Gopakumar–Vafa Invariants Do Not Determine Flops

Brown, Gavin; Wemyss, Michael

doi:10.1007/s00220-017-3038-z

Cited by 21 publications

(32 citation statements)

References 16 publications

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“…For the D-series, we will study two families of threefolds: the so-called Brown-Wemyss family [18], and the family of Laufer's examples [19]. These are the so-called 'flops of length two'.…”

Section: Jhep10(2021)018mentioning

confidence: 99%

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Higgs branches of 5d rank-zero theories from geometry

Collinucci

Marco

Sangiovanni

et al. 2021

J. High Energ. Phys.

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We study the Higgs branches of five-dimensional $$ \mathcal{N} $$ N = 1 rank-zero theories obtained from M-theory on two classes non-toric non-compact Calabi-Yau threefolds: Reid’s pagodas, and Laufer’s examples. Our approach consists in reducing to IIA with D6-branes and O6-planes, and computing the open-string spectra giving rise to hypermultiplets. Starting with the seven-dimensional worldvolume theories, we switch on T-brane backgrounds to give rise to bound states with angles. We observe that the resulting partially Higgsed 5d theories have discrete gauge groups, from which we readily deduce the geometry of the Higgs branches as orbifolds of quaternionic varieties.

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Section: Jhep10(2021)018mentioning

confidence: 99%

“…This threefold was introduced in [18]. It is singular at the origin, where the ALE fiber develops a D 4 singularity.…”

Section: Brown-wemyss Threefoldmentioning

confidence: 99%

Higgs branches of 5d rank-zero theories from geometry

Collinucci

Marco

Sangiovanni

et al. 2021

J. High Energ. Phys.

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“…In particular, Brown and Wemyss have managed to give greater insight into these Gopakumar-Vafa curve invariants using explicit calculations with the contraction algebra [BW17].…”

Section: Introductionmentioning

confidence: 99%

“…For completeness we include Magma code that can calculate the dimension of contraction algebras. We also refer the reader to [BW17,Appendix], where examples of code for understanding flops and contraction algebras are given in Singular [CKP11] and Magma.…”

mentioning

confidence: 99%

The length classification of threefold flops via noncommutative algebras

Karmazyn¹

2019

Advances in Mathematics

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Smooth threefold flops with irreducible centres are classified by the length invariant, which takes values 1, 2, 3, 4, 5 or 6. This classification by Katz and Morrison identifies 6 possible partial resolutions of Kleinian singularities that can occur as generic hyperplane sections, and the simultaneous resolutions associated to such a partial resolution produce the universal flop of length l.In this paper we translate these ideas into noncommutative algebra. We introduce the universal flopping algebra of length l from which the universal flop of length l can be recovered by a moduli construction, and we present each of these algebras as the path algebra of a quiver with relations. This explicit realisation can then be used to construct examples of NCCRs associated threefold flops of any length as quiver with relations defined by superpotentials, to recover the matrix factorisation description of the universal flop conjectured by Curto and Morrison, and to realise examples of contraction algebras.

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“…† The Jacobian algebra of this quiver with potential appears as a contraction algebra in[23,29]. We thank Osamu Iyama for providing these two references.…”

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confidence: 99%

Discreteness of silting objects and t-structures in triangulated categories

Adachi

Mizuno

Yang

2018

Proc. London Math. Soc.

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We introduce the notion of ST‐pairs of triangulated subcategories, a prototypical example of which is the pair of the bound homotopy category and the bound derived category of a finite‐dimensional algebra. For an ST‐pair (C,D), we construct an injective order‐preserving map from silting objects in sans-serifC to bounded t‐structures on sans-serifD and show that the map is bijective if and only if sans-serifC is silting‐discrete if and only if sans-serifD is t‐discrete. Based on the work of Qiu and Woolf, the above result is applied to show that if sans-serifC is silting‐discrete then the stability space of sans-serifD is contractible. This is used to obtain the contractibility of the stability spaces of some Calabi–Yau triangulated categories associated to Dynkin quivers.

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Gopakumar–Vafa Invariants Do Not Determine Flops

Cited by 21 publications

References 16 publications

Higgs branches of 5d rank-zero theories from geometry

Higgs branches of 5d rank-zero theories from geometry

The length classification of threefold flops via noncommutative algebras

Discreteness of silting objects and t-structures in triangulated categories

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