ESI Program on
Higher Structures in Mathematics and Physics

Vienna, September 1 - November 7, 2010

Minicourse Abstracts

Introduction to gauged super gravivity , H. Samtleben (ENS, Lyon)
The low-energy effective theories describing string compactifications are so-called gauged supergravities: non-abelian deformations of the standard abelian supergravity theories. I give an introduction to the construction of these theories whose structure is to a large extent determined by the underlying symmetry groups. In particular, I will discuss in detail the hierarchy of non-abelian higher rank tensor gauge fields which generically show up in these theories

Supergeometry and differential graded manifolds , D. Roytenberg (Utrecht) and T. Voronov (Manchester)
Supermanifolds have proved to be as indispensable a tool for geometry and physics as the standard tensor language. In these lectures we shall give an introduction to the basics of supermanifold geometry and show examples of their applications. Roughly, differentiable supermanifolds are similar to the familiar smooth manifolds, the main peculiarity being a Z/2-grading in the "algebras of functions", which locally look like extensions of ordinary smooth functions by Grassmann generators. In many important cases there is an extra Z-grading (e.g., ghost number in physics). Such supermanifolds are known as "graded manifolds". Non-negatively graded manifolds have a nice bundle structure generalizing vector bundles. After introducing some generalities on supermanifolds and graded manifolds, we shall discuss "Q-manifolds" (or "differential graded manifolds"), which are supermanifolds / graded manifolds endowed with an odd vector field of square zero. Such a structure is of great interest on its own and it also allows one to encode various geometric and algebraic information. For example, it is the most efficient language for discussing homotopy Lie algebras and Lie algebroids.

Introduction to higher gauge theories , U. Schreiber (Utrecht), T. Strobl (Lyon), C. Laurent-Gengoux (Coimbra)
Higher gauge theories is a generalization of Yang-Mills theories to higher form degrees of the gauge fields. This may be motivated by the Kalb-Ramond B-field in string theory, the supergravity C-field, or by pure curiosity for the construction of a general framework for such theories including nonabelian gerbes.

Strobl: The perspective of differential graded manifolds
We show that a tower of differential forms as gauge fields leads to the notion of differential graded manifolds as a generalization of the Lie algebras present in a Yang-Mills theory. We discuss the notion of gauge fields, gauge transformations, nontrivial bundels, and characteristic classes within this setting and present a gauge invariant functional for nonabelian gerbes.

Schreiber: The categorical perspective
A connection on an ordinary bundle is equivalently encoded in a functor that sends a groupoid of paths in the base space to the classifying groupoid of the gauge group ("Wilson lines" in physics language). We describe how this perspective generalizes to a formulation of higher connections in terms of functors out of the path infinity-Lie groupoid of a space.

Laurent-Gengoux: Non-Abelian gerbes with Lie groupoids
We describe non-Abelian gerbes and their various connections and curvatures with the language of (ordinary) classical gemeotry, namely Lie groupoids and Lie groupoid extensions. For gerbes over manifold, we claim that it gives short and practical manners to interpret various cocycle relations and higher Bianchi identities.

An introduction to modern category theory , C. Lazaroiu (Dublin)
This is a brief introduction to a few key concepts of modern category theory which might be of interest to those working on physics-inspired mathematical problems.
Lecture contents:
1. Standard categories. Functors and natural transformations; the strong 2-category Cat; monads and adjunct pairs. Weak 2-categories (a.k.a. "bicategories").
2. Multicategories and operads. The role of symmetries.
3. Monoidal categories and representable multicategories.
4. Some key constructions on Cat.
5. Higher categories and other extensions.

Categorified symplectic geometry , C. Rogers (Riverside)
Higher analogues of algebraic and geometric structures studied in symplectic geometry naturally arise on manifolds equipped with a closed non-degenerate form of degree > 2. Traditionally, such manifolds (which we will call `n-plectic') have been used to describe field theories within a formalism known as multisymplectic geometry. In these lectures, we'll introduce the point of view that multisymplectic geometry is `categorified' symplectic geometry. Along the way, we will encounter familiar higher structures such as Lie n- algebras, Courant algebroids, and U(1)-gerbes, and we will describe their roles in classical field theory and (pre)quantization.
Lecture 1: Introduction to n-plectic manifolds and their associated algebraic structures (Lie n-algebras, dg Leibniz algebras).
Lecture 2: Examples for n=2 (compact Lie groups, "phase spaces" for 1+1-dimensional field theories) and higher Dirac structures.
Lecture 3: Prequantization for n=2, the role of Courant algebroids and gerbes, and central extensions of Lie 2-algebras.