2024 IHES Summer School - Symmetries and Anomalies: a modern take

24 juin 2024, 08:30 → 5 juil. 2024, 17:00 Europe/Paris

Marilyn and James Simons Conference Center (IHES)

2024 IHES SUMMER SCHOOL

Organizing Committee: Zohar Komargodski (SCGP), Bruno Le Floch (CNRS & LPTHE), Elli Pomoni (DESY), and Masahito Yamazaki (IPMU).

Scientific Committee: Anton Kapustin (Caltech), Yuji Tachikawa (IPMU), Xiao-Gang Wen (MIT).

The Summer School will be held at the Institut des Hautes Études Scientifiques (IHES) from June 24 to July 05, 2024. IHES is located in Bures-sur-Yvette, south of Paris (40 minutes by train from Paris) - Access map

This school is open to everybody but intended primarily for young participants, including Ph.D. students and postdoctoral fellows. 

Applications are closed. All candidates have received an email stating whether they were accepted. 

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2024 IHES Summer School - Symmetries and Anomalies: a Modern Take

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.

Speakers:

• Clay CÓRDOVA (University of Chicago)
• Clément DELCAMP (IHES)
• Thomas DUMITRESCU (UCLA)
• Iñaki GARCÍA ETXEBARRIA (Durham University)
• Max METLITSKI (MIT)
• Shu-Heng SHAO (Stony Brook University)
• Đàm Thanh SƠN (Univ. of Chicago)
• Yifan WANG (New York U)

lundi 24 juin

08:30 Welcome coffee and registration

1 Generalized Symmetries and Phases of Gauge Theory (1/4)

These lectures will introduce higher-form and higher-group symmetries (as well as related concepts such as anomalies and SPTs) through the lens of gauge theory phases in 3+1 dimensions.

Orateur: Prof. Thomas DUMITRESCU (UCLA)

Vidéo

10:45 Coffe break

2 Gong show 1

12:30 Lunch break (buffet at IHES)

3 Introduction to anomalies in condensed matter physics (1/4)

1) General "definition" of topological phases and of invertible phases.
2) Illustration of invertible phases with a 1+1d Majorana chain and connection to continuum field theory (massive Majorana fermion).
3) Introduction of bordism invariance and classification of invertible phases.
4) If time allows, invertible phases with Z classification (Chern simons response) as illustrated by 2+1d p+ip superconductor.
5) Phases protected by symmetry. Symmetry on the lattice.
6) In cohomology SPT and Dijkgraaf-Witten theories. If time allows, a detailed analysis of the 2+1D Levin-Gu bosonic Z 2 SPT.

Orateur: Prof. Max METLITSKI (MIT)

Vidéo

16:00 Coffee break

4 Speed-dating / Exercices

mardi 25 juin

08:45 Welcome coffee

5 Generalized Symmetries and Phases of Gauge Theory (2/4)

These lectures will introduce higher-form and higher-group symmetries (as well as related concepts such as anomalies and SPTs) through the lens of gauge theory phases in 3+1 dimensions.

Orateur: Prof. Thomas DUMITRESCU (UCLA)

Vidéo

10:45 Coffee break

6 Introduction to anomalies in condensed matter physics (2/4)

1) General "definition" of topological phases and of invertible phases.
2) Illustration of invertible phases with a 1+1d Majorana chain and connection to continuum field theory (massive Majorana fermion).
3) Introduction of bordism invariance and classification of invertible phases.
4) If time allows, invertible phases with Z classification (Chern simons response) as illustrated by 2+1d p+ip superconductor.
5) Phases protected by symmetry. Symmetry on the lattice.
6) In cohomology SPT and Dijkgraaf-Witten theories. If time allows, a detailed analysis of the 2+1D Levin-Gu bosonic Z 2 SPT.

Orateur: Max METLITSKI (MIT)

Vidéo

12:30 Lunch break

7 Generalized Symmetries and Phases of Gauge Theory (3/4)

These lectures will introduce higher-form and higher-group symmetries (as well as related concepts such as anomalies and SPTs) through the lens of gauge theory phases in 3+1 dimensions.

Orateur: Prof. Thomas DUMITRESCU (UCLA)

Vidéo

16:00 Coffee break

8 Exercices

mercredi 26 juin

08:45 Welcome coffee

9 Generalized Symmetries and Phases of Gauge Theory (4/4)

These lectures will introduce higher-form and higher-group symmetries (as well as related concepts such as anomalies and SPTs) through the lens of gauge theory phases in 3+1 dimensions.

Orateur: Prof. Thomas DUMITRESCU (UCLA)

Vidéo

10:45 Coffee break

10 Introduction to anomalies in condensed matter physics (3/4)

1) General "definition" of topological phases and of invertible phases.
2) Illustration of invertible phases with a 1+1d Majorana chain and connection to continuum field theory (massive Majorana fermion).
3) Introduction of bordism invariance and classification of invertible phases.
4) If time allows, invertible phases with Z classification (Chern simons response) as illustrated by 2+1d p+ip superconductor.
5) Phases protected by symmetry. Symmetry on the lattice.
6) In cohomology SPT and Dijkgraaf-Witten theories. If time allows, a detailed analysis of the 2+1D Levin-Gu bosonic Z 2 SPT.

Orateur: Prof. Max METLITSKI (MIT)

Vidéo

jeudi 27 juin

08:45 Welcome coffee

11 Introduction to anomalies in condensed matter physics (4/4)

1) General "definition" of topological phases and of invertible phases.
2) Illustration of invertible phases with a 1+1d Majorana chain and connection to continuum field theory (massive Majorana fermion).
3) Introduction of bordism invariance and classification of invertible phases.
4) If time allows, invertible phases with Z classification (Chern simons response) as illustrated by 2+1d p+ip superconductor.
5) Phases protected by symmetry. Symmetry on the lattice.
6) In cohomology SPT and Dijkgraaf-Witten theories. If time allows, a detailed analysis of the 2+1D Levin-Gu bosonic Z 2 SPT.

Orateur: Prof. Max METLITSKI (MIT)

Vidéo

10:45 Coffee break

12 Higher Symmetry in Particle Physics (1/4)

These lectures will focus on higher symmetry in the standard model and beyond, illustrating how new symmetry principles can be a powerful organizing tool for well motivated models of particle physics.

Orateur: Prof. Clay CÓRDOVA (University of Chicago)

Vidéo

12:30 Lunch break

13 Higher Symmetry in Particle Physics (2/4)

These lectures will focus on higher symmetry in the standard model and beyond, illustrating how new symmetry principles can be a powerful organizing tool for well motivated models of particle physics.

Orateur: Prof. Clay CÓRDOVA (University of Chicago)

Vidéo

16:00 Coffee break

14 Exercices

vendredi 28 juin

08:45 Welcome coffee

15 Higher Symmetry in Particle Physics (3/4)

These lectures will focus on higher symmetry in the standard model and beyond, illustrating how new symmetry principles can be a powerful organizing tool for well motivated models of particle physics.

Orateur: Prof. Clay CÓRDOVA (University of Chicago)

Vidéo

10:45 Coffee break

16 Gong Show 2

12:30 Lunch break

17 Higher Symmetry in Particle Physics (4/4)

These lectures will focus on higher symmetry in the standard model and beyond, illustrating how new symmetry principles can be a powerful organizing tool for well motivated models of particle physics.

Orateur: Prof. Clay CÓRDOVA (University of Chicago)

Vidéo

16:00 Coffee break

18 Exercices

lundi 1 juillet

08:45 Welcome coffee

19 Non-Invertible Symmetries (1/3)

I will review aspects of non-invertible symmetries and their relations to anomalies on the lattice. I will start with the Kramers-Wannier symmetry in the quantum Ising lattice model, and discuss its dynamical consequences. I will then discuss topological phases protected by non-invertible symmetries, with the cluster model as the simplest example.

Orateur: Prof. Shu-Heng SHAO (Stony Brook University)

10:45 Coffee break

20 Generalized Symmetries and their Gauging in 2d CFTs (1/4)

After a brief review of 2d CFT basics, we introduce generalized, non-invertible symmetries in terms of explicit CFT observables. We describe how such symmetries are formalized by fusion categories and how to implement gauging of such symmetries. We also discuss the physical consequences of these mathematical structures.

Orateur: Prof. Yifan WANG (New York University)

12:30 Lunch break

21 Nonrelativistic Conformal Field Theory (1/2)

We will review the notions of Schrödinger symmetry and nonrelativistic conformal field theory, in particular the restrictions that Schrödinger symmetry imposes on correlation function and the operator-state correspondence. We will then consider the most important example of NRCFT --- fermions at unitarity, and derive physical consequences of the formalism.

Orateur: Prof. Đàm Thanh SƠN (Univ. of Chicago)

16:00 Coffee break

22 Exercices

mardi 2 juillet

08:45 Welcome coffee

23 Non-Invertible Symmetries (2/3)

I will review aspects of non-invertible symmetries and their relations to anomalies on the lattice. I will start with the Kramers-Wannier symmetry in the quantum Ising lattice model, and discuss its dynamical consequences. I will then discuss topological phases protected by non-invertible symmetries, with the cluster model as the simplest example.

Orateur: Prof. Shu-Heng SHAO (Stony Brook University)

10:45 Coffee break

24 Generalized Symmetries and their Gauging in 2d CFTs (2/4)

After a brief review of 2d CFT basics, we introduce generalized, non-invertible symmetries in terms of explicit CFT observables. We describe how such symmetries are formalized by fusion categories and how to implement gauging of such symmetries. We also discuss the physical consequences of these mathematical structures.

Orateur: Prof. Yifan WANG (New York University)

12:30 Lunch break

25 String Theory, Symmetries and Anomalies (1/4)

I will review some aspects of anomalies of d-dimensional QFTs from a modern viewpoint (in terms of a d+1 dimensional anomaly theory, the "anomaly theory"), how symmetries can be described in a related way in terms of a d+1 dimensional topological field theory (the "SymTFT") and how these d+1 theories arise from string theory.

Orateur: Prof. Iñaki GARCÍA ETXEBARRIA (Durham University)

16:00 Coffee break

26 Exercices

mercredi 3 juillet

08:45 Welcome coffee

27 String Theory, Symmetries and Anomalies (2/4)

I will review some aspects of anomalies of d-dimensional QFTs from a modern viewpoint (in terms of a d+1 dimensional anomaly theory, the "anomaly theory"), how symmetries can be described in a related way in terms of a d+1 dimensional topological field theory (the "SymTFT") and how these d+1 theories arise from string theory.

Orateur: Prof. Iñaki GARCÍA ETXEBARRIA (Durham University)

10:45 Coffee break

28 Non-Invertible Symmetries (3/3)

I will review aspects of non-invertible symmetries and their relations to anomalies on the lattice. I will start with the Kramers-Wannier symmetry in the quantum Ising lattice model, and discuss its dynamical consequences. I will then discuss topological phases protected by non-invertible symmetries, with the cluster model as the simplest example.

Orateur: Prof. Shu-Heng SHAO (Stony Brook University)

12:30 Lunch break

29 Participant Seminars

16:00 Coffee break

30 Participant Seminars

jeudi 4 juillet

08:45 Welcome coffee

31 Generalized Symmetries and their Gauging in 2d CFTs (3/4)

After a brief review of 2d CFT basics, we introduce generalized, non-invertible symmetries in terms of explicit CFT observables. We describe how such symmetries are formalized by fusion categories and how to implement gauging of such symmetries. We also discuss the physical consequences of these mathematical structures.

Orateur: Prof. Yifan WANG (New York University)

10:45 Coffee break

32 String Theory, Symmetries and Anomalies (3/4)

I will review some aspects of anomalies of d-dimensional QFTs from a modern viewpoint (in terms of a d+1 dimensional anomaly theory, the "anomaly theory"), how symmetries can be described in a related way in terms of a d+1 dimensional topological field theory (the "SymTFT") and how these d+1 theories arise from string theory.

Orateur: Prof. Iñaki GARCÍA ETXEBARRIA (Durham University)

12:30 Lunch break (buffet at IHES)

33 Nonrelativistic Conformal Field Theory (2/2)

We will review the notions of Schrödinger symmetry and nonrelativistic conformal field theory, in particular the restrictions that Schrödinger symmetry imposes on correlation function and the operator-state correspondence. We will then consider the most important example of NRCFT --- fermions at unitarity, and derive physical consequences of the formalism.

Orateur: Prof. Đàm Thanh SƠN (Univ. of Chicago)

16:00 Coffee break

34 Exercices

vendredi 5 juillet

08:45 Welcome coffee

35 Generalized Symmetries and their Gauging in 2d CFTs (4/4)

After a brief review of 2d CFT basics, we introduce generalized, non-invertible symmetries in terms of explicit CFT observables. We describe how such symmetries are formalized by fusion categories and how to implement gauging of such symmetries. We also discuss the physical consequences of these mathematical structures.

Orateur: Prof. Yifan WANG (New York University)

10:45 Coffee break

36 Non-Invertible Symmetries in One-Dimensional Quantum Lattice Models

We will study a lattice realisation of the Symmetry Topological Field Theory (SymTFT) picture for one-dimensional quantum models. We will exploit this picture to study lattice models with non-invertible symmetries, discuss generalised gauging, and construct renormalisation group fixed points of gapped symmetric phases.

Orateur: Prof. Clément DELCAMP (IHES)

12:30 Lunch break

37 String Theory, Symmetries and Anomalies (4/4)

I will review some aspects of anomalies of d-dimensional QFTs from a modern viewpoint (in terms of a d+1 dimensional anomaly theory, the "anomaly theory"), how symmetries can be described in a related way in terms of a d+1 dimensional topological field theory (the "SymTFT") and how these d+1 theories arise from string theory.

Orateur: Prof. Iñaki GARCÍA ETXEBARRIA (Durham University)