XIX Modave Summer School in Mathematical Physics

**Introduction to AKSZ sigma models** by Thomas BASILE (UMONS).

The AKSZ construction (for Alexandrov, Kontsevich, Schwarz and Zaboronsky, authors of the seminal paper [hep-th/9502010] in which it was first introduced), allows one to define an action functional for a topological sigma model out of any graded symplectic manifold equipped with a compatible cohomological vector field, action which satisfies the classical master equation of the BV-BRST formalism. We will review this construction, along with the necessary background in graded geometry, and illustrate it by detailing the examples of the Poisson sigma model and Chern-Simons theory from this standpoint.

**Carrollian Physics: Flat Spacetime as the Hologram of the Wonderworld** by Adrien FIORUCCI (Technische Universität Wien).

This series of lectures aims at introducing key concepts of Carrollian physics that emerges as the limit of Einstein's relativity in the vanishing speed of light regime. On the geometric side, the contraction induces the closure of light-cones along a timelike direction and the resulting manifold is equipped with a degenerate metric. Such geometric structures, which can be conveniently described as fibre bundles, are analysed in detail, as well as convenient choices of coordinates and connections. Carrollian isometries, which correspond to the contraction of the PoincarĂ© algebra, are discussed and some elements of their representation theory are provided. Next, a catalogue of Carrollian physical theories of particles and fields is compiled to further explore the interesting and peculiar features of this regime at both classical and quantum levels. Finally, the exploration of the Carrollian world culminates in a briefly discussion of its most important field of application, namely the formulation of a holographic description of asymptotically flat spacetime in the spirit of the celebrated AdS/CFT correspondence.

**Field exploration in phases of matter** by Antoine PASTERNAK (INFN Milan).

A new look has been given to symmetries in quantum field theory in the last ten years. Generalized symmetries broadened our vision of their role in constraining the renormalization group flow. Together with the knowledge of their anomalies, they can be used to classify phases of matter that fall outside the scope of Landauâ€™s paradigm with ordinary symmetries. Examples include topological field theories, symmetry-protected topological phases, or deconfined phases of gauge theories. These lectures will introduce these notions and work out examples of quantum field theories that exhibit such phases.

**Introduction to Exceptional field theory and its applications** by Colin STERCKX (ULB, Oviedo U.).

Supergravities are the low energy effective theories of the different possible superstrings. These supergravities are 10 or 11-dimensional theories, locally supersymmetric and with fluxes degree of freedom. The compactification of these theories, à la Kaluza-Klein, reveals unexpected exceptional symmetries descending from the superstring U-duality. It is a famously hard problem to consistently compactify supergravities in presence of fluxes to build new solutions in supergravity. In these lectures, we will see how Exceptional Field Theory makes these hidden symmetries explicit by expressing the degrees of freedom of supergravity in a U-duality covariant way and how it can be used to build consistent compactifications and truncations of different supergravities. More precisely, we will focus on compactifications of type IIB supergravity down to four dimensions maximal supergravity using E_{7(7)}-Exceptional Field Theory.

**Thermodynamics and Holography for de Sitter Space** by Manus VISSER (Cambridge U., DAMTP).

In this course I will survey some classical and quantum features of de Sitter space. The first part of the lectures is devoted to the thermodynamics of de Sitter horizons. In pioneering work from 1977 Gibbons and Hawking demonstrated that the static patch of de Sitter space has an associated entropy and temperature. I will review their paper and recent interpretations of their results in terms of operator algebras and thermodynamic ensembles. Secondly, I will discuss two different proposals for holography of de Sitter space: the dS/CFT correspondence and static patch holography. Even though there is not yet a precise dual description of de Sitter space, I will lay out some features that it has to satisfy.

Please find below the schedule of the school. To download a PDF version, please click here.