[Paper Review] Note on mirror symmetry and coisotropic D-branes on tori

Christian van Enckevort|ArXiv.org|Nov 6, 2001

Black Holes and Theoretical Physics7 references3 citations

TL;DR

This paper provides a concrete description of mirror symmetry for coisotropic D-branes on even-dimensional tori using the Ooguri-Oz-Yin framework, showing that semi-homogeneous vector bundles—non-Lagrangian D-branes—map consistently under mirror symmetry. It confirms that mirror symmetry intertwines isogenies and preserves structure across A- and B-models, offering a complete picture of mirror symmetry on D-brane objects in toroidal Calabi-Yau manifolds.

ABSTRACT

We describe mirror symmetry on higher dimensional tori, paying special attention to the behaviour of D-branes under mirror symmetry. To find the mirror D-branes the description of mirror symmetry on D-branes due to Ooguri, Oz en Yin is used. This method allows us to deal with the coisotropic D-branes recently introduced by Kapustin and Orlov. We compare this to the description of mirror symmetry on D-branes using the Fourier-Mukai transform of charges.

Motivation & Objective

• To provide a concrete realization of mirror symmetry for coisotropic D-branes on higher-dimensional tori, which are not Lagrangian and thus not captured by standard A-brane descriptions.
• To extend the Ooguri-Oz-Yin method for mirror symmetry on D-branes to include non-Lagrangian objects such as semi-homogeneous vector bundles.
• To verify that the mirror map preserves key geometric and categorical structures, including isogenies and the interplay between A- and B-models.
• To establish a functorial correspondence between D-brane categories on mirror tori, particularly for objects beyond Lagrangian submanifolds.
• To lay the groundwork for a full categorical mirror symmetry description by showing consistency with derived categories and Fourier-Mukai-type transforms.

Proposed method

• Uses the Ooguri-Oz-Yin framework for mirror symmetry on D-branes, which describes boundary conditions via orthogonal matrices R with respect to the metric g.
• Applies the description of D-branes as affine subspaces in R^n × R^n with gluing matrices R satisfying R^t g R = g and R^2 = I.
• Introduces semi-homogeneous vector bundles as objects that are coisotropic but not Lagrangian, using their classification via isogenies and complex structures.
• Employs the SYZ fibration via projection to x-coordinates, treating T^n × T^n and its mirror T^n × T̂n as dual tori.
• Uses the mirror map μ to relate objects on X = T^n × T^n to those on Y = T^n × T̂n, showing μ(i_*(L)) = i_*(μ(L)) for isogenies i.
• Applies Gaussian integrals on tori to compute topological invariants, using formulas for ∫ e^{½⟨dy,Ady⟩} = √det(A) · vol(T^{2m}) and completing the square for linear terms.

Experimental results

Research questions

• RQ1 How do coisotropic D-branes, which are not Lagrangian, transform under mirror symmetry on tori?
• RQ2 Can the Ooguri-Oz-Yin method for mirror symmetry on D-branes be extended to include non-Lagrangian objects like semi-homogeneous vector bundles?
• RQ3 How do isogenies on the base and fiber tori interact with the mirror map, and does the mirror symmetry functor preserve these structures?
• RQ4 To what extent does the mirror map on D-brane objects align with the derived category equivalence in homological mirror symmetry?
• RQ5 What role do B-fields and general background fields play in the mirror correspondence, and how can the framework be generalized beyond the B-field-free case?

Key findings

• Coisotropic D-branes, specifically semi-homogeneous vector bundles, are consistently mapped under mirror symmetry via the Ooguri-Oz-Yin framework, confirming their role in mirror symmetry.
• The mirror map μ intertwines isogenies i_* and ī^*, satisfying μ(ī^*(L)) = i_*(μ(L)) and μ(ī_*(L)) = i^*(μ(L)), showing functorial consistency.
• The mirror symmetry map preserves the structure of D-brane categories across A- and B-models, with the A-model on X and B-model on Y being dual under μ.
• The paper confirms that the mirror of a semi-homogeneous vector bundle is a coisotropic D-brane in the mirror manifold, consistent with expectations from physics and geometry.
• Gaussian integrals on tori yield ∫ e^{½⟨dy,Ady⟩} = √det(A) · vol(T^{2m}), providing a computational tool for topological invariants in the mirror map.
• The framework is consistent with the Fourier-Mukai transform of D-brane charges, and the results support the conjecture that semi-homogeneous bundles generate the derived category.