Proceedings of the Centre for Mathematics and its Applications

Riemannian geometry and mathematical physics: vector bundles and gauge theories

Michael K. Murray

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Abstract

The mathematical motivation for studying vector bundles comes from the example of the tangent bundle $TM$ of a manifold $M$. Recall that the tangent bundle is the union of all the tangent spaces $T_mM$ for every $m$ in $M$. As such it is a collection of vector spaces, one for every point of $M$.

The physical motivation comes from the realisation that the fields in physics may not just be maps $\phi : M \rightarrow C^N$ say, but may take values in different vector spaces at each point. Tensors do this for example. The argument for this is partly quantum mechanics because, if $\phi$ is a wave function on a space-time $M$ say, then what we can know about are expectation values, that is things like: \[\int_M \langle\phi(x), \phi(x)\rangle dx \] and to define these all we need to know is that $\phi(x)$ takes its values in a one-dimensional complex vector space with Hermitian inner product. There is no reason for this to be the same one-dimensional Hermitian vector space here as on Alpha Centuari. Functions like $\phi$, which are generalisations of complex valued functions, are called sections of vector bundles.

We will consider first the simplest theory of vector bundles where the vector space is a one-dimensional complex vector space -line bundles.

Article information

Source
Instructional Workshop on Analysis and Geometry, Part 2. Tim Cranny and John Hutchinson, eds. Proceedings of the Centre for Mathematics and its Applications, v. 34. (Canberra AUS: Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, 1996), 165-184

Dates
First available in Project Euclid: 18 November 2014

Permanent link to this document
https://projecteuclid.org/ euclid.pcma/1416323492

Mathematical Reviews number (MathSciNet)
MR1394690

Zentralblatt MATH identifier
0849.53021

Citation

Murray, Michael K. Riemannian geometry and mathematical physics: vector bundles and gauge theories. Instructional Workshop on Analysis and Geometry, Part 2, 165--184, Centre for Mathematics and its Applications, Mathematical Sciences Institute, The Australian National University, Canberra AUS, 1996. https://projecteuclid.org/euclid.pcma/1416323492


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