​Landau-Ginzburg systems and HMS

​toric Fano manifold
Let X be a toric Fano manifold given by a Fano polytope Δ and let Δ​0 be the corresponding polar polytope. The polytope Δ​0 determines the space L(Δ​0​) of all Laurent polynomials of the form f(z)=∑a​nz​n​, where the sum runs over all n∈Δ​0(0). To any element f∈L(Δ​0(0)) we associate the corresponding Landau-Ginzburg system z​i·(∂f/∂z​i)=0 for i=1,...,n. We denote by Crit(f)⊂(ℂ∖0)​n the solution scheme of the system. Landau-Ginzburg potentials were first introduced by Batyrev in the description of quantum cohomology of toric Fano manifolds in [Ba1] and in the description of the middle homology of mirrors of Calabi-Yau hypersurfaces, in [Ba2]. In the context of HMS of toric Fano manifolds Landau-Ginzburg systems play a fundamental role and the mirror of X is given by a pair ((ℂ∖0)​n​,f). Let us mention the following two fundamental works:

Picard-Lefschetz theory, vanishing cycles and Seidel's approach to mirror symmetry

If f∈L(Δ​0) is a generic element it has the structure of a Lefschetz fibration. Concretely, f is an algebraic fibration with fiber W​z=f​-1(z)⊂(ℂ∖0)​n for z∈ℂ. A fiber is singular exactly when Crit(f)∩W​z is not empty. If f is generic all the singularities are isolated and quadratic (which is the Lefschetz condition). A fundamental property of Lefschetz fibrations, from the symplectic view-point, is that the vanishing cycles of the fibration in H​n-1​(W​z​), for a smooth fiber, can be represented by Lagranagian spheres, see [Ar,Se1]. An important feature is that the vanishing Lagrangian spheres are ​not uniquely determined by their corresponding vanishing cycles. In fact, a vanishing Lagrangian sphere L(z;γ) is determined by a curve γ⊂f(Crit(f))  connecting a singular value w​i​=f(z​i​) where Crit(f)={z​1​,...,z​n​} to the regular value w=f(z). Seidel's remarkable view-point (see, for instance [Se2,Se3,Se4,Se5]) is that a collection of vanishing Lagrangian spheres determined by a choice of paths {γ​1,...,γ​N} has properties analog to that of exceptional collections in D​b​(X). In fact, the resulting collection {L​1,...,L​N​} is an exceptional collection in the corresponding Fukaya-Seidel category FS((ℂ∖0)​n,f)associated to the Lefschetz fibration f, we refer the reader to the book [Se5]. In particular, one obtains a homological mirror symmetry functor by finding an equivalent (full strongly) exceptional collection {E​1,...,E​N}⊂D​​b(X) analog to a collection of Lagrangian vanishing spheres {L​1,...,L​N} in the mirror. This was done in various cases, for instance, Auroux, Katzarkov and Orlov describe such a functor in the case of Del-Pezzo surfaces, in [AKO]. In [Se3] the case of the quartic surface is studied in detail. It should be noted that the exceptional objects in D​b​(X) corresponding to vanishing cycles are not necessarily given in terms of vector bundles.

Abouzaid's homological mirror symmetry for toric Fano manifolds

Abouzaid's results on homological mirror symmetry for toric Fano manifolds were developed in [Ab1,Ab2]. On the algebraic side, it is noted in [Ab2] that the derived category of coherent sheaves D​b​(X) of a toric manifold X is generated as a triangulated category by the elements of Pic(X). On the other hand, Abouzaid describes a class of Lagrangian submanifolds in ((ℂ∖0)​n,Φ) which are mirror analogs of line bundles in Pic(X), called tropical Lagrangian sections (Abouzaid works with a certain mild deformation Φ of a Laurent polynomial f). This correspondence is the basis of Abouzaid's homological mirror symmetry functor. More concretely, a ​Lagrangian brane L⊂(ℂ∖0)​n is an embedded compact graded Lagrangian submanifold, which is spin and exact. A Lagrangian brane is said to be admisible if ∂L⊂M, where M:={f=0}, and there exists a small neighborhood of ∂L in L which agrees with the parallel transport of ∂L along a segment in ℂ. Based on ideas of Kontsevich, the admissible Lagrangians are taken to be the objects of the Fukaya
A​-pre-category Fuk((ℂ∖0)​n,Φ).
One has the following two maps (T-fibrations): the moment map μ:(ℂ∖0)​n→ℝ​n and the logarithm map Log:(ℂ∖0)​n→ℝ. Recall that the image of the moment map coincides with the polytope, μ((ℂ∖0)​n)=Δ. On the other hand, it is noted by Abouzaid that there exists a component of the complement of the amobea of M under the map Log which coincides with (a -smoothning of) Δ. A ​tropical Lagrangian section is an admissible Lagrangian brane which is a section of Log restricted to (a smoothning of) Δ. We denote by Fuk​trop(Φ)⊂Fuk(Φ) the collection of tropical Lagrangian sections. It is shown by Abouzaid in [Ab2] that up to Hamiltonian isotopy torpical Lagrangian sections are in one to one correspondence with elements of Pic(X).


[Ab1] M. Abouzaid. Homogeneous coordinate rings and mirror symmetry for toric varieties. Geometry and Topology 10 (2006) 1097-1156.
[Ab2] M. Abouzaid. Morse homology, tropical geometry, and homological mirror symmetry for toric varieties. Selecta Mathematica August 2009, Volume 15, Issue 2, pp 189-270.
[AKO] D. Auroux, L. Katzarkov, D. Orlov. Mirror symmetry for del Pezzo surfaces: vanishing cycles and coherent sheaves. Invent. Math. 166 (2006), no. 3, 537-582.
[Ar] V. I. Arnold, Some remarks on symplectic monodromy of Milnor fibrations, Progress in mathematics, volume 133, the Floer memorial volume, 99-103.
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[Ba2] V. Batyrev. Dual polyhedra and mirror symmetry for Calabi-Yau hypersurfaces in toric varieties. J. Algebraic Geom. 3 (1994), no. 3, 493-535.
[HV] K. Hori, C. Vafa. Mirror Symmetry. arXiv:hep-th/0002222
[K] M. Kontsevich. Homological algebra of mirror symmetry. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zurich, 1994), 120-139, Birkhauser, Basel, 1995.
[K2] M. Kontsevich. Course at ENS. 1998.
[Se1] P. Seidel, Floer homology and the symplectic isotopy problem, thesis, Oxford, 1997.
[Se2] P, Seidel, Vanishing cycles and mutation, European congress of mathematics , volume 202 of the series Progress in Mathematics, 65-85.
[Se3] P. Seidel. More about vanishing cycles and mutation. Symplectic geometry and mirror symmetry (Seoul, 2000), 429-465, World Sci. Publ., River Edge, NJ, 2001.
[Se4] P. Seidel, Homological mirror symmetry for the quartic surface, Memoirs of the AMS, volume 236, 2015.
[Se5] P. Seidel, Fukaya categories and Picad-Lefschetz theory, Zurich lectures in advanced mathematics , European mathemathical society, 2008.