T-duality and homological mirror symmetry for toric varieties

Bohan Fang*, Chiu Chu Melissa Liu, David Treumann, Eric Zaslow

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

22 Scopus citations


Let X σ be a complete toric variety. The coherent-constructible correspondence κ of Fang et al. (2011) [14] equates PerfT(Xσ) with a subcategory Shcc(M R{double-struck}Λσ) of constructible sheaves on a vector space MR. The microlocalization equivalence μ of Nadler and Zaslow (2009) [27] and Nadler (2009) [25] relates these sheaves to a subcategory Fuk(T*M R{double-struck};Λσ) of the Fukaya category of the cotangent T*M R{double-struck}. When X σ is nonsingular, taking the derived category yields an equivariant version of homological mirror symmetry, DCoh T(Xσ)*DFuk(T*M R{double-struck}Λσ), which is an equivalence of triangulated tensor categories.The nonequivariant coherent-constructible correspondence κΛ of Treumann (preprint) [33] embeds Perf(Xσ) into a subcategory Shc(T R{double-struck}VΛ̄σ) of constructible sheaves on a compact torus TRV. When X σ is nonsingular, the composition of κΛ and microlocalization yields a version of homological mirror symmetry, DCoh(Xσ)=DFuk(T*T R{double-struck}Λσ), which is a full embedding of triangulated tensor categories.When X σ is nonsingular and projective, the composition τ=μ{ring operator}κ is compatible with T-duality, in the following sense. An equivariant ample line bundle L has a hermitian metric invariant under the real torus, whose connection defines a family of flat line bundles over the real torus orbits. This data produces a T-dual Lagrangian brane L{double-struck} on the universal cover T*M R{double-struck} of the dual real torus fibration. We prove L*τ(L{double-struck}) in Fuk(T*MR{double-struck}Λσ). Thus, equivariant homological mirror symmetry is determined by T-duality.

Original languageEnglish (US)
Pages (from-to)1873-1911
Number of pages39
JournalAdvances in Mathematics
Issue number3
StatePublished - Feb 15 2012


  • Homological mirror symmetry
  • T-duality
  • Toric varieties

ASJC Scopus subject areas

  • Mathematics(all)

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