  In this thesis, we explore the aspects of symmetry, topology and anomalies in
quantum matter with entanglement from both condensed matter and high energy
theory viewpoints. The focus of our research is on the gapped many-body quantum
systems including symmetry-protected topological states and topologically
ordered states. Chapter 1. Introduction. Chapter 2. Geometric phase,
wavefunction overlap, spacetime path integral and topological invariants.
Chapter 3. Aspects of Symmetry. Chapter 4. Aspects of Topology. Chapter 5.
Aspects of Anomalies. Chapter 6. Quantum Statistics and Spacetime Surgery.
Chapter 7. Conclusion: Finale and A New View of Emergence-Reductionism. (Thesis
supervisor: Prof. Xiao-Gang Wen)
