Symmetries in Classical and Quantum Field Theory

Symmetries are instrumental in all of physics. They prove particularly useful for constructing quantum field theories and classifying phases of matter. In recent years the standard notion of symmetry has been generalized in a way which encompasses all former results and extends them to more exotic setups. The aim of this course is to introduce these recent developments in a concrete way, followed by more abstract tools used in modern literature. We will also highlight key recent developments in the field, such as the Symmetry Topological Field Theory (SymTFT) framework.

(Preliminary) Plan

1. Generalized Landau paradigm. Ising model, toric code, BF theory. (2 weeks). Literature: McGreevy 2204.03045.
2. Global/local symmetries in classical and quantum field theory. (2 weeks). Literature: Deligne-Freed, Paugam.
3. Generalized symmetries, gauging, symmetry breaking and examples. (2 weeks). Literature: Bhardwaj et al. 2307.07547.
4. Gauge fields and differential cohomology. (2 weeks). Literature: Hopkins–Singer 0211216, Hsieh–Tachikawa–Yonekura 2003.11550, Moore’s 2023 TASI lectures.
5. Modern developments. (3 weeks). Literature: ?

Prerequisites

• Thorough understanding of classical field theory, differential geometry and representation theory;
• Basic notions of quantum field theory and algebraic topology.

Credits and Evaluation

The course is listed as Master “specialization module” in Mathematics, and cross-listed as such also in Physics. Receiving credit for the course requires regular attendance and participation, and completion of a mini-project at the end of the term. The language of instruction is English.

Organisation

• The course will meet on Tuesday and Thursday from 14-16h in the Mathematikon SR 6.
• Instructors: Dr. Hulik Email, Dr. Tatitscheff Email, Prof. Walcher Email
• Participation “for credit” will require registration in heiCO, but we are not planning to use either Müsli or MaMpf.