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REVIEW 3 major objections 4 minor 25 references

Supersymmetry for brane diagrams and bow varieties

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Pith's one-line read The paper proves that an affine type A bow variety is non-empty exactly when its brane diagram is supersymmetric, and that this can be decided by finitely many inequalities.

desk verdict A serious paper with a real result—first combinatorial criterion for non-emptiness of affine type A bow varieties—but the load-bearing Step 1 rests on Lemma 5.7, stated without proof and deferred to the author's own preprint. read the letter →

arxiv 2504.19226 v1 pith:3DJFU4UZ submitted 2025-04-27 math.RT hep-thmath.AG

Tiziano Gaibisso This is my paper · ORCID
classification math.RThep-thmath.AG MSC 14D2116G2017B6714L24
keywords bowvarietiesbranediagramssupersymmetryHanany-WittentransitionsaffinetypeAnon-emptinesscriterionquiverLiealgebraweights
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks when a bow diagram—the combinatorial data, built from arrows, points, and a dimension vector, that encodes a configuration of D3-, D5-, and NS5-branes in type IIB string theory—produces a non-empty bow variety, a moduli space of instanton-type objects. It establishes that non-emptiness is equivalent to the brane configuration being supersymmetric in the physical sense: between any two 5-branes of different types, no more than one D3-brane of each orientation winds around any number of full loops. The main theorem states six equivalent conditions, including a purely numerical one: after separating the diagram, a finite list of explicit inequalities (the supersymmetry inequalities) must hold. This converts a previously delicate existence question into a finitely checkable calculation, and it implies that when the answer is yes, the ordinary non-deformed bow variety $M_{0,0}$ is non-empty and a point of it can be constructed by the paper's algorithm.

What carries the argument

The proof is carried by three interlocking devices. The Hanany-Witten transition is a local rewriting of a bow diagram that swaps an adjacent arrow and x-point and changes the dimension vector by $v_+ + v' = v_- + v_+ + 1$; a theorem on these transitions makes them isomorphisms of bow varieties, so any condition invariant under Hanany-Witten equivalence is a candidate geometric invariant. The supersymmetry inequalities $cD^t_{s,k} \geq 0$ and $aD^t_{n+1-s,w+1-k} \geq 0$ are explicit quadratic expressions in the dimension entries of a separated diagram; they encode exactly the requirement that moving x-points around arrows any number of full loops never produces a negative dimension. The stratum condition reformulates the same requirement in the language of balanced bow diagrams and affine Lie algebra weights, with dominant weight inequalities playing the role of non-negativity. The reduction from affine to finite type A is achieved by supersymmetric increments—adding unfixed D3-branes between branes of the same type—which are shown to preserve both supersymmetry and non-emptiness of the moment fiber.

What would settle it

Compute $M_{0,0}$ for a small affine type A bow diagram with all dimension entries non-negative whose separated Hanany-Witten equivalent has a negative dimension after one transition; if such a diagram has a non-empty $M_{0,0}$, Theorem 5.1 is false, and in particular Lemma 5.7 fails. The paper's own Section 6 algorithm provides the finite inequality list whose violation is the candidate witness.

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Extended reading notes

Core claim

The central discovery is Theorem 5.1: for a bow diagram $(B,\Lambda,v)$ of affine type A, the following are equivalent: (a) $M_{0,0}$ is non-empty; (b) some bow variety $M_{\lambda,\theta}$ with arbitrary deformation and stability parameters is non-empty; (c) every Hanany-Witten equivalent bow diagram originates from a brane diagram, i.e. no sequence of Hanany-Witten transitions creates a negative dimension; (d) every Hanany-Witten equivalent separated diagram satisfies the supersymmetry inequalities; (e) the diagram is supersymmetric; and (f) it satisfies the stratum condition, which can be expressed in terms of dominant weights of affine Lie algebras. In particular, the algebro-geometric question 'does this bow variety exist?' is the same as the string-theory question 'does this brane system preserve supersymmetry?', and both are answered by finitely many inequalities.

Load-bearing premise

The chain from non-empty bow variety to supersymmetric brane system hangs on Lemma 5.7, whose proof is omitted here and deferred to the author's separate preprint [Gai24]; if that lemma is false, condition (a) need not imply condition (c), and the six-way equivalence collapses.

Editorial extensions

If this is right

  • Non-emptiness of an affine type A bow variety is decidable: the Section 6 algorithm reduces the question to finitely many supersymmetry inequalities, and if they all pass, $M_{0,0} \neq \emptyset$.
  • Every supersymmetric bow diagram has a non-empty non-deformed bow variety, and the paper's construction yields an explicit point of $M_{0,0}$, not just an existence certificate.
  • Supersymmetric increments between arrows or between x-points preserve the property of having a non-empty bow variety, so adding D3-branes of the same type never destroys existence.
  • The stratum condition gives a representation-theoretic test: non-emptiness is equivalent to the existence of a dominant affine Lie algebra weight $\kappa$ lying between two weights attached to the diagram, generalizing the classical stratification of balanced bow varieties.
  • For types B, C, and D brane systems, the same algorithm detects supersymmetry after reflecting through the orientifold to reduce to type A.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the theorem equates supersymmetry with non-emptiness, a computational census of small separated diagrams could map the exact boundary of the non-emptiness region in dimension-vector space; the paper gives the inequalities but does not carry out such a census.
  • The deferred Lemma 5.7 is the single bridge from geometry to combinatorics: if its proof in [Gai24] does not go through for general $\lambda$, the implication '(a) implies (c)' is unsupported, although the remaining equivalences among (c)-(f) would still stand.
  • The S-duality transposition of generalized Young diagrams suggests a geometric pairing between strata and slices of bow varieties; proving that correspondence would give a mirror-symmetric reading of the same numerical criterion.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper gives combinatorial and numerical criteria for deciding whether an affine type A bow diagram gives rise to a non-empty bow variety. The main theorem, Theorem 5.1, asserts the equivalence of six conditions: non-emptiness of the non-deformed bow variety, existence of some non-empty deformation/stability bow variety, non-negativity of all dimension vectors reachable by Hanany–Witten transitions, validity of the supersymmetry inequalities for every separated Hanany–Witten equivalent diagram, supersymmetry of the diagram in the sense of brane systems, and the stratum condition formulated in terms of affine Lie algebra weights. The proof is organized into five steps, and additional results include the preservation of non-emptiness under supersymmetric increments (Theorem 5.16) and a direct combinatorial proof of one implication in Appendix A. A finite-step algorithm for detecting supersymmetry is presented in Section 6, with worked examples in Section 7.

Significance. If Theorem 5.1 is correct, the paper provides a substantial and useful characterization: non-emptiness of affine type A bow varieties becomes a finite, checkable numerical condition, and it links the representation-theoretic non-emptiness question to supersymmetry in type IIB brane systems and to weights of affine Lie algebras. The explicit algorithm in Section 6 and the construction of brane diagrams and moment-map solutions in Sections 6.2 and 6.3 are concrete contributions that go beyond a bare equivalence statement. The proof of Theorem 5.16 on supersymmetric increments is a self-contained result of independent interest, and Appendix A contains a clever combinatorial argument for a nontrivial direction. However, the central theorem is not self-contained: the key implication from non-emptiness to the combinatorial conditions rests on Lemma 5.7, whose proof is omitted and deferred to the author's own preprint [Gai24], and one step in Section 5.5 invokes Theorem 5.1 in the course of proving it. These issues are fixable, but they make the main claim conditional as written.

major comments (3)
  1. Lemma 5.7 is load-bearing for Step 1, and therefore for the implications (a)⇒(c) and (b)⇒(c) in Theorem 5.1. The manuscript states 'We omit the proof here' and refers the reader to the author's own preprint [Gai24, Corollary 4.17]. No statement of the proof or of the supporting argument is included. Since condition (c) is the bridge from non-emptiness of bow varieties to the combinatorial supersymmetry criteria, this omitted proof is not a local detail: as written, the main theorem is conditional on an external result. The manuscript should either include a complete proof of Lemma 5.7 or explicitly state Theorem 5.1 as conditional on [Gai24, Corollary 4.17].
  2. In the first direction of Proposition 5.24, the proof says 'By Theorem 5.1, we know that it originates from a supersymmetric brane diagram' while Theorem 5.1 is precisely the statement being proved. The intended argument is presumably to use the already established Step 3, specifically Proposition 5.13, which shows that supersymmetry inequalities imply supersymmetry in the finite type A case. However, as written the proof is circular. This needs to be rewritten so that every invocation in the proof of Proposition 5.24 refers only to implications already proved independently of Theorem 5.1.
  3. Remark 3.4 asserts that [NT17, Proposition 7.1], the isomorphism of bow varieties under Hanany–Witten transitions, extends from nonnegative dimension vectors to arbitrary bow diagrams, including those with negative entries. This extension is used in Step 1, but no proof is given in §5.1; the text only says that the result extends and cites [Gai24]. Since negative dimensions appear precisely in the transitions whose effects Lemma 5.7 must control, the isomorphism theorem cannot be treated as a black box at this point. A proof or a precise statement of the needed extension should be included.
minor comments (4)
  1. The sentence 'The implication (e)⇒(c) follows from the definition of Hanany–Witten transitions for supersymmetric brane systems...' appears twice in the outline; one occurrence should be deleted.
  2. The label 'Example 4.4' is used twice: once for Proposition 4.4 and once for the example following it. The example should be renumbered to avoid confusion.
  3. The sentence 'Since the relation between brane diagrams and bow diagrams of affine type A, one may wonder...' is missing a word; it should say 'Since the relation between brane diagrams and bow diagrams of affine type A is established' or similar.
  4. The notation switches between lowercase 'cdt' and uppercase 'cDt' in Lemma 5.9 and the surrounding text; although the intended meaning is clear, consistent notation would improve readability.

Circularity Check

2 steps flagged · score 4.0 of 10

Theorem 5.1's Step 1 rests on an omitted lemma deferred to the author's own preprint [Gai24], and a Step 5 proof cites the theorem under proof.

  1. self citation load bearing [Section 5.1, Lemma 5.7 (Step 1)]
    "We omit the proof here. For λ = 0, this was established in [SW23, Corollary 3.14], and for general λ, it can be derived from Nakajima and Takayama's proof of Theorem 3.3 ([NT17, Proposition 7.1]). For a proof, we refer the reader to [Gai24, Corollary 4.17]."

    Step 1 must show that non-emptiness (a)/(b) implies (c), i.e. no Hanany–Witten equivalent diagram has a negative dimension. Lemma 5.7 is exactly that local non-negativity assertion, and it is explicitly not proved in this paper. For general λ, the only proof offered is the author's own preprint [Gai24], which is not machine-checked and is not otherwise established in the present text. Since Step 1 is the unique bridge from non-emptiness to the combinatorial supersymmetry criterion, the general-λ direction (b)⇒(c) of Theorem 5.1 is load-bearing on this self-citation. The λ=0 case relevant to (a) is attributed externally to [SW23], so the primary M_{0,0} criterion has independent support; this limits the severity.

  2. other [Section 5.5, proof of Proposition 5.24]
    "Suppose the bow diagram satisfies supersymmetry inequalities. Let us use the intuition coming from the relation between brane diagrams and bow diagrams to construct a weight κ as in (2). By Theorem 5.1, we know that it originates from a supersymmetric brane diagram with v0 fixed D3-branes."

    This is inside the proof of Step 5, which is part of the proof of Theorem 5.1. From supersymmetry inequalities it invokes the theorem being proved to obtain a supersymmetric brane diagram; the needed implication is (d)⇒(e), which had already been proved independently in Step 3 (§5.3). Thus, as written, Proposition 5.24 uses Theorem 5.1 to prove Theorem 5.1. Because Step 3 supplies an alternative proof of (d)⇒(e), this circular citation is replaceable and not fatal, but it is a genuine in-proof circular reference. Proposition 5.21 similarly offers 'Theorem 5.1 and Remark 2.12' as one option in the same Step 5.

full rationale

The paper contains no empirical fitting and no prediction-by-construction; Steps 2–4 are explicit computational arguments (Lemma 5.9, Propositions 5.12–5.13, Theorems 5.16 and 5.20), and Appendix A proves (d)⇒(c). The circularity burden is therefore concentrated in self-reference. The strongest issue is Lemma 5.7: the text says 'We omit the proof here' and sends the reader to [Gai24, Corollary 4.17], the author's own preprint. This lemma is exactly the assertion needed for Step 1, so the general-λ condition (b)⇒(c) in Theorem 5.1 is supported only by that self-citation. The λ=0 case is independently attributed to [SW23], and the M_{0,0} criterion (a)⇔(e) can be reached without [Gai24], which keeps the central claim from collapsing entirely. There is also a local circular reference in the Step 5 proof: Proposition 5.24 cites Theorem 5.1 while proving it, although the needed (d)⇒(e) implication was already proved as Step 3. Overall the theorem is not a renaming of its inputs and most of the derivation is self-contained, but the omitted self-cited lemma is load-bearing for one equivalence in the main theorem. Score 4.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on several background results from the existing bow variety literature and from string theory. The most fragile is Lemma 5.7, whose proof is not given here. No free parameters are fitted, and no new entities are postulated.

assumptions (5)
  • domain assumption Hanany-Witten transitions induce isomorphisms of bow varieties (Theorem 3.3, from [NT17, Prop 7.1]).
    Used throughout to pass between Hanany-Witten equivalent diagrams; the paper relies on this for the equivalence of non-emptiness across transitions.
  • domain assumption Lemma 5.7: non-emptiness of a bow variety implies a single Hanany-Witten transition cannot produce a negative dimension.
    Stated without proof in §5.1, deferred to [Gai24, Cor 4.17]. It is load-bearing for Step 1 and the (a)⇒(c) and (b)⇒(c) directions.
  • domain assumption Hanany-Witten transitions preserve supersymmetry of brane diagrams (physics input from [HW97]).
    Used in (e)⇒(c) and in the algorithm; not proved mathematically in this paper.
  • domain assumption The Nakajima-Takayama quiver description of affine type A bow varieties and their stratification (Theorem 7.26, Remark 7.27 of [NT17]) is valid.
    Used in Section 4 to connect the stratum condition to strata, and to define the stratum condition.
  • domain assumption Every bow diagram is Hanany-Witten equivalent to a separated bow diagram (all x-points on one wavy line).
    Assumed in Definition 3.5 and used throughout; the paper does not prove this explicitly, but it follows from moving x-points past arrows.

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Pith. "Pith review of Supersymmetry for brane diagrams and bow varieties." pith.science (2026). https://pith.science/paper/3DJFU4UZ

@misc{pith2026250419226,
  author       = {Pith},
  title        = {Pith review of: Supersymmetry for brane diagrams and bow varieties},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3DJFU4UZ}},
  note         = {Machine review of arXiv:2504.19226}
}
read the original abstract

We provide combinatorial and numerical criteria to characterize affine type A bow diagrams giving rise to a non-empty bow variety. The key idea is to prove that such diagrams correspond to supersymmetric brane systems in type IIB string theory, allowing us to reformulate the problem in purely combinatorial terms. To achieve this, we characterize supersymmetry for affine type A brane systems (and, by extension, for types B, C, and D) using Hanany--Witten transitions. This leads to a finite-step algorithm that decides whether a given affine type A bow or brane diagram is supersymmetric, which consists in checking a finite set of inequalities, so providing a numerical criterion for non-emptiness. Finally, we provide a different perspective by introducing a further criterion in terms of weights of affine Lie algebras. Along the way, we also prove that increasing dimension vectors between two consecutive x-points or arrows in a bow diagram (not necessarily of type A) preserves the properties of generating non-empty bow varieties.

Figures

Figures reproduced from arXiv: 2504.19226 by the authors.

Figure 1
Figure 1. Vertical segments are NS5-branes, diagonal segments are D5-branes and horizontal segments are D3-branes. Black dots denote extreme points of D3-branes. Definition 2.1. A brane diagram, denoted by B, is given by the following data. (1) A graph defined by: • two types of vertices given by either | or × and called, respectively, NS5-branes and D5-branes; • edges connecting vertices (independently from the types). (2) I… view at source ↗
Figure 2
Figure 2. A fine type A brane diagram on the left and an affine type A brane diagram on the right. In particular, there exists a correspondence between brane system of finite type A and brane diagrams immersed in a segment by replacing NS5-branes with |, D5-branes with × and D3-branes with edges. Although affine type A brane systems can be constructed similarly, for our purposes it will be sufficient to work solely with brane… view at source ↗
Figure 3
Figure 3. Bow diagram associated with a brane diagram. × × × × 3 1 1 2 2 2 1 × × × × 3 1 1 2 2 2 1 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Bow diagram of affine type A on the left-hand side and its simplified version on the right-hand side. Definition 2.2. A bow diagram is a triple (B,Λ, v), where B is a bow, Λ is a set of x-points on wavy lines, and v ∈ Z I s is a vector called the dimension vector. If v…
Figure 5
Figure 5. Figure 5: Equivalence between affine and finite type A bow dia￾grams. We notice that, a priori, there is not a unique way to open the circle into a line since there may be different segments with dimension 0 that one could choose to cut the circle. However, for our purposes, all…
Figure 6
Figure 6. Figure 6: Separated bow diagram of affine type A. 0 vn−1 · · · v1 v0 v−1 · · · v−(w−1) 0 × × n · · · 1 1 · · · w [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: Separated bow diagram of finite type A. Remark 3.6. In the cobalanced case (i.e. when for every x-point x, v − x = v + x ), there is a correspondence between bow diagrams and pairs of quivers and framed dimension vectors, see [Che11], [NT17]. In this case, the number o…
Figure 6
Figure 6. Figure 6: We prove that if it satisfies supersymmetry inequalities (3.3) then [PITH_FULL_IMAGE:figures/full_fig_p052_6.png]

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Works this paper leans on

25 extracted references · 22 canonical work pages

  1. [1]

    The Coulomb branch of 3d N= 4 theories

    Mathew Bullimore, Tudor Dimofte, and Davide Gaiotto. The Coulomb branch of 3d N= 4 theories . Communications in Mathematical Physics , 354:671--751, 2017

    work page 2017

  2. [2]

    Towards a mathematical definition of Coulomb branches of 3 -dimensional N = 4 gauge theories, II

    Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. Towards a mathematical definition of Coulomb branches of 3 -dimensional N = 4 gauge theories, II . Advances in Theoretical and Mathematical Physics , 22(5):1071–1147, 2018

    work page 2018

  3. [3]

    Coulomb branches of 3 d N =4 quiver gauge theories and slices in the affine Grassmannian

    Alexander Braverman, Michael Finkelberg, and Hiraku Nakajima. Coulomb branches of 3 d N =4 quiver gauge theories and slices in the affine Grassmannian . Adv. Theor. Math. Phys. , 23(1):75--166, 2019

    work page 2019

  4. [4]

    Branes, quivers, and the affine grassmannian

    Antoine Bourget, Julius F Grimminger, Amihay Hanany, Marcus Sperling, and Zhenghao Zhong. Branes, quivers, and the affine grassmannian. arXiv preprint arXiv:2102.06190 , 2023

    arXiv 2023

  5. [5]

    Of conical symplectic resolutions quantizations of conical symplectic resolutions ii: category o and symplectic duality

    Tom Braden, Anthony Licata, Nicholas Proudfoot, and Ben Webster. Of conical symplectic resolutions quantizations of conical symplectic resolutions ii: category o and symplectic duality. Ast \'e risque , 384:75--179, 2016

    work page 2016

  6. [6]

    Moduli spaces of instantons on the Taub-NUT space

    Sergey A Cherkis. Moduli spaces of instantons on the Taub-NUT space . Communications in Mathematical Physics , 290:719--736, 2009

    work page 2009

  7. [7]

    Instantons on gravitons

    Sergey A Cherkis. Instantons on gravitons. Communications in mathematical physics , 306:449--483, 2011

    work page 2011

  8. [8]

    Super Yang-Mills theory with impurity walls and instanton moduli spaces

    Sergey A Cherkis, Clare O’Hara, and Christian S \"a mann. Super Yang-Mills theory with impurity walls and instanton moduli spaces . Physical Review D—Particles, Fields, Gravitation, and Cosmology , 83(12):126009, 2011

    work page 2011

Show all 25 references
  1. [9]

    Quiver description of Cherkis bow varieties and Nakajima quiver varieties

    Tiziano Gaibisso. Quiver description of Cherkis bow varieties and Nakajima quiver varieties . arXiv preprint arXiv:2407.19267 , 2024

    2024 arXiv

  2. [10]

    Brane dynamics and gauge theory

    Amit Giveon and David Kutasov. Brane dynamics and gauge theory. Reviews of Modern Physics , 71(4):983, 1999

    1999

  3. [11]

    Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics

    Amihay Hanany and Edward Witten. Type IIB superstrings, BPS monopoles, and three-dimensional gauge dynamics . Nuclear Physics B , 492(1-2):152--190, 1997

    1997

  4. [12]

    Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

    Hiraku Nakajima. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras . Duke Mathematical Journal , 76(2):365 -- 416, 1994

    1994

  5. [13]

    Quiver varieties and Kac-Moody algebras

    Hiraku Nakajima. Quiver varieties and Kac-Moody algebras . Duke Mathematical Journal , 91, 02 1998

    1998

  6. [14]

    Quiver varieties and branching

    Hiraku Nakajima. Quiver varieties and branching. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications , 5:003, 2009

    2009

  7. [15]

    Towards a mathematical definition of Coulomb branches of 3 -dimensional N = 4 gauge theories, I

    Hiraku Nakajima. Towards a mathematical definition of Coulomb branches of 3 -dimensional N = 4 gauge theories, I . Adv. Theor. Math. Phys. , 20(3):595--669, 2016

    2016

  8. [16]

    Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N= 4 gauge theories

    Hiraku Nakajima. Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N= 4 gauge theories . In Modern Geometry: A Celebration of the Work of Simon Donaldson, Proceedings of Symposia in Pure Mathematics , volume 99, pages 193--211, 2018

    2018

  9. [17]

    Towards geometric Satake correspondence for Kac-Moody algebras, Cherkis bow varieties and affine Lie algebras of type A

    Hiraku Nakajima. Towards geometric Satake correspondence for Kac-Moody algebras, Cherkis bow varieties and affine Lie algebras of type A. Ann. Sci. \'E c. Norm. Sup \'e r.(4) , 56(arXiv: 1810.04293):1777--1824, 2023

    2023 arXiv

  10. [18]

    Level-rank duality of WZW models in conformal field theory

    Tomoki Nakanishi and Akihiro Tsuchiya. Level-rank duality of WZW models in conformal field theory . Communications in mathematical physics , 144(2):351--372, 1992

    1992

  11. [19]

    Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A

    Hiraku Nakajima and Yuuya Takayama. Cherkis bow varieties and Coulomb branches of quiver gauge theories of affine type A . Selecta Mathematica , 23:2553--2633, 2017

    2017

  12. [20]

    Bow varieties---geometry, combinatorics, characteristic classes

    R Rim \'a nyi and Y Shou. Bow varieties---geometry, combinatorics, characteristic classes . arXiv preprint arXiv:2012.07814 , 2020

    2012 arXiv

  13. [21]

    Stratified symplectic spaces and reduction

    Reyer Sjamaar and Eugene Lerman. Stratified symplectic spaces and reduction. Annals of Mathematics , pages 375--422, 1991

    1991

  14. [22]

    Seiberg and E

    N. Seiberg and E. Witten. Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD . Nuclear Physics B , 431(3):484–550, 1994

    1994

  15. [23]

    Existence and orthogonality of stable envelopes for bow varieties

    Catharina Stroppel and Till Wehrhan. Existence and orthogonality of stable envelopes for bow varieties. arXiv preprint arXiv:2312.03144 , 2023

    2023 arXiv

  16. [24]

    Nahm's equations, quiver varieties and parabolic sheaves

    Yuuya Takayama. Nahm's equations, quiver varieties and parabolic sheaves . Publications of the Research Institute for Mathematical Sciences , 52(1):1--41, 2016

    2016

  17. [25]

    3-dimensional mirror symmetry

    Ben Webster and Philsang Yoo. 3-dimensional mirror symmetry. arXiv preprint arXiv:2308.06191 , 2023

    2023 arXiv

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