Teaching

The school will include five courses, each comprising approximately six hours of instruction, divided into lecture and workshop components. The precise balance between lectures and workshops will be determined by the individual instructors; however, the general structure will involve lectures in the mornings and problem-solving workshops in the afternoons. 

1. Geometric Mechanics – differential-geometric methods in classical physics (Katarzyna Grabowska, Faculty of Physics, University of Warsaw) – this is intended as an introductory course. Assuming basic knowledge of differential calculus on smooth manifolds, we will study geometric structures important from the point of view of Geometric Mechanics, such as symplectic and Poisson structures, Lie algebroids, Dirac structures, and contact structures, illustrating all the concepts by examples coming from physics.

2. Graded differential geometry and applications (Janusz Grabowski, IM PAS). Various graded structures on manifolds canonically arise in differential geometry, to mention vector bundles or supergeometry. They have found important applications in physics, especially geometric mechanics and field theory. We will discuss the main examples, together with an introduction to supergeometry.

3. The gauge principle – from group-modelled to categorified symmetries, and defects (Rafał R. Suszek, Faculty of Physics, University of Warsaw). The notion of gauge symmetry is omnipresent in modern theoretical and mathematical physics, as well as pure mathematics: from σ-model dynamics of charged probes in external gauge fields, through the Standard Model of fundamental interactions, all the way to topological field theories such as the Chern-Simons theory and the Poisson σ-model. The course shall begin with a gentle introduction to the classic concepts and techniques based on the transformation-group model of symmetry–such as Cartan's mixing construction and Crittenden's connection–also in the presence of non-tensorial couplings in the field theory, determined by Cheeger-Simons differential characters. It shall subsequently develop towards a generalisation to categorified symmetry models: Lie groupoids (with the corresponding principaloid and derived bundles, with twisted-equivariant gerbes over their fibres) and more general (differential) categories whose action is realised through defects in the space-time of the field theory. The abstract constructions shall be illustrated amply on the example of the gauging of groupoidal symmetries in the Polyakov-Alvarez-Gawędzki σ-model of charged loops, in which the stacky descent of higher geometric structures–such as the gerbe of the σ-model–to groupoidal Godement quotients shall be encountered and explained. In the symplectic setting, the gauging shall lead us to a novel conceptualisation of the Poisson σ-model, and its generalisations.

4. Topological quantum field theory with defects (Nils Carqueville, Faculty of Physics, University of Vienna). In the first part we would review basic category theory and introduce monoidal categories and their graphical calculus. This would be illustrated by several examples, including those of categories of vector spaces and bordisms. The second part would start with the definition of closed TQFTs as well as several concrete examples in low dimension – in particular appearing as topological twists of supersymmetric field theories. Then we would introduce defect TQFTs as symmetric monoidal functors on stratified bordism categories.

5. From Poisson sigma models to different kinds of novel gauge theories and geometrical structures (Thomas Strobl, Camille Jordan Institute, Claude Bernard University Lyon1). Poisson sigma models (PSMs) were introduced to unify various coupled gravity and Yang-Mills (YM) gauge theories in two spacetime dimensions. They are also of interest since their quantisation is related to the quantisation of the target Poisson manifold, and, last but not least, they constitute a new kind of gauge theory, which, in contrast to YM or Chern-Simons theories, is not based on ordinary principal bundles. In this lecture series, we focus on the latter aspect in particular. After introducing the PSM, we generalise it to the AKSZ sigma model, a tower of topological field theories (see also: the course of Carqueville), labelled by the dimension of the base manifold and a tower of algebroids (see: also the course of Grabowska). In this context, we make use of Z-graded differential geometry (see: the course of Grabowski). We introduce a generalisation of ordinary characteristic classes to dg-bundles and show their relation to the AKSZ models. Time permitting, we also present curved Yang-Mills-Higgs gauge theories, a generalisation of ordinary YM theories to the realm of Lie groupoids, and relate the dg-bundles of degree 1 to the principaloid bundles (see: the course of Suszek).