Gauge transformation of local bundle data

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

s_i' := s_i\cdot u_i.

This is the local form of a local gauge transformation.

Transformation of transition functions

On an overlap $U_i\cap U_j$ ,

s_j' = s_j u_j = s_i g_{ij} u_j = s_i u_i\,(u_i^{-1} g_{ij} u_j).

By uniqueness of transition functions, the new transition functions satisfy

g_{ij}' = u_i^{-1}\, g_{ij}\, u_j.

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:

A_i' = (s_i')^*\omega = \mathrm{Ad}_{u_i^{-1}} A_i + u_i^{-1} d u_i.

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

F_i' = \mathrm{Ad}_{u_i^{-1}} F_i,

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

Examples

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, such as $U(1)$ , $\mathrm{Ad}$ is trivial and this reduces to $A^u=A+u^{-1}du$ .

A pure gauge is gauge-equivalent to zero. Starting with $A=0$ , gauge transformation by $u$ gives $A^u=u^{-1}du$ , the pure gauge connection. Applying the inverse change of gauge recovers $(A^u)^{u^{-1}}=0,$ the flat connection in the original gauge.

Changing local sections changes the clutching data by cocycle equivalence. If a principal bundle is presented by a transition function $g$ on a two-set cover, replacing the local sections by $s_1'=s_1u_1$ and $s_2'=s_2u_2$ changes it to $g'=u_1^{-1}gu_2.$ This does not change the bundle's isomorphism class. When $u_1=u_2$ on the overlap, the formula specializes to conjugation.

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