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 P n , a Lagrangian in the mirror Landau-Ginzburg model is constructed. Under this correspondence, the full strong exceptional collection O P n (−n − 1), . . . , O P n (−1) is mapped to standard Lagrangians in the sense of [23]. 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 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 P n , which is a kind of homological mirror symmetry.

Dual torus fibrations and homological mirror symmetry for $A_n$-singularities

“…There are also works by Fang [15] and Fang, Liu, Treumann and Zaslow [16,17] on homological mirror symmetry for toric varieties, which are also motivated by the SYZ conjecture but different from the work of Abouzaid.…”

SYZ transforms for immersed Lagrangian multisections

“…The SYZ transform has been constructed and applied to understand mirror symmetry in the semi-flat case [5,34,33,20] and the toric case [1,3,22,23,24,12,14,16,13,19,18,17,21]. But in all of these works the primary focus was on Lagrangian sections and the mirror holomorphic line bundles the SYZ program produces.…”

Homological Mirror Symmetry for Local Calabi-Yau Manifolds via SYZ

“…More generally, one can consider manifolds with an effective anti-canonical divisor [10] or even general type manifolds [57,55,47], for which mirrors are again given by LG models. In this setting, the HMS conjecture has been verified for P 2 and P 1 × P 1 [68, 67], toric del Date: December 22, 2016. Pezzo surfaces [77,78], weighted projective planes and Hirzebruch surfaces [13], del Pezzo surfaces [12], projective spaces [34] and more general projective toric varieties [3,4,35,36, 37], 1 higher genus Riemann surfaces [71,32] and Fano hypersurfaces in projective spaces [73]. 2 The proofs of these HMS statements, though often involve deep and ingenious arguments (e.g.…”