XVII Modave Summer School in Mathematical Physics

**Classical Integrability in 2 d Field Theories** by Sibylle DRIEZEN (Universidad de Santiago de Compostela).

In these lectures, I will introduce the salient features of 2*d* classical integrable field theories. These models are distinguished by the rare property of having an infinite tower of conserved charges that are in involution and, therefore, in a certain sense completely solvable. In particular, I will cover the notions of Lax connections, monodromy matrices and Maillet brackets. I will illustrate these concepts with the Principal Chiral Model which is a two-dimensional integrable sigma model relevant for string theory and the AdS/CFT correspondence. At the end I will give a broad overview of deformations of sigma models of which the integrable structure is preserved.

**Horizon-Size Microstructure, Fuzzballs and Observations** by Daniel MAYERSON (CEA Saclay).

In these lectures, we will discuss various aspects of the fuzzball paradigm. We will briefly introduce the information paradox and discuss how the fuzzball paradigm aims to resolve it. Then, we will explore in some detail the two most important and well-known families of explicit fuzzball solutions in supergravity (generically called microstate geometries): multicentered geometries and superstrata. Finally, we will review some very recent developments of the gravitational phenomenological study of fuzzballs and delineate the exciting opportunities as well as the limitations of studying fuzzballs as alternatives to black holes in current and future observations.

**Cosmology, String Theory and the Swampland** by Alex COLE (University of Amsterdam).

The swampland program studies universal aspects of quantum gravity, with particular interest in how these aspects descend to constraints on low-energy physics. In these we will introduce, motivate, and study the consequences of several conjectured swampland constraints. We will see how perturbative string theory, extremal black holes, asymptotic limits in moduli spaces, and holography can be used to test conjectures.

A rough breakdown of topics:

→ introduction, EFT, global symmetries and completeness;

→ weak gravity conjecture;

→ swampland distance conjecture;

→ cosmological consequences?

→ the weak gravity conjecture and black hole corrections.

**Goldstone Boson Physics and Effective Field Theories** by Daniel NAEGELS (ULB, Brussels).

The lecture will start by briefly stating Goldstone theorem and emphasize the motivations behind Goldstone physics; the main asset being the universality of spontaneous symmetry breaking (SSB), the fundamental hypothesis of Goldstone theorem. Once we clarified/reviewed the different notions of SSB, Goldstone theorem will be presented and proved. A prediction of this theorem is the existence of gapless particles, called Nambu-Goldstone (NG) modes. From the discussion on Goldstone results, some aspects of the NG modes will emerge. Beside to be massless, there are systematically weakly coupled at low energy. Therefore, an effective field theory (EFT) building tool called “coset construction” will be presented to explicitly display these specific features of NG modes. Coset construction suits for our goal since it is mainly based on the symmetry realizations of the perturbed theory around the background inducing SSB. From the general obtained EFT, a counting rule for the NG modes will be derived. The limitations of this rule as well as the still ongoing research generalization will be discussed (cf. spacetime symmetries). This will allow to a smooth transition to a brief state of the art of Goldstone physics. Finally, if time allows it, the tools developed during this course will be illustrated with a concrete example in physics.

*N.B.* No prerequisites are required beside the standard courses of a Master in theoretical physics.

**Holography and Quantum Information** by Alice BERNAMONTI (Università degli Studi, Firenze).

— Canceled —
❶ Basics of AdS/CFT correspondence;

❷ Holographic dictionary: IR/UV connection, field/operator correspondence, holographic correlation functions, eternal black hole and thermofield double state;

❸ Basic notions from quantum information theory: entanglement and relative entropy;

❹ Entanglement entropy in Quantum Field Theory;

❺ Entanglement in holography: the Ryu-Takayanagi formula and applications to strongly coupled field theories and quantum gravity.

Here is the schedule of lectures.

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