Localized mirror functor constructed from a Lagrangian torus

Abstract

Fixing a weakly unobstructed Lagrangian torus in a symplectic manifold X, we define a holomorphic function W known as the Floer potential. We construct a canonical A ∞ -functor from the Fukaya category of X to the category of matrix factorizations of W . It provides a unified way to construct matrix factorizations from Lagrangian Floer theory. The technique is applied to toric Fano manifolds to transform Lagrangian branes to matrix factorizations. Using the method, we also obtain an explicit expression of the matrix factorization mirror to the real locus of the complex projective space. arXiv:1406.4597v3 [math.SG] 30 Sep 2016 1≤j≤n z δ(a j ,−1) j and m pq = T k/2 1≤j≤n z δ(a j ,1) j if i = 0, m qp = T k/2 z i and m pq = T k/2 if i = 0. δ(a, b) = 1 when a = b and zero otherwise. (2) m qp = m pq = 0 otherwise.

Twin Lagrangian fibrations in mirror symmetry

“…commutes, where Φ • is the equivalence (11), and Φ CHL is some variation of the localized mirror functor defined in [16,17].…”

Pairings in Mirror Symmetry Between a Symplectic Manifold and a Landau–Ginzburg B-Model

“…In this section, we recall the formalism of localized mirror functors developed in [6] and [7]. Later we will use this to understand Kapustin-Li pairing in terms of Lagrangian Floer theory.…”

“…We recall the construction of [7] to which we refer readers for details. In the toric case, the idea in the immersed case does not immediately generalize.…”

“…The second, more serious issue is that in this case, we need a new variable y = e x and matrix factorizations in y variable, but naive generalization of the above idea produces some factorizations in x-variables, which does not provide matrix factorizations in y-variables. In [7] the following approach was introduced to overcome these difficulties. The idea is to consider a flat line bundle over the Lagrangian, whose holonomy is concentrated near co-dimension one tori (which is called hypertori).…”

“…For closed string mirror symmetry, we consider the Kodaira-Spencer map ks (as in [17]). For open string mirror symmetry, we use the homological mirror functor F L from localized mirror construction of [6] and [7]. There an explicit homological mirror functor has been constructed by studying formal deformation space of a particular (immersed) Lagrangian submanifold L, which is called reference Lagrangian.…”