group describing the (continuous) symmetries of internal degrees of freedom. The corresponding Lie algebra through the Lie group–Lie algebra correspondence is denoted . This is referred to as the gauge group. A principal -bundle , otherwise known as a -torsor. This is sometimes written as where is the canonical projection map on and is the base manifold. Connections and gauge fields Here we take the view of the connection as a principal connection. In field theory this connection is also viewed as a covariant derivative whose action on various fields is defined later. A principal connection denoted is a -valued 1-form on P satisfying technical conditions of 'projection' and 'right-equivariance': details found in the principal connection article. Under a trivialization this can be written as a local gauge field , a -valued 1-form on a trivialization patch . It is this local form of the connection which is identified with gauge fields in physics. When the base manifold is flat, there are simplifications which remove this subtlety. Associated vector bundles and matter content An associated vector bundle associated to the principal bundle through a representation For completeness, given a representation , the fiber of is . A field or matter field is a section of an associated vector bundle. The collection of these, together with gauge fields, is the matter content of the theory. Lagrangian A Lagrangian : given a fiber bundle , the Lagrangian is a function . Suppose that the matter content is given by sections of with fibre from above. Then for example, more concretely we may consider to be a bundle where the fibre at is . This then allows to be viewed as a functional of a field. This completes the mathematical prerequisites for a large number of interesting theories, including those given in the examples section above. Theories on flat spacetime When the base manifold is flat, that is, (Pseudo-)Euclidean space, there are many useful simplifications that make theories less conceptually difficult to deal with. The simplifications come from the observation that flat spacetime is contractible: it is then a theorem in algebraic topology that any fibre bundle over flat is trivial. In particular, this allows us to pick a global trivialization of , and therefore identify the connection globally as a gauge field Furthermore, there is a trivial connection which allows us to identify associated vector bundles as , and then we need