Pith. sign in

REVIEW 4 major objections 4 minor 56 references

On holographic duals of certain isolated weighted Gorenstein cDV singularities

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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

Pith's one-line read A mirror-symmetry computation rules out crepant resolutions for 24 cDV singularities.

desk verdict A careful mirror-symmetry computation whose no-go conclusion rests on an unproved conjecture the authors explicitly concede; worth a referee, but the abstract oversells it. read the letter →

arxiv 2412.10698 v4 pith:O2K5QE4U submitted 2024-12-14 hep-th math-phmath.AGmath.MP

Yuanyuan Fang , Zekai Yu This is my paper · ORCID
classification hep-thmath-phmath.AGmath.MP MSC 14J1753D3716E4014E1581T60
keywords compoundDuValsingularitiescrepantresolutionsymplecticcohomologyhomologicalmirrorsymmetryHochschildmatrixfactorizationsN=1quivergaugetheoryAdS/CFTcorrespondence
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

The paper asks which weighted homogeneous Gorenstein compound Du Val (cDV) threefold singularities admit crepant resolutions, and therefore which can serve as backgrounds for 4d $\mathcal{N}=1$ superconformal quiver gauge theories on D3-branes. It argues that among the K-stable $E_6$ and $E_7$ families, only four types up to weight-one deformations are resolvable, and that none of the remaining unresolved cases listed in Problem 2.1 admit a crepant resolution. The route is indirect: compute a symplectic invariant of the Milnor fiber, the negative-degree symplectic cohomology, through homological mirror symmetry and Hochschild cohomology of equivariant matrix factorizations. A companion enumeration of all consistent two- and three-node quiver gauge theories finds no candidate dual, matching the geometric no-go result. The geometric conclusion is stated under the assumption of the unproved 'only if' direction of the conjecture of [28].

What carries the argument

The load-bearing object is the negative-degree symplectic cohomology $SH^{<0}(M_X)$ of the Milnor fiber of the singularity; the criterion from [28] says a crepant resolution with $l$ irreducible exceptional curves forces this invariant to have constant rank $l$ in every negative degree. Since direct computation is hard, the paper computes it on the mirror side: homological mirror symmetry identifies $SH^\bullet$ with the Hochschild cohomology $HH^\bullet$ of the category of equivariant matrix factorizations of the mirror superpotential, and the formula of [34] reduces each rank to counting solutions of explicit integral linear congruences associated to group elements fixing coordinate subspaces. Constant rank across all negative degrees is read off from tables; nonconstant rank is the no-go signal.

What would settle it

An explicit crepant resolution of any one of the 24 singularities in Problem 2.1, for example a small resolution of $x^2+y^3+z^4+y w^5$ with one exceptional curve, would falsify the main claim. Short of that, extending the Hochschild computations in Tables 3 and 4 to degrees below $-10$ and finding a previously excluded $k$ whose negative-degree ranks become constant would break the stabilization argument.

Watch

Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is a dichotomy for the K-stable weighted homogeneous cDV singularities with $J=E_n$ in Table 1: up to weight-one deformations, exactly four equations admit a crepant resolution, namely $x^2+y^3+z^4+w^{12}=0$, $x^2+y^3+yz^3+w^{18}=0$, $x^2+y^3+z^5+w^{30}=0$, and $x^2+y^3+yz^3+w^{2}z=0$. Consequently, none of the 24 singularities collected in Problem 2.1 admit a crepant resolution, and according to the AdS/CFT dictionary they cannot be holographic backgrounds for a 4d $\mathcal{N}=1$ superconformal quiver gauge theory. The argument runs through the symplectic-cohomology criterion of [28]: after mirror symmetry, the negative-degree ranks are computed as Hochschild cohomology ranks of equivariant matrix factorizations, and they are constant across all negative degrees exactly in the four known resolvable cases. The paper verifies the conclusion from the field-theory side by enumerating all consistent two- and three-node quiver gauge theories and finding none with the correct Hilbert series.

Load-bearing premise

The load-bearing premise is the unproved 'only if' direction of the conjecture of [28]: a cDV singularity with a crepant resolution containing $l$ irreducible exceptional curves must have negative-degree symplectic cohomology of constant rank $l$, and a counterexample to that direction would remove the force of the no-go conclusion.

Editorial extensions

If this is right

  • The 24 singularities in Problem 2.1 cannot serve as the transverse geometry for D3-branes in a 4d $\mathcal{N}=1$ superconformal quiver gauge theory, under the standard holographic dictionary.
  • Within the K-stable range, the only $E_n$-type singularities in the paper's list that admit a crepant resolution are the four displayed equations, up to weight-one deformations.
  • Constant negative-degree rank is a workable sufficient signal for crepant resolvability, and nonconstant rank is an obstruction, so the same Hochschild-side computation can be reused on other invertible cDV families.
  • The exhaustive quiver search finds no two- or three-node candidate with matching Hilbert series, so the field-theoretic side independently reproduces the geometric no-go.

Reading between the lines

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

  • If the 'only if' conjecture from [28] is true in full, the stabilization computation becomes a general finite arithmetic test for crepant resolutions of invertible cDV singularities, no resolution construction required.
  • The tables only probe degrees down to $-10$; proving a lower bound on the degrees where new contributions can appear, or extending the search much deeper, would close the residual arithmetic loophole in the stabilization claim.
  • The candidate quiver dualities found in the last section suggest that matching quiver Hilbert series can generate new gauge-theory dual pairs; comparing their large-N superconformal indices is a concrete check the paper leaves open.
  • The no-go statement targets quiver gauge theories; it does not rule out other 4d $\mathcal{N}=1$ SCFTs, such as non-Lagrangian or non-quiver theories, as holographic duals of these singularities.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies isolated weighted homogeneous Gorenstein compound Du Val (cDV) singularities of cE6 and cE7 type in the K-stable range, and claims that none of the singularities listed in Problem 2.1 admit a crepant resolution; the only exceptions are four known singularities (Claim 1.1). The method is homological: the negative-degree symplectic cohomology of the Milnor fiber is identified, via homological mirror symmetry, with the Hochschild cohomology of equivariant matrix factorizations of the Berglund–Hübsch mirror, and the Evans–Lekili criterion is used to convert constant negative-degree rank ℓ into existence of a crepant resolution with ℓ exceptional curves. The rank computations are worked out for the cE6 and cE7 families and are tabulated for degrees −10 ≤ d ≤ 2, with k = 8 and k = 2, 14 identified as the only cases whose ranks appear to stabilize. Section 4 attempts a field-theoretic cross-check by enumerating quiver gauge theories with matching Hilbert series. The paper is clearly written and contains a detailed worked example, but its main no-go conclusion is explicitly conditional on an unproved conjecture and on finite truncation of the degree range.

Significance. If fully established, the paper would resolve an open question about crepant resolutions of a family of K-stable cDV singularities and would rule out 4d N=1 superconformal quiver duals for those geometries, with direct consequences for AdS/CFT applications. The paper has genuine strengths: the translation of the symplectic-cohomology computation into finite linear congruences is explicit, the k=8 example is worked out in detail, and the attached Mathematica notebook makes the enumeration reproducible. However, the central non-existence claim is not proved as stated: it relies on the unproved 'only if' direction of the Evans–Lekili conjecture, the stabilization claims are read off from tables truncated at d = −10, and the physics-side 'confirmation' is a constrained search rather than an exhaustive enumeration. The conditional computational framework is interesting, but the significance of the unconditional claims is not currently supported.

major comments (4)
  1. [§2.3 and Claim 1.1] The main no-existence conclusion is conditional on the unproved 'only if' direction of the Evans–Lekili conjecture. The paper states this explicitly: 'Under the assumption that the “only if” direction of the above conjecture holds, this result leads to the main claim of our paper in the introduction.' Since the inference from nonconstant negative-degree symplectic cohomology ranks to absence of a crepant resolution is precisely the contrapositive of that conjecture direction, and since the conjecture is proved only for cA_n singularities, Claim 1.1 and the abstract's 'demonstrate' overstate what is established. The paper should either prove or cite a proof of the missing direction, or present the main result as explicitly conditional.
  2. [§3.2] The cE7 computation contains a self-flagged gap: the text notes that if γ fixes y and z but not w, then the restricted singularity Ẇ_γ is not isolated and 'the simplification below (2.11) may break down,' with only the expectation that the caveat is immaterial. The simplified counting of A/B/C monomials relies on the cohomology of the Koszul complex being concentrated in low degrees with a Jacobian-ring basis, and this is exactly what can fail for non-isolated Ẇ_γ. Since Table 4 and the resulting no-crepant-resolution conclusion depend on those counts, an argument showing that the non-isolated cases do not affect negative-degree ranks or HH^2=0 is needed.
  3. [§3.1–3.2, Tables 3 and 4] The stabilization statements are inferred from finite tables: ranks are listed only for −10 ≤ d ≤ 2, and the conclusion that rank stabilizes only for k=8 (cE6) and k=2,14 (cE7) is read off from that finite window. The argument requires knowledge of negative-degree symplectic cohomology for all d < 0; a sequence that is nonconstant down to d = −10 could in principle become constant at d < −10. No generating-function expression, closed-form recurrence, or upper bound on the vanishing range is supplied. The assertion that stabilization occurs only in the listed cases is therefore not supported by the data presented.
  4. [§4.2 and Abstract] The physics-side verification is not an enumeration of all consistent quiver gauge theories. The search imposes three additional constraints: R-charges in [0,2], equal R-charges for each pair of bifundamentals, and R-charges restricted to multiples of 1/m. The body of §5 acknowledges that 'the search is subject to additional constraints,' but the abstract says the absence of duals is 'confirmed by enumerating all consistent quiver gauge theories.' These constraints are introduced for simplicity and are not derived from consistency, so the search cannot exclude superconformal quivers outside the restricted class. The abstract and Claim 1.1 should be softened, or the constraints should be justified as consequences of consistency.
minor comments (4)
  1. [§1 and §2.3] The introduction calls the Evans–Lekili statement an 'important conjecture established in [28],' while §2.3 correctly describes it as a conjecture whose 'only if' direction is unproved except for cA_n; this inconsistency should be fixed.
  2. [§5, footnote 16] The footnote says 'The non-isolated singularities considered in this paper admit at least two C* actions,' but the paper explicitly restricts to isolated cDV singularities; either the footnote refers to examples from [13] or it is misworded.
  3. [Throughout] There are numerous typos and infelicities: 'accosiated', 'Berlund-Hübsch', 'stablizes', 'T able', 'Pinckham', 'kernal', and 'mulitplication' are examples. A careful proofreading pass is needed.
  4. [§4.1] The Futaki-invariant formula (4.4) is stated for test configurations generated by coordinate vector fields, but the paper does not discuss whether these exhaust the relevant test configurations for K-stability; if not, the K-stable range in Table 1 should be cited as an external result rather than derived from formula (4.4) alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the derivation is conditional on an external conjecture and has technical gaps, but it does not reduce to its own inputs.

full rationale

I walked the claimed derivation chain: the paper reduces the existence question for cE6/cE7 singularities to a symplectic cohomology criterion from [28], then computes the relevant Hochschild cohomology ranks on the mirror side via (2.11), and finally cross-checks by enumerating quiver candidates. None of these steps defines the target answer in terms of itself. The symplectic cohomology criterion is an external conjecture, not a quantity constructed from the paper's own desired conclusion; the paper explicitly concedes in Section 2.3: 'Under the assumption that the “only if” direction of the above conjecture holds, this result leads to the main claim of our paper in the introduction.' That is a load-bearing unproved assumption, hence a correctness/rigor concern, but it is not a circular reduction. The same holds for the Section 3.2 caveat that the simplification below (2.11) 'may break down' when the restricted singularity is non-isolated, and for the Section 4 search, which the paper itself describes as 'subject to additional constraints' (Section 5). An under-constrained or over-claimed numerical search is not circularity. The positive benchmarks (cA1, Morrison–Pinkham, k=8, k=2,14) are reproduced from independent computation rather than fitted. The one self-citation, [27] by author Fang, is used for quiver Hilbert series conventions and known dual quiver data, but the central no-go conclusion does not rest on it. I therefore find no specific circular step that can be exhibited as an equation reducing to its own input.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central claim rests on two large conjectural frameworks: the Evans-Lekili symplectic cohomology criterion and Berglund-Hubsch-Krawitz homological mirror symmetry. The paper also uses hand-chosen truncation and R-charge restrictions that limit the reach of the physics-side enumeration. No new particles or forces are introduced.

free parameters (3)
  • HH degree truncation bound = -10
    The stabilization claim for all negative degrees is inferred from table entries down to d=-10; no analytic argument is given for d<-10, so the cutoff is load-bearing for the conclusion that negative-degree ranks stabilize only for k=8 or k=2,14.
  • R-charge range upper bound = [0,2]
    Section 4.2 restricts all trial R-charges to [0,2]; outside this range the Hilbert-series matching is not searched, so the no-go conclusion is conditional on this hand-chosen window.
  • R-charge discretization = multiples of 1/m
    Section 4.2 restricts candidate R-charges to {0,1/m,...,2m/m}; this discretization is not derived from the singularity and could miss admissible quivers with other R-charges.
assumptions (7)
  • domain assumption Evans-Lekili conjecture: a cDV singularity has a crepant resolution with l irreducible exceptional curves iff the negative-degree symplectic cohomology of its Milnor fiber has rank l.
    Central criterion used throughout Section 3; explicitly stated as conjectural in Section 2.3 and proved only for cA_n in [35]. The 'only if' direction is the load-bearing assumption for non-existence.
  • domain assumption Berglund-Hubsch-Krawitz homological mirror symmetry conjectures (2.6) and (2.7): quasi-equivalences between Fukaya-Seidel and wrapped Fukaya categories and categories of equivariant matrix factorizations.
    Used in Section 2.5 to replace symplectic cohomology computations with Hochschild cohomology computations; these are conjectures, not proven theorems.
  • domain assumption Isomorphism SH^*(M_X) ≅ HH^*(W(M_X)) and the isomorphism (2.9) once HH^2 vanishes.
    Invoked in Section 2.5 to identify negative-degree symplectic cohomology with Hochschild cohomology of the mirror; this relies on the HMS conjectures.
  • domain assumption AdS/CFT correspondence and matching of quiver Hilbert series H_00 with the singularity Hilbert series as a necessary condition for a holographic dual.
    Section 4 uses absence of a quiver with matching Hilbert series to conclude absence of a dual; this is a widely believed but unproven correspondence, and the paper acknowledges it remains conjectural.
  • standard math For isolated cDV singularities, existence of an NCCR is equivalent to existence of a crepant resolution.
    Cited theorem from Van den Bergh [14]; used to translate the physical quiver question into a crepant resolution question.
  • standard math Morrison's classification of dual graphs of small crepant resolutions of isolated Gorenstein threefold singularities.
    Used in Section 4.1 to restrict the allowed quiver topologies to a finite set.
  • standard math Brieskorn's criterion: for F=f_ADE+w^k=0, a crepant resolution exists precisely when k is a multiple of the Coxeter number.
    Used in Section 2.2 to reduce the candidate list of singularities.

how reviews work

0 comments
Cite this review

Pith. "Pith review of On holographic duals of certain isolated weighted Gorenstein cDV singularities." pith.science (2026). https://pith.science/paper/O2K5QE4U

@misc{pith2026241210698,
  author       = {Pith},
  title        = {Pith review of: On holographic duals of certain isolated weighted Gorenstein cDV singularities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/O2K5QE4U}},
  note         = {Machine review of arXiv:2412.10698}
}
read the original abstract

We employ a novel approach,based on homological mirror symmetry for Landau-Ginzburg models,to demonstrate the non-existence of crepant resolutions for certain weighted homogeneous Gorenstein compound Du Val singularities.Physically,this implies that such singularities cannot serve as holographic backgrounds for four dimensional N=1 superconformal quiver gauge theories realized on the worldvolume of a large number of D3 branes placed at the singular locus.This is confirmed by enumerating all consistent quiver gauge theories.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 43 canonical work pages

  1. [6]

    De Marco, A

    M. De Marco, A. Sangiovanni and R. Valandro,5d higgs branches from m-theory on quasi-homogeneous cdv threefold singularities,Journal of High Energy Physics2022(2022)

    work page 2022

  2. [1]

    Wang and D

    Y. Wang and D. Xie,Classification of argyres-douglas theories from m5 branes,Physical Review D94(2016)

    work page 2016

  3. [2]

    Xie and S.-T

    D. Xie and S.-T. Yau,4d n=2 scft and singularity theory part i: Classification,1510.01324

    arXiv

  4. [3]

    Intriligator, D.R

    K. Intriligator, D.R. Morrison and N. Seiberg,Five-dimensional supersymmetric gauge theories and degenerations of calabi-yau spaces,Nuclear Physics B497(1997) 56–100

    work page 1997

  5. [4]

    Apruzzi, C

    F. Apruzzi, C. Lawrie, L. Lin, S. Sch¨ afer-Nameki and Y.-N. Wang,5d superconformal field theories and graphs,Physics Letters B800(2020) 135077

    work page 2020

  6. [5]

    Closset, S

    C. Closset, S. Sch¨ afer-Nameki and Y.-N. Wang,Coulomb and higgs branches from canonical singularities. part i. hypersurfaces with smooth calabi-yau resolutions,Journal of High Energy Physics2022(2022)

    work page 2022

  7. [7]

    Maldacena,The Large N limit of superconformal field theories and supergravity,Adv

    J.M. Maldacena,The Large N limit of superconformal field theories and supergravity,Adv. Theor. Math. Phys.2(1998) 231 [hep-th/9711200]

    arXiv 1998

  8. [8]

    Klebanov and E

    I.R. Klebanov and E. Witten,Superconformal field theory on three-branes at a Calabi-Yau singularity,Nucl. Phys. B536(1998) 199 [hep-th/9807080]

    arXiv 1998

Show all 56 references
  1. [9]

    Eager, J

    R. Eager, J. Schmude and Y. Tachikawa,Superconformal indices, sasaki-einstein manifolds, and cyclic homologies,arXiv preprint arXiv:1207.0573(2012) . – 34 –

    2012 arXiv

  2. [10]

    Xie and S.-T

    D. Xie and S.-T. Yau,Singularity, Sasaki-Einstein manifold, Log del Pezzo surface and N= 1AdS/CFT correspondence: Part I,1903.00150

    1903 arXiv

  3. [11]

    Collins, D

    T.C. Collins, D. Xie and S.-T. Yau,K stability and stability of chiral ring,arXiv preprint arXiv:1606.09260(2016)

    2016 arXiv

  4. [12]

    Aspinwall and D.R

    P.S. Aspinwall and D.R. Morrison,Quivers from matrix factorizations,Communications in Mathematical Physics313(2012) 607–633

    2012

  5. [13]

    Fazzi and A

    M. Fazzi and A. Tomasiello,Holography, matrix factorizations and k-stability,Journal of High Energy Physics2020(2020)

    2020

  6. [14]

    Van den Bergh,Non-commutative crepant resolutions, inThe Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002, pp

    M. Van den Bergh,Non-commutative crepant resolutions, inThe Legacy of Niels Henrik Abel: The Abel Bicentennial, Oslo, 2002, pp. 749–770, Springer (2004)

    2004

  7. [15]

    Franco, A

    S. Franco, A. Hanany, D. Vegh, B. Wecht and K.D. Kennaway,Brane dimers and quiver gauge theories,Journal of High Energy Physics2006(2006) 096

    2006

  8. [16]

    Franco, A

    S. Franco, A. Hanany, D. Martelli, J. Sparks, D. Vegh and B. Wecht,Gauge theories from toric geometry and brane tilings,Journal of High Energy Physics2006(2006) 128

    2006

  9. [17]

    Benvenuti, S

    S. Benvenuti, S. Franco, A. Hanany, D. Martelli and J. Sparks,An Infinite family of superconformal quiver gauge theories with Sasaki-Einstein duals,JHEP06(2005) 064 [hep-th/0411264]

    2005 arXiv

  10. [18]

    Berenstein and M.R

    D. Berenstein and M.R. Douglas,Seiberg duality for quiver gauge theories,hep-th/0207027

    arXiv

  11. [19]

    Collinucci, M.D

    A. Collinucci, M.D. Marco, A. Sangiovanni and R. Valandro,Higgs branches of 5d rank-zero theories from geometry,2105.12177

    arXiv

  12. [20]

    Collinucci, A

    A. Collinucci, A. Sangiovanni and R. Valandro,Genus zero gopakumar-vafa invariants from open strings,Journal of High Energy Physics2021(2021)

    2021

  13. [21]

    Collinucci, M

    A. Collinucci, M. De Marco, A. Sangiovanni and R. Valandro,Flops of any length, gopakumar-vafa invariants and 5d higgs branches,Journal of High Energy Physics2022 (2022)

    2022

  14. [22]

    De Marco and A

    M. De Marco and A. Sangiovanni,Higgs branches of rank-0 5d theories from m-theory on (aj, al) and (ak, dn) singularities,Journal of High Energy Physics2022(2022)

    2022

  15. [23]

    BRIESKORN, ¨Uber die aufl¨ osung gewisser singularit¨ aten von holomorphen abbildungen., Mathematische Annalen166(1966) 76

    E. BRIESKORN, ¨Uber die aufl¨ osung gewisser singularit¨ aten von holomorphen abbildungen., Mathematische Annalen166(1966) 76

    1966

  16. [24]

    BRIESKORN,Die aufl¨ osung der rationalen singularit¨ aten holomorpher abbildungen., Mathematische Annalen178(1968) 255

    E. BRIESKORN,Die aufl¨ osung der rationalen singularit¨ aten holomorpher abbildungen., Mathematische Annalen178(1968) 255

    1968

  17. [25]

    Karmazyn,The length classification of threefold flops via noncommutative algebras, 2017

    J. Karmazyn,The length classification of threefold flops via noncommutative algebras, 2017

    2017

  18. [26]

    Gubser, N

    S. Gubser, N. Nekrasov and S. Shatashvili,Generalized conifolds and 4-Dimensional N=1 SuperConformal Field Theory,JHEP05(1999) 003 [hep-th/9811230]

    1999 arXiv

  19. [27]

    Y. Fang, J. Feng and D. Xie,Three dimensional quotient singularity and 4dN= 1ads/cft correspondence,2310.15792

    arXiv

  20. [28]

    Evans and Y

    J.D. Evans and Y. Lekili,Symplectic cohomology of compound du val singularities, 2104.11713

    arXiv

  21. [29]

    Grassi, T

    A. Grassi, T. Weigand and with an Appendix by V. Srinivas,On topological invariants of algebraic threefolds with (q-factorial) singularities,1804.02424. – 35 –

    arXiv

  22. [30]

    Peters,On isolated hypersurface singularities: algebra-geometric and symplectic aspects, 2405.03475

    C. Peters,On isolated hypersurface singularities: algebra-geometric and symplectic aspects, 2405.03475

    arXiv

  23. [31]

    Flenner,Divisorenklassengruppen quasihomogener singularit¨ aten.,Journal f¨ ur die reine und angewandte Mathematik328(1981) 128

    H. Flenner,Divisorenklassengruppen quasihomogener singularit¨ aten.,Journal f¨ ur die reine und angewandte Mathematik328(1981) 128

    1981

  24. [32]

    Caibar,Minimal models of canonical singularities and their cohomology, Ph.D

    M. Caibar,Minimal models of canonical singularities and their cohomology, Ph.D. thesis, University of Warwick, U.K

  25. [33]

    Giacomelli,Rg flows with supersymmetry enhancement and geometric engineering,Journal of High Energy Physics2018(2018)

    S. Giacomelli,Rg flows with supersymmetry enhancement and geometric engineering,Journal of High Energy Physics2018(2018)

    2018

  26. [34]

    Lekili and K

    Y. Lekili and K. Ueda,Homological mirror symmetry for milnor fibers of simple singularities,2004.07374

    2004 arXiv

  27. [35]

    Adaloglou, F

    N. Adaloglou, F. Pasquotto and A. Zanardini,Symplectic cohomology of quasihomogeneous can singularities,2404.17301

    arXiv

  28. [36]

    Seidel,A biased view of symplectic cohomology,0704.2055

    P. Seidel,A biased view of symplectic cohomology,0704.2055

    arXiv

  29. [37]

    Seidel,Disjoinable lagrangian spheres and dilations,Inventiones mathematicae197(2013) 299–359

    P. Seidel,Disjoinable lagrangian spheres and dilations,Inventiones mathematicae197(2013) 299–359

    2013

  30. [38]

    Orlov,Triangulated categories of singularities and d-branes in landau-ginzburg models, math/0302304

    D. Orlov,Triangulated categories of singularities and d-branes in landau-ginzburg models, math/0302304

    arXiv

  31. [39]

    Witten,Phases of n = 2 theories in two dimensions,Nuclear Physics B403(1993) 159–222

    E. Witten,Phases of n = 2 theories in two dimensions,Nuclear Physics B403(1993) 159–222

    1993

  32. [40]

    Vafa and N.P

    C. Vafa and N.P. Warner,Catastrophes and the Classification of Conformal Theories,Phys. Lett. B218(1989) 51

    1989

  33. [41]

    Berglund and T

    P. Berglund and T. H¨ ubsch,A generalized construction of mirror manifolds,Nuclear Physics B393(1993) 377

    1993

  34. [42]

    Krawitz,Fjrw rings and landau-ginzburg mirror symmetry,0906.0796

    M. Krawitz,Fjrw rings and landau-ginzburg mirror symmetry,0906.0796

    arXiv

  35. [43]

    Greene and M.R

    B.R. Greene and M.R. Plesser,Duality in Calabi-Yau Moduli Space,Nucl. Phys. B338 (1990) 15

    1990

  36. [44]

    Gukov, C

    S. Gukov, C. Vafa and E. Witten,Cft’s from calabi–yau four-folds,Nuclear Physics B584 (2000) 69–108

    2000

  37. [45]

    Eager,Equivalence of a-maximization and volume minimization,Journal of High Energy Physics2014(2014) 1

    R. Eager,Equivalence of a-maximization and volume minimization,Journal of High Energy Physics2014(2014) 1

    2014

  38. [46]

    Novikov, M.A

    V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov,Exact Gell-Mann-Low Function of Supersymmetric Yang-Mills Theories from Instanton Calculus,Nucl. Phys. B 229(1983) 381

    1983

  39. [47]

    Anselmi, D.Z

    D. Anselmi, D.Z. Freedman, M.T. Grisaru and A.A. Johansen,Nonperturbative formulas for central functions of supersymmetric gauge theories,Nucl. Phys. B526(1998) 543 [hep-th/9708042]

    1998 arXiv

  40. [48]

    Martelli, J

    D. Martelli, J. Sparks and S.-T. Yau,The Geometric dual ofa-maximisation for Toric Sasaki-Einstein manifolds,Commun. Math. Phys.268(2006) 39 [hep-th/0503183]

    2006 arXiv

  41. [49]

    Collins and G

    T. Collins and G. Sz´ ekelyhidi,Sasaki–einstein metrics and k–stability,Geometry & Topology 23(2019) 1339. – 36 –

    2019

  42. [50]

    Morrison,The birational geometry of surfaces with rational double points., Mathematische Annalen271(1985) 415

    D.R. Morrison,The birational geometry of surfaces with rational double points., Mathematische Annalen271(1985) 415

    1985

  43. [51]

    August,The tilting theory of contraction algebras,1802.10366

    J. August,The tilting theory of contraction algebras,1802.10366

    arXiv

  44. [52]

    de Celis and Y

    A.N. de Celis and Y. Sekiya,Flops and mutations for crepant resolutions of polyhedral singularities,1108.2352

    arXiv

  45. [53]

    Bocklandt,Graded calabi yau algebras of dimension 3,Journal of Pure and Applied Algebra212(2008) 14

    R. Bocklandt,Graded calabi yau algebras of dimension 3,Journal of Pure and Applied Algebra212(2008) 14

    2008

  46. [54]

    Bajc,Kutasov-Seiberg dualities and cyclotomic polynomials,JHEP06(2019) 083 [1901.02846]

    B. Bajc,Kutasov-Seiberg dualities and cyclotomic polynomials,JHEP06(2019) 083 [1901.02846]

    2019 arXiv

  47. [55]

    B. Chen, D. Xie, S.S.T. Yau, S.-T. Yau and H. Zuo,4d n=2 scft and singularity theory part iii: Rigid singularity,1712.00464

    arXiv

  48. [56]

    Gadde, L

    A. Gadde, L. Rastelli, S.S. Razamat and W. Yan,On the Superconformal Index ofN= 1IR Fixed Points: A Holographic Check,JHEP03(2011) 041 [1011.5278]. – 37 –

    2011 arXiv

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.