PhD Comprehensive Exam | Christopher Pollack, Poisson-Lie Groups & Quantum Groups in Gravity

M3 3103

Candidate

Christopher Pollack | Applied Mathematics, University of Waterloo

Title

Poisson-Lie Groups & Quantum Groups in Gravity

Abstract

This research proposal explores the role of Poisson-Lie and quantum group symmetries in gravity and physics. We review the previously established appearances of Poisson-Lie symmetries (the semi-classical picture of quantum group symmetries) and quantum group symmetries in 3D gravity as well as our novel advancements in elucidating such structures in 4D gravity. The situation in both 3D and 4D naturally splits into two cases, depending on the value of the cosmological constant. A key difference being, however, that no symmetry analogous to the shift symmetry of 3D gravity exists in 4D. We extend Noether's first theorem to symmetries which are not symplectomorphisms. To do so, we use the framework of the covariant phase space method, and focus on the symmetries of the Euler-Lagrange equations which generalize the typical Lagrangian symmetries. We then show how under appropriate assumptions we can construct a dynamically conserved current and scalar charge from these general symmetries. Poisson-Lie symmetries provide a natural example of such generalized symmetries. We illustrate our framework with the Klimčík-Ševera non-linear sigma-model which possesses Poisson-Lie symmetries. Speculation is then made on how the role of such symmetries in 4D gravity can be further understood despite no apparent shift symmetry, as well as how our novel framework can be applied to study integrable systems, asymptotic symmetries, the infrared triangle, and 2-Poisson-Lie groups.