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Homological mirror symmetry for \(A_{n}\)-resolutions as a \(T\)-duality. (English) Zbl 1279.53079

This paper treats a version of homological mirror symmetry (HMS) for \(A_n\)-resolutions, from the view point of \(T\)-duality (SYZ transformation). It approaches this particular HMS about the resolutions of \(A_n\)-singularities by the explicit geometric method of \(T\)-dualiy. Starting from a torus fibration, the author constructs its B-model mirror by the SYZ transformation with instanton corrections. The resulting B-model complex manifold is precisely the \(A_n\)-resolution minus a hypersurface. The author further performs this \(T\)-duality process for several spherical Lagrangian branes and produces the corresponding B-model sheaves supported at the exceptional locus of the \(A_n\)-resolution. The objects are the generating objects in the HMS for \(A_n\)-resolutions [A. Ishii et al., J. Differ. Geom. 84, No. 1, 87–126 (2010; Zbl 1198.14020)]. Thus, this paper shows that the HMS for \(A_n\)-resolutions is compatible with \(T\)-duality, both for the mirror space construction and on the level of objects.

MSC:

53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
14J33 Mirror symmetry (algebro-geometric aspects)
53D12 Lagrangian submanifolds; Maslov index
14M25 Toric varieties, Newton polyhedra, Okounkov bodies

Citations:

Zbl 1198.14020
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