Mirror Symmetry for Truncated Cluster Varieties

Gammage, Benjamin; Le, Ian

doi:10.3842/sigma.2022.055

Cited by 3 publications

(4 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We also note that whereas the left-hand side of both of our equivalences is a geometrically defined category, the right-hand sides are defined by picking certain generators inside the ambient category of deformation-quantization modules. This is in contrast to the equivalence proven in the sequel to this paper [21], which equates coherent sheaves on M C with the wrapped Fukaya category of its mirror. A more direct geometric definition of M and its grading, in particular, would be of great interest.…”

Section: Michael Mcbreen and Ben Webstermentioning

confidence: 59%

“…In the sequel [21] to this paper, joint with Ben Gammage, we show that the core C D becomes the Liouville skeleton of B, thought of as a Liouville manifold with respect to the affine Liouville structure. Microlocal sheaves on this skeleton compute the wrapped Fukaya category of B.…”

Section: Michael Mcbreen and Ben Webstermentioning

confidence: 89%

“…We expect the relevant A-model category to be a subcategory of a Fukaya category of the Dolbeault hypertoric manifold D, built from Lagrangian branes endowed with an extra structure corresponding to the conical G m -action on M. However, rather than working directly with the Fukaya category, we will replace it below by a category of DQ-modules on D. The calculations presented there should also be valid in the Fukaya category. The reader is referred to the sequel [21] to this paper for more discussion of this point.…”

Section: Mirror Symmetry Via Microlocal Sheavesmentioning

confidence: 99%

“…This helps place our main results in the usual context of homological mirror symmetry. The relationship between the two papers is explained in more detail in [21].…”

Section: Michael Mcbreen and Ben Webstermentioning

confidence: 99%

See 3 more Smart Citations

Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves

McBreen,

Webster

2024

Geom. Topol.

View full text Add to dashboard Cite

Following [35], in this section we exploit the large center of quantizations in characteristic p so as to relate modules over A K with coherent sheaves on M .1/ K . Roughly speaking, upon restriction to fibers of W M.1/ K ! N .1/ K , the quantization becomes the algebra of endomorphisms of a vector bundle, and thus Morita-equivalent to the structure sheaf of the fiber.Theorem 2. 4 [35, Theorems 4.3.1 and 4.3.4] For any 2 t F p , there exists a coherent sheaf A of algebras Azumaya over the structure sheaf on M .1/ K such that .M

show abstract

Section: Michael Mcbreen and Ben Webstermentioning

confidence: 59%

Section: Michael Mcbreen and Ben Webstermentioning

confidence: 89%

Section: Mirror Symmetry Via Microlocal Sheavesmentioning

confidence: 99%

“…This helps place our main results in the usual context of homological mirror symmetry. The relationship between the two papers is explained in more detail in [21].…”

Section: Michael Mcbreen and Ben Webstermentioning

confidence: 99%

See 2 more Smart Citations

Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves

McBreen,

Webster

2024

Geom. Topol.

View full text Add to dashboard Cite

show abstract

Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

Argüz

Bousseau

2023

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

View full text Add to dashboard Cite

Cluster varieties come in pairs: for any 𝒳 {\mathcal{X}} cluster variety there is an associated Fock–Goncharov dual 𝒜 {\mathcal{A}} cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. In this paper we bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. Particularly, we show that the mirror to the 𝒳 {\mathcal{X}} cluster variety is a degeneration of the Fock–Goncharov dual 𝒜 {\mathcal{A}} cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross, Hacking, Keel and Kontsevich compares with the canonical scattering diagram defined by Gross and Siebert to construct mirror duals in arbitrary dimensions. Consequently, we derive an enumerative interpretation of the cluster scattering diagram. Along the way, we prove the Frobenius structure conjecture for a class of log Calabi–Yau varieties obtained as blow-ups of toric varieties.

show abstract

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Bardwell-Evans¹,

Cheung²,

Hong³

et al. 2021

Preprint

View full text Add to dashboard Cite

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert [28] and the canonical scattering diagrams of Gross-Hacking-Keel [24]. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.

show abstract

Mirror Symmetry for Truncated Cluster Varieties

Cited by 3 publications

References 30 publications

Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves

Homological mirror symmetry for hypertoric varieties, I: Conic equivariant sheaves

Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

Scattering Diagrams from Holomorphic Discs in Log Calabi-Yau Surfaces

Contact Info

Product

Resources

About