[Paper Review] On N=1 Mirror Symmetry for Open Type II Strings

W. Lerche, Peter Mayr|ArXiv.org|Nov 13, 2001

Black Holes and Theoretical PhysicsPhysics and Astronomy36 references63 citations

TL;DR

This paper establishes a differential system—generalized Picard-Fuchs equations—for N=1 mirror symmetry in open type II string theory on non-compact Calabi-Yau manifolds with D-branes. It derives the mirror map for flat coordinates and the exact disc instanton-corrected superpotential via holomorphic chain integrals, using B-model techniques without relying on open-closed duality. The key contribution is a systematic framework for N=1 special geometry with multiple holomorphic functions, validated through toric geometry and explicit instanton number computations.

ABSTRACT

We study the open string extension of the mirror map for N=1 supersymmetric type II vacua with D-branes on non-compact Calabi-Yau manifolds. Its definition is given in terms of a system of differential equations that annihilate certain period and chain integrals. The solutions describe the flat coordinates on the N=1 parameter space, and the exact disc instanton corrected superpotential on the D-brane world-volume. A gauged linear sigma model for the combined open-closed string system is also given. It allows to use methods of toric geometry to describe D-brane phase transitions and the N=1 Kähler cone. Applications to a variety of D-brane geometries are described in some detail.

Motivation & Objective

• To extend closed string mirror symmetry to open strings in N=1 supersymmetric type II compactifications with D-branes.
• To define a consistent differential system for flat coordinates and the superpotential on the N=1 moduli space without assuming a closed string dual.
• To generalize the concept of special geometry to N=1 theories using multiple holomorphic functions instead of a single prepotential.
• To construct a gauged linear sigma model for the A-model that captures D-brane phase transitions and the N=1 Kähler cone via toric geometry.
• To compute disc instanton numbers explicitly for various D-brane configurations, providing quantitative checks on the mirror map.

Proposed method

• Derives a generalized Picard-Fuchs system from the B-model, where differential operators annihilate period and chain integrals associated with holomorphic 3-forms and D-brane cycles.
• Uses chain integrals ∫Γ Ω(z) over 3-chains Γ with boundary C−C* to define the superpotential, with z0 as open string modulus.
• Applies the differential operator D to the chain integral, yielding a boundary term proportional to dη(z), which leads to the generalized PF system.
• Constructs a gauged linear sigma model (GLSM) for the A-model to describe the mirror D-brane geometry and study phase transitions in the N=1 moduli space.
• Employs toric geometry to define the Kähler cone and analyze classical vacua and transitions in the N=1 setting.
• Performs explicit computations of disc instanton numbers N_{n1,n2,n0} for D-branes on F2 and other non-compact CY 3-folds, providing closed-form expressions.

Experimental results

Research questions

• RQ1 How can the mirror map for N=1 open string compactifications be defined independently of open-closed duality?
• RQ2 What differential system governs the flat coordinates and superpotential in the presence of D-branes on non-compact Calabi-Yau manifolds?
• RQ3 How does N=1 special geometry differ from N=2 special geometry in terms of holomorphic data and geometric structure?
• RQ4 What role does toric geometry play in describing the Kähler cone and phase transitions in the N=1 moduli space?
• RQ5 What are the exact disc instanton numbers for D-branes ending on specific cycles, such as the inner edge of F2?

Key findings

• The generalized Picard-Fuchs system derived from the B-model fully determines the flat coordinates and the exact superpotential for D-branes on non-compact Calabi-Yau manifolds.
• The superpotential is given by ∫Γ Ω(z), where Γ is a 3-chain with boundary C−C*, and the differential operator D applied to this integral yields a non-zero boundary term, leading to the system of equations.
• For the D-brane on the inner edge of F2, the disc instanton numbers N_{n1,n2,n0} are explicitly computed, with N_{1,1,0} = -1 and N_{1,0,0} = -1.
• Closed-form expressions are derived for N_{n1,1,n0}, showing dependence on n0 and n1, with N_{n1,1,n0} = -n0 - 1 for 0 ≤ n0 < n1.
• The instanton number N_{4,2,n0} is given by (1/4)(57 - 2n0 + n0²) - (1/4)ε(2,n0) for n0 > 4, providing a non-trivial check on the mirror map.
• The gauged linear sigma model construction allows for the systematic study of D-brane phase transitions and the N=1 Kähler cone using toric techniques.