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 journalArticle

17 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1873-1911
Number of pages39
JournalAdvances in Mathematics
Volume229
Issue number3
DOIs
StatePublished - Feb 15 2012

Fingerprint

Mirror Symmetry
Constructible
Toric Varieties
Torus
Duality
Sheaves
Equivariant
Tensor Category
Triangulated Category
Line Bundle
Correspondence
Equivalence
Equate
Cotangent
Universal Cover
Invariant Metric
Derived Category
Fibration
Branes
Vector space

Keywords

  • Homological mirror symmetry
  • T-duality
  • Toric varieties

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Fang, Bohan ; Liu, Chiu Chu Melissa ; Treumann, David ; Zaslow, Eric. / T-duality and homological mirror symmetry for toric varieties. In: Advances in Mathematics. 2012 ; Vol. 229, No. 3. pp. 1873-1911.
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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 journalArticle

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

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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

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