Thematic Program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics

卡拉比-丘品种专题项目：算术、几何和物理

• 批准号: 1247441
• 负责人: Mark Gross
• 金额: $ 3万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-15 至 2014-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1247441&HistoricalAwards=false
• 关键词: Thematic; Program; Calabi; Yau; Varieties

This proposal will help support the workshop on "Modular forms around string theory", scheduled for September 16-20 2013 at the Fields Institute in Toronto, Canada. The workshop is part of the thematic program on Calabi-Yau varieties at the Fields Institute. The NSF support will complement the existing funding and will be used to cover some of the travel expenses of the participants based in the United States. The Fields Institute is a major international research center with extensive experience in organizing similar events. The workshop in question focuses on the appearance of modular forms in many different contexts: in the physics of Calabi-Yau manifolds, such as generating functions for certain physical quantities; in the arithmetic of Calabi-Yau manifolds, as part of the general Langlands philosophy in the study of Galois representations of Calabi-Yau varieties defined over number fields; in the structure of geometric invariants on Calabi-Yau varieties, such as Donaldson-Thomas invariants. All of these represent extremely active topics of research, and the workshop aims to synthesize the different points of view brought by each of these perspectives.Calabi-Yau manifolds play a vital role at the crossroads of mathematics and physics. While they first arose as important objects in geometry, Calabi-Yau manifolds really came to prominence after they appeared naturally in string theory. String theory replaces the traditional notion of the point particle with a small loop of string, moving through space-time. To make string theory compatible with quantum mechanics, space-time must be ten-dimensional. Since space-time appears four-dimensional, one expects six of these dimensions to be a very small "curled up" geometric object. These objects are Calabi-Yau manifolds. Their introduction into physics led to a great deal of new mathematics. A crucial property of Calabi-Yau manifolds is that they both have very delicate flatness properties (satisfying so-called Ricci flatness) and they can be defined using systems of polynomials equations. As a result, they can be viewed as both algebro-geometric and arithmetic objects, the latter if the polynomial equations have coefficients in a number field. Intimate connections have been found between the physical, algebro-geometric and artihmetic features of Calabi-Yau manifolds. The goal of this workshop is to explore these relationships, making new connections between researchers with these three different perspectives.This award is co-funded by the Algebra and Number Theory and the Geometric Analysis programs.

该提案将有助于支持定于2013年9月16日至20日在加拿大多伦多菲尔兹研究所举行的“弦理论的模块化形式”研讨会。该研讨会是菲尔兹研究所卡拉比-丘品种专题计划的一部分。 国家科学基金的支助将补充现有的资金，并将用于支付在美国的与会者的部分旅费。菲尔兹研究所是一个重要的国际研究中心，在组织类似活动方面拥有丰富的经验。讨论中的研讨会侧重于模形式在许多不同背景下的出现：在Calabi-Yau流形的物理学中，例如某些物理量的生成函数;在Calabi-Yau流形的算术中，作为一般Langlands哲学的一部分，研究在数域上定义的Calabi-Yau变体的伽罗瓦表示;在Calabi-Yau变种的几何不变量的结构中，例如Donaldson-Thomas不变量。所有这些都代表了非常活跃的研究主题，研讨会旨在综合这些观点带来的不同观点。卡-丘流形在数学和物理的十字路口发挥着至关重要的作用。虽然它们最初是作为几何学中的重要对象出现的，但卡-丘流形真正开始突出是在它们自然地出现在弦理论中之后。弦理论用一个在时空中运动的小弦环取代了传统的点粒子概念。为了使弦理论与量子力学相容，时空必须是10维的。既然时空看起来是四维的，那么我们可以预期其中的六个维度是一个非常小的“卷曲”的几何对象。这些物体是卡-丘流形。他们引入物理学导致了大量的新数学。卡-丘流形的一个重要性质是它们都具有非常微妙的平坦性（满足所谓的里奇平坦性），并且它们可以使用多项式方程组来定义。因此，它们可以被视为代数几何和算术对象，后者如果多项式方程的系数在数域中。在Calabi-Yau流形的物理、代数几何和艺术特征之间已经发现了密切的联系。本次研讨会的目标是探索这些关系，使研究人员与这三个不同的观点之间的新的连接。这个奖项是由代数和数论和几何分析程序共同资助。