Gauge transformation of local bundle data

Let $\pi:P\to M$ be a principal G-bundle with a principal connection given by a connection 1-form $\omega$.

Fix an open cover $\{U_i\}$ and local sections $s_i:U_i\to P$. Let

• $g_{ij}:U_i\cap U_j\to G$ be the transition functions defined by $s_j=s_i\cdot g_{ij}$ (as in the construction of transition functions from local sections ),
• $A_i := s_i^*\omega \in \Omega^1(U_i;\mathfrak g)$ be the corresponding local connection 1-forms .

A local gauge change is a choice of smooth maps $u_i:U_i\to G$ and new sections

This is the local form of a local gauge transformation .

Transformation of transition functions

On an overlap $U_i\cap U_j$,

By uniqueness of transition functions, the new transition functions satisfy

Thus, changing local sections replaces the cocycle $\{g_{ij}\}$ by an equivalent cocycle .

Transformation of local connection forms

The local connection forms transform by the usual gauge rule:

This is the statement recorded in the local gauge transformation law for A and matches gauge-transforming a local connection form .

If $F_i$ denotes the local curvature 2-form on $U_i$, then

as in the local curvature transformation law .

These formulas are the standard local expression of a global gauge transformation acting on the space of connections.

Examples

1. Trivial bundle on one chart. On $U$ with a trivialization, a local gauge function $u:U\to G$ sends a $\mathfrak g$-valued 1-form $A$ to

   $$A^u = \mathrm{Ad}_{u^{-1}}A + u^{-1}du.$$

   For an abelian group (e.g. $U(1)$), $\mathrm{Ad}$ is trivial and this reduces to $A^u = A + u^{-1}du$.

2. Pure gauge becomes zero. On a trivial bundle, take $A=u^{-1}du$, the pure gauge connection . Gauge transforming by $u$ gives

   $$A^u = 0,$$

   recovering the flat connection in that gauge.

3. Changing local sections changes the clutching data by conjugation. If a principal bundle is presented by transition functions on a two-set cover (a “clutching function” presentation as in clutching functions ), replacing the chosen local sections by $s_i'=s_i u_i$ modifies the clutching map by $g' = u^{-1} g\, u$ on the overlap. This does not change the isomorphism class of the bundle, precisely because it is an instance of cocycle equivalence.