Homological mirror symmetry is T-duality for \mathbb Pⁿ

2008/04/04 by Fang, Bohan
#Algebraic Geometry (math.AG) #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.0804.0646

Abstract

In this paper, we apply the idea of T-duality to projective spaces. From a connection on a line bundle on \mathbb Pⁿ, a Lagrangian in the mirror Landau-Ginzburg model is constructed. Under this correspondence, the full strong exceptional collection \mathcal O_{\mathbb Pⁿ}(-n-1),...,\mathcal O_{\mathbb Pⁿ}(-1) is mapped to standard Lagrangians in the sense of \citenz. 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 Pⁿ. In this way, T-duality provides quasi-equivalence of the Fukaya category generated by these Lagrangians and the category of coherent sheaves on \mathbb Pⁿ, which is a kind of homological mirror symmetry.

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