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

In this paper, we apply the idea of T-duality to projectivespaces. From a connection on a line bundle on $\mathbbP^n$, a Lagrangian in the mirror Landau–Ginzburg model isconstructed. Under this correspondence, the full strongexceptional collection $\mathcal O_{\mathbbP^n}(-n-1),\ldots,$ $\mathcal O_{\mathbb P^n}(-1)$ ismapped to standard Lagrangians in the sense of \cite{nz}.Passing to constructible sheaves, we explicitly compute thequiver structure of these Lagrangians, and find that theymatch the quiver structure of this exceptional collectionof $\mathbb P^n$. In this way, T-duality providesquasi-equivalence of the Fukaya category generated by theseLagrangians and the category of coherent sheaves on$\mathbb P^n$, which is a kind of homological mirrorsymmetry.

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Published 1 January 2008