Topics and Lectures
Geometry, reduction and quantization
•Geometry of Dirac structures
Henrique Bursztyn
(IMPA, Brazil)
•Cohomological formulae for the equivariant index of a transversally
elliptic operator
Paul-Emile Paradan
(Montpellier, France)
• Holomorphic structures and unitary connections on Hermitian vector bundles
Florent Schaffhauser
(Keio, Japan)
Multizeta, polylogarithms and periods in quantum field theory
• Iterated integrals in quantum field theory
Francis Brown
(Paris VII, France)
• A Prologomon to Renormalization
Sylvie Paycha
(Clermont-Ferrand, France)
• Introduction to Feynman integrals
Stefan Weinzierl
(Mainz, Germany)
Geometry of quantum fields and the standard model
• Geometric issues in Quantum Field Theory and String Theory
Luis J. Boya
(Zaragoza, Spain)
• Geometric Aspects of the Standard Model and the Mysteries of Matter
Florian Scheck
(Mainz, Germany)
Abstract:
The basic structure of gauge theories seems to
distinguish radiation from matter as two categories of
different origin. The vector or tensor bosons which are the carriers of the
fundamental forces, belong to what may be termed radiation. They
are described by geometric theories, i.e. Yang-Mills theories or, in the
case of general relativity, by semi-Riemannian geometry in dimension four.
To a large extent,
they are classical theories.
Matter, i.e. quarks and leptons and composites thereof, a priori,
seems to belong to a different kind of physics which, at first sight, does
not exhibit an underlying geometrical structure. As soon as one enters the quantum world, however, the two categories start mingling their waters.
The Higgs particle plays a rather enigmatic role. Its phenomenological role in
providing mass terms for some of the vector bosons and for the fermions of the
theory suggests that it be another form of matter. Models based on
noncommutative geometry, in turn, classify the Higgs field in the generalized
Yang-Mills connection, besides the gauge bosons, and hence declare it to be
part of radiation. Quarks and leptons are described by a Dirac
operator which in its mass sector exhibits a significant, though mysterious
structure. Dirac operators, in turn, are the driving vehicles in constructing
noncommutative geometries designed to generalize Yang-Mills theory.
We work out several of these themes, both by way of construction and by means
of instructive examples. We start with a schematic description of Yang-Mills
theories including spontaneous symmetry breaking (SSB) within the classical
geometric framework, and including matter particles. In a first excursion to
quantum field theory we describe the stratification of the space of
connections and its relevance to anomalies. In order to clarify the
phenomenological basis on which Yang-Mills theories of fundamental
interactions are built, we describe some of the most pertinent
phenomenological features of leptons and of quarks. Via the Dirac operator
describing leptons and quarks we turn to constructions of the standard model
in the framework of noncommutative geometry. This, in turn leads us to a
closer analysis of the mass sector and state mixing phenomena of fermions. The
intricacies of quantization are illustrated by a semi-realistic model for
massive and massless vector bosons.
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