Looking back at the IHES Summer School 2024

This year, IHES hosted yet another summer school in physics, organized by Bruno Le Floch, Elli Pomoni, and Masahito Yamazaki. Following the 2018 edition on supersymmetric localization and exact results, they joined forces with Zohar Komargodski to organize two weeks focused on symmetries and anomalies.

While the 2018 school aimed to overview calculation methods in supersymmetric gauge theories—a field kick-started by the computation of the instanton partition function for four-dimensional theories with N=2 supersymmetry by Nikita Nekrasov and of their sphere partition function by Vasily Pestun, and which has links to matrix models, integrable systems, conformal field theory and holography—the situation for this year’s school was quite different. “We were thinking about organizing a school on modern notions of symmetry for quite some time now. If I remember correctly, our first email exchanges on this idea go back to 2021,” says Bruno Le Floch. “Back then, the subject was just starting, and we didn’t imagine that it would generate the amount of activity it did in the last two to three years. This is, of course, very fortunate, and the Symmetries and Anomalies summer school couldn’t have come at a better time,” explains Masahito Yamazaki.

A symmetry is a certain invariance of a physical system that can be helpful in its analysis. For instance, a symmetry may reduce the number of parameters one has to consider. It has recently become apparent that the usual notion of symmetry can be widely extended with the introduction of more general notions of symmetries such as higher-form, higher-group or non-invertible symmetries. Just as ordinary symmetries are understood in terms of groups, the language of these generalized symmetries is that of fusion n-categories.

An anomaly appears when a symmetry does not carry over from the classical to the quantum world. 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 a model’s consistency. This can give important clues on possible extensions to the Standard Model or new topological phenomena in quantum materials. Anomalies have also played an essential role in the modern developments of supersymmetric quantum field theories and string theory.

The 2024 summer school aimed to 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 was to have courses from three points of view that build upon each other: that of a mathematician, a high-energy physicist, and a condensed matter physicist.

A notable feature of the symmetries and anomalies community is that its current leaders are often still at an early stage of their careers. “Knowing that many top contributors are not much older than us is really motivating. It shows that we could come up with new results as well,” remarked a participant.

During the two weeks at IHES, participants not only had the opportunity to enjoy lectures from leading researchers in the field, but also to meet one of the field’s originators, Xiao-Gang Wen, who introduced topological order in the 1980s and was present at the school as a member of the scientific committee. “I really enjoyed the mix of intense lectures and exercise classes with more informal sessions, such as the panel discussion with some of the organizers and lecturers. The summer school was a very welcome opportunity to catch up with my research community in the wonderful setting of IHES,” said another student.

The organizers wish to thank the sponsors, ENGIE, FMJH, QRT, and Société Générale, as well as the entire IHES staff for their work and dedication in making this summer school a success.

Photo credit: © Chris Peus / IHES