Symmetries in string theory and quantum gravity

弦理论和量子引力中的对称性

• 批准号: SAPIN-2017-00025
• 负责人: Harrison, Sarah
• 金额: $ 2.91万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Subatomic Physics Envelope - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=653110
• 关键词: Symmetries; string; theory; quantum; gravity

Symmetries are a powerful tool which helps us organize our understanding of the most basic physical systems. I propose to use fascinating new mathematical insights to investigate fundamental aspects of string theory and quantum gravity. These insights relate large, discrete symmetry groups to basic structures underlying string theory, algebra, geometry, and number theory . ***The kinds of symmetries I propose to focus on underlie a fascinating and mysterious relation between modular objects and finite groups known as “moonshine.” The first example of this relation, uncovered in the 1980s and dubbed monstrous moonshine, denotes a connection between certain modular forms in number theory and the representation theory of the monster group, the largest of the sporadic finite simple groups. Many aspects of this relationship are elucidated by the existence of a "monster module," which is intimately connected to string theory and 2d conformal field theory. Yet many mysteries remain.***A recent and as of yet unexplained discovery suggests that moonshine may have a fundamental relation to aspects of string theory and quantum gravity--from holography to black holes. In 2010, three physicists observed that dimensions of representations of M24, one of the sporadic finite simple groups, appear as coefficients of a mock modular form counting BPS states in the elliptic genus of string theory on K3 surfaces. K3 surfaces, long important objects in algebraic geometry, also underlie many important constructions in string theory, from supersymmetric string vacua to examples of holography, to microscopic descriptions of extremal black holes.***I propose to investigate what these deep mathematical connections can teach us about three aspects of string theory and quantum gravity: string vacua, holographic theories in three dimensions, and supersymmetric black holes. Firstly, I propose to ask whether there is a new way to formulate string vacua based on symmetries or underlying mathematical and geometric structure, shedding light on fundamental aspects of string theory and the physical origin of many fascinating results in mathematics.***Secondly, I propose to investigate recently uncovered connections between moonshine modules and holographic theories of gravity in three dimensions. In particular, I propose to investigate the physical interpretation of the underlying group- and number-theoretic structures, and understand to what extent these structures can lead to a general description of families holographic theories of gravity in three dimensions, elucidating universal aspects of quantum gravity and black hole physics. ***Finally, I propose to study relationships between mock modular forms, geometry, and moonshine modules which arise in the context of string-theoretic constructions of extremal black holes. This can lead to new ways of thinking about quantum black holes and their microstates.

对称性是一个强大的工具，它帮助我们组织我们对最基本的物理系统的理解。我打算用令人着迷的新数学见解来研究弦理论和量子引力的基本方面。这些见解将大的离散对称群与弦论、代数、几何和数论的基本结构联系起来。* 我建议关注的对称性类型是模对象和有限群之间一种迷人而神秘的关系的基础，这种关系被称为“月光”。这种关系的第一个例子在20世纪80年代被发现，并被称为怪物月光，表示数论中某些模形式与怪物群（最大的零星有限单群）的表示理论之间的联系。这种关系的许多方面都可以通过一个与弦论和二维共形场论密切相关的“怪物模块”的存在来阐明。但仍有许多未解之谜。最近的一项尚未解释的发现表明，月光可能与弦理论和量子引力的各个方面有着根本的关系-从全息术到黑洞。在2010年，三位物理学家观察到，M24（一个零星的有限单群）的表示维数，在K3表面上的弦理论的椭圆亏格中，表现为计算BPS态的模拟模形式的系数。K3曲面是代数几何中长期以来的重要对象，也是弦论中许多重要构造的基础，从超对称弦真空到全息术的例子，再到极端黑洞的微观描述。我打算研究这些深刻的数学联系能教会我们关于弦理论和量子引力的三个方面：弦真空、三维全息理论和超对称黑洞。首先，我想问的是，是否有一种基于对称性或基本的数学和几何结构的新方法来表述弦真空，从而阐明弦理论的基本方面以及数学中许多有趣结果的物理起源。其次，我建议调查最近发现的月光模块和三维重力全息理论之间的联系。特别是，我建议调查的基础群和数论结构的物理解释，并了解到这些结构在何种程度上可以导致家庭的一般性描述全息理论的重力在三维空间，阐明量子引力和黑洞物理学的普遍方面。*** 最后，我建议研究在极端黑洞的弦理论构造中出现的模拟模形式、几何和月光模之间的关系。这可能会导致思考量子黑洞及其微观状态的新方法。