Calabi-Yau Manifolds and Mirror Symmetry

卡拉比-丘流形和镜像对称

• 批准号: RGPIN-2019-04000
• 负责人: YUI, NORIKO
• 金额: $ 1.09万
• 依托单位: Queen's University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=713282
• 关键词: Calabi; Yau; Manifolds; Mirror; Symmetry

The proposal is concerned with problems at the crossroads of number theory and string theory. String theory is a physics prediction that what the universe is made of, is, strings (one-dimensional objects), rather than point particles (zero-dimensional objects). String theory demands ten-dimensional space-time (as opposed to the dimension four of our real world). This is, roughly speaking, because more dimensions can accommodate more possible string vibrations. The extra six-dimensional objects in string theory are known as Calabi-Yau threefolds, and they are the main objects of this investigation. The goal of the proposed research is to understand the physical prediction of mirror symmetry for Calabi-Yau manifolds, and its consequences, from a mathematical point of view. A Calabi-Yau manifold is a compact complex Kaehler manifold with vanishing first Chern class and zero first Betti number. Calabi-Yau manifolds of dimension one, two, and three are, respectively, elliptic curves, K3 surfaces, and Calabi-Yau threefolds. Mirror symmetry is a prediction in string theory that certain "mirror pairs" of Calabi-Yau threefolds yield identical physical theories. Modular forms, Siegel and Jacobi modular forms, and automorphic forms appear, either as generating functions of geometric invariants, or as partition functions. One of my research goals is to interpret mirror symmetry in terms of arithmetic invariants such as zeta-functions and L-series of the Calabi-Yau manifolds. In this connection, the automorphy question for the L-series will be vigorously pursued. Here, "automorphy" refers to the fact that the (motivic) L-series arising from Calabi-Yau manifolds are automorphic L-series as in the context of the Langlands program. At the moment, the most pressing issue is to address the automorphy question for families of Calabi-Yau manifolds with all Hodge numbers of the third cohomology equal to one. Such families give rise to four-dimensional Galois representations. A very crude conjecture is that these irreducible four-dimensional Galois representations should correspond to Siegel modular forms of weight 3 and genus 2 on some paramodular subgroups of Sp(4, Z). Another central goal is the conceptual understanding of the appearance of various modular forms in the partition functions, and of the generating functions of the Gromov-Witten invariants, the Donaldson-Thomas invariants, the Gopakumar-Vafa invariants and other geometric or physical invariants (e.g, BPS state counting numbers), for Calabi-Yau manifolds. It is imperative to lay solid mathematical foundations for string theory, for the benefit of both mathematicians and string theorists. I plan to train HQP (postdoctoral fellows and graduate students) through this project. My approach will be to assign concrete examples to each of them to work out and understand the main goal of this project. and eventually lead to new mathematical discoveries.

该提案涉及数论和弦论交叉的问题。弦理论是一种物理学预言，宇宙是由弦（一维）构成的