Symmetries in string theory and quantum gravity

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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.