Noncommutative Homological Mirror Functor

Cho, Cheol-Hyun; Hong, Hyunkee; Lau, Siu-Cheong

doi:10.1090/memo/1326

Cited by 5 publications

(16 citation statements)

References 83 publications

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“…(compare also many related papers [21][22][23]). In conclusion, as the points in S 2 are missed, the Maurer-Cartan picture fails over the singular locus.…”

Section: Generalizationsmentioning

confidence: 95%

Family Floer mirror space for local SYZ singularities

Yuan¹

2022

Preprint

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We state a non-archimedean and Floer-theoretic version of the SYZ conjecture and verify it for the Gross's special Lagrangian fibration in any toric Calabi-Yau manifold. By combining both the ideas of Gross-Hacking-Keel and Kontsevich-Soibelman, the dual analytic fibration with singularities is explicitly written down and is proved to be compatible with the family Floer mirror construction. As a surprising outcome, the Maurer-Cartan set of a singular Lagrangian is naturally only a strict subset of the corresponding dual singular fiber. This responds negatively to the previous expectation and leads to new perspectives of SYZ singularities. As extra evidence, we also check some computations for a well-known folklore conjecture.

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“…(compare also many related papers [21][22][23]). In conclusion, as the points in S 2 are missed, the Maurer-Cartan picture fails over the singular locus.…”

Section: Generalizationsmentioning

confidence: 95%

Family Floer mirror space for local SYZ singularities

Yuan¹

2022

Preprint

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“…Given any superfiltered 𝐴 ∞ -deformation  of , we can lift the given identification Φ 1 ∶ gr  →  to a superfiltered module isomorphism 𝜑 ∶  →  as in Example 1.5. The requirement for  to be a superfiltered 𝐴 ∞ -deformation of  is then that under the identification 𝜑 the operations on  are of the form (5). This condition is independent of the choice of lift 𝜑 of Φ 1 .…”

Section: 𝑨 ∞ -Algebras and Deformationsmentioning

confidence: 99%

“…Viewing this as an

A_\infty

‐category with a single object we apply the localised mirror functor of Cho–Hong–Lau [6]. This is a powerful tool for proving mirror symmetry and has been used with great success by these authors and others (see, for example, [4, 5, 7]). It is a variant of Fukaya's

A_\infty

‐Yoneda embedding [14, 15], and some history and related constructions are discussed in the introduction to [6].…”

Section: Introductionmentioning

confidence: 99%

Superfiltered A∞$A_\infty$‐deformations of the exterior algebra, and local mirror symmetry

Smith

2022

Journal of London Math Soc

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The exterior algebra E$E$ on a finite‐rank free module V$V$ carries a double-struckZ/2$\mathbb {Z}/2$‐grading and an increasing filtration, and the double-struckZ/2$\mathbb {Z}/2$‐graded filtered deformations of E$E$ as an associative algebra are the familiar Clifford algebras, classified by quadratic forms on V$V$. We extend this result to A∞$A_\infty$‐algebra deformations A$\mathcal {A}$, showing that they are classified by formal functions on V$V$. The proof translates the problem into the language of matrix factorisations, using the localised mirror functor construction of Cho–Hong–Lau, and works over an arbitrary ground ring. We also compute the Hochschild cohomology algebras of such A$\mathcal {A}$. By applying these ideas to a related construction of Cho–Hong–Lau, we prove a local form of homological mirror symmetry: the Floer A∞$A_\infty$‐algebra of a monotone Lagrangian torus is quasi‐isomorphic to the endomorphism algebra of the expected matrix factorisation of its superpotential.

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“…For weakly unobstructed Lagrangian L in symplectic manifold M , localized mirror functor formalism [CHL17] gives an A ∞ -functor from Fukaya category of M to the matrix factorization category of W L (here we follow the convention in [CHL15]).…”

Section: They Satisfy Similar Amentioning

confidence: 99%

Homological mirror symmetry of indecomposable Cohen-Macaulay modules for some degenerate cusp singularities

Cho¹,

Jeong²,

Kim³

et al. 2022

Preprint

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Burban-Drozd showed that the degenerate cusp singularities have tame Cohen-Macaulay representation type, and classified all indecomposable Cohen-Macaulay modules over them. One of their main example is the non-isolated singularity W = x y z. On the other hand, Abouzaid-Auroux-Efimov-Katzarkov-Orlov showed that W = x y z is mirror to a pair of pants. In this paper, we investigate homological mirror symmetry of these indecomposable Cohen-Macaulay modules for x y z.Namely, we show that closed geodesics (with a flat C-bundle) of a hyperbolic pair of pants have a one-toone correspondence with indecomposable Cohen-Macaulay modules for x y z with multiplicity one that are locally free on the punctured spectrum. In particular, this correspondence is established first by a geometric A ∞ -functor from the Fukaya category of the pair of pants to the matrix factorization category of x y z, and next by the correspondence between Cohen-Macaulay modules and matrix factorizations due to Eisenbud. For the latter, we compute explicit Macaulayfications of modules from Burban-Drozd's classification and find a canonical form of the corresponding matrix factorizations. In the sequel, we will show that indecomposable modules with higher multiplicity correspond to twisted complexes of closed geodesics.We also find mirror images of rank 1 indecomposable Cohen-Macaulay modules (of band type) over the singularity W = x 3 + y 2 − x y z as closed loops in the orbifold sphere P 1 3,2,∞ .

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Noncommutative Homological Mirror Functor

Cited by 5 publications

References 83 publications

Family Floer mirror space for local SYZ singularities

Family Floer mirror space for local SYZ singularities

Superfiltered A∞$A_\infty$‐deformations of the exterior algebra, and local mirror symmetry

Homological mirror symmetry of indecomposable Cohen-Macaulay modules for some degenerate cusp singularities

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