Topology, Geometry and Gauge fields Foundations
“Topology, Geometry and Gauge Fields Foundations” by Gregory L. Naber is a comprehensive exploration of the interplay between mathematics and physics, published by Springer New York in September 2010. This second edition, spanning 437 pages, invites readers to engage with the intricate connections between classical gauge theory and the topological and geometrical concepts that underpin these theories. The book is presented in English and is designed for those with a solid foundation in real analysis, linear algebra, and some familiarity with modern algebra.
Readers will find a structured journey that begins with fundamental definitions in topology and progresses to advanced topics, including the moduli space of anti-self-dual SU(2) connections on S4. The text emphasizes the importance of fostering the relationship between mathematics and physics, encouraging readers to consider concepts such as electromagnetic fields and elementary quantum mechanics. This edition serves as a resource for those interested in the mathematical frameworks that support various scientific disciplines, including physics and quantum theory.
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Like any books on a subject as vast as this, this book has to have a point-of-view to guide the selection of topics. Naber takes the view that the rekindled interest that mathematics and physics have shown in each other of late should be fostered, and that this is best accomplished by allowing them to cohabit. The book weaves together rudimentary notions from the classical gauge theory of physics with the topological and geometrical concepts that became the mathematical models of these notions. The reader is asked to join the author on some vague notion of what an electromagnetic field might be, to be willing to accept a few of the more elementary pronouncements of quantum mechanics, and to have a solid background in real analysis and linear algebra and some of the vocabulary of modern algebra. In return, the book offers an excursion that begins with the definition of a topological space and finds its way eventually to the moduli space of anti-self-dual SU(2) connections on S4 with instanton number -1.
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