Donaldson–Thomas transformations of moduli spaces of G-local systems

Goncharov, A. B.; Shen, Linhui

doi:10.1016/j.aim.2017.06.017

Cited by 58 publications

(68 citation statements)

References 50 publications

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“…The positive space A scat prin,s 0 is isomorphic to A prin,s 0 . In section 4, we build a positive spaceÃ scat prin,s 0 Ą A scat prin,s 0 using all chambers in ∆s 0 Ť ∆ś 0 if the path ordered product p s 0 ,´, or equivalently the Donaldson-Thomas transformation of A prin,s 0 as defined in [GS16], is rational. Let A scat,ṕ rin,s 0 ĂÃ scat prin,s 0 be the subspace built by gluing all tori attached to negative chambers.…”

Section: Introductionmentioning

confidence: 99%

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Cluster Structures and Subfans in Scattering Diagrams

Zhou

2020

SIGMA

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We give more precise statements of Fock-Goncharov duality conjecture for cluster varieties parametrizing SL2{P GL2-local systems on the once punctured torus. Then we prove these statements. Along the way, using disjoint subfans in the scattering diagram, we produce an example of a cluster variety with two non-equivalent cluster structures. To overcome the technical difficulty of infinite non-cluster wall-crossing in the scattering diagram, we introduce quiver folding into the machinery of scattering diagrams and give a quotient construction of scattering diagrams. 29 5.1. The Full Fock-Goncharov conjecture for A prinT 2 1 and X T 2 1 . 29 5.2. A cluster variety with two non-equivalent cluster structures. 33 5.3. The Full Fock-Goncharov conjecture for A T 2 1 . 35 References 38 1 T 2 1

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Section: Introductionmentioning

confidence: 99%

“…Then p ś ,`˝Σ˚i s the Donaldson-Thomas transformation of A prin,S 2 4 with respect to the seed s, as defined in [GS16]. By the existence of maximal green sequence, DT transformation of A prin,S 2 4 can be written as a composition of finitely many cluster mutations.…”

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confidence: 99%

Cluster Structures and Subfans in Scattering Diagrams

Zhou

2020

SIGMA

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“…Many cluster varieties A associated to a marked bordered surface with at least two punctures also have a maximal green sequence, see [CLS], §1.3 for a summary of known results on this. The recent [GS16], Theorem 1.12,1.17, shows that (4) holds for the Fock-Goncharov cluster varieties of PGL m local systems on most decorated surfaces. Together with Proposition 0.14 these results imply the full Fock-Goncharov theorem in any of these cases.…”

Section: Note This Implies Mid(v ) = Up(v ) = Can(v )mentioning

confidence: 99%

“…Note that Ξ is convex in our generalized sense. We show, making use of recent results of Magee and Goncharov-Shen, [Ma15], [Ma16], [GS16], that in the representation theoretic examples which were the original motivation for the definition of cluster algebras our polyhedral cones Ξ specialize to the piecewise linear parameterizations of canonical bases of Berenstein and Zelevinsky [BZ01], Knutson and Tao [KT99], and Goncharov and Shen [GS13]:…”

Section: Note This Implies Mid(v ) = Up(v ) = Can(v )mentioning

confidence: 99%

Canonical bases for cluster algebras

Gross¹,

Hacking²,

Keel³

et al. 2017

J. Amer. Math. Soc.

363

968

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Abstract. In [GHK11], Conjecture 0.6, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral tropical points of the mirror. Further, the structure constants for the multiplication rule in this basis should be given by counting broken lines (certain combinatorial objects, morally the tropicalisations of holomorphic discs).Here we prove the conjecture in the case of cluster varieties, where the statement is a more precise form of the Fock-Goncharov dual basis conjecture, [FG06], Conjecture 4.3. In particular, under suitable hypotheses, for each Y the partial compactification of an affine cluster variety U given by allowing some frozen variables to vanish, we obtain canonical bases for H 0 (Y, O Y ) extending to a basis of H 0 (U, O U ). Each choice of seed canonically identifies the parameterizing sets of these bases with integral points in a polyhedral cone. These results specialize to basis results of combinatorial representation theory. For example, by considering the open double Bruhat cell U in the basic affine space Y we obtain a canonical basis of each irreducible representation of SL r , parameterized by a set which each choice of seed identifies with integral points of a lattice polytope. These bases and polytopes are all constructed essentially without representation theoretic considerations. Along the way, our methods prove a number of conjectures in cluster theory, including positivity of the Laurent phenomenon for cluster algebras of geometric type.

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“…The proof of this result is based on ideas of Musiker, Schiffler, and Williams [48] who defined canonical bases for cluster algebras arising from triangulated surfaces. A more general method of constructing canonical bases using tools from mirror symmetry was recently developed by Gross, Hacking, Keel, and Kontsevich [40], and their results have been used by Goncharov and Shen [38] to describe canonical bases for the coordinate rings of cluster varieties associated to more general surfaces. In this paper, rather than work with the abstractly defined basis of [40], we focus on examples where the bases are defined concretely in terms of trace functions on moduli spaces of local systems.…”

Section: Canonical Bases and Categorificationmentioning

confidence: 99%

Categorified canonical bases and framed BPS states

Allegretti¹

2019

Sel. Math. New Ser.

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We consider a cluster variety associated to a triangulated surface without punctures. The algebra of regular functions on this cluster variety possesses a canonical vector space basis parametrized by certain measured laminations on the surface. To each lamination, we associate a graded vector space, and we prove that the graded dimension of this vector space gives the expansion in cluster coordinates of the corresponding basis element. We discuss the relation to framed BPS states in N = 2 field theories of class S.

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Donaldson–Thomas transformations of moduli spaces of G-local systems

Cited by 58 publications

References 50 publications

Cluster Structures and Subfans in Scattering Diagrams

Cluster Structures and Subfans in Scattering Diagrams

Canonical bases for cluster algebras

Categorified canonical bases and framed BPS states

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