Symmetries play an outsized role in understanding physical phenomena. In quantum systems ranging from condensed matter to high-energy particle physics, symmetries can feature different types of anomalies, which may constrain the dynamics or ruin the model's consistency. This gives important clues on extensions to the Standard Model, or new topological phenomena in quantum materials. Anomalies have played an essential role in the modern developments of supersymmetric quantum field theories as well as string theory. Last but not least, their study has influenced and benefitted from different areas of mathematics and in particular algebraic topology.

This school will introduce students to the physical and mathematical underpinnings of anomalies including its more mathematical aspects on topological quantum field theory and characteristic classes, with a view toward recent applications to topological phases of matter and strongly coupled gauge theories. The overarching idea is to have courses from three points of view that build upon each other: that of a mathematician (TFT, category theory, characteristic classes), a high-energy physicist (chiral anomalies and Hooft anomaly matching), and a condensed matter physicist (symmetry-protected and symmetry-enhanced topological order). The school would be suited to PhD students and postdocs coming from these three fields. We will ensure that with several tracks of exercise sessions revisiting background knowledge in math/hep-th/cond-mat as necessary.

Courses will range from basic aspects of anomalies of continuous flavour symmetries to cutting-edge topics: conformal anomalies, lattice symmetries, CPT symmetries, higher-form symmetries, higher-group symmetries, as well as a categorical point of view thereon.