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.

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.

Reading material:

Introduction to graded bundles, Andrew J. Bruce, Katarzyna Grabowska, Janusz Grabowski

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 with 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.

Reading material:

Gaugings of Groupoids, Strings in Shadows, and Emergent Poisson σ-Models, Rafał R. Suszek

Global Gauge Anomalies in Two-Dimensional Bosonic Sigma Models, Krzysztof Gawędzki, Konrad Waldorf and Rafał R. Suszek

Principaloid Bundles, Thomas Strobl and Rafał R. Suszek

Gerbe-holonomy for surfaces with defect networks, Ingo Runkel and Rafał R. Suszek

Lie Groupoids and Lie Algebroids. Lecture Notes, Eckhard Meinrenken

An Introduction to Bundle Gerbes, Michael K. Murray

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.

Reading material:

Orbifolds of topological quantum field theories, Nils Carqueville

Topological defects, Nils Carqueville, Michele Del Zotto and Ingo Runkel

Lecture notes on 2-dimensional defect TQFT, Nils Carqueville

Introductory lectures on topological quantum field theory, Nils Carqueville and Ingo Runkel

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).