[Paper Review] Mirror symmetry and T-duality in the complement of an anticanonical divisor

Denis Auroux|ArXiv.org|Jun 21, 2007

Nonlinear Waves and Solitons25 references295 citations

TL;DR

This paper investigates mirror symmetry and T-duality in the complement of an anticanonical divisor in a compact Kähler manifold, proposing a mirror construction via moduli spaces of special Lagrangian tori with flat U(1) connections and a superpotential defined by the m₀ obstruction. It shows that quantum corrections from Maslov index 0 discs are essential, and provides evidence for a relative homological mirror symmetry conjecture linking the Fukaya category of the mirror to the derived category of coherent sheaves on the divisor.

ABSTRACT

We study the geometry of complexified moduli spaces of special Lagrangian submanifolds in the complement of an anticanonical divisor in a compact Kahler manifold. In particular, we explore the connections between T-duality and mirror symmetry in concrete examples, and show how quantum corrections arise in this context.

Motivation & Objective

• To understand the geometric relationship between mirror symmetry and T-duality in non-Calabi-Yau settings, particularly in the complement of an anticanonical divisor.
• To investigate how quantum corrections from Maslov index 0 holomorphic discs affect the mirror construction in this context.
• To formulate and provide evidence for a relative homological mirror symmetry conjecture linking the Fukaya category of the mirror to coherent sheaves on the divisor.
• To extend the Strominger-Yau-Zaslow conjecture beyond Calabi-Yau manifolds by analyzing special Lagrangian fibrations in non-compact, non-Calabi-Yau settings.
• To demonstrate the role of the superpotential m₀ in defining the mirror Landau-Ginzburg model and its connection to quantum cohomology and wall-crossing phenomena.

Proposed method

• Constructs a mirror manifold M as a moduli space of special Lagrangian tori in X\D with flat U(1) connections.
• Defines a superpotential W: M → ℂ using the m₀ obstruction from Fukaya-Oh-Ohta-Ono theory, which measures the obstruction to defining Floer homology.
• Analyzes the geometry of the complexified moduli space of special Lagrangians, focusing on ψ-harmonic 1-forms and their role in deformation theory.
• Introduces admissible Lagrangians in the mirror with boundary in the fiber M_D = {z_δ = 1}, ensuring well-defined Floer homology despite wall-crossing.
• Uses a rescaling limit (Conjecture 4.4) to simplify the superpotential to W = z_δ + o(1), removing complications from wall-crossing.
• Proposes a restriction functor ρ from the Fukaya category of M to that of M_D, linking A-model on the mirror to B-model on the divisor D.

Experimental results

Research questions

• RQ1 How does T-duality relate to mirror symmetry in the complement of an anticanonical divisor in a Fano or Kähler manifold?
• RQ2 What role do quantum corrections from Maslov index 0 holomorphic discs play in the mirror construction when the manifold is not Calabi-Yau?
• RQ3 How can the superpotential m₀ be consistently defined in the presence of wall-crossing phenomena in the moduli space of special Lagrangians?
• RQ4 What is the relationship between the critical values of the superpotential and the quantum cohomology of the original manifold X?
• RQ5 To what extent does the Fukaya category of the mirror Landau-Ginzburg model categorically mirror the derived category of coherent sheaves on the anticanonical divisor D?

Key findings

• The superpotential W defined via the m₀ obstruction is multivalued due to wall-crossing, necessitating modifications to the naive mirror conjecture.
• Quantum corrections from Maslov index 0 holomorphic discs are expected to play a role analogous to the Calabi-Yau case, modifying the mirror geometry.
• The mirror manifold M constructed from special Lagrangian tori is incomplete, suggesting the need for a renormalization limit as proposed in Conjecture 4.4.
• Admissible Lagrangians in the mirror M with boundary in M_D allow for a well-defined Floer homology theory, avoiding issues from wall-crossing.
• The restriction functor ρ: F(M, M_D) → F(M_D) is well-defined and matches expected behavior in known examples like Del Pezzo surfaces.
• Evidence is provided for Conjecture 7.7: a commutative diagram relating derived categories of coherent sheaves on X and D to the Fukaya categories of M and M_D, suggesting relative homological mirror symmetry.