Abstract
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 language | English (US) |
|---|---|
| Pages (from-to) | 1873-1911 |
| Number of pages | 39 |
| Journal | Advances in Mathematics |
| Volume | 229 |
| Issue number | 3 |
| DOIs | |
| State | Published - Feb 15 2012 |
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Keywords
- Homological mirror symmetry
- T-duality
- Toric varieties
ASJC Scopus subject areas
- Mathematics(all)
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T-duality and homological mirror symmetry for toric varieties. / Fang, Bohan; Liu, Chiu Chu Melissa; Treumann, David; Zaslow, Eric.
In: Advances in Mathematics, Vol. 229, No. 3, 15.02.2012, p. 1873-1911.Research output: Contribution to journal › Article
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
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
VL - 229
SP - 1873
EP - 1911
JO - Advances in Mathematics
JF - Advances in Mathematics
SN - 0001-8708
IS - 3
ER -