectic resolutions II : category O and symplectic duality. Ast\'erisque , (384):75--179, 2016. with an appendix by I. Losev

work page 2016
[8]

Steven B. Bradlow. Special metrics and stability for holomorphic bundles with global sections. J. Differential Geom. , 33(1):169--213, 1991

work page 1991
[9]

Symplectic resolutions of quiver varieties

Gwyn Bellamy and Travis Schedler. Symplectic resolutions of quiver varieties. Selecta Math. (N.S.) , 27(3):Paper No. 36, 50, 2021

work page 2021
[10]

Bia ynicki Birula

A. Bia ynicki Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2) , 98:480--497, 1973

work page 1973
[11]

F-theory on singular spaces

Andr\' e s Collinucci and Raffaele Savelli. F-theory on singular spaces. J. High Energy Phys. , (9):100, front matter+38, 2015

work page 2015
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T-branes as branes within branes

Andr\'es Collinucci and Raffaele Savelli. T-branes as branes within branes. J. High Energy Phys. , (9):161, front matter+27, 2015

work page 2015
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S. K. Donaldson. Anti self-dual Y ang- M ills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3) , 50(1):1--26, 1985

work page 1985
[14]

S. K. Donaldson. Infinite determinants, stable bundles and curvature. Duke Math. J. , 54(1):231--247, 1987

work page 1987
[15]

S. K. Donaldson. Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3) , 55(1):127--131, 1987

work page 1987
[16]

Hyperpolygons and H itchin systems

Jonathan Fisher and Steven Rayan. Hyperpolygons and H itchin systems. Int. Math. Res. Not. IMRN , (6):1839--1870, 2016

work page 2016
[17]

G. W. Gibbons and S. W. Hawking. Classification of gravitational instanton symmetries. Comm. Math. Phys. , 66(3):291--310, 1979

work page 1979
[18]

Grothendieck

A. Grothendieck. Sur la classification des fibr\'es holomorphes sur la sph\`ere de R iemann. Amer. J. Math. , 79:121--138, 1957

work page 1957
[19]

Heckman, Max H\"ubner, Ethan Torres, and Hao Y

Jonathan J. Heckman, Max H\"ubner, Ethan Torres, and Hao Y. Zhang. The branes behind generalized symmetry operators. Fortschr. Phys. , 71(1):Paper No. 2200180, 13, 2023

work page 2023
[20]

N. J. Hitchin. The self-duality equations on a R iemann surface. Proc. London Math. Soc. (3) , 55(1):59--126, 1987

work page 1987
[21]

Arithmetic harmonic analysis on character and quiver varieties

Tam\'as Hausel, Emmanuel Letellier, and Fernando Rodriguez-Villegas. Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. , 160(2):323--400, 2011

work page 2011
[22]

Intriligator and N

K. Intriligator and N. Seiberg. Mirror symmetry in three-dimensional gauge theories. Phys. Lett. B , 387(3):513--519, 1996

work page 1996
[23]

Kronheimer and Hiraku Nakajima

Peter B. Kronheimer and Hiraku Nakajima. Yang- M ills instantons on ALE gravitational instantons. Math. Ann. , 288(2):263--307, 1990

work page 1990
[24]

Differential geometry of complex vector bundles , volume 15 of Publications of the Mathematical Society of Japan

Shoshichi Kobayashi. Differential geometry of complex vector bundles , volume 15 of Publications of the Mathematical Society of Japan . Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kan\^o Memorial Lectures, 5

work page 1987
[25]

Instantons on ALE spaces, quiver varieties, and K ac- M oody algebras

Hiraku Nakajima. Instantons on ALE spaces, quiver varieties, and K ac- M oody algebras. Duke Math. J. , 76(2):365--416, 1994

work page 1994
[26]

Quiver varieties and K ac- M oody algebras

Hiraku Nakajima. Quiver varieties and K ac- M oody algebras. Duke Math. J. , 91(3):515--560, 1998

work page 1998
[27]

M. S. Narasimhan and C. S. Seshadri. Stable bundles and unitary bundles on a compact R iemann surface. Proc. Nat. Acad. Sci. U.S.A. , 52:207--211, 1964

work page 1964
[28]

Schaposnik

Steven Rayan and Laura P. Schaposnik. Moduli spaces of generalized hyperpolygons. Q. J. Math. , 72(1-2):137--161, 2021

work page 2021
[29]

Carlos T. Simpson. Constructing variations of H odge structure using Y ang- M ills theory and applications to uniformization. J. Amer. Math. Soc. , 1(4):867--918, 1988

work page 1988
[30]

Carlos T. Simpson. Higgs bundles and local systems. Inst. Hautes \'Etudes Sci. Publ. Math. , (75):5--95, 1992

work page 1992
[31]

Uhlenbeck and S.-T

K. Uhlenbeck and S.-T. Yau. On the existence of H ermitian- Y ang- M ills connections in stable vector bundles. volume 39, pages S257--S293. 1986. Frontiers of the mathematical sciences: 1985 (New York, 1985)

work page 1986

[1] [1]

Hitchin- K obayashi correspondence, quivers, and vortices

Luis \' A lvarez C\' o nsul and Oscar Garc\' a-Prada. Hitchin- K obayashi correspondence, quivers, and vortices. Comm. Math. Phys. , 238(1-2):1--33, 2003

work page 2003

[2] [2]

M. F. Atiyah, N. J. Hitchin, V. G. Drinfel'd, and Y. I. Manin. Construction of instantons. Phys. Lett. A , 65(3):185--187, 1978

work page 1978

[3] [3]

Moduli stacks of quiver bundles with applications to H iggs bundles, ar X iv:2407.11958, 2024

Mahmud Azam and Steven Rayan. Moduli stacks of quiver bundles with applications to H iggs bundles, ar X iv:2407.11958, 2024

work page arXiv 2024

[4] [4]

All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces

Gwyn Bellamy, Alastair Craw, Steven Rayan, Travis Schedler, and Hartmut Weiss. All 81 crepant resolutions of a finite quotient singularity are hyperpolygon spaces. J. Algebraic Geom. , 33(4):757--793, 2024

work page 2024

[5] [5]

Towards a mathematical definition of C oulomb branches of 3-dimensional =4 gauge theories, II

Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. Towards a mathematical definition of C oulomb branches of 3-dimensional =4 gauge theories, II . Adv. Theor. Math. Phys. , 22(5):1071--1147, 2018

work page 2018

[6] [6]

Bradlow, Oscar Garc\'ia-Prada, and Peter B

Steven B. Bradlow, Oscar Garc\'ia-Prada, and Peter B. Gothen. Moduli spaces of holomorphic triples over compact R iemann surfaces. Math. Ann. , 328(1-2):299--351, 2004

work page 2004

[7] [7]

Quantizations of conical symplectic resolutions II : category O and symplectic duality

Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster. Quantizations of conical symplectic resolutions II : category O and symplectic duality. Ast\'erisque , (384):75--179, 2016. with an appendix by I. Losev

work page 2016

[8] [8]

Steven B. Bradlow. Special metrics and stability for holomorphic bundles with global sections. J. Differential Geom. , 33(1):169--213, 1991

work page 1991

[9] [9]

Symplectic resolutions of quiver varieties

Gwyn Bellamy and Travis Schedler. Symplectic resolutions of quiver varieties. Selecta Math. (N.S.) , 27(3):Paper No. 36, 50, 2021

work page 2021

[10] [10]

Bia ynicki Birula

A. Bia ynicki Birula. Some theorems on actions of algebraic groups. Ann. of Math. (2) , 98:480--497, 1973

work page 1973

[11] [11]

F-theory on singular spaces

Andr\' e s Collinucci and Raffaele Savelli. F-theory on singular spaces. J. High Energy Phys. , (9):100, front matter+38, 2015

work page 2015

[12] [12]

T-branes as branes within branes

Andr\'es Collinucci and Raffaele Savelli. T-branes as branes within branes. J. High Energy Phys. , (9):161, front matter+27, 2015

work page 2015

[13] [13]

S. K. Donaldson. Anti self-dual Y ang- M ills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. (3) , 50(1):1--26, 1985

work page 1985

[14] [14]

S. K. Donaldson. Infinite determinants, stable bundles and curvature. Duke Math. J. , 54(1):231--247, 1987

work page 1987

[15] [15]

S. K. Donaldson. Twisted harmonic maps and the self-duality equations. Proc. London Math. Soc. (3) , 55(1):127--131, 1987

work page 1987

[16] [16]

Hyperpolygons and H itchin systems

Jonathan Fisher and Steven Rayan. Hyperpolygons and H itchin systems. Int. Math. Res. Not. IMRN , (6):1839--1870, 2016

work page 2016

[17] [17]

G. W. Gibbons and S. W. Hawking. Classification of gravitational instanton symmetries. Comm. Math. Phys. , 66(3):291--310, 1979

work page 1979

[18] [18]

Grothendieck

A. Grothendieck. Sur la classification des fibr\'es holomorphes sur la sph\`ere de R iemann. Amer. J. Math. , 79:121--138, 1957

work page 1957

[19] [19]

Heckman, Max H\"ubner, Ethan Torres, and Hao Y

Jonathan J. Heckman, Max H\"ubner, Ethan Torres, and Hao Y. Zhang. The branes behind generalized symmetry operators. Fortschr. Phys. , 71(1):Paper No. 2200180, 13, 2023

work page 2023

[20] [20]

N. J. Hitchin. The self-duality equations on a R iemann surface. Proc. London Math. Soc. (3) , 55(1):59--126, 1987

work page 1987

[21] [21]

Arithmetic harmonic analysis on character and quiver varieties

Tam\'as Hausel, Emmanuel Letellier, and Fernando Rodriguez-Villegas. Arithmetic harmonic analysis on character and quiver varieties. Duke Math. J. , 160(2):323--400, 2011

work page 2011

[22] [22]

Intriligator and N

K. Intriligator and N. Seiberg. Mirror symmetry in three-dimensional gauge theories. Phys. Lett. B , 387(3):513--519, 1996

work page 1996

[23] [23]

Kronheimer and Hiraku Nakajima

Peter B. Kronheimer and Hiraku Nakajima. Yang- M ills instantons on ALE gravitational instantons. Math. Ann. , 288(2):263--307, 1990

work page 1990

[24] [24]

Differential geometry of complex vector bundles , volume 15 of Publications of the Mathematical Society of Japan

Shoshichi Kobayashi. Differential geometry of complex vector bundles , volume 15 of Publications of the Mathematical Society of Japan . Princeton University Press, Princeton, NJ; Princeton University Press, Princeton, NJ, 1987. Kan\^o Memorial Lectures, 5

work page 1987

[25] [25]

Instantons on ALE spaces, quiver varieties, and K ac- M oody algebras

Hiraku Nakajima. Instantons on ALE spaces, quiver varieties, and K ac- M oody algebras. Duke Math. J. , 76(2):365--416, 1994

work page 1994

[26] [26]

Quiver varieties and K ac- M oody algebras

Hiraku Nakajima. Quiver varieties and K ac- M oody algebras. Duke Math. J. , 91(3):515--560, 1998

work page 1998

[27] [27]

M. S. Narasimhan and C. S. Seshadri. Stable bundles and unitary bundles on a compact R iemann surface. Proc. Nat. Acad. Sci. U.S.A. , 52:207--211, 1964

work page 1964

[28] [28]

Schaposnik

Steven Rayan and Laura P. Schaposnik. Moduli spaces of generalized hyperpolygons. Q. J. Math. , 72(1-2):137--161, 2021

work page 2021

[29] [29]

Carlos T. Simpson. Constructing variations of H odge structure using Y ang- M ills theory and applications to uniformization. J. Amer. Math. Soc. , 1(4):867--918, 1988

work page 1988

[30] [30]

Carlos T. Simpson. Higgs bundles and local systems. Inst. Hautes \'Etudes Sci. Publ. Math. , (75):5--95, 1992

work page 1992

[31] [31]

Uhlenbeck and S.-T

K. Uhlenbeck and S.-T. Yau. On the existence of H ermitian- Y ang- M ills connections in stable vector bundles. volume 39, pages S257--S293. 1986. Frontiers of the mathematical sciences: 1985 (New York, 1985)

work page 1986