Pith. sign in

REVIEW 2 major objections 4 minor 78 references

Elliptic genera and Witten indices equal a sum over crystal molecules of weights fixed only by the molecules' boundaries.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · grok-4.5

2026-07-12 01:09 UTC pith:CZV7Z3YJ

load-bearing objection Solid combinatorial reduction of JK residues to molecule-boundary products; the main formulae look correct and recover Nekrasov as a special case. the 2 major comments →

arxiv 2607.03617 v1 pith:CZV7Z3YJ submitted 2026-07-03 hep-th math-phmath.AGmath.COmath.MP

Quiver BPS Indices from Crystal Profiles

Jiakang Bao , Masahito Yamazaki This is my paper
classification hep-th math-phmath.AGmath.COmath.MP
keywords crystal meltingelliptic genusJeffrey-Kirwan residuequiver gauge theoryNekrasov partition functionbrane brick modelRonkin functionBPS index
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the elliptic genus of a two-dimensional supersymmetric gauge theory (and its one- and zero-dimensional analogues) can be rewritten exactly as a sum over finite pieces of an infinite crystal. Each piece, called a molecule, contributes a weight that depends only on the atoms sitting on its boundary. The same language that once produced the Nekrasov instanton sum over Young diagrams now works for far more general quivers, including those that come from toric Calabi-Yau fourfolds. In the thermodynamic limit the molten crystal settles into a smooth profile whose shape is the Ronkin function of the associated Newton polynomial, so a projection of the original Calabi-Yau geometry reappears from discrete combinatorics. The result therefore turns an abstract Jeffrey-Kirwan residue into a concrete crystal-melting rule and simultaneously generalises the classic combinatorics of Young diagrams.

Core claim

For a broad class of quiver gauge theories the grand-canonical partition function equals a discrete sum over all molecules M inside a crystal C, where the weight of each molecule is a product of elementary zeta (or sine or linear) factors evaluated solely on the boundary atoms of M. Explicit formulae are given for both N=4 and N=2 theories; when the quiver is the Jordan quiver the sum reduces to the ordinary Nekrasov partition function.

What carries the argument

Boundary-only cancellation of one-loop determinants inside the Jeffrey-Kirwan residue: after all internal numerators and denominators cancel, the surviving factors are completely determined by two (or more) combinatorial boundaries of the molecule, written as products of the functions E_ij(x).

Load-bearing premise

Atoms that arise from different gauge nodes or different colour indices are never allowed to sit at the same point in flavour space; if two atoms ever coincide the whole identification of poles with crystal sites collapses.

What would settle it

Compute the elliptic genus of any concrete N=2 or N=4 quiver by direct Jeffrey-Kirwan residue and by the crystal-boundary formula; any mismatch for a single dimension vector falsifies the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper derives exact formulae for the elliptic genera of 2d N≥2 quiver gauge theories (and their 1d Witten indices and 0d matrix-model partition functions) by evaluating Jeffrey-Kirwan residues and organising the non-vanishing poles into crystals and molecules. The grand-canonical partition function is expressed as a sum over molecules M of a weight that, after systematic cancellation of one-loop factors, depends only on boundary data of M (eqs. (3.18)–(3.19) for N=4 and (4.21)–(4.26) for N=2). The formulae recover the Nekrasov partition function for Young diagrams and are illustrated on the Jordan quiver and Q^{1,1,1}. A thermodynamic-limit analysis for toric Calabi-Yau fourfolds proposes that the limit shape of the molten crystal is governed by the Ronkin function of the Newton polynomial of the associated brane brick model.

Significance. If the boundary formulae hold, the work supplies a broad combinatorial generalisation of the Nekrasov instanton sum and of earlier crystal-melting results for toric Calabi-Yau three-folds, extending them to a large class of N=2 and N=4 quivers (including those dual to toric CY4s). The careful cancellation analysis that reduces the JK integrand to boundary zeta factors, together with explicit recovery of known special cases, constitutes a concrete computational advance. The thermodynamic discussion, while more provisional, offers a concrete proposal for how a projection of the Calabi-Yau geometry can emerge from discrete crystal data, potentially linking BPS counting to amoebae and Ronkin functions in higher dimensions.

major comments (2)
  1. [Section 2.1] Section 2.1 introduces the no-overlap condition as a working hypothesis: every admissible JK pole u* must produce exactly rank(G) distinct atoms. All subsequent identifications of poles with atoms, the melting rule, and the residual boundary factors after cancellation (Sections 3.2 and 4.2) rely on this condition. The paper never proves that the condition holds for a general quiver; it only verifies it for the examples treated later. The central claims should therefore be explicitly restricted to quivers satisfying no-overlap, or a proof (or a clear sufficient criterion) should be supplied.
  2. [Section 6] In Section 6 the true molecule weights w(M;q,ϵ) that appear in the exact formulae are replaced by ordinary brick-matching (dimer) weights. The authors note that this changes the partition function in general and leave the equivariant correction σ_equiv unspecified (eq. (6.29)). Consequently the claim that “a projection of the Calabi-Yau geometry emerges” is established only for the modified statistical model, not for the BPS indices derived in Sections 3–4. The status of the thermodynamic statements relative to the exact formulae should be clarified.
minor comments (4)
  1. Numerous typographical errors appear throughout (e.g. “cyrstal” in §5.3, “conditinons”, “pricesely”, “occassionally”, “unqiue”, “non-overlap conditinons”). A careful proof-reading pass is needed.
  2. [Section 3.3] The notation E_{ij}(x) is introduced in (3.17) and then used with slightly varying index conventions; a single consistent convention (or a short glossary) would improve readability.
  3. [Sections 4 and 6] Figures 4.1 and 6.1–6.2 are referenced but lack captions that fully explain the colour coding and the dashed circles; expanding the captions would help the reader follow the cancellation arguments.
  4. [Section 4.3] In the N=2 rewriting (4.23)–(4.26) the outer-boundary sets ∂_a M and ∂_{-a} M are defined; a short remark confirming that these sets are independent of the arbitrary choice of which chiral realises a given uncancelled factor would remove a potential ambiguity.

Circularity Check

1 steps flagged

No significant circularity: JK cancellations produce boundary formulae independently of the crystal definition supplied by self-citation.

specific steps
  1. self citation load bearing [Section 2.1, definitions of crystals and molecules; abstract and Introduction]
    "Following our previous paper [BY25], we use the JK residue formula to compute the elliptic genera … the contributions of the poles can be organised nicely in the language of crystals [BY25, Bao25]. … The crystal is an oriented weighted graph C=(A,I) with the set of vertices A and the set of arrows I. … Given the crystal C=(A,I), we can define the molecule as a finite subset M⊂A such that the following condition (melting rule) is satisfied."

    The combinatorial objects (atoms, chemical bonds, molecules, melting rule) that label the summands of the final formulae are defined by citation to the authors’ own earlier paper rather than re-derived from first principles inside the present work. The citation is not load-bearing for the algebraic cancellation that produces the boundary weights, so the circularity is minor.

full rationale

The central claim (grand-canonical Z equals a sum over molecules of products of zeta-functions evaluated only on boundary atoms) is obtained by direct algebraic cancellation of one-loop factors inside the standard Jeffrey-Kirwan residue formula (eqs. 3.8–3.14 for N=4 and 4.5–4.20 for N=2). The residual factors are then collected into the explicit products (3.19) and (4.22)/(4.26). The crystal and molecule language is taken from the authors’ prior work [BY25], but that citation only supplies the combinatorial scaffolding that organises the poles; it does not dictate the form of the surviving weights. The thermodynamic-limit discussion replaces the true BPS weight by an auxiliary dimer weight and therefore does not feed back into the main formulae. The only mild self-reference is the definition of crystals themselves; it is not load-bearing for the cancellation identities. Hence the derivation is self-contained against the JK starting point and scores 1.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 2 invented entities

The paper is a pure derivation inside supersymmetric localization and combinatorial representation theory. It inherits the Jeffrey-Kirwan residue theorem, the definition of crystals from the authors’ previous work, and standard one-loop determinants. The only non-standard working hypotheses are the no-overlap condition, the requirement that every Fermi multiplet participates in a J/E-term, and the restriction to the cyclic chamber η=(1,…,1). No numerical free parameters are fitted.

axioms (5)
  • standard math Jeffrey-Kirwan residue formula evaluates the supersymmetric partition function as a sum of residues at admissible poles (Appendix A, eq. A.1).
    Standard localization result used as the starting point for all subsequent combinatorics.
  • domain assumption No-overlap condition: every admissible JK pole produces a set of distinct atoms (Section 2.1).
    Required for atoms to be well-defined lattice points and for the boundary cancellations to be unambiguous; not proved for arbitrary quivers.
  • domain assumption All Fermi multiplets appear in J- or E-term interactions (Section 2.1 and 4.2).
    Ensures that Fermi factors cancel chiral factors rather than remaining as free floating contributions.
  • domain assumption Restriction to the cyclic chamber η=(1,…,1) (Section 2.2).
    Simplifies crystal growth to single-atom steps and places the crystal inside a positive cone; wall-crossing for other chambers is deferred.
  • ad hoc to paper In the thermodynamic limit the true BPS molecule weights may be replaced by ordinary dimer/brick-matching weights (Section 6).
    Explicitly introduced to obtain a tractable Ronkin analysis; the authors note that equivariant corrections remain open.
invented entities (2)
  • Molecule (finite substructure of a crystal obeying the melting rule) no independent evidence
    purpose: Organizes the contributing JK poles into combinatorial objects whose boundary data determine the partition-function weight.
    Defined in Section 2.1 from the crystal of [BY25]; the boundary formulae that follow are new.
  • Boundary sets ∂±M, ∂_χM, ∂_{-χ-Λ}M, etc. no independent evidence
    purpose: Locate the uncancelled one-loop factors after pairwise cancellation of numerators and denominators.
    Introduced in Sections 3.2 and 4.2; their precise definitions differ between N=4 and N=2 and are essential to the final product formulae.

pith-pipeline@v1.1.0-grok45 · 36393 in / 3095 out tokens · 29330 ms · 2026-07-12T01:09:10.622105+00:00 · methodology

0 comments
read the original abstract

We derive new exact formulae for elliptic genera in two spacetime dimensions with $\mathcal{N}\ge 2$ supercharges, as well as their one- and zero-dimensional counterparts (for Witten indices and matrix-model partition functions). Our results are written as a discrete sum over geometric/combinatorial structures of crystals introduced previously by the authors. The contribution from each crystal may be expressed in terms of the boundary data of a finite substructure of the crystal called the molecules. Our results provide vast generalisations of the celebrated Nekrasov partition function enumerated by the Young diagrams, and uncover new combinatorics underlying the crystal melting. We also analyse the thermodynamic limit of crystals arising from two-dimensional $\mathcal{N}=(0,2)$ theories associated with toric Calabi-Yau fourfolds, where a projection of the Calabi-Yau geometry emerges in the limit.

Figures

Figures reproduced from arXiv: 2607.03617 by Jiakang Bao, Masahito Yamazaki.

Figure 4
Figure 4. Figure 4 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p028_6.png] view at source ↗
Figure 6
Figure 6. Figure 6 [PITH_FULL_IMAGE:figures/full_fig_p037_6.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

78 extracted references · 19 canonical work pages · 14 internal anchors

  1. [1]

    Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues

    Bao, Jiakang and Yamazaki, Masahito. Crystals and Double Quiver Algebras from Jeffrey-Kirwan Residues. 2025. arXiv:2501.03365

    Pith/arXiv arXiv 2025

  2. [2]

    Shifted quiver Yangians and representations from BPS crystals

    Galakhov, Dmitry and Li, Wei and Yamazaki, Masahito. Shifted quiver Yangians and representations from BPS crystals. JHEP. 2021. doi:10.1007/JHEP08(2021)146. arXiv:2106.01230

    Pith/arXiv arXiv doi:10.1007/jhep08(2021)146 2021

  3. [3]

    Elliptic analogue of the Vershik-Kerov limit shape

    Grekov, Andrey and Nekrasov, Nikita. Elliptic analogue of the Vershik-Kerov limit shape. Funct. Anal. Appl. 2023. doi:10.1134/S0016266324020059. arXiv:2403.07168

    Pith/arXiv arXiv doi:10.1134/s0016266324020059 2023

  4. [4]

    Gauge origami and quiver W-algebras

    Kimura, Taro and Noshita, Go. Gauge origami and quiver W-algebras. JHEP. 2024. doi:10.1007/JHEP05(2024)208. arXiv:2310.08545

    Pith/arXiv arXiv doi:10.1007/jhep05(2024)208 2024

  5. [5]

    Seiberg-Witten theory and random partitions

    Nekrasov, Nikita and Okounkov, Andrei. Seiberg-Witten theory and random partitions. Prog. Math. 2006. doi:10.1007/0-8176-4467-9_15. arXiv:hep-th/0306238

    Pith/arXiv arXiv doi:10.1007/0-8176-4467-9_15 2006

  6. [6]

    Seiberg-Witten prepotential from instanton counting

    Nekrasov, Nikita A. Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 2003. doi:10.4310/ATMP.2003.v7.n5.a4. arXiv:hep-th/0206161

    Pith/arXiv arXiv doi:10.4310/atmp.2003.v7.n5.a4 2003

  7. [7]

    and Marshakov, Andrei and Nekrasov, Nikita A

    Losev, Andrei S. and Marshakov, Andrei and Nekrasov, Nikita A. Small instantons, little strings and free fermions. From Fields to Strings: Circumnavigating Theoretical Physics: A Conference in Tribute to Ian Kogan. 2003. arXiv:hep-th/0302191

    Pith/arXiv arXiv 2003

  8. [8]

    Instanton counting on blowup

    Nakajima, Hiraku and Yoshioka, Kota. Instanton counting on blowup. 1. Invent. Math. 2005. doi:10.1007/s00222-005-0444-1. arXiv:math/0306198

    Pith/arXiv arXiv doi:10.1007/s00222-005-0444-1 2005

  9. [9]

    Shell formulas for instantons and gauge origami

    Jiang, Jiaqun. Shell formulas for instantons and gauge origami. 2025. arXiv:2512.21606

    Pith/arXiv arXiv 2025

  10. [10]

    Emergent Calabi-Yau Geometry

    Ooguri, Hirosi and Yamazaki, Masahito. Emergent Calabi-Yau Geometry. Phys. Rev. Lett. 2009. doi:10.1103/PhysRevLett.102.161601. arXiv:0902.3996

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.102.161601 2009

  11. [11]

    Dimers and amoebae

    Kenyon, Richard and Okounkov, Andrei and Sheffield, Scott. Dimers and amoebae. 2003. arXiv:math-ph/0311005

    Pith/arXiv arXiv 2003

  12. [12]

    math/0008220

    Cohn, Henry and Kenyon, Richard and Propp, James , eprint = "math/0008220", archivePrefix = "arXiv", journal=. A variational principle for domino tilings

    Pith/arXiv arXiv

  13. [13]

    2d (0,2) Quiver Gauge Theories and D-Branes

    Franco, Sebastian and Ghim, Dongwook and Lee, Sangmin and Seong, Rak-Kyeong and Yokoyama, Daisuke. 2d (0,2) Quiver Gauge Theories and D-Branes. JHEP. 2015. doi:10.1007/JHEP09(2015)072. arXiv:1506.03818

    Pith/arXiv arXiv doi:10.1007/jhep09(2015)072 2015

  14. [14]

    Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0,2) Quivers

    Franco, Sebastian and Lee, Sangmin and Seong, Rak-Kyeong. Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0,2) Quivers. JHEP. 2016. doi:10.1007/JHEP02(2016)047. arXiv:1510.01744

    Pith/arXiv arXiv doi:10.1007/jhep02(2016)047 2016

  15. [15]

    Brane brick models and 2d (0, 2) triality

    Franco, Sebastian and Lee, Sangmin and Seong, Rak-Kyeong. Brane brick models and 2d (0, 2) triality. JHEP. 2016. doi:10.1007/JHEP05(2016)020. arXiv:1602.01834

    Pith/arXiv arXiv doi:10.1007/jhep05(2016)020 2016

  16. [16]

    Brane Brick Models in the Mirror

    Franco, Sebastian and Lee, Sangmin and Seong, Rak-Kyeong and Vafa, Cumrun. Brane Brick Models in the Mirror. JHEP. 2017. doi:10.1007/JHEP02(2017)106. arXiv:1609.01723

    Pith/arXiv arXiv doi:10.1007/jhep02(2017)106 2017

  17. [17]

    Elliptic Genera of 2d (0,2) Gauge Theories from Brane Brick Models

    Franco, Sebastian and Ghim, Dongwook and Lee, Sangmin and Seong, Rak-Kyeong. Elliptic Genera of 2d (0,2) Gauge Theories from Brane Brick Models. JHEP. 2017. doi:10.1007/JHEP06(2017)068. arXiv:1702.02948

    Pith/arXiv arXiv doi:10.1007/jhep06(2017)068 2017

  18. [18]

    2d Supersymmetric Gauge Theories, D-branes and Trialities

    Franco, Sebasti \'a n. 2d Supersymmetric Gauge Theories, D-branes and Trialities. 2022. arXiv:2201.10987

    Pith/arXiv arXiv 2022

  19. [19]

    Fano 3-folds, reflexive polytopes and brane brick models

    Franco, Sebastian and Seong, Rak-Kyeong. Fano 3-folds, reflexive polytopes and brane brick models. JHEP. 2022. doi:10.1007/JHEP08(2022)008. arXiv:2203.15816

    Pith/arXiv arXiv doi:10.1007/jhep08(2022)008 2022

  20. [20]

    Brane brick models for the Sasaki-Einstein 7-manifolds Y ^ p,k ( C P ^ 1 C P ^ 1 ) and Y ^ p,k ( C P ^ 2 )

    Franco, Sebastian and Ghim, Dongwook and Seong, Rak-Kyeong. Brane brick models for the Sasaki-Einstein 7-manifolds Y ^ p,k ( C P ^ 1 C P ^ 1 ) and Y ^ p,k ( C P ^ 2 ). JHEP. 2023. doi:10.1007/JHEP03(2023)050. arXiv:2212.02523

    Pith/arXiv arXiv doi:10.1007/jhep03(2023)050 2023

  21. [21]

    2d (0,2) Gauge Theories from Branes: Recent Progress in Brane Brick Models

    Franco, Sebasti \'a n. 2d (0, 2) gauge theories from branes: Recent progress in brane brick models. Int. J. Mod. Phys. A. 2024. doi:10.1142/S0217751X24460059. arXiv:2402.06993

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217751x24460059 2024

  22. [22]

    Quantum Calabi-Yau and classical crystals

    Okounkov, Andrei and Reshetikhin, Nikolai and Vafa, Cumrun. Quantum Calabi-Yau and classical crystals. Prog. Math. 2006. doi:10.1007/0-8176-4467-9_16. arXiv:hep-th/0309208

    Pith/arXiv arXiv doi:10.1007/0-8176-4467-9_16 2006

  23. [23]

    D-brane gauge theories from toric singularities and toric duality

    Feng, Bo and Hanany, Amihay and He, Yang-Hui. D-brane gauge theories from toric singularities and toric duality. Nucl. Phys. B. 2001. doi:10.1016/S0550-3213(00)00699-4. arXiv:hep-th/0003085

    Pith/arXiv arXiv doi:10.1016/s0550-3213(00)00699-4 2001

  24. [24]

    alg-geom/9706005

    Maillot, VINCENT , eprint = "alg-geom/9706005", archivePrefix = "arXiv", year=. G\'eom\'etrie d'Arakelov des vari\'et\'es toriques et fibr\'es en droites int\'egrables

    Pith/arXiv arXiv

  25. [25]

    2308.04601

    Roy, Subham , eprint = "2308.04601", archivePrefix = "arXiv", primaryClass = "math.NT", journal=. Generalized Mahler measures of Laurent polynomials. 2024 , publisher=

    Pith/arXiv arXiv 2024

  26. [26]

    Laurent determinants and arrangements of hyperplane amoebas

    Forsberg, Mikael and Passare, Mikael and Tsikh, August , journal=. Laurent determinants and arrangements of hyperplane amoebas. 2000 , publisher=

    2000

  27. [27]

    Amoebas, Monge-Ampere measures and triangulations of the Newton polytope

    Passare, Mikael and Rullg. Amoebas, Monge-Ampere measures and triangulations of the Newton polytope. 2000 , publisher=

    2000

  28. [28]

    math/0108225

    Mikhalkin, Grigory , eprint = "math/0108225", archivePrefix = "arXiv", year=. Amoebas of algebraic varieties

    Pith/arXiv arXiv

  29. [29]

    BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters

    Nekrasov, Nikita. BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters. JHEP. 2016. doi:10.1007/JHEP03(2016)181. arXiv:1512.05388

    Pith/arXiv arXiv doi:10.1007/jhep03(2016)181 2016

  30. [30]

    Quiver W-algebras

    Kimura, Taro and Pestun, Vasily. Quiver W-algebras. Lett. Math. Phys. 2018. doi:10.1007/s11005-018-1072-1. arXiv:1512.08533

    Pith/arXiv arXiv doi:10.1007/s11005-018-1072-1 2018

  31. [31]

    The Origin of Calabi-Yau Crystals in BPS States Counting

    Bao, Jiakang and Seong, Rak-Kyeong and Yamazaki, Masahito. The origin of Calabi-Yau crystals in BPS states counting. JHEP. 2024. doi:10.1007/JHEP03(2024)140. arXiv:2401.02792

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep03(2024)140 2024

  32. [32]

    4d crystal melting, toric Calabi-Yau 4-folds and brane brick models

    Franco, Sebasti \'a n. 4d crystal melting, toric Calabi-Yau 4-folds and brane brick models. JHEP. 2024. doi:10.1007/JHEP03(2024)091. arXiv:2311.04404

    Pith/arXiv arXiv doi:10.1007/jhep03(2024)091 2024

  33. [33]

    Elliptic Genera of 2d N = 2 Gauge Theories

    Benini, Francesco and Eager, Richard and Hori, Kentaro and Tachikawa, Yuji. Elliptic Genera of 2d N = 2 Gauge Theories. Commun. Math. Phys. 2015. doi:10.1007/s00220-014-2210-y. arXiv:1308.4896

    Pith/arXiv arXiv doi:10.1007/s00220-014-2210-y 2015

  34. [34]

    Witten Index and Wall Crossing

    Hori, Kentaro and Kim, Heeyeon and Yi, Piljin. Witten Index and Wall Crossing. JHEP. 2015. doi:10.1007/JHEP01(2015)124. arXiv:1407.2567

    Pith/arXiv arXiv doi:10.1007/jhep01(2015)124 2015

  35. [35]

    An Index Formula for Supersymmetric Quantum Mechanics

    Cordova, Clay and Shao, Shu-Heng. An Index Formula for Supersymmetric Quantum Mechanics. J. Singul. 2016. doi:10.5427/jsing.2016.15b. arXiv:1406.7853

    Pith/arXiv arXiv doi:10.5427/jsing.2016.15b 2016

  36. [36]

    General instanton counting and 5d SCFT

    Hwang, Chiung and Kim, Joonho and Kim, Seok and Park, Jaemo. General instanton counting and 5d SCFT. JHEP. 2015. doi:10.1007/JHEP07(2015)063. arXiv:1406.6793

    Pith/arXiv arXiv doi:10.1007/jhep07(2015)063 2015

  37. [37]

    alg-geom/9307001

    Jeffrey, Lisa C and Kirwan, Frances C , eprint = "alg-geom/9307001", archivePrefix = "arXiv", journal=. Localization for nonabelian group actions. 1995 , publisher=

    Pith/arXiv arXiv 1995

  38. [38]

    Two-dimensional gauge theories revisited

    Witten, Edward. Two-dimensional gauge theories revisited. J. Geom. Phys. 1992. doi:10.1016/0393-0440(92)90034-X. arXiv:hep-th/9204083

    Pith/arXiv arXiv doi:10.1016/0393-0440(92)90034-x 1992

  39. [39]

    Toric reduction and a conjecture of Batyrev and Materov

    Szenes, Andr. Toric reduction and a conjecture of Batyrev and Materov. arXiv:math/0306311

    Pith/arXiv arXiv

  40. [40]

    Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups

    Benini, Francesco and Eager, Richard and Hori, Kentaro and Tachikawa, Yuji. Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups. Lett. Math. Phys. 2014. doi:10.1007/s11005-013-0673-y. arXiv:1305.0533

    Pith/arXiv arXiv doi:10.1007/s11005-013-0673-y 2014

  41. [41]

    Elliptic Genera and Quantum Field Theory

    Witten, Edward. Elliptic Genera and Quantum Field Theory. Commun. Math. Phys. 1987. doi:10.1007/BF01208956

    work page doi:10.1007/bf01208956 1987

  42. [42]

    Constraints on Supersymmetry Breaking

    Witten, Edward. Constraints on Supersymmetry Breaking. Nucl. Phys. B. 1982. doi:10.1016/0550-3213(82)90071-2

    doi:10.1016/0550-3213(82)90071-2 1982

  43. [43]

    Elliptic Genera of 2d N =(0,1) Gauge Theories

    Bao, Jiakang and Yamazaki, Masahito and Zhou, Dongao. Elliptic Genera of 2d N =(0,1) Gauge Theories. 2025. arXiv:2508.06865

    Pith/arXiv arXiv 2025

  44. [44]

    Crystal Melting and Toric Calabi-Yau Manifolds

    Ooguri, Hirosi and Yamazaki, Masahito. Crystal Melting and Toric Calabi-Yau Manifolds. Commun. Math. Phys. 2009. doi:10.1007/s00220-009-0836-y. arXiv:0811.2801

    Pith/arXiv arXiv doi:10.1007/s00220-009-0836-y 2009

  45. [45]

    Non-commutative Donaldson-Thomas theory and the conifold

    Szendroi, Balazs. Non-commutative Donaldson Thomas invariants and the conifold. Geom. Topol. 2008. doi:10.2140/gt.2008.12.1171. arXiv:0705.3419

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.2140/gt.2008.12.1171 2008

  46. [46]

    Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions

    Chuang, Wu-yen and Jafferis, Daniel L. Wall Crossing of BPS States on the Conifold from Seiberg Duality and Pyramid Partitions. Commun. Math. Phys. 2009. doi:10.1007/s00220-009-0832-2. arXiv:0810.5072

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00220-009-0832-2 2009

  47. [47]

    Refined, Motivic, and Quantum

    Dimofte, Tudor and Gukov, Sergei. Refined, Motivic, and Quantum. Lett. Math. Phys. 2010. doi:10.1007/s11005-009-0357-9. arXiv:0904.1420

    Pith/arXiv arXiv doi:10.1007/s11005-009-0357-9 2010

  48. [48]

    The Non-commutative Topological Vertex and Wall Crossing Phenomena

    Nagao, Kentaro and Yamazaki, Masahito. The Non-commutative Topological Vertex and Wall Crossing Phenomena. Adv. Theor. Math. Phys. 2010. doi:10.4310/ATMP.2010.v14.n4.a3. arXiv:0910.5479

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4310/atmp.2010.v14.n4.a3 2010

  49. [49]

    Open BPS Wall Crossing and M-theory

    Aganagic, Mina and Yamazaki, Masahito. Open BPS Wall Crossing and M-theory. Nucl. Phys. B. 2010. doi:10.1016/j.nuclphysb.2010.03.019. arXiv:0911.5342

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2010.03.019 2010

  50. [50]

    Wall Crossing As Seen By Matrix Models

    Ooguri, Hirosi and Sulkowski, Piotr and Yamazaki, Masahito. Wall Crossing As Seen By Matrix Models. Commun. Math. Phys. 2011. doi:10.1007/s00220-011-1330-x. arXiv:1005.1293

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s00220-011-1330-x 2011

  51. [51]

    Instantons, Quivers and Noncommutative Donaldson-Thomas Theory

    Cirafici, Michele and Sinkovics, Annamaria and Szabo, Richard J. Instantons, Quivers and Noncommutative Donaldson-Thomas Theory. Nucl. Phys. B. 2011. doi:10.1016/j.nuclphysb.2011.08.002. arXiv:1012.2725

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.nuclphysb.2011.08.002 2011

  52. [52]

    Crystal Melting and Wall Crossing Phenomena

    Yamazaki, Masahito. Crystal Melting and Wall Crossing Phenomena. Int. J. Mod. Phys. A. 2011. doi:10.1142/S0217751X11051482. arXiv:1002.1709

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0217751x11051482 2011

  53. [53]

    Geometry and Combinatorics of Crystal Melting

    Yamazaki, Masahito. Geometry and Combinatorics of Crystal Melting. RIMS Kokyuroku Bessatsu B. 2011. arXiv:1102.0776

    Pith/arXiv arXiv 2011

  54. [54]

    Magnificent four

    Nekrasov, Nikita. Magnificent four. Adv. Theor. Math. Phys. 2020. doi:10.4310/ATMP.2020.v24.n5.a4. arXiv:1712.08128

    Pith/arXiv arXiv doi:10.4310/atmp.2020.v24.n5.a4 2020

  55. [55]

    Magnificent Four with Colors

    Nekrasov, Nikita and Piazzalunga, Nicol \`o. Magnificent Four with Colors. Commun. Math. Phys. 2019. doi:10.1007/s00220-019-03426-3. arXiv:1808.05206

    Pith/arXiv arXiv doi:10.1007/s00220-019-03426-3 2019

  56. [56]

    Global Magni4icence, or: 4G Networks

    Nekrasov, Nikita and Piazzalunga, Nicol \`o. Global Magni4icence, or: 4G Networks. SIGMA. 2024. doi:10.3842/SIGMA.2024.106. arXiv:2306.12995

    doi:10.3842/sigma.2024.106 2024

  57. [57]

    Charging solid partitions

    Galakhov, Dmitry and Li, Wei. Charging solid partitions. JHEP. 2024. doi:10.1007/JHEP01(2024)043. arXiv:2311.02751

    Pith/arXiv arXiv doi:10.1007/jhep01(2024)043 2024

  58. [58]

    Tetrahedron Instantons on Orbifolds

    Szabo, Richard J. and Tirelli, Michelangelo. Tetrahedron instantons on orbifolds. Lett. Math. Phys. 2025. doi:10.1007/s11005-025-01903-6. arXiv:2405.14792

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11005-025-01903-6 2025

  59. [59]

    An overview of crystals and double quiver Yangians

    Bao, Jiakang. An overview of crystals and double quiver Yangians. Int. J. Mod. Phys. A. 2026. doi:10.1142/S0217751X25300182. arXiv:2509.16918

    work page doi:10.1142/s0217751x25300182 2026

  60. [60]

    Quiver Yangian from Crystal Melting

    Li, Wei and Yamazaki, Masahito. Quiver Yangian from Crystal Melting. JHEP. 2020. doi:10.1007/JHEP11(2020)035. arXiv:2003.08909

    Pith/arXiv arXiv doi:10.1007/jhep11(2020)035 2020

  61. [61]

    Quiver Yangian and Supersymmetric Quantum Mechanics

    Galakhov, Dmitry and Yamazaki, Masahito. Quiver Yangian and Supersymmetric Quantum Mechanics. Commun. Math. Phys. 2022. doi:10.1007/s00220-022-04490-y. arXiv:2008.07006

    Pith/arXiv arXiv doi:10.1007/s00220-022-04490-y 2022

  62. [62]

    Toroidal and elliptic quiver BPS algebras and beyond

    Galakhov, Dmitry and Li, Wei and Yamazaki, Masahito. Toroidal and elliptic quiver BPS algebras and beyond. JHEP. 2022. doi:10.1007/JHEP02(2022)024. arXiv:2108.10286

    Pith/arXiv arXiv doi:10.1007/jhep02(2022)024 2022

  63. [63]

    A Note on Quiver Quantum Toroidal Algebra

    Noshita, Go and Watanabe, Akimi. A note on quiver quantum toroidal algebra. JHEP. 2022. doi:10.1007/JHEP05(2022)011. arXiv:2108.07104

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2022)011 2022

  64. [64]

    Quiver Yangians and crystal meltings: A concise summary

    Yamazaki, Masahito. Quiver Yangians and crystal meltings: A concise summary. J. Math. Phys. 2023. doi:10.1063/5.0089785. arXiv:2203.14314

    Pith/arXiv arXiv doi:10.1063/5.0089785 2023

  65. [65]

    Quiver elliptic W-algebras

    Kimura, Taro and Pestun, Vasily. Quiver elliptic W-algebras. Lett. Math. Phys. 2018. doi:10.1007/s11005-018-1073-0. arXiv:1608.04651

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/s11005-018-1073-0 2018

  66. [66]

    On Quiver W-algebras and Defects from Gauge Origami

    Koroteev, Peter. On Quiver W-algebras and Defects from Gauge Origami. Phys. Lett. B. 2020. doi:10.1016/j.physletb.2019.135101. arXiv:1908.04394

    work page doi:10.1016/j.physletb.2019.135101 2020

  67. [67]

    Gauge origami and quiver W-algebras II: Vertex function and beyond quantum q -Langlands correspondence

    Kimura, Taro and Noshita, Go. Gauge origami and quiver W-algebras II: Vertex function and beyond quantum q -Langlands correspondence. 2024. arXiv:2404.17061

    Pith/arXiv arXiv 2024

  68. [68]

    Gauge origami and quiver W-algebras

    Kimura, Taro and Noshita, Go. Gauge origami and quiver W-algebras. Part III. Donaldson-Thomas qq-characters. JHEP. 2025. doi:10.1007/JHEP03(2025)050. arXiv:2411.01987

    Pith/arXiv arXiv doi:10.1007/jhep03(2025)050 2025

  69. [69]

    Gauge origami and quiver W-algebras

    Kimura, Taro and Noshita, Go. Gauge origami and quiver W-algebras. Part IV. Pandharipande-Thomas qq-characters. JHEP. 2026. doi:10.1007/JHEP01(2026)063. arXiv:2508.12125

    work page doi:10.1007/jhep01(2026)063 2026

  70. [70]

    The 4-fold Pandharipande Thomas vertex and Jeffrey Kirwan residue

    Kimura, Taro and Noshita, Go. The 4-fold Pandharipande Thomas vertex and Jeffrey Kirwan residue. Lett. Math. Phys. 2026. doi:10.1007/s11005-026-02071-x. arXiv:2508.12128

    work page doi:10.1007/s11005-026-02071-x 2026

  71. [71]

    BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters

    Nekrasov, Nikita. BPS/CFT Correspondence III: Gauge Origami partition function and qq-characters. Commun. Math. Phys. 2018. doi:10.1007/s00220-017-3057-9. arXiv:1701.00189

    Pith/arXiv arXiv doi:10.1007/s00220-017-3057-9 2018

  72. [72]

    Crystal Melting, Triality and Partition Functions for Toric Calabi-Yau Fourfolds

    Carcamo, Mario and Franco, Sebasti \'a n. Crystal Melting, Triality and Partition Functions for Toric Calabi-Yau Fourfolds. 2026. arXiv:2603.08815

    arXiv 2026

  73. [73]

    Quantum foam and topological strings

    Iqbal, Amer and Nekrasov, Nikita and Okounkov, Andrei and Vafa, Cumrun. Quantum foam and topological strings. JHEP. 2008. doi:10.1088/1126-6708/2008/04/011. arXiv:hep-th/0312022

    Pith/arXiv arXiv doi:10.1088/1126-6708/2008/04/011 2008

  74. [74]

    Wall-crossing effects on quiver BPS algebras

    Galakhov, Dmitry and Morozov, Alexei and Tselousov, Nikita. Wall-crossing effects on quiver BPS algebras. JHEP. 2024. doi:10.1007/JHEP05(2024)118. arXiv:2403.14600

    Pith/arXiv arXiv doi:10.1007/jhep05(2024)118 2024

  75. [75]

    Quivers, YBE and 3-manifolds

    Yamazaki, Masahito. Quivers, YBE and 3-manifolds. JHEP. 2012. doi:10.1007/JHEP05(2012)147. arXiv:1203.5784

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/jhep05(2012)147 2012

  76. [76]

    Ronkin, L. I. , TITLE =. Complex analysis in modern mathematics (Russian) , PAGES =. 2001 , MRCLASS =

    2001

  77. [77]

    Amoebas,

    Passare, Mikael and Rullg. Amoebas,. Duke Math. J. , FJOURNAL =. 2004 , NUMBER =

    2004

  78. [78]

    P. W. Kasteleyn. The statistics of dimers on a lattice. Physica. 1961

    1961