References for prerequisite material for understanding papers on Generalized Global Symmetries

Question

I want to understand the papers https://arxiv.org/abs/1412.5148 and https://arxiv.org/abs/1703.00501. Assuming that I understand basics of gauge theories, could someone suggest some references explaining Wilson lines and 't Hooft lines? Also, some more references relating to discrete symmetries would be helpful. Thanks!

Answer 1

The paper

• Symmetries in Quantum Field Theory and Quantum Gravity by Harlow and Ooguri (https://arxiv.org/abs/1810.05338)

is a gold mine of insights about symmetries in general. Prerequisites for understanding the paper include some experience with quantum field theory, including both the path-integral and canonical formulations. The paper is exceptionally well written, and it is written for the purpose of teaching and clarifying things in addition to reporting new perspectives and new results. The paper's relevance to the question is noted in the abstract, which says:

We extend all of these results to the case of higher-form symmetries.

Here's a subset of the table of contents, with my own annotation about which sections are expository:

• 2 Global symmetry

• 2.1 Splittability — expository. This section coins the name "splittable symmetry," which is an ancient-but-important concept that really needed a good name.

• 2.2 Unsplittable theories and continuous symmetries without currents — expository

• 2.3 Background gauge fields — expository

• 2.4 ’t Hooft anomalies — expository

• 2.5 ABJ anomalies and splittability — expository. You can read about 't Hooft loops in this section.

• 2.6 Towards a classification of ’t Hooft anomalies — expository

• 3 Gauge symmetry

• 3.1 Definitions — expository. Don't be fooled by this section's boring name. This section re-introduces key concepts like Wilson loops, even though loops/lines were already used in the preceding sections.

• 3.2 Hamiltonian lattice gauge theory for general compact groups — expository

• 3.3 Phases of gauge theory — expository

• 3.4 Comments on the topology of the gauge group — expository

• 3.5 Mixing of gauge and global symmetries — expository

• 7 Spacetime symmetries - expository

• 8 $p$-form symmetries - expository. This is why the paper is relevant to the question.

• 8.1 $p$-form global symmetries - expository

• 8.2 $p$-form gauge symmetries - expository