Speaker: Brano Jurco
Noncommutative gauge theories, deformation quantization and formality. (2 lectures)
An elementary introduction to noncommutative gauge theory of the type that
arises in string theory with background B-field is given. We discuss the
mathematics of gauge fields from the point of view of Kontsevich's
deformation quantization and the related notions of a noncommutative line
bundle and of a noncommutative gerbe.
Speaker: Brano Jurco
Nonabelian bundle gerbes
Bundle gerbes are a higher version of line bundles, we present nonabelian
bundle gerbes as a higher version of principal bundles. Connection,
curving, curvature and gauge transformations are studied both in a global
coordinate independent formalism and in local coordinates. These are the
gauge fields needed for the construction of Yang-Mills theories with
2-form gauge potential. As an application the anomaly of M5-branes is
Title: Dixmier traces and their applications. (2 lectures)
Abstract: A Dixmier trace is a non-normal singular trace vanishing
on all finite rank operators. It was discovered by J.Dixmier in
1966 as an example of a non-normal trace on the algebra B(H) of
all bounded linear operators on an infinite dimensional Hilbert
space. Later, Alain Connes discovered that although Dixmier trace
is not normal, nevertheless it has many important applications to
non-commutative geometry. In fact, a Dixmier trace considered on
the algebra of pseudodifferential operators on a compact smooth
manifold can be viewed as a non-commutative integral in the sense
of differential geometry, whereas the ordinary trace on B(H) is a
non-commutative integral in the sense of measure theory.
Furthermore, Dixmier traces are strongly related to the so-called
Wodzicki residue of pseudo-differential operators, and to the
Speaker: Hisham Sati
Title: The M-theory partition function and topology.
Abstract: Witten has shown that the topological part of the
M-theory partition function is encoded in an index of an
E8 bundle in eleven dimensions. Diaconescu-Moore-Witten related
this to the K-theoretic partition function of type IIA string
theory obtained via dimensional reduction, and later Mathai
and I generalized part of the construction to twisted K-theory.
In this talk, after reviewing the above, I report on my recent
work with Igor Kriz on the appearance of elliptic cohomology in
this context. I will try to focus on the general idea rather than
on the technical construction.