XVI Modave Summer School in Mathematical Physics

** Useful topics in Lie algebras** by Victor LEKEU (Imperial College, London).

In these lectures, I will explain a variety of topics in the theory of Lie algebras: real forms of (complex, finite-dimensional) simple Lie algebras, Chevalley-Eilenberg cohomology, central extensions, Kac-Moody and loop algebras. They appear in many areas of theoretical physics and I therefore hope that the title is not a complete lie. Plenty of exercises will be provided; familiarity with roots and Dynkin diagrams is useful but not absolutely necessary.

**Differential geometry of gauge theory** by Jordan FRANCOIS (UMONS).

We propose an introduction to the differential geometry of connections on fiber bundles underlying the physics of (classical) gauge theories. Our first aim is to show how Ehresmann connections are behind Yang-Mills/Utiyama theory. We will then be prepared to appreciate why Cartan connections provide a compelling geometric framework for gauge theories of gravity (the tetrad formulation of GR being the most relevant special case). An important takeaway will be the recipe for cooking a gauge theory.

If time allows it, we will end by a presentation of the "dressing field method" which can be seen either as a way to achieve gauge symmetry reduction, or as a diagnosis tool to detect fictitious gauge symmetries in a theory.

**Quantum Information** by Aidan CHATWIN-DAVIES (KUL, Leuven).

Quantum information science sits at an intersection point of physics, mathematics, and computer science. The field concerns itself with the information contained in quantum mechanical systems, how that information can be encoded, manipulated, and retrieved, and how these operations’ properties, capabilities, and limitations can be quantified. As we sit at the cusp of the era of quantum computers, the practical importance of quantum information science only continues to increase. In parallel, quantum information science continues to drive new discoveries and further our theoretical understanding of questions in high energy physics.

The aim of these lectures is to introduce a handful of core ideas from quantum information science that figure prominently in modern research on quantum gravity. The central concept that will form the base of our studies is that of a quantum channel; that is, the most general map between quantum states and between operators on Hilbert space. A large part of the lectures will be honest-to-goodness quantum information theory, and we will try to avoid cutting too many corners in a rush to get to gravity. Nevertheless, we will still manage to see how a few problems in high energy physics, such as the black hole information problem or bulk reconstruction in AdS/CFT, can be cast in the information theoretic language that we will have set up.

1. Quantum information basics

→ States and partitions

→ Entropy, entropic quantities, interpretations

→ The black hole information problem

2. Quantum channels

→ Superoperators and their properties

→ Kraus representation

→ Stinespring dilation

3. Exercise session

→ Properties of entropy

→ Properties of channels

4. Quantum error correction

→ General features and definitions

→ Criteria for recovery from errors

→ Recovery maps

5. Some topics in high energy physics

→ Bulk reconstruction, complementary error correcting codes

→ AdS/CFT as quantum error correction.

→ States and partitions

→ Entropy, entropic quantities, interpretations

→ The black hole information problem

2. Quantum channels

→ Superoperators and their properties

→ Kraus representation

→ Stinespring dilation

3. Exercise session

→ Properties of entropy

→ Properties of channels

4. Quantum error correction

→ General features and definitions

→ Criteria for recovery from errors

→ Recovery maps

5. Some topics in high energy physics

→ Bulk reconstruction, complementary error correcting codes

→ AdS/CFT as quantum error correction.

Notes : day 1 (PDF, 3 243 Ko)

Notes : day 2 (PDF, 3 486 Ko)

Notes : day 3 (part 1) (PDF, 2 618 Ko)

Notes : day 3 (part 2) (PDF, 2 576 Ko)

Problem set (PDF, 147 Ko)

Notes : day 2 (PDF, 3 486 Ko)

Notes : day 3 (part 1) (PDF, 2 618 Ko)

Notes : day 3 (part 2) (PDF, 2 576 Ko)

Problem set (PDF, 147 Ko)

Here is the schedule of lectures.

Please remember that due to the sanitary crisis, the school will be held in Auditorium D.0.08 [ Brussels, Pleinlaan/Boulevard de la Plaine, 2 - 1050 Elsene/Ixelles ] !

Wednesday 09/09 | Thursday 10/09 | Friday 11/09 | |
---|---|---|---|

09:45-10:45 | Topics in Lie Algebras
(V. Lekeu) |
Geometry of gauge theory
(J. François) |
Quantum Information
(A. Chatwin-Davies) |

10:45-11:45 | Topics in Lie Algebras
(V. Lekeu) |
Geometry of gauge theory
(J. François) |
Quantum Information
(A. Chatwin-Davies) |

11:45-13:15 | Lunch |
Lunch |
Lunch |

13:15-14:15 | Geometry of gauge theory
(J. François) |
Quantum Information
(A. Chatwin-Davies) |
Topics in Lie Algebras
(V. Lekeu) |

14:15-15:15 | Geometry of gauge theory
(J. François) |
Quantum Information
(A. Chatwin-Davies) |
Topics in Lie Algebras
(V. Lekeu) |

15:15-15:45 | Break |
Break |
Break |

15:45-16:45 | Quantum Information
(A. Chatwin-Davies) |
Topics in Lie Algebras
(V. Lekeu) |
Geometry of gauge theory
(J. François) |

16:45-17:45 | Quantum Information
(A. Chatwin-Davies) |
Topics in Lie Algebras
(V. Lekeu) |
Geometry of gauge theory
(J. François) |