Yu-An Chen: A generalized “spin-statistics theorem” for loop excitations

In this talk, we explore an extension of the spin-statistics theorem, traditionally linked to the worldline representation of particles as a \(2\pi\)-twisted ribbon. We extend this conceptual framework to encompass loops, proposing a novel interpretation through a Klein bottle with twisted framing. This theoretical development aims to deepen our understanding of fundamental physical principles and suggests a pathway toward a generalized spin-statistics theorem applicable to loops. Our approach uses the second and third Stiefel-Whitney classes, \(w_2\) and \(w_3\), to define the fermionic self-statistics of loops and articulate their field theory representations. Furthermore, we report on constructing a vast array of new exactly solvable local commuting projector lattice Hamiltonian models for beyond group cohomology invertible bosonic topological phases across various spacetime dimensions. These models are distinguished by their boundaries, characterized by gravitational anomalies. We specifically highlight the "\(w_2 w_3 \)" phase in \(4+1\)D, which features an anomalous \(3+1\)D boundary topological order with fermionic particle and fermionic loop excitations that exhibit mutual statistics. We will present examples of these fermionic loop excitations within \(3+1\)D toric codes, illustrating their distinct properties. In addition, we have developed new Pauli stabilizer models in \(4+1\) dimensions, such as \(\mathbb{Z}_4\) loop-only toric codes following \(e^2m^2\)-loop condensation. These models serve to demonstrate fermionic statistics as characterized by \(w_3\). This presentation aims to consolidate existing research and introduce an innovative framework to advance the theoretical understanding of fermionic loop statistics within topological field theories.