Donaldson-Thomas invariants from tropical disks

Cheung, Man-Wai; Mandel, Travis

doi:10.48550/arxiv.1902.05393

Cited by 6 publications

(6 citation statements)

References 18 publications

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“…As noted in Footnote 4, our definition of a wall is slightly different from that used in [KS14,GHKK18], and our quantum torus Lie algebras are different from the Lie algebras used in [GS11], so we briefly explain how to recover Theorem 2.13 in our setup. First, we note that the uniqueness statement follows from the the exact same argument used in the proof of [CM,Thm. 3.5] (just replace the Λ ∨ R there with Λ R , which in our notation is L R -also, g k there is g k−1 here).…”

Section: 34mentioning

confidence: 95%

Positivity for quantum cluster algebras

Davison

2018

Ann. of Math. (2)

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Abstract. Building on work by Kontsevich, Soibelman, Nagao and Efimov, we prove the positivity of quantum cluster coefficients for all skew-symmetric quantum cluster algebras, via a proof of a conjecture first suggested by Kontsevich on the purity of mixed Hodge structures arising in the theory of cluster mutation of spherical collections in 3-Calabi-Yau categories. The result implies positivity, as well as the stronger Lefschetz property conjectured by Efimov, and also the classical positivity conjecture of Fomin and Zelevinsky, recently proved by Lee and Schiffler. Closely related to these results is a categorified "no exotics" type theorem for cohomological Donaldson-Thomas invariants, which we discuss and prove in the appendix.

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Section: 34mentioning

confidence: 95%

Positivity for quantum cluster algebras

Davison

2018

Ann. of Math. (2)

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“…Travis Mandel and the author have given a more explicit construction of stability scattering diagram in [CM19] by using idea from [GPS10].…”

Section: Stability Conditions and Scattering Diagramsmentioning

confidence: 99%

Theta functions and quiver Grassmannians

Cheung

2019

Preprint

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In this article, we use the relationship between cluster scattering diagrams and stability scattering diagrams to relate quiver representations with these diagrams. With a notion of positive crossing of a path γ, we show that if γ has positive crossing in the scattering diagram, then it goes in the opposite direction of the Auslander-Reiten quiver of Q. We then give the Hall algebra theta functions which recover the cluster character formula by the Euler characteristic map. At last, we define the Hall algebra broken lines and then are able to give the stratification of the quiver Grassmannians by the bending of the broken lines.

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“…Essentially the same algebraic structure appears in an a priori completely different context: the wall-crossing behavior of Donaldson-Thomas counts of semistable objects in a Calabi-Yau triangulated category of dimension 3, upon variation of the stability condition [KS08][JS12] [KS14]. Some precise connection between scattering diagrams and spaces of stability conditions, for quivers with potential, is established in [Bri17], recently followed by [CM19].…”

Section: Introductionmentioning

confidence: 99%

Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$

Bousseau

2019

Preprint

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We show that a purely algebraic structure, a two-dimensional scattering diagram, describes a large part of the wall-crossing behavior of moduli spaces of Bridgeland semistable objects in the derived category of coherent sheaves on P 2 . This gives a new algorithm computing the Hodge numbers of the intersection cohomology of the classical moduli spaces of Gieseker semistable sheaves on P 2 . Contents0. Introduction 0.1. Description of S(D in u,v ) and D P 2 u,v 0.2. Structure of the proof of Theorem 0.1 0.3. Applications to moduli spaces of Gieseker semistable sheaves 0.4. Relations with previous and future works 0.5. Plan of the paper 0.6. Acknowledgments 1. The scattering diagram S(D in u,v ) 1.1. Local scattering diagrams 1.2. Scattering diagrams 1.3. The scattering diagrams S(D in ) 1.4. The scattering diagrams S(D in u,v ), S(D in q ± ) and S(D in cl ± ) 1.5. Action of ψ(1) on scattering diagrams 2. The scattering diagram D P 2 u,v 2.1. Coherent sheaves on P 2 2.2. Stability conditions 2.3. Moduli spaces and walls 2.4. Intersection cohomology invariants 2.5. Scattering diagrams from stability conditions 2.6. Action of ψ(1) on D P 2 u,v 3. Consistency of D P 2 u,v 3.1. Statement of Theorem 3.1 3.2. Local structure near a wall 3.3. Numerical invariants from mixed Hodge theory 3.4. Donaldson-Thomas formalism 3.5. Wall-crossing formula 3.6. End of the proof of Theorem 3.1

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Donaldson-Thomas invariants from tropical disks

Cited by 6 publications

References 18 publications

Positivity for quantum cluster algebras

Positivity for quantum cluster algebras

Theta functions and quiver Grassmannians

Scattering diagrams, stability conditions, and coherent sheaves on $\mathbb{P}^2$

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