## Proceedings of the Centre for Mathematics and its Applications

### Riemannian geometry and mathematical physics: vector bundles and gauge theories

Michael K. Murray

#### 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

Dates
First available in Project Euclid: 18 November 2014