Vector bundles and gauge theories

Line bundles occur in physics in electromagnetism. The electro-magnetic tensor can be interpreted as the curvature form of a line bundle. A very nice account of this and related material is given by Bott in [3]. More interesting however are so-called non-abelian gauge theories which involve vector bundles.

To generalize the previous sections to a vector bundles $E$ one needs to work through replacing $\mathbb{C}$ by $\mathbb{C}^n$ and $\mathbb{C}^\times$ by $GL(n, \mathbb{C})$. Now non-vanishing sections and local trivialisations are not the same thing. A local trivialisation corresponds to a local frame, that is $n$ local sections $s_{1}, ..., s_{n}$ such that $s_{1}(m), ..., s_{n}(m)$ are a basis for $E_m$ all $m$. The transition function is then matrix valued

$$g_{\alpha \beta} : U_{\alpha} \cap U_{\beta} \to GL(n, \mathbb{C}).$$

The clutching construction still works.

A connection is defined the same way but locally corresponds to matrix valued one-forms $A_\alpha$. That is

$$\nabla\vert _{U \alpha} (\Sigma_i \xi^{i}s_{i}) = \Sigma_{i} (d \xi i + \Sigma_{j} A^{i}_{\alpha j} \xi^{j}) s_{i}$$

and the relationship between $A_{\beta}$ and $A_\alpha$ is

$$A_{\beta} = g^{- 1}_{\alpha \beta} \ A_{\alpha} \ g_{\alpha \beta} + g^{- 1}_{\alpha \beta} \ d g _{\alpha \beta}.$$

The correct definition of curvature is

$$F_{\alpha} = d A_{\alpha} + A_{\alpha} \wedge A_{\alpha}$$

where the wedge product involves matrix multiplication as well as wedging of one forms. We find that

$$F_{\beta} = g^{-1}_{\alpha \beta} \ F_{\alpha} \ g_{\alpha \beta}$$

and that $F$ is properly thought of as a two-form with values in the linear operators on $E$. That is if $X$ and $Y$ are vectors in the tangent space to $M$ at $m$ then $F(X, Y)$ is a linear map from $E_m$ to itself.

We have no time here to even begin to explore the rich geometrical theory that has been built out of gauge theories and instead refer the reader to some references [1,2,6,7].

We conclude with some remarks about the relationship of the theory we have developed here and classical Riemannian differential geometry. This is of course where all this theory began not where it ends! There is no reason in the above discussion to work with complex vector spaces, real vector spaces would do just as well. In that case we can consider the classical example of tangent bundle $TM$ of a Riemannian manifold. For that situation there is a special connection, the Levi-Civita connection. If $(x^1, \dots, x^n)$ are local co-ordinates on the manifold then the Levi-Civita connection is often written in terms of the Christoffel symbols as

$$\nabla_{\frac{\partial\phantom{x^i}}{\partial x^i}}(\frac{\partia... ...rtial x^j}) = \sum_k \Gamma^k_{ij} \frac{\partial\phantom{x^k}}{\partial x^k}.$$

The connection one-forms are supposed to be matrix valued and they are

$$\sum_i \Gamma^k_{ij} dx^i.$$

The curvature $F$ is the Riemann curvature tensor $R$. As a two-form with values in matrices it is

$$\sum_{ij} R_{ijk}^k dx^i \wedge dx^j.$$