Topological Aspects of Symmetry in Low Dimensions
1
Introduction
1.1
Lecture guide
1.1.1
Usage of this note
1.1.2
Objective
1.1.3
Prerequisite
1.1.4
Useful references
1.2
Motivation
1.2.1
Example without explanation: 4d pure YM at
\(\theta =\pi\)
2
Quantum Anomaly in Quantum Mechanics
2.1
Basics about quantum mechanics
2.1.1
operator algebra*
2.2
Wigner’s theorem
2.3
Bloch sphere and projective representation
2.3.1
Projective phase of
\(SO(3)\)
2.3.2
\(SO(3)\)
or
\(SU(2)\)
?
2.3.3
Anti-Unitary symmetry*
2.3.4
General finite states model*
2.4
\(\mathbb{Z}_2\times \mathbb{Z}_2\)
projective representation
2.5
Charged particle on the Aharonov-Bohm ring
2.6
Second group cohomology
2.7
Anomalous symmetry and 1D SPT
2.7.1
Symmetry Protected Topological Phase
2.7.2
Example of 1+1d SO(3) SPT
2.7.3
SPT for a general projective representation*
3
Quantum Anomaly in 1+1d QFT
3.1
QFT on a manifold and symmetry
3.2
No projective phase on S^1
3.3
1+1-dimensional periodic scalar theory
3.3.1
The model
3.3.2
Quantization on
\(S^1\)
and
\(U(1)^2\)
symmetry
3.3.3
Projective phases on the Hilbert spaces on an interval
3.4
Third group cohomology
3.4.1
Projective phases on Hilbert spaces on intervals
3.4.2
Group cohomology
3.4.3
Boundary condition independence
3.5
General properties of anomaly in 1+1d
3.5.1
Anomaly-BC relationship
3.5.2
Countinuous symmetry and anomalous Ward identity
3.5.3
Anomaly as the obsturction to gauging
3.6
\(\mathbb{CP}^1\)
model
3.7
Prospects
3.7.1
Background formalism
3.7.2
Higher dimensions
3.7.3
Fermions
References
Published with bookdown
Quantum Anomalies as Projective Phases
3.7
Prospects
3.7.1
Background formalism
3.7.2
Higher dimensions
3.7.3
Fermions