Stability on contraction algebras

August, Jenny; Wemyss, Michael

doi:10.48550/arxiv.1907.12756

Cited by 2 publications

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“…Remark 5.12. The matrix N i appears via moduli tracking in the HomMMP [Wem18, 5.4], and via the K-theory of contraction algebras [AW,2.4]. In contrast, M i from (5.A) is the dual, and it arises from the change in dimension vector [Wem18,5.4], or in the K-theory of projective modules [HW, 3.2].…”

Section: And Thus ωI(i)mentioning

confidence: 99%

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GV and GW invariants via the enhanced movable cone

Nabijou,

Wemyss

2021

Preprint

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Given any smooth germ of a threefold flopping contraction, we first give a combinatorial characterisation of which Gopakumar-Vafa (GV) invariants are nonzero, by prescribing multiplicities to the walls in the movable cone. This allows us to describe, and even draw, the critical locus of the associated quantum potential. We prove that the critical locus is the infinite hyperplane arrangement of [IW], and moreover that the quantum potential can be reconstructed from a finite fundamental domain. We then iterate, obtaining a combinatorial description of the matrix which controls the transformation of the non-zero GV invariants under a flop. There are three main ingredients and applications: (1) a construction of flops from simultaneous resolution via cosets, which describes how the dual graph changes, (2) a closed formula which describes the change in dimension of the contraction algebra under flop, and (3) a direct and explicit isomorphism between quantum cohomologies of different crepant resolutions, giving a Coxeter-style, visual proof of the Crepant Transformation Conjecture for isolated cDV singularities.

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Section: And Thus ωI(i)mentioning

confidence: 99%

“…In contrast, M i from (5.A) is the dual, and it arises from the change in dimension vector [Wem18,5.4], or in the K-theory of projective modules [HW, 3.2]. For more details, see [AW,2.4] and references therein. 5.3.…”

Section: And Thus ωI(i)mentioning

confidence: 99%

GV and GW invariants via the enhanced movable cone

Nabijou,

Wemyss

2021

Preprint

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“…The key examples are (Calabi-Yau) ADE Dynkin quiver case and new classes of examples are studied in [AW19]. • The Calabi-Yau-3 affine type A case in [Qiu16].…”

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Contractibility of space of stability conditions on the projective plane via global dimension function

Fan¹,

Li²,

Liu³

et al. 2020

Preprint

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We compute the global dimension function gldim on the principal component Stab † (P 2 ) of the space of Bridgeland stability conditions on P 2 . We describe the preimage gldim −1 (2) of the minimal value 2 of gldim, which is contained in the closure Stab Geo (P 2 ) of the subspace consisting of geometric stability conditions. We show that gldim −1 [2, x) contracts to gldim −1 (2) for any real number x ≥ 2 and that gldim −1 (2) is contractible.

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Stability on contraction algebras

Cited by 2 publications

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GV and GW invariants via the enhanced movable cone

GV and GW invariants via the enhanced movable cone

Contractibility of space of stability conditions on the projective plane via global dimension function

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