ESI Program on Higher Structures in Mathematics and Physics
Vienna, September 1 -
November 7, 2010
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
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.
1. Standard categories. Functors and natural transformations; the
strong 2-category Cat; monads and adjunct pairs. Weak 2-categories
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
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.