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REVIEW 3 major objections 4 minor 74 references

Tilting Mutations and Quiver-Invariant Dualities in Brane Tilings

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper shows that quiver-invariant dualities among toric phases of the Calabi-Yau 3-fold $H^{1,1,2,1}$ are realized by tilting mutations, producing five doublets and one triplet of phases that share identical quivers but different…

desk verdict A useful catalogue of quiver-invariant dualities, but the key winding-number verification is asserted, not shown. read the letter →

arxiv 2608.07269 v1 pith:RLX4UGVV submitted 2026-08-07 hep-th math-phmath.AGmath.MP

Seong-Jin Lee , Rak-Kyeong Seong , Benjamin Suzzoni This is my paper · ORCID
classification hep-thmath-phmath.AGmath.MP
keywords branetilingsdimermodelsquivergaugetheoriesSeibergdualitytoricCalabi-Yau3-foldsquiver-invarianttiltingmutationszig-zagpaths
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 aims to prove that quiver-invariant dualities—distinct 4d N=1 supersymmetric gauge theories that share the same quiver up to relabelling but have different superpotentials—are produced by tilting mutations of brane tilings. The authors work with the toric Calabi-Yau 3-fold $H^{1,1,2,1}$, which admits 42 toric phases, and find that five pairs and one triple of these phases have identical quivers. They show that each such relation is realized by one of three families of tilting mutations, all of which reverse the orientation of zig-zag paths inside the mutation region. For each example, they verify that the mutated tiling has the same toric Calabi-Yau by checking that the zig-zag winding numbers are preserved up to a common $GL(2,\mathbb{Z})$ transformation. If correct, this gives a local, single-step geometric mechanism behind duality chains that were previously only described as sequences of Seiberg dualities.

What carries the argument

The central object is the tilting mutation, a local operation on a brane tiling (or its dual periodic quiver) that exchanges two oriented paths $U_a,V_a$ through the mutation region while keeping the complementary paths $P_a,Q_a$ fixed, and flips the signs of the internal terms $\Omega_{\rm mut}$. The accompanying criterion is the winding-number set (3.5)--(3.6): the zig-zag paths of the tiling, each carrying a primitive winding vector $(p_i,q_i)\in\mathbb{Z}^2$, must map under the mutation to the same multiset up to one common $GL(2,\mathbb{Z})$ transformation. This condition guarantees that the toric diagram, and hence the mesonic moduli space, is unchanged. In the three families studied, the distinguishing feature is that zig-zag paths inside the mutation region reverse their orientation (one path in family A, two in family B), while the rest of the tiling's zig-zag data is adjusted by choosing the untouched superpotential terms $W_0$ so that (3.6) holds.

What would settle it

Recompute, for the five doublets and the triplet, the zig-zag winding numbers of the brane tilings in Appendix A and test whether a single $G\in GL(2,\mathbb{Z})$ maps one set onto the other; the claim fails if any pair violates condition (3.6), or if the toric diagram computed from the perfect matchings of either mutated tiling differs from $H^{1,1,2,1}$.

Watch

Extended reading notes

Core claim

The paper's central claim is that the quiver-invariant dualities observed among the 42 toric phases of $H^{1,1,2,1}$ are not accidents of long Seiberg-duality chains but are realized directly by local tilting mutations. A tilting mutation acts on a region of the periodic quiver containing two paths $U_a$ and $V_a$ with the same endpoints; it exchanges them, replaces every closed superpotential cycle $U_a P_a - V_a Q_a$ by $V_a P_a - U_a Q_a$, and reverses the signs of the superpotential terms $\Omega_{\rm mut}$ supported inside the mutation region. The paper identifies three general families (A, B, C) of such mutations, distinguished by which zig-zag paths inside the mutation region reverse orientation, and shows that in $H^{1,1,2,1}$ they account for five doublets and one triplet of toric phases with identical quivers but distinct superpotentials. The verification that the mutated tiling describes the same Calabi-Yau 3-fold is the winding-number condition (3.6): the complete set of zig-zag winding numbers is preserved up to a common $G\in GL(2,\mathbb{Z})$. The paper also proves that both the original and mutated superpotentials satisfy the toric condition, so every chiral field still appears exactly once with a plus sign and once with a minus sign.

Load-bearing premise

The whole argument rests on the computed fact that, for each of the six examples, the set of winding numbers of the zig-zag paths is unchanged up to a single common change of two-dimensional lattice basis; that check is done case by case, not proven once for the full families, and an error in any one check would break the claimed duality.

Editorial extensions

If this is right

  • If the central claim is correct, then quiver-invariant dualities lift from long Seiberg-duality chains to single local mutations of brane tilings.
  • The winding-number condition gives a computable signature for recognizing quiver-invariant dual pairs in other toric Calabi-Yau geometries.
  • The three mutation families provide templates for constructing new quiver-invariant dualities within the $H^{a,b,c,d}$ family of toric models.
  • In the $H^{1,1,2,1}$ duality tree, every phase that appears as an intermediate in a quiver-invariant chain is itself a member of one of the doublets or the triplet, so these relations form a closed substructure of the duality tree.

Reading between the lines

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

  • Beyond the paper, the winding-number criterion looks like it could serve as an algorithmic detector: scan brane tilings for pairs with equal zig-zag winding multisets up to $GL(2,\mathbb{Z})$, and the tilting mutation that connects them is then a candidate quiver-invariant duality.
  • The fact that all three families reverse zig-zag orientations suggests the reversal itself may be the mechanism; testing whether any quiver-invariant tilting mutation in the larger $H^{a,b,c,d}$ family also reverses a zig-zag path would show whether this is a general law.
  • The paper notes that similar examples appeared in orientifold settings; if tilting mutations survive orientifold projections, quiver-invariant dualities would also appear in orientifolded brane tilings.
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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 studies quiver-invariant dualities — pairs or triples of 4d N=1 gauge theories that share the same quiver up to relabelling but have distinct superpotentials — in the brane tiling model associated with the toric Calabi-Yau 3-fold H^{1,1,2,1}. The authors introduce three families of local "tilting mutations" (Section 3) that act on a region of the brane tiling while leaving the quiver invariant, and they claim that these mutations realize quiver-invariant dualities. Among the 42 toric phases of H^{1,1,2,1} they identify five doublets and one triplet of phases with identical quivers, and for each pair/triple they assert that the tilting mutation preserves the toric Calabi-Yau by checking condition (3.6), the equality of zig-zag winding-number multisets up to a common GL(2,Z) transformation. Explicit superpotentials, quivers, and global symmetry charges are presented for the examples in Sections 4.1–4.6, and Appendix A lists all 42 phases.

Significance. If the winding-number checks are correct, the paper provides the first systematic realization of quiver-invariant dualities via a single local mutation, going beyond the single example of the authors' prior work [23]. The three mutation families and the explicit catalogue of phases are potentially useful for further studies of Seiberg duality trees on large toric diagrams. The verification is based on the externally established zig-zag criterion, not on fitted parameters, and no circularity is apparent. The strength of the work lies in the detailed, explicit model data; its main weakness is that the load-bearing verification of condition (3.6) is not exhibited for any of the six cases, so the central empirical claim cannot currently be audited by the reader.

major comments (3)
  1. [§4.1–4.6, Eq. (3.6)] The central claim that the six listed doublets/triplet are quiver-invariant dualities rests on the asserted equality Sw(W') = G Sw(W) for some G in GL(2,Z). The paper does not display the multiset Sw(W), the mutated multiset Sw(W'), or the matrix G for any of the six examples, nor does it provide an ancillary file or code. Since an error in any one of these unexhibited checks would remove the corresponding pair or triple from the catalogue, this is a load-bearing gap. Please add the full winding data for each example, or provide a reproducible script that computes these multisets and verifies (3.6) for all cases.
  2. [§2.3, Table 2, Appendix A] The statement that H^{1,1,2,1} admits exactly 42 toric phases, and that these split into 29 singlets, 5 doublets, and 1 triplet, is asserted without derivation or a reproducible enumeration. If the enumeration is incomplete, the paper's claim to have identified all quiver-invariant dualities among the phases would be unsupported. Please provide the method used to exhaust the Seiberg-duality tree (e.g., generation from a known phase by iterated spider moves, with a termination criterion) or a reference that allows the reader to reproduce the count.
  3. [§3.1–3.3] The paper calls the three constructions "general families" of tilting mutations, but the preservation of the mesonic moduli space is only verified for the specific instances in Section 4, and the text itself notes that condition (3.6) 'restricts the choice of W0' unaffected by the mutation. It is therefore not established that every mutation of family A, B, or C preserves the Calabi-Yau for arbitrary W0; the families are characterized by the local sign structure and the orientation reversal of zig-zag paths, but the sufficient condition on W0 is not given. Please state clearly which statements hold for the whole family and which are only checked example-wise, or derive the general condition that W0 must satisfy for (3.6) to hold in each family.
minor comments (4)
  1. [Abstract and §1] The abbreviation "H1121" in the abstract and the notation "H^{1,1,2,1}" used in the body should be used consistently throughout the paper.
  2. [Figures 8, 10, 12] The winding numbers and path labels in the zig-zag figures are difficult to read at the current resolution; vector graphics or larger insets would make the claimed orientation reversals easier to verify.
  3. [§3, Eq. (3.1)] The definition of the paths P_a, Q_a, U_a, V_a is informal; it would help to specify whether these are unique shortest paths in the periodic quiver or chosen representatives, and to state how the signs of Ω_mut are fixed.
  4. [§2.1, Eq. (1.1)] For the H^{a,b,c,d} family, the convexity condition d + a(c−b)>0 is stated without derivation; a brief explanation or reference would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the winding-number criterion (3.6) is an external consistency check applied example-wise, and the cited [23] result is background, not the load-bearing input.

full rationale

The paper's central mechanism is a local tilting mutation that exchanges paths U_a and V_a and flips signs in Omega_mut (eqs. 3.1 and 3.2); this is an explicit transformation of the superpotential, not a definition that presupposes the final toric phase. The claim that a mutated brane tiling describes the same toric Calabi-Yau 3-fold is checked through the independently established zig-zag winding condition (3.6), namely the existence of a single G in GL(2,Z) with Sw(W') = G Sw(W). The six examples in Sections 4.1-4.6 are presented as concrete verifications of this condition; the paper does not fit any parameter to the target conclusion, and the criterion is not derived from the examples. The only notable self-reference is to the authors' earlier work [23], which introduced one tilting mutation and one doublet; here that prior result is used as background and context, while the three mutation families and the five additional doublets plus one triplet are new content checked against (3.6). The sceptical concern that the winding-number verifications are not displayed in full is a reproducibility or auditability issue, not circularity: nothing in the argument defines the predicted quiver-invariant duality in terms of the inputs of the mutation, and the paper does not rename a fit as a prediction. No self-definitional, fitted-input, self-citation-load-bearing, or ansatz-smuggling step can be exhibited from the text. The paper is therefore not circular in the sense used here; at most it would invite an independent re-check of the example-wise computations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted in this work; the model coordinates a,b,c,d and the R-charges are determined by geometry and a-maximization. The paper builds entirely on the standard brane tiling framework and its own prior definition of tilting mutation, with no new physical entities postulated.

assumptions (4)
  • domain assumption Brane tiling / dimer model dictionary maps faces to gauge groups, edges to bifundamentals, nodes to superpotential terms; Seiberg duality is realized by urban renewal (spider move).
    Invoked in Sections 2.1 and 2.2 without proof; standard in the field.
  • domain assumption Two consistent brane tilings describe the same toric Calabi-Yau 3-fold iff their zig-zag path winding numbers coincide up to a common GL(2,Z) transformation.
    Used as the criterion (3.6) to verify moduli space preservation; cited from refs [31-34].
  • domain assumption The list of 42 toric phases of H^{1,1,2,1} is complete and correct.
    The classification in Section 2.3 and Appendix A is presented as computed; no derivation or independent verification is shown.
  • ad hoc to paper For the tilting mutations considered, the mutated superpotential W' satisfies the toric condition and the quiver is unchanged after integrating out massive fields.
    Stated in Section 3 (after Eq. 3.2) and checked for the examples, but not proven for the general families.

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Pith. "Pith review of Tilting Mutations and Quiver-Invariant Dualities in Brane Tilings." pith.science (2026). https://pith.science/paper/RLX4UGVV

@misc{pith2026260807269,
  author       = {Pith},
  title        = {Pith review of: Tilting Mutations and Quiver-Invariant Dualities in Brane Tilings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLX4UGVV}},
  note         = {Machine review of arXiv:2608.07269}
}
read the original abstract

Quiver-invariant dualities relate distinct 4d N=1 supersymmetric gauge theories that arise as worldvolume theories on a D3-brane probing the same toric Calabi-Yau 3-fold. These dual theories share an identical quiver and differ only in their superpotentials. We show that quiver-invariant dualities are realized by tilting mutations on the brane tilings that realize these theories. Among the 42 toric phases of the H1121 model, we identify three general families of tilting mutations, all characterized by the reversal of the orientations of zig-zag paths within the mutation region of the brane tiling. We present new examples of quiver-invariant dualities, given by five doublets and one triplet of toric phases with identical quivers, for which we verify that the brane tilings related by the tilting mutations correspond to the same toric Calabi-Yau 3-fold H1121.

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Works this paper leans on

74 extracted references · 17 canonical work pages

  1. [23]

    Kho, S.-J

    M. Kho, S.-J. Lee and R.-K. Seong,Quiver-invariant dualities between brane tilings, Phys. Rev. D113(2026) 126007, [2601.20936]

    arXiv 2026

  2. [1]

    M. R. Douglas, B. R. Greene and D. R. Morrison,Orbifold resolution by D-branes, Nucl. Phys. B506(1997) 84–106, [hep-th/9704151]

    arXiv 1997

  3. [2]

    M. R. Douglas and G. W. Moore,D-branes, quivers, and ALE instantons, hep-th/9603167

    arXiv

  4. [3]

    B. Feng, A. Hanany and Y.-H. He,D-brane gauge theories from toric singularities and toric duality,Nucl. Phys. B595(2001) 165–200, [hep-th/0003085]

    arXiv 2001

  5. [4]

    B. Feng, A. Hanany and Y.-H. He,Phase structure of D-brane gauge theories and toric duality,JHEP08(2001) 040, [hep-th/0104259]

    arXiv 2001

  6. [5]

    Franco, A

    S. Franco, A. Hanany, K. D. Kennaway, D. Vegh and B. Wecht,Brane dimers and quiver gauge theories,JHEP01(2006) 096, [hep-th/0504110]

    arXiv 2006

  7. [6]

    Hanany and K

    A. Hanany and K. D. Kennaway,Dimer models and toric diagrams,hep-th/0503149

    arXiv

  8. [7]

    Franco, A

    S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht,Gauge theories from toric geometry and brane tilings,JHEP01(2006) 128, [hep-th/0505211]

    arXiv 2006

Show all 74 references
  1. [8]

    Kenyon,An introduction to the dimer model,arXiv Mathematics e-prints(Oct.,

    R. Kenyon,An introduction to the dimer model,arXiv Mathematics e-prints(Oct.,

  2. [9]

    Kasteleyn,Graph theory and crystal physics,Graph theory and theoretical physics (1967) 43–110

    P. Kasteleyn,Graph theory and crystal physics,Graph theory and theoretical physics (1967) 43–110

    1967

  3. [10]

    Seiberg,Electric - magnetic duality in supersymmetric nonAbelian gauge theories, Nucl

    N. Seiberg,Electric - magnetic duality in supersymmetric nonAbelian gauge theories, Nucl. Phys. B435(1995) 129–146, [hep-th/9411149]

    1995 arXiv

  4. [11]

    B. Feng, A. Hanany, Y.-H. He and A. M. Uranga,Toric duality as Seiberg duality and brane diamonds,JHEP12(2001) 035, [hep-th/0109063]

    2001 arXiv

  5. [12]

    B. Feng, S. Franco, A. Hanany and Y.-H. He,Symmetries of toric duality,JHEP12 (2002) 076, [hep-th/0205144]

    2002 arXiv

  6. [13]

    Butti, D

    A. Butti, D. Forcella, A. Hanany, D. Vegh and A. Zaffaroni,Counting Chiral Operators in Quiver Gauge Theories,JHEP11(2007) 092, [0705.2771]. – 58 –

    2007 arXiv

  7. [14]

    Forcella, A

    D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni,The Master Space of N=1 Gauge Theories,JHEP08(2008) 012, [0801.1585]

    2008 arXiv

  8. [15]

    Forcella, A

    D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni,Mastering the Master Space,Lett. Math. Phys.85(2008) 163–171, [0801.3477]

    2008 arXiv

  9. [16]

    Fulton,Introduction to toric varieties

    W. Fulton,Introduction to toric varieties. No. 131. Princeton university press, 1993

    1993

  10. [17]

    D. Cox, J. Little and H. Schenck,Graduate studies in mathematics, 2011

    2011

  11. [18]

    A. B. Goncharov and R. Kenyon,Dimers and cluster integrable systems,arXiv e-prints (July, 2011) arXiv:1107.5588, [1107.5588]

    2011 arXiv

  12. [19]

    Ciucu,A complementation theorem for perfect matchings of graphs having a cellular completion,Journal of Combinatorial Theory, Series A81(1998) 34–68

    M. Ciucu,A complementation theorem for perfect matchings of graphs having a cellular completion,Journal of Combinatorial Theory, Series A81(1998) 34–68

    1998

  13. [20]

    R. W. Kenyon, J. G. Propp and D. B. Wilson,Trees and Matchings,arXiv Mathematics e-prints(Mar., 1999) math/9903025, [math/9903025]

    1999 arXiv

  14. [21]

    Franco, A

    S. Franco, A. Hanany, Y.-H. He and P. Kazakopoulos,Duality walls, duality trees and fractional branes,Int. J. Mod. Phys. A40(2025) 2450143

    2025

  15. [22]

    Hanany and R.-K

    A. Hanany and R.-K. Seong,Brane Tilings and Reflexive Polygons,Fortsch. Phys.60 (2012) 695–803, [1201.2614]

    2012 arXiv

  16. [24]

    Amariti, M

    A. Amariti, M. Bianchi, M. Fazzi, S. Mancani, F. Riccioni and S. Rota, Multi-planarizable quivers, orientifolds, and conformal dualities,JHEP09(2023) 094, [2212.03913]

    2023 arXiv

  17. [25]

    Witten,Phases of N=2 theories in two-dimensions,Nucl

    E. Witten,Phases of N=2 theories in two-dimensions,Nucl. Phys. B403(1993) 159–222, [hep-th/9301042]

    1993 arXiv

  18. [26]

    Kenyon, A

    R. Kenyon, A. Okounkov and S. Sheffield,Dimers and amoebae,math-ph/0311005

    arXiv

  19. [27]

    Hanany, C

    A. Hanany, C. P. Herzog and D. Vegh,Brane tilings and exceptional collections,JHEP 07(2006) 001, [hep-th/0602041]

    2006 arXiv

  20. [28]

    K. A. Intriligator and B. Wecht,The Exact superconformal R symmetry maximizes a, Nucl. Phys. B667(2003) 183–200, [hep-th/0304128]

    2003 arXiv

  21. [29]

    Butti and A

    A. Butti and A. Zaffaroni,R-charges from toric diagrams and the equivalence of a-maximization and Z-minimization,JHEP11(2005) 019, [hep-th/0506232]

    2005 arXiv

  22. [30]

    Butti and A

    A. Butti and A. Zaffaroni,From toric geometry to quiver gauge theory: The Equivalence of a-maximization and Z-minimization,Fortsch. Phys.54(2006) 309–316, [hep-th/0512240]

    2006 arXiv

  23. [31]

    Hanany and D

    A. Hanany and D. Vegh,Quivers, tilings, branes and rhombi,JHEP10(2007) 029, [hep-th/0511063]. – 59 –

    2007 arXiv

  24. [32]

    Kenyon and J.-M

    R. Kenyon and J.-M. Schlenker,Rhombic embeddings of planar graphs with faces of degree 4,arXiv e-prints(May, 2003) math–ph/0305057, [math-ph/0305057]

    2003

  25. [33]

    Jejjala, S

    V. Jejjala, S. Ramgoolam and D. Rodriguez-Gomez,Toric CFTs, Permutation Triples and Belyi Pairs,JHEP03(2011) 065, [1012.2351]

    2011 arXiv

  26. [34]

    Hanany and R.-K

    A. Hanany and R.-K. Seong,Brane Tilings and Specular Duality,JHEP08(2012) 107, [1206.2386]

    2012 arXiv

  27. [35]

    Aharony, A

    O. Aharony, A. Hanany and B. Kol,Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams,JHEP01(1998) 002, [hep-th/9710116]

    1998 arXiv

  28. [36]

    Aharony and A

    O. Aharony and A. Hanany,Branes, superpotentials and superconformal fixed points, Nucl. Phys. B504(1997) 239–271, [hep-th/9704170]

    1997 arXiv

  29. [37]

    N. C. Leung and C. Vafa,Branes and toric geometry,Adv. Theor. Math. Phys.2 (1998) 91–118, [hep-th/9711013]

    1998 arXiv

  30. [38]

    Cremonesi, A

    S. Cremonesi, A. Hanany and R.-K. Seong,Double Handled Brane Tilings,JHEP10 (2013) 001, [1305.3607]

    2013 arXiv

  31. [39]

    Akhtar, T

    M. Akhtar, T. Coates, S. Galkin and A. M. Kasprzyk,Minkowski Polynomials and Mutations,arXiv e-prints(Dec., 2012) arXiv:1212.1785, [1212.1785]

    2012 arXiv

  32. [40]

    Higashitani and Y

    A. Higashitani and Y. Nakajima,Deformations of Dimer Models,SIGMA18(Apr.,

  33. [41]

    D. Ghim, M. Kho and R.-K. Seong,Combinatorial and algebraic mutations of toric Fano 3-folds and mass deformations of 2d(0,2) quiver gauge theories,Phys. Rev. D 110(2024) 086001, [2407.19924]

    2024 arXiv

  34. [42]

    D. Ghim, M. Kho and R.-K. Seong,Birational transformations and 2d (0, 2) quiver gauge theories beyond toric Fano 3-folds,JHEP06(2025) 032, [2502.08741]

    2025 arXiv

  35. [43]

    M. Kho, N. Lee and R.-K. Seong,Birational transformations on dimer integrable systems,Phys. Rev. D112(2025) L041901, [2504.21081]

    2025 arXiv

  36. [44]

    Eager, S

    R. Eager, S. Franco and K. Schaeffer,Dimer Models and Integrable Systems,JHEP06 (2012) 106, [1107.1244]

    2012 arXiv

  37. [45]

    M. Kho, N. Lee and R.-K. Seong,Classification and Birational Equivalence of Dimer Integrable Systems for Reflexive Polygons,2510.12290

  38. [46]

    M. Kho, K. Lee, N. Lee and R.-K. Seong,Integrable Systems for Generalized Toric Polygons and Higgsed 5d N=1 Theories,2603.19353

  39. [47]

    Benini, S

    F. Benini, S. Benvenuti and Y. Tachikawa,Webs of five-branes and N=2 superconformal field theories,JHEP09(2009) 052, [0906.0359]

    2009 arXiv

  40. [48]

    van Beest, A

    M. van Beest, A. Bourget, J. Eckhard and S. Schafer-Nameki,(Symplectic) Leaves and – 60 – (5d Higgs) Branches in the Poly(go)nesian Tropical Rain Forest,JHEP11(2020) 124, [2008.05577]

    2020 arXiv

  41. [49]

    Bourget, A

    A. Bourget, A. Collinucci and S. Schafer-Nameki,Generalized toric polygons, T-branes, and 5d SCFTs,SciPost Phys.18(2025) 079, [2301.05239]

    2025 arXiv

  42. [50]

    Carre˜ no Bolla, S

    I. Carre˜ no Bolla, S. Franco and D. Rodr ´ ıguez-G´ omez,The 5d tangram: brane webs, 7-branes and primitive T-cones,JHEP05(2025) 175, [2411.01510]

    2025 arXiv

  43. [51]

    Franco and R.-K

    S. Franco and R.-K. Seong,Twin theories, polytope mutations and quivers for GTPs, JHEP07(2023) 034, [2302.10951]

    2023 arXiv

  44. [52]

    Arias-Tamargo, S

    G. Arias-Tamargo, S. Franco and D. Rodr ´ ıguez-G´ omez,The geometry of GTPs and 5d SCFTs,JHEP07(2024) 159, [2403.09776]

    2024 arXiv

  45. [53]

    Franco,Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes, JHEP11(2012) 141, [1207.0807]

    S. Franco,Bipartite Field Theories: from D-Brane Probes to Scattering Amplitudes, JHEP11(2012) 141, [1207.0807]

    2012 arXiv

  46. [54]

    Franco, D

    S. Franco, D. Galloni and R.-K. Seong,New Directions in Bipartite Field Theories, JHEP06(2013) 032, [1211.5139]

    2013 arXiv

  47. [55]

    Kho and R.-K

    M. Kho and R.-K. Seong,Hilbert series of bipartite field theories,JHEP09(2024) 020, [2405.19165]

    2024 arXiv

  48. [56]

    Franco, D

    S. Franco, D. Ghim, S. Lee, R.-K. Seong and D. Yokoyama,2d (0,2) Quiver Gauge Theories and D-Branes,JHEP09(2015) 072, [1506.03818]

    2015 arXiv

  49. [57]

    Franco, S

    S. Franco, S. Lee and R.-K. Seong,Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0,2) Quivers,JHEP02(2016) 047, [1510.01744]

    2016 arXiv

  50. [58]

    Franco, S

    S. Franco, S. Lee and R.-K. Seong,Brane brick models and 2d (0, 2) triality,JHEP05 (2016) 020, [1602.01834]

    2016 arXiv

  51. [59]

    Franco, S

    S. Franco, S. Lee, R.-K. Seong and C. Vafa,Brane Brick Models in the Mirror,JHEP 02(2017) 106, [1609.01723]

    2017 arXiv

  52. [60]

    Franco, S

    S. Franco, S. Lee, R.-K. Seong and C. Vafa,Quadrality for Supersymmetric Matrix Models,JHEP07(2017) 053, [1612.06859]

    2017 arXiv

  53. [61]

    Franco, D

    S. Franco, D. Ghim, S. Lee and R.-K. Seong,Elliptic Genera of 2d (0,2) Gauge Theories from Brane Brick Models,JHEP06(2017) 068, [1702.02948]

    2017 arXiv

  54. [62]

    Franco and R.-K

    S. Franco and R.-K. Seong,Fano 3-folds, reflexive polytopes and brane brick models, JHEP08(2022) 008, [2203.15816]

    2022 arXiv

  55. [63]

    Franco, D

    S. Franco, D. Ghim and R.-K. Seong,Brane brick models for the Sasaki-Einstein 7-manifolds Y p,k(CP1×CP 1) and Y p,k(CP2),JHEP03(2023) 050, [2212.02523]

    2023 arXiv

  56. [64]

    Kho and R.-K

    M. Kho and R.-K. Seong,On the master space for brane brick models,JHEP09(2023) 150, [2306.16616]. – 61 –

    2023 arXiv

  57. [65]

    Franco, D

    S. Franco, D. Ghim, G. P. Goulas and R.-K. Seong,Mass deformations of brane brick models,JHEP09(2023) 176, [2307.03220]

    2023 arXiv

  58. [66]

    Carcamo, S

    M. Carcamo, S. Franco, D. Ghim, G. P. Goulas and R.-K. Seong,Relevant deformations, brane brick models and triality,JHEP04(2026) 049, [2510.05517]

    2026

  59. [67]

    Kwon and R.-K

    J. Kwon and R.-K. Seong,Abelian Orbifolds for Brane Brick Models,2606.28110

    arXiv

  60. [68]

    Krefl and R.-K

    D. Krefl and R.-K. Seong,Machine Learning of Calabi-Yau Volumes,Phys. Rev. D96 (2017) 066014, [1706.03346]

    2017 arXiv

  61. [69]

    Choi and R.-K

    E. Choi and R.-K. Seong,Machine learning regularization for the minimum volume formula of toric Calabi-Yau 3-folds,Phys. Rev. D109(2024) 046015, [2310.19276]

    2024 arXiv

  62. [70]

    Seong,Unsupervised machine learning techniques for exploring tropical coamoeba, brane tilings and Seiberg duality,Phys

    R.-K. Seong,Unsupervised machine learning techniques for exploring tropical coamoeba, brane tilings and Seiberg duality,Phys. Rev. D108(2023) 106009, [2309.05702]

    2023 arXiv

  63. [71]

    Seong,Generative AI for brane configurations and coamoeba,Phys

    R.-K. Seong,Generative AI for brane configurations and coamoeba,Phys. Rev. D111 (2025) 086013, [2411.16033]

    2025 arXiv

  64. [72]

    J. Bao, S. Franco, Y.-H. He, E. Hirst, G. Musiker and Y. Xiao,Quiver Mutations, Seiberg Duality and Machine Learning,Phys. Rev. D102(2020) 086013, [2006.10783]

    2020 arXiv

  65. [73]

    Capuozzo, T

    P. Capuozzo, T. Schettini Gherardini and B. Suzzoni,Machine learning toric duality in brane tilings,Adv. Theor. Math. Phys.30(2026) 149–190, [2409.15251]. – 62 –

    2026 arXiv

  66. [2003]

    math/0310326, [math/0310326]

    arXiv

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.