Localized Mirror Functor

• Localized Mirror Functor is an explicit A∞-functor that bridges the Fukaya category and matrix factorizations using deformation theory of a Lagrangian.
• It constructs local Landau–Ginzburg superpotentials via Maurer–Cartan equations and computes pairings between open and closed string theories.
• The approach enables gluing of local mirrors into a global framework, thereby achieving a computable version of homological mirror symmetry in diverse geometric settings.

A localized mirror functor is an explicit $A_\infty$–functor constructed from the deformation theory of a fixed Lagrangian submanifold (often an immersed Lagrangian) in a symplectic manifold. It maps the Fukaya $A_\infty$–category of unobstructed Lagrangians to a dg or $A_\infty$–category of matrix factorizations associated to a localized Landau–Ginzburg model. This construction provides a functorial bridge between open-string (Fukaya category) and closed-string (complex or algebraic geometry) data and enables a precise, computable version of homological mirror symmetry (HMS) for both classical and singular (non-toric, orbifold, or noncommutative) ambient spaces. Localized mirror functors have been rigorously developed and applied in a series of foundational papers, including the construction and comparison of pairings, the systematic glueing of local mirrors, explicit computation for toric/orbifold and quiver varieties, and in the context of toric and noncommutative mirror symmetry (Cho et al., 2018, Lee, 2019, Cho et al., 2018, Bai et al., 11 Jan 2025, Lau et al., 20 Jan 2026, Cho et al., 2014, Cho et al., 2013, Lee, 2020).

1. Definition and General Construction

Given a compact symplectic manifold $M$ and a reference Lagrangian $L \subset M$ (possibly immersed), the weak Maurer–Cartan solutions $b \in \MC(L)$ satisfying $m(e^b) = W(b)1_L$ produce a formal Landau–Ginzburg superpotential $W = W_L(x_1, ..., x_n)$. One defines an explicit $A_\infty$–functor

$\mathcal{F}^L : \Fuk(M) \to \MF(W)$

from the Fukaya category of $M$ to the dg-category of matrix factorizations of $W$. The assignment on objects sends an unobstructed Lagrangian $(K, \xi)$ (with the same potential value) to a matrix factorization

$$\mathcal{F}^L(K,\xi) = (E, Q), \qquad E = CF(K, L), \quad Q = -m_1^{\xi, b}, \quad Q^2 = (W(b) - W(\xi)) \,\text{Id}.$$

On morphisms, the functor's higher components are

$$\mathcal{F}^L_k(x_1, ..., x_k)(p) = m_{k+1}^{\xi, ..., \xi, b}(x_1, ..., x_k, p),$$

which satisfy the $A_\infty$–functor equations by the underlying Fukaya category relations (Cho et al., 2018, Cho et al., 2014, Cho et al., 2013).

2. Deformation Theory, Mirror Potentials, and Chart Construction

The foundation of the localized mirror approach is the unobstructed deformation theory of the chosen Lagrangian $L$. Odd-degree generators $X_1, ..., X_n \in CF^1(L,L)$, often corresponding to immersed points or torus directions, parameterize a formal deformation $b = \sum x_i X_i$. The weak Maurer–Cartan equation

$$m_0^b = m_0(1) + \sum_{k \geq 1} m_k(b, ..., b) = W(b) \cdot 1_L$$

defines the mirror chart as the formal locus where deformations are unobstructed. The superpotential $W$ is computed as a formal power (or Laurent) series, depending on the geometry and brane data, via enumeration of holomorphic discs or polygons. In the toric case, $W$ is a Laurent polynomial in SYZ coordinates, while for immersed spheres or orbifold loci it typically includes quantum corrections from higher-genus or orbifolded holomorphic discs (Cho et al., 2014, Cho et al., 2013, Lee, 2019).

3. Algebraic and Functorial Properties

The localized mirror functor is a curved Yoneda-type $A_\infty$–functor that detects objects in $\Fuk(M)$ which intersect $L$. The construction ensures:

• The $A_\infty$–functor property by direct computation from the associated Maurer–Cartan relations.
• On morphisms, for composable sequences, higher compositions are computed via insertion of the bounding cochain on the reference Lagrangian.
• In well-behaved settings, the functor is a quasi-equivalence onto its essential image, provided the image split-generates the derived category of matrix factorizations.
• The functor extends over local charts defined by different Lagrangians (possibly with nontrivial wall-crossing structure) and can be glued over overlaps to produce a global mirror functor, achieving full HMS for certain classes of spaces (e.g., punctured surfaces, toric Fano, orbifold spheres) (Cho et al., 2018, Cho et al., 2013, Cho et al., 2018).

4. Comparison of Pairings and Conformal Factor

Under the localized mirror functor, open and closed string pairings in Floer theory are related to corresponding pairings on the $B$-model side (the Kapustin–Li and residue pairings on matrix factorizations and the Jacobian ring). Concretely: $$\langle v, w \rangle_{PD_L} = c_L^2 \langle \mathcal{F}^L(v), \mathcal{F}^L(w) \rangle_{KL},$$ where the conformal factor

$c_L = \frac{vol^{\mathrm{Floer}}}{vol_L}$

is the ratio of the Floer-theoretically defined “volume” class and the classical volume form. This conformal factor encodes obstructions to perfect matching of pairings and is crucial for correcting the Kodaira–Spencer isomorphism in closed-string theory. Notably, for nef toric manifolds, $c_L = 1$, resulting in an isometry of pairings. In complex or orbifold settings, $c_L$ may be a nontrivial power series, computable from Lagrangian enumerative invariants (Cho et al., 2018).

Table: Correspondence of Pairings

Floer data	Mirror data (MF)	Correction
Poincaré Duality	Kapustin–Li	Conformal $c_L^2$
Quantum cup	Residue pairing	Conformal $c_L^2$

5. Gluing, Global Mirrors, and Applications

For a collection of Lagrangians $\{L_i\}$ furnishing a cover by local charts, the construction of the global mirror involves glueing along overlaps (where isomorphism data exist in Floer cohomology). The homotopy-fiber product of local categories of matrix factorizations realizes the global mirror geometry and a global $A_\infty$–functor: $\mathcal{F}^{gl}: \Fuk(X) \to MF_{\Gamma}(W)$ where $\Gamma$ encodes the gluing graph (Cho et al., 2018). This framework applies to:

• Punctured Riemann surfaces, where the tropical skeleton and toric charts glue to the rigid analytic open Calabi–Yau mirror (Cho et al., 2018).
• Toric manifolds, orbifold spheres, and their quantum-corrected mirrors (Cho et al., 2014, Cho et al., 2013, Cho et al., 2018).
• Plumbing of spheres (Nakajima quiver varieties), where the functor realizes Hecke correspondences and modules over quiver algebras (Lau et al., 20 Jan 2026, Hu et al., 2024).
• Elliptic curves and their noncommutative (Sklyanin) mirrors, related by Orlov's LG/CY correspondence and comparison with the Polishchuk–Zaslow Fourier–Mukai functor (Lee, 2020, Lee, 2019).

6. Worked Examples

Toric Manifolds: For a Lagrangian torus, the localized mirror functor reduces to the canonical functor to MF(W), where $W$ is the toric superpotential, and on the Clifford torus in projective space CP$^{n}$, this reproduces the Hori–Vafa mirror (Cho et al., 2014).

Orbifold Sphere $\mathbb{P}^1_{3,3,3}$: Using Seidel’s immersed circle, the mirror potential is

$$W(x, y, z) = \phi(q)(x^3 + y^3 + z^3) + \psi(q)\,xyz,$$

with explicit generating series $\phi, \psi$, and the localized mirror functor produces matrix factorizations encoding all quantum corrections. The conformal factor

$c_L = \sum_{k\in\mathbb{Z}}(-1)^k q^{(6k+1)^2}$

governs the pairings (Cho et al., 2018, Cho et al., 2013).

Nakajima Quiver Varieties: The functor from Floer theory of plumbings to dg-modules over the completed quiver path algebra (with relations from holomorphic disc counts) recovers fully faithful embedding and, in the case of non-ADE graphs, achieves a quasi-equivalence to perfect complexes on the quiver variety (Lau et al., 20 Jan 2026, Hu et al., 2024).

Elliptic Curves: The functor constructed from the Seidel-immersed Lagrangian recovers the noncommutative Landau–Ginzburg mirror and agrees with the canonical Polishchuk–Zaslow equivalence up to explicit symplectic automorphism and Orlov's LG/CY correspondence (Lee, 2019, Lee, 2020).

7. Theoretical and Structural Consequences

The localized mirror functor construction:

• Provides a computable, categorical realization of the mirror correspondence at the level of $A_\infty$–categories, matching not only objects and morphisms but also higher algebraic and geometric structures.
• Tracks the precise relationship of open- and closed-string pairings via the conformal factor, correcting shortcomings of naïve isomorphisms.
• Is compatible with geometric gluing data, allowing patching of local charts into global mirrors and functors.
• Extends to noncommutative and stacky targets, capturing new phenomena in higher genus, non-toric, and ADE/non-ADE degeneration cases (Bai et al., 11 Jan 2025, Hu et al., 2024).
• Enables new approaches to quantum corrections, wall-crossing, and wall-tracking, as well as the study of pairings and their enumerative implications, including explicit calculation of mirror maps and primitive form choices (Cho et al., 2018, Cho et al., 2013).

The approach is now central in the categorical and enumerative geometry aspects of mirror symmetry and is foundational for recent work in global and higher genus mirror symmetry, quiver varieties, and microlocal sheaf–theoretic perspectives.