TY - JOUR

T1 - Local Calabi–Yau manifolds of type A˜ via SYZ mirror symmetry

AU - Kanazawa, Atsushi

AU - Lau, Siu Cheong

N1 - Funding Information:
The authors are grateful to Naichung Conan Leung and Shing-Tung Yau for useful discussions and encouragement. The first author benefited from many conversations with Amer Iqbal, Charles Doran, Shinobu Hosono and Yuecheng Zhu. The second author is very thankful to Mark Gross and Eric Zaslow for useful discussions and explaining the toric description of X ( 1 ) dated back to 2011. He also thanks Helge Ruddat for explaining his work on mirrors of general-type varieties using toric Calabi–Yau geometries and discussing the 3 -dimensional honeycomb tiling also in 2011. The first author was supported by the Harvard CMSA, USA and the Kyoto University Hakubi Project. The second author was supported by Boston University, USA and the Simons Collaboration, USA Grant for Mathematicians at the time of publication of this paper. Appendix We consider an m -dimensional complex torus X = ℂ m ∕ Λ . Here e 1 , … , e m be a complex basis of ℂ m and Λ be the lattice generated by the 2 m independent vectors λ i = ∑ ω ˜ α i e α in R 2 m ≅ ℂ m . We define the differentials d z α and d x i in such a way that ∫ e β d z α = δ α β and ∫ λ j d x i = δ i j hold. The m × 2 m matrix Ω ˜ = ( ω ˜ α i ) is called the period matrix and the lattice Λ is generated by the 2 m columns of Ω ˜ . The Kodaira embedding theorem asserts that the complex torus X is an abelian variety if and only if it admits a Hodge form (an integral closed positive ( 1 , 1 ) -form) ω = i ∑ α , β h α , β d z α ∧ d z ¯ β . We may pick a new basis of ℂ m and Λ , not in a unique way, such that Ω ˜ = δ 1 0 ⋱ Ω 0 δ m , where Ω = ( ω i j ) ∈ ℌ m and integers δ i ≥ 1 ( 1 ≤ i ≤ m ) such that δ i | δ i + 1 . Here ℌ m is the Siegel upper half-space of degree m defined as ℌ m ≔ { Ω ∈ M m ( ℂ ) | Ω t = Ω , Im ( Ω ) > 0 } . In these new coordinates, ω takes of the form ω = ∑ α δ α d x α ∧ d x m + α . The cohomology class [ ω ] of the Hodge form, or equivalently the sequence of integers ( δ 1 , … , δ m ) , provides the so-called ( δ 1 , … , δ m ) -polarization of the abelian variety X . The sequence ( δ 1 , … , δ m ) is an invariant of the cohomology class [ ω ] and independent of the choice of a basis. When δ 1 = ⋯ = δ m = 1 , the abelian variety X is called principally polarized. By abuse of notation, we do not impose the divisibility condition δ i | δ i + 1 in this article, but you can always find a new basis with respect to which the corresponding sequence ( δ 1 ′ , … , δ m ′ ) satisfies the divisibility condition. For a , b ∈ R m , the genus m Riemann theta function with characteristic a b is defined by Θ m a b ( z ; Ω ) ≔ ∑ n ∈ Z m exp 2 π i 1 2 ( n + a ) ⋅ Ω ( n + a ) + ( n + a ) ⋅ ( z + b ) , where z ∈ ℂ m , Ω ∈ ℌ m . We allow the shift b to be in ℂ m for simplicity of notations in this article. We also denote Θ m 0 0 ( z ; Ω ) by Θ m ( z ; Ω ) . Let L be the line bundle associated to the Hodge form ω . It is known that H 0 ( X , L ) has a basis given by the theta functions Θ m ( i 1 δ 1 , … , i m δ m ) 0 ( z ; Ω ) , ( 0 ≤ i k ≤ δ k − 1 ) . It is also useful to realize X as ( ℂ × ) m ∕ Z m via the shifted exponential map exp : ℂ m → ( ℂ × ) m , ( z 1 , z 2 , … , z m ) ↦ ( e 2 π i δ 1 z 1 , e 2 π i δ 2 z 2 , … , e 2 π i δ m z m ) . Then X can be thought as a quotient of ( ℂ × ) m by the equivalent relations for ( y 1 , y 2 , … , y m ) ∈ ( ℂ × ) m : ( y 1 , y 2 , … , y m ) ∼ ( e 2 π i ω i 1 δ 1 y 1 , e 2 π i ω i 2 δ 2 y 2 , … , e 2 π i ω i m δ m y m ) , ( 1 ≤ i ≤ m ) .
Funding Information:
The authors are grateful to Naichung Conan Leung and Shing-Tung Yau for useful discussions and encouragement. The first author benefited from many conversations with Amer Iqbal, Charles Doran, Shinobu Hosono and Yuecheng Zhu. The second author is very thankful to Mark Gross and Eric Zaslow for useful discussions and explaining the toric description of X
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2019/5

Y1 - 2019/5

N2 - We carry out the SYZ program for the local Calabi–Yau manifolds of type A˜ by developing an equivariant SYZ theory for the toric Calabi–Yau manifolds of infinite-type. Mirror geometry is shown to be expressed in terms of the Riemann theta functions and generating functions of open Gromov–Witten invariants, whose modular properties are found and studied in this article. Our work also provides a mathematical justification for a mirror symmetry assertion of the physicists Hollowood–Iqbal–Vafa (Hollowood et al., 2008).

AB - We carry out the SYZ program for the local Calabi–Yau manifolds of type A˜ by developing an equivariant SYZ theory for the toric Calabi–Yau manifolds of infinite-type. Mirror geometry is shown to be expressed in terms of the Riemann theta functions and generating functions of open Gromov–Witten invariants, whose modular properties are found and studied in this article. Our work also provides a mathematical justification for a mirror symmetry assertion of the physicists Hollowood–Iqbal–Vafa (Hollowood et al., 2008).

KW - Abelian varieties

KW - Calabi–Yau manifolds

KW - Riemann theta functions

KW - SYZ mirror symmetry

KW - Toric geometry

KW - open Gromov–Witten invariants

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U2 - 10.1016/j.geomphys.2018.12.015

DO - 10.1016/j.geomphys.2018.12.015

M3 - Article

AN - SCOPUS:85061610212

SN - 0393-0440

VL - 139

SP - 103

EP - 138

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

ER -