2005 Fiscal Year Final Research Report Summary
Geometry of mirror symmetry and string theory
| Project/Area Number |
15540010
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Graduate School of Mathematical Sciences, University of Tokyo |
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Principal Investigator |
HOSONO Shinobu University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (60212198)
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| Project Period (FY) |
2003 – 2005
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| Keywords | Calabi-Yau manifold / Mirror symmetry / Hypergeometric series / Period integral / Singularity / GKZ system / GKZ方程式系 |
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| Research Abstract |
In this research project, the following results are obtained regarding non-compact Calabi-Yau varieties, their period integrals and differential equations satisfied them.
Firstly, when a Calabi-Yau variety is given as the c_1 = 0 resolution of the singularity C^2/Z_<μ+1>, it is found that the period integrals are very close to the so-called primitive forms introduced by K.Saito. Since the period integrals are invariant under certain torus actions, we found that they satisfy a system of differential equations called Gel'fand-Kapranov-Zelvinski (GKZ) system. As a result we obtained a way to rephrase the theory of the primitive forms in terms of the GKZ system. It is known that the monodromy of the primitive forms is given by the Weyl group of the A_n root system. In our case, it is found that this Weyl group action is extended too the corresponding affine Weyl group actions on the period integrals.
Secondly, we studied the cases of three dimensional singularity C^3/G (G⊂SL(3,C) : a finite abelian group) and its c_1 = 0 resolutions. We obtained a precise definition of the period integrals and their characterization in terms of the GKZ systems. We observed that the monodromy of the period integrals is closely related to the McKay correspondence which connects the representation theory to algebraic geometry. Namely, under mirror symmetry, the McKay correspondence is transformed to the theory of transcendental cycles, and for example, Fourier-Mukai transforms on the derived category of coherent sheaves appear as the monodromy of the period integrals. We verified this 'mirror monodromy relations' in explicit examples. This monodromy property has been made precise as a mathematical conjecture in terms of certain hypergeometric series taking its values in the relevant cohomology group.
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