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.

Rare macroscopic fluctuations leading to large deviations in many-body systems have attracted significant attention in recent years. In this talk we present an exact solution to such a problem - the emptiness formation in one-dimensional quantum polytropic gases characterized by an arbitrary polytropic index \gamma, which defines the equation of state P ~ \rho^\gamma, where P is the pressure and \rho is the density. The problem consists of determining the probability of spontaneous formation of an empty interval in the ground state of the gas. In the limit of a macroscopically large interval, this probability is dominated by an instanton configuration. By solving the hydrodynamic equations in imaginary time, we derive the analytic form of the emptiness instanton. This solution is expressed as an integral representation analogous to those used for correlation functions in Conformal Field Theory. Prominent features of the spatiotemporal profile of the instanton are obtained directly from this representation and will be discussed in the context of limit shape phenomena. [1] A.G. Abanov and D.M. Gangardt, arXiv:2412.1168

We obtain the Hawking spectrum by exponentiating a series of Feynman diagrams describing a scalar field scattering through a collapse background. Our approach is rooted in semiclassical methods of scattering amplitudes which have recently been developed for application to gravitational-wave physics. The diagrams we encounter do not compute a standard amplitude, but rather an in-in generalisation of an amplitude which is closely connected to the Bogoliubov coefficients. We also compute the subdominant one-loop correction in our perturbative approach, analogous to the triangle correction to Schwarzschild scattering. This term can be interpreted as a finite-size correction sensitive to the radius of the black hole.

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].