This book provides an introduction to noncommutative geometry and
presents a number of its recent applications to particle physics.
It is intended for graduate students in mathematics/theoretical
physics who are new to the field of noncommutative geometry, as
well as for researchers in mathematics/theoretical physics with an
interest in the physical applications of noncommutative geometry.
In the first part, we introduce the main concepts and techniques by
studying finite noncommutative spaces, providing a “light”
approach to noncommutative geometry. We then proceed with the
general framework by defining and analyzing noncommutative spin
manifolds and deriving some main results on them, such as the local
index formula. In the second part, we show how noncommutative spin
manifolds naturally give rise to gauge theories, applying this
principle to specific examples. We subsequently geometrically
derive abelian and non-abelian Yang-Mills gauge theories, and
eventually the full Standard Model of particle physics, and
conclude by explaining how noncommutative geometry might indicate
how to proceed beyond the Standard Model.
General
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