Noncommutative mirror symmetry for punctured surfaces

Abstract

ABSTRACT. In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov showed that the wrapped Fukaya Categories of punctured spheres and finite unbranched covers of punctured spheres are derived equivalent to the categories of singularities of a superpotential on certain crepant resolutions of toric 3 dimensional singularities. We generalize this result to other punctured Riemann surfaces and reformulate it in terms of certain noncommutative algebras coming from dimer models. In particular, given any consistent dimer model we can look at a subcategory of noncommutative matrix factorizations and show that this category is A∞-isomorphic to a subcategory of the wrapped Fukaya category of a punctured Riemann surface. The connection between the dimer model and the punctured Riemann surface then has a nice interpretation in terms of a duality on dimer models. INTRODUCTIONOriginally homological mirror symmetry was developed by Kontsevich [31] as a framework to explain the similarities between the symplectic geometry and algebraic geometry of certain Calabi-Yau manifolds. More precisely its main conjecture states that for any compact Calabi-Yau manifold with a complex structure X, one can find a mirror CalabiYau manifold X ′ equipped with a symplectic structure such that the derived category of coherent sheaves over X is equivalent to the zeroth homology of the triangulated envelop of the split closure of the Fukaya category of X ′ . The latter is a category that represents the intersection theory of Lagrangian submanifolds of X ′ .Over the years it turned out that this conjecture is part of a set of equivalences which are much broader than the compact Calabi-Yau setting [27,21,1,6,7]. Removing the compactness or Calabi-Yau condition often makes the mirror a singular object, which physicists call a Landau-Ginzburg model [37,38]. A Landau-Ginzburg model (X, W ) is a pair of a smooth space X and a complex valued function W : X → C, which is called the potential. On the algebraic side we associate to it the dg-category of matrix factorizations MF(X, W ). Its objects are diagrams P 0 G G P 1 o o where P i are vector bundles and the composition of the maps results in multiplication with W . The morphisms are morphisms between these vector bundles equipped with a natural differential.On the other hand if X ′ is noncompact we need to tweak the notion of the Fukaya category, by imposing certain conditions on the behaviour of the Lagrangians near infinity and using a Hamiltonian flow to adjust the intersection theory. This gives us the notion of a wrapped Fukaya category [3].In [2] Abouzaid, Auroux, Efimov, Katzarkov and Orlov proved an instance of mirror symmetry between such objects. On the symplectic side they considered a sphere with k punctures and on the algebraic side they considered a special Landau Ginzburg model on a certain toric quasiprojective noncompact Calabi Yau threefold and they proved an equivalence between the derived wrapped Fukaya category of the former and the derived category of matrix factorizations of the lat...

On Extensions for Gentle Algebras

“…where f L , f ′ and f R are nontrivial. By Remark 2.15 (5), length(σ C ) > 1 and length(τ C ) > 1. In particular, since σ is a homotopy string or band corresponding to a projective resolution, σ C is a homotopy letter occurring between degrees −1 and 0.…”

“…If σ 1 , σ R = ∅, then neither is incident with dir(v) or inv(v), in which case v = d q · · ·d 1 c p · · · c 1 c, where c ∈ Q 1 is as above, is a substring of v. Now suppose we are in case (b) of part (1). If σ R = ∅ then using Remark 2.15(1) again we have σ s is incident with inv(v) and v =d q · · ·d 2 is a substring of v. If σ R = ∅, then by Remark 2.15 (5), length(σ R ) = 1 and σ R is incident with inv(v), in which case v =d q · · ·d 2 is again a substring of v.…”

“…Recall the notation and unfolded diagram for a singleton single map f • : Q • σ → ΣQ • τ from Section 1.4.2 (5). Throughout this section, whenever σ R = ∅ or τ R = ∅ in (5) we assume, without loss of generality, that f R ∈ Q 1 and f L ∈ Q 1 , respectively.…”

Mutation of frozen Jacobian algebras

“…Jacobian algebras, defined via the data of a quiver with potential, play an important role in the theory of cluster algebras, particularly that relating to their categorification by triangulated categories. However, the concept predates this subject, appearing for example in the mathematical physics of dimer models on closed surfaces [20], which has then found applications in algebraic and noncommutative geometry [5,8,9,13] and mirror symmetry [7]. More recently, it has been fruitful to enhance the data of a quiver with potential by declaring a subquiver to be frozen, leading to the more general notion of a frozen Jacobian algebra.…”

A dimer ABC:

“…The result is called the mirror quiver or twisted quiver and we denote it by Q.Example 0.4.2. We illustrate this with some examples from[8]:…”