Week 07.04.2025 – 13.04.2025

Symmetry plays an important role in quantum systems: it can constrain the dynamics, give rise to selection rules, and provide computation methods in quantum computers. In recent years there are also new types of symmetries called generalized symmetries discovered in many quantum systems, including non-invertible symmetry and higher group symmetry. These lectures will be about symmetries in various quantum systems and their applications such as constraints on the low energy dynamics. Examples will be discussed in the lectures include quantum mechanics systems, gauge theories, lattice models, and the symmetry includes ordinary and higher form symmetry as well non-invertible symmetry.

In this talk, I will revisit results on the construction of Hamiltonian surface charges in general relativity in the presence of a finite timelike boundary, with an emphasis on how different boundary conditions influence the definition of conserved quantities. The analysis, originally published a few years ago [2109.02883], focuses on Dirichlet, Neumann, and York's mixed boundary conditions, and demonstrates how each leads to consistent, integrable charges using canonical methods. These results are shown to match those obtained via a covariant phase space formalism enhanced by a boundary Lagrangian. A key outcome of the study is the identification of an integrable charge for the Einstein-Hilbert action that differs from Komar's and remains well-defined even without Killing symmetries. We also analyze how the charge depends on the choice of boundary conditions, demonstrating that both quasi-local and asymptotic expressions are affected. These findings are relevant to current efforts to understand gravitational dynamics in finite regions and may have implications for the thermodynamics of black holes.

Th-Cosmo-talk on April 10, 2025, at KCL room K3.11.

Title: Dark Matter as Topological Order.

Abstract: We propose that topological order can replace sterile neutrinos as dark matter candidates to cancel the Standard Model global gravitational anomalies. Standard Model (SM) with 15 Weyl fermions per family (lacking the 16th, the sterile right-handed neutrino nuR) suffers from mixed gauge-gravitational anomalies tied to baryon number plus or minus lepton number B+(-)L symmetry. Including nuR per family can cancel these anomalies, but when B+(-)L symmetry is preserved as discrete finite subgroups rather than a continuous U(1), the perturbative local anomalies become nonperturbative global anomalies. We systematically enumerate these gauge-gravitational global anomalies involving discrete B+(-)L that are enhanced from the fermion parity Z2F to Z2NF. The discreteness of B+(-)L is constrained by multi-fermion deformations beyond-the-SM and the family number Nf. Unlike the free quadratic nuR Majorana mass gap preserving the minimal Z2F, we explore novel scenarios canceling (B+(-)L)-gravitational anomalies while preserving the Z2NF discrete symmetries, featuring 4-dimensional interacting gapped topological orders or gapless sectors (e.g., conformal field theories). We propose symmetric anomalous sectors as quantum dark matter to cancel SM global anomalies. We find the uniqueness of the family number at Nf = 3, such that when the representation of Z2NF from the faithful B+L for baryons at both Nf and N equal to 3 is extended to the faithful Q + NcL for quarks at N = NcNf = 9, this symmetry extension ZNc=3 to ZNcNf =9 to ZNf =3 matches with the topological order dark matter construction. Key implications include: (1) a 5th force mediating between SM and dark matter via discrete B+(-)L gauge fields, (2) dark matter as topological order quantum matter with gapped anyon excitations at ends of extended defects, and (3) Ultra Unification and topological leptogenesis. [Based on arXiv:2502.21319, arXiv:2501.00607, arXiv:2412.21196, arXiv:2411.05786, arXiv:2012.15860, arXiv:2112.14765, arXiv:2204.08393, arXiv:2302.14862, arXiv:2312.14928].