### 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) |
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Pages (from-to) | 1873-1911 |

Number of pages | 39 |

Journal | Advances in Mathematics |

Volume | 229 |

Issue number | 3 |

DOIs | |

State | Published - Feb 15 2012 |

### Fingerprint

### Keywords

- Homological mirror symmetry
- T-duality
- Toric varieties

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*229*(3), 1873-1911. https://doi.org/10.1016/j.aim.2011.10.022

}

*Advances in Mathematics*, vol. 229, no. 3, pp. 1873-1911. https://doi.org/10.1016/j.aim.2011.10.022

**T-duality and homological mirror symmetry for toric varieties.** / Fang, Bohan; Liu, Chiu Chu Melissa; Treumann, David; Zaslow, Eric.

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

UR - http://www.scopus.com/inward/record.url?scp=84455173842&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84455173842&partnerID=8YFLogxK

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 -