Symmetries in string theory and quantum gravity

对称性是一个强大的工具，可以帮助我们组织对最基本的物理系统的理解。我建议使用引人入胜的新数学见解来研究弦理论和量子引力的基本方面。这些见解将大型离散对称群与弦论、代数、几何和数论的基本结构联系起来。*我所要关注的对称性是模对象和有限群之间一种迷人而神秘的关系，这种关系被称为“月光”。这种关系的第一个例子是在20世纪80年代发现的，被称为怪物月光，它表示数论中的某些模形式与怪物群的表示理论之间的联系，怪物群是零星有限单群中最大的。这种关系的许多方面都被“怪物模”的存在所阐明，它与弦理论和二维共形场理论密切相关。然而，许多谜团仍然存在。*一项最近尚未解释的发现表明，月光可能与弦理论和量子引力--从全息术到黑洞--有根本的联系。2010年，三位物理学家观察到，M24的表示的维度显示为K3曲面上的椭圆弦理论亏格中计算BPS态的模拟模形式的系数。K3曲面是代数几何中长期重要的对象，也是弦理论中许多重要结构的基础，从超对称弦真空到全息例子，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可以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，我提出了一个问题，即是否有一种新的方法来建立基于对称性或潜在的数学和几何结构的弦真空，从而揭示了弦理论的基本方面和许多令人着迷的数学结果的物理起源。*其次，我建议在三维空间中研究最近发现的月光模块和全息引力理论之间的联系。特别是，我建议研究基本的群论和数论结构的物理解释，并了解这些结构在多大程度上可以导致对三维全息引力理论的一般描述，阐明量子引力和黑洞物理的普遍方面。*最后，我建议研究在极值黑洞的弦理论构造的背景下出现的模拟模形式、几何和月光模之间的关系。这可能导致对量子黑洞及其微态的新的思考方式。

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.