Homological mirror symmetry is T-duality for ${\mathbb{P}}^n$

Abstract

In this paper, we apply the idea of T-duality to projective spaces. From a connection on a line bundle on ${\mathbb{P}}^n$, a Lagrangian in the mirror Landau-Ginzburg model is constructed. Under this correspondence, the full strong exceptional collection ${\mathcal{O}}_{{\mathbb{P}}^n}(-n-1),...,{\mathcal{O}}_{{\mathbb{P}}^n}(-1)$ is mapped to standard Lagrangians in the sense of \cite{nz}. Passing to constructible sheaves, we explicitly compute the quiver structure of these Lagrangians, and find that they match the quiver structure of this exceptional collection of ${\mathbb{P}}^n$. In this way, T-duality provides quasi-equivalence of the Fukaya category generated by these Lagrangians and the category of coherent sheaves on ${\mathbb{P}}^n$, which is a kind of homological mirror symmetry.

21 pages, 4 figures, submitted version

References in corpus (5)

• D-Branes And Mirror Symmetry
• Mirror symmetry and T-duality in the complement of an anticanonical divisor
• Homogeneous coordinate rings and mirror symmetry for toric varieties
• Mirror symmetry for toric Fano manifolds via SYZ transformations
• Mirror symmetry for weighted projective planes and their noncommutative deformations

Cited by in corpus (4)

• The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties
• T-Duality and Homological Mirror Symmetry of Toric Varieties
• On SYZ mirror transformations
• Exact Lefschetz fibrations associated with dimer models