Morse homology, tropical geometry, and homological mirror symmetry for toric varieties

Abstract

Given a smooth projective toric variety X, we construct an A∞ category of Lagrangians with boundary on a level set of the Landau-Ginzburg mirror of X. We prove that this category is quasi-equivalent to the DG category of line bundles on X.

Noncommutative mirror symmetry for punctured surfaces

“…The paper also includes a second appendix by Mohammed Abouzaid which explains how one can simplify the construction of a wrapped fukaya category in the case of punctured Riemann surfaces. 1 Note that in this example the Jacobi algebra Jac Q ∼ = C [X, Y, Z| is not noncommutative, but for all other dimer models it is.…”

“…, β l ρ k ) which is equal to its µ-version by the previous case. For the last terms, lemma A.10 implies that the only combination of 2 consecutive terms for which the ordinary product is nonzero is ρ i , β 1 Proof. Clearly the µ is also homogeneous for deg because by the condition above every positive cycle β 1 .…”

“…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].…”

“…Commutatively this corresponds to the category of singularities of the three standard planes in affine 3-space. Noncommutatively 1 we can see this as a dimer model on the torus with a superpotential ℓ coming from the faces.…”

“…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].…”