TY - JOUR
T1 - T-duality and homological mirror symmetry for toric varieties
AU - Fang, Bohan
AU - Liu, Chiu Chu Melissa
AU - Treumann, David
AU - Zaslow, Eric
N1 - Funding Information:
We thank M. Abouzaid, A. Bondal and P. Seidel for explaining relevant aspects of their work, and D. Nadler for helpful suggestions. The work of E.Z. is supported in part by NSF/DMS-0707064. B.F. and E.Z. would like to thank the Kavli Institute for Theoretical Physics and the Pacific Institute for the Mathematical Sciences, where some of this work was performed.
PY - 2012/2/15
Y1 - 2012/2/15
N2 - 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.
AB - 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.
KW - Homological mirror symmetry
KW - T-duality
KW - Toric varieties
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U2 - 10.1016/j.aim.2011.10.022
DO - 10.1016/j.aim.2011.10.022
M3 - Article
AN - SCOPUS:84455173842
VL - 229
SP - 1873
EP - 1911
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
IS - 3
ER -