[Paper Review] Mirror Symmetry, D-Branes and Counting Holomorphic Discs

Mina Aganagic, Cumrun Vafa|ArXiv.org|Dec 5, 2000

Geometry and complex manifoldsMathematics20 references331 citations

TL;DR

This paper establishes a mirror symmetry correspondence between special Lagrangian submanifolds in Calabi-Yau manifolds and holomorphic submanifolds in their mirror geometries, enabling the counting of holomorphic disc instantons via the Abel-Jacobi map on the mirror. It confirms a universal 1/n² multi-covering formula and derives new integer-valued disc instanton numbers, passing integrality checks and providing non-trivial predictions for A-brane systems on O(-1)⊕O(-1) over ℙ¹ and degenerate ℙ¹×ℙ¹.

ABSTRACT

We consider a class of special Lagrangian subspaces of Calabi-Yau manifolds and identify their mirrors, using the recent derivation of mirror symmetry, as certain holomorphic varieties of the mirror geometry. This transforms the counting of holomorphic disc instantons ending on the Lagrangian submanifold to the classical Abel-Jacobi map on the mirror. We recover some results already anticipated as well as obtain some highly non-trivial new predictions.

Motivation & Objective

• To extend mirror symmetry to special Lagrangian submanifolds in Calabi-Yau manifolds and their mirror holomorphic counterparts.
• To develop a method for counting holomorphic disc instantons ending on these Lagrangian submanifolds using mirror geometry.
• To verify the 1/n² multi-covering formula for disc amplitudes and derive new integer-valued instanton numbers.
• To investigate the role of the mirror map in relating quantum-corrected areas and boundary fields in the A-model.
• To provide non-trivial, integrality-preserving predictions for disc instanton numbers in specific toric Calabi-Yau geometries.

Proposed method

• Utilizes the recent derivation of mirror symmetry via T-duality in the linear sigma model to map special Lagrangian submanifolds to holomorphic cycles in the mirror manifold.
• Applies the Abel-Jacobi map on the mirror Calabi-Yau to compute holomorphic disc amplitudes, transforming the A-model disc counting problem into a classical complex geometry computation.
• Constructs special Lagrangians in toric Calabi-Yau geometries using rational linear subspaces of the toric skeleton with constraints ∑qᵢ^α = 0.
• Derives the superpotential in the A-model via mirror map relations, incorporating quantum-corrected Kähler moduli and boundary fields.
• Uses the mirror map to relate classical moduli t₁, t₂ to quantum Kähler parameters T₁, T₂, enabling computation of disc instanton contributions.
• Expands the superpotential in boundary variables to extract disc instanton degeneracies dₖ,ₘ and checks their integrality.

Experimental results

Research questions

• RQ1 How does mirror symmetry map special Lagrangian submanifolds in a Calabi-Yau to holomorphic submanifolds in the mirror geometry?
• RQ2 Can the counting of holomorphic disc instantons ending on special Lagrangians be reduced to a classical Abel-Jacobi map computation on the mirror manifold?
• RQ3 Do the disc instanton numbers derived via mirror symmetry satisfy the integrality constraints predicted in prior work?
• RQ4 Is the 1/n² multi-covering formula for disc amplitudes universally valid in these geometries, and can it be derived from mirror symmetry?
• RQ5 What are the explicit disc instanton degeneracies for A-branes on O(-1)⊕O(-1) over ℙ¹ and in the degenerate limit of ℙ¹×ℙ¹?

Key findings

• The paper confirms the 1/n² multi-covering formula for holomorphic disc instantons in the A-model, consistent with earlier predictions.
• For the O(-1)⊕O(-1) over ℙ¹ in phase II, the disc instanton numbers dₙ are integers and grow as dₙ ∼ n² for large n, matching the 1/n² formula.
• In the degenerate limit of ℙ¹×ℙ¹, the disc instanton degeneracies dₖ,ₘ are integers for all k, m, with dₖ,ₘ growing as m²ᵏ⁻¹ for fixed k and large m.
• The superpotential expansion in phase II yields coefficients Cₖ,ₘ that are rational functions, and the resulting dₖ,ₘ are integers, passing the integrality test.
• The mirror map for boundary fields is modified to e^û = -(1+q)eᵘ and e^v̂ = -(1+q)eᵛ, ensuring consistency with the Lagrangian intersection at the equator.
• The method successfully computes disc instanton numbers for multiple geometries, including ℙ¹×ℙ¹ in both phases, with explicit tables of dₖ,ₘ provided for k ≤ 16 and m ≤ 16.