[Paper Review] Moduli of Lagrangian immersions with formal deformations

Hansol Hong, Siu-Cheong Lau|arXiv (Cornell University)|Jan 22, 2018

Algebraic structures and combinatorial models36 references3 citations

TL;DR

This paper introduces a Floer-theoretic gluing framework to construct quantum-corrected moduli spaces of Lagrangian immersions, enabling the compactification of moduli of Lagrangian torus fibers by filling punctures via immersed spheres. The key contribution is a noncommutative deformation space that resolves singularities in SYZ mirror symmetry and wall-crossing phenomena through A∞-structure compatibility and cocycle conditions on morphism complexes.

ABSTRACT

We introduce a joint project with Cheol-Hyun Cho on the construction of quantum-corrected moduli of Lagrangian immersions. The construction has important applications to mirror symmetry for pair-of-pants decompositions, SYZ and wall-crossing. The key ingredient is Floer-theoretical gluing between local moduli spaces of Lagrangians with different topologies.

Motivation & Objective

• To develop a moduli theory for Lagrangian immersions that incorporates quantum corrections and complex structures, extending classical moduli of vector bundles to symplectic geometry.
• To address the challenge of singular Lagrangians in SYZ mirror symmetry by using immersed Lagrangians, which retain well-defined Floer cohomology.
• To construct a noncommutative deformation space for singular fibers by gluing local moduli via A∞-algebra isomorphisms and Floer-theoretic compatibility conditions.
• To resolve punctures in the moduli space of Lagrangian tori by filling them with the deformation space of an immersed two-sphere, achieving partial compactification.
• To generalize the construction to higher-dimensional fibrations by taking products with torus factors, enabling wall-crossing and mirror symmetry applications.

Proposed method

• Uses Floer-theoretic gluing between local moduli spaces of Lagrangians with different topologies, particularly leveraging immersed Lagrangians to extend moduli beyond smooth tori.
• Applies the Fukaya trick in a generalized form via Hamiltonian isotopies to relate Floer complexes of non-diffeomorphic Lagrangians, using pearl trajectories to compute differential maps.
• Implements A∞-structure compatibility through cocycle conditions on morphism complexes, ensuring consistency across overlapping Lagrangian pairs.
• Employs the Novikov ring Λ₀ and its units to parameterize deformations, with formal parameters in Λ₊ for immersed sectors, enabling richer deformation spaces than flat connections.
• Computes the differential d in Floer complexes by counting pearl trajectories with symplectic area contributions T^Δ, incorporating contributions from immersed points via hypertori (x,y-parameters).
• Derives consistency conditions via composition of isomorphisms between triples of Lagrangians, leading to algebraic relations such as x = uv − 1 and y⁻¹ = u, derived from symplectic area matching and coordinate changes.

Experimental results

Research questions

• RQ1 How can moduli spaces of Lagrangian immersions be constructed to include quantum corrections and complex structures, enabling compactification of classical moduli?
• RQ2 What is the role of Floer-theoretic gluing in relating Lagrangians of different topologies when the Fukaya trick fails due to non-diffeomorphism?
• RQ3 How do immersed Lagrangians resolve singularities in the moduli space of SYZ fibers, particularly in the context of wall-crossing and mirror symmetry?
• RQ4 What algebraic constraints (e.g., cocycle conditions) arise from the compatibility of A∞-morphisms across overlapping Lagrangian pairs?
• RQ5 Can the deformation space of a singular Lagrangian fiber be noncommutative, and how does this affect the global moduli structure?

Key findings

• The moduli space of smooth Lagrangian tori has a puncture at u = v = 0, which is filled by the deformation space of an immersed two-sphere, achieving partial compactification.
• The deformation space of the immersed sphere is noncommutative, with parameters satisfying x = uv − 1 and y⁻¹ = u, derived from symplectic area matching and isomorphism composition.
• The differential d in the Floer complex is computed as d(α₀) = T^Δ(1 − uy)β₁ + (xh(uv) + g(uv))α₁, with h and g power series of leading term ±1.
• The isomorphism class of the morphism α₀ is preserved if and only if the cocycle conditions x + H(uv) = 0 and y⁻¹ = u are satisfied, where H = g/h.
• The composition of isomorphisms between (L₁, (x,y)) → (L, uU + vV) → (L₂, (x′,y′)) yields a consistent triple with x′ = x and y′ = y(x + 1), leading to the identity 1 − H(uv) = uv.
• The construction generalizes to higher dimensions via direct products with torus factors, suggesting a framework for resolving generic singular fibers in Lagrangian fibrations.