While reading through [Brylinski](http://books.google.com/books/about/Loop_Spaces_Characteristic_Classes_and_G.html?id=ta5UB1D64_gC), as in all of my posts, I am trying to understand the following equation:



> 
> $ g\_\* \tilde{\theta} = \tilde{\theta} - g^{-1} dg$
> 
> 
> 


Setting
-------


I have a principal $B$-bundle, $Q$, over my space M, with connection $\theta$ (values in the Lie Algebra $\mathfrak{b}$ of the lie group $B$), and a central group extension by $\mathbb{C}^\*$, 


$$ 0 \to \mathbb{C}^\* \to \tilde{B} \to B \to 0$$ where the map $p:\tilde{B} \to B$ is a principal $\mathbb{C}^\*$ fibration, yielding an exact sequence of sheaves of groups:


$$ 0 \to \underline{\mathbb{C}}\_M^\* \to \underline{\tilde{B}}\_M \to \underline{B}\_M \to 0$$
where $M$ is a paracompact space.


Long story short, I restrict my attention of the space to an open set $ U \subset M$, small enough so that I can "lift" the bundle to a $\tilde{B}$-bundle, $\tilde{Q}$, and the connection to a $\tilde{\mathfrak{b}}$-valued connection $\tilde{\theta}$ on the bundle $\tilde{Q}$. The lifted bundle must satisy the condition that we have a bundle isomorphism $f: \tilde{Q}/\mathbb{C}^\* \tilde{\to} Q\_{U} $. The lifted connection must satisfy $$q \circ \tilde{\theta} = f^\*\theta$$ as 1-forms with values in `$\mathfrak{b} = \tilde{\mathfrak{b}}/\mathbb{C}$; where` $q$ `is simply the map which quotients out by` $\mathbb{C}$.


Finally, $g$ is simply a $\mathbb{C}^\*$-valued function, which can be thought of as a bundle isomorphism given our local picture is a trivialization. By $g\_\*$ we simply mean the pullback of 
$g^{-1}$.


Warning: Matrices Beware!
-------------------------


I know that this formula is true for vector bundles, or when the group automorphisms are represented by invertible matrices; I have read that literature already. **Unless you are prepared to help me understand why this situation is most certainly in the land of finite vector bundles or matrix groups, such an answer would be redundant.**
The formula also makes sense up to the fact that the transformed connection and original connection must differ by a complex-valued form, which is exactly what this formula prescribes. 



Connection Transformation Formula; Degree 3 Cech Cohomology
The equality follows directly from the definition of a connection, and is independent of the context of lifting structure groups, or degree three cohomology.


Recall that a connection on a principal $G$-bundle $\pi:P \to M$ is a 1-form $\omega \in \Omega^1(P,\mathfrak{g})$ such that
$$
p\_2^{\ast}\omega = Ad\_g^{-1} (p\_1^{\ast}\omega) + g^{\ast}\theta
$$
over $P \times\_M P$, where $p\_1,p\_2: P \times\_M P \to P$ are the two projections, and $g: P \times\_M P \to G$ is the difference map defined by $g(p,p')\cdot p=p'$. Here $g^\ast\theta=g^{-1}dg$, whatever notation is preferred. 


Now, if $b: P \to G$ is a smooth map, it induces a map 
$$
\tilde b: P \to P \times\_M P: p \mapsto (p\cdot b(p),p).
$$
Pullback of above defining equation along $\tilde b$ produces
$$
\omega = Ad\_b(b\_{\ast}\omega)+b^\*\theta.
$$
If $b$ is central, this is your equation.


This works of course also, if $b:M \to G$ is defined on the base (you didn't say clearly where $b$ is defined). Then use $b' := b \circ \pi$ instead. 


