Analytic Torsion on Calabi-Yau Moduli

摘要/ abstract摘要：dms -0606721项目负责人：郝方主要研究Calabi-Yau流形的解析扭转。受beshadsky - coccoti - ooguri - vafa的启发，首席研究员与Lu和Yoshikawa共同构造了Calabi-Yau流形的一个新的全纯不变量，可以计算一维模空间的不变量。总的来说，主要研究者提出了一种方法来分析新的不变量的渐近行为，通过使用局部指标理论技术。该项目将有助于更好地理解扭转不变量和Calabi-Yaumoduli。这项研究可以看作是用微分几何方法研究代数几何问题的一次尝试。通过解析地研究这些奇异点附近的局部几何，主要研究者还希望给出显式可计算的扭转不变量的新例子，并揭示扭转不变量与数论的关系。该项目还将产生更广泛的影响。扭转不变量来源于拓扑学和高能物理理论。这项提议的研究将为理解镜像对称提供新的思路，镜像对称是当前理论物理学的核心问题之一。理解这一理论将为弦理论提供数学基础，弦理论是高能物理的潜在统一理论，自爱因斯坦时代以来一直受到追捧。例如，拟议的研究可能会导致对宇宙（数量和形状）的更好理解。期望与其他数学家和物理学家进行互动，这将是本研究项目成功的必要条件。主要研究者将参与提高国家教育水平的综合研究和教育活动。

AbstractAward: DMS-0606721Principal Investigator: Hao FangThe principal investigator proposes to study analytic torsion onthe Calabi-Yau manifolds. Inspired byBeshadsky-Coccoti-Ooguri-Vafa, the principal investigator , jointwith Lu and Yoshikawa, has constructed a new holomorphicinvariant of Calabi-Yau manifolds, which can be computed theinvariant for one dimensional moduli spaces. In general, theprincipal investigator suggests an approach to analysis theasymptotic behavior of the new invariant, by using the localindex theory techniques. This project should give insight onbetter understanding of the torsion invariants and the Calabi-Yaumoduli. The proposed study can be viewed as an attempt to studyproblems in algebraic geometry using methods of differentialgeometry. By studying local geometry analytically near thesingularities, the principal investigator also hopes to give newexamples of the explicitly computable torsion invariants andreveal relations of torsion invariants to number theory.This project also will have broader impact. Torsion invariantshave come from topology and high energy physics theories. Theproposed study will shed new light on understanding MirrorSymmetry, which is one of the central problems in currenttheoretical physics. Understanding this theory will givemathematical foundation of the String Theory, which is apotential unified theory for high energy physics long soughtafter since Einstein's time. For example, the proposed researchmight lead to better understanding of (the number and the shape)of the universe. Interaction with fellow mathematicians andphysicists is expected and will be necessary for the success ofthis research project. The principal investigator willparticipate the integrated research and education activities thatwill promote the education level of the nation.