About the curvature of a connection? [closed]

Question

In "Lectures on gauge theory and integrable systems" of M.Audin, she identifies the space of connections $\mathcal{A}$ on the trivial bundle $G\times S$ ($G$ Lie group, $S$ surface with boundary) with the vector space $\Omega^1(S,\mathfrak{g})$. Then, for each $A\in\mathcal{A}$ define

$$d_A:\Omega^0(S,\mathfrak{g})\rightarrow\Omega^1(S,\mathfrak{g});\alpha\mapsto d\alpha+[A,\alpha]$$

where $[A,\alpha](X)=[A(X),\alpha]$ for a vector $X$ tangent to $S$.

Similarly $$d_A:\Omega^1(S,\mathfrak{g})\rightarrow\Omega^2(S,\mathfrak{g});\phi\mapsto d\phi+[A,\phi]$$ with $[A,\phi](X,Y)=[A(X),\phi(Y)]-[A(Y),\phi(X)]$.

1. For $\alpha\in\Omega^0(S,\mathfrak{g})$, she claims $d_A\circ d_A(\alpha)=[d_AA,\alpha]$, then she defines the curvature form as $F(A)=d_AA$. Why does this hold? (I think should be $d_A\circ d_A(\alpha) = [dA+\frac{1}{2}[A,A].\alpha]$).

2. With her definition of $F(A)$ she asks to show that $(T_AF)(\phi)=d_A\phi$, but I couldn't. So, should I consider $F(A)=dA+\frac{1}{2}[A,A]?$ I would like to understand this.

Answer 1

1. The calculation goes as follows:

\begin{align} d_A \circ d_A(\alpha) & = d_A(d\alpha + [A, \alpha]) \\ & = d(d\alpha + [A,\alpha]) + [A, d\alpha + [A,\alpha]] \\ & = d^2 \alpha + d[A, \alpha] + [A, d\alpha] + [A, [A, \alpha]] \\ & = 0 + [dA, \alpha] - [A, d\alpha] + [A, d\alpha] + \tfrac{1}{2}[[A, A], \alpha] \\ & = [dA + \tfrac{1}{2}[A,A], \alpha] \\ & = [d_A A, \alpha]. \end{align}

There's a couple basic identities you need to check in the process, but it's nothing difficult. The $\tfrac{1}{2}$ shows up when you use the Jacobi identity, so it seems that perhaps Audin is missing it, but I don't have access to the book currently and can't look at what she does. For matrix Lie groups, $\tfrac{1}{2}[A, A] = A \wedge A$ where we consider $A \wedge A$ as a matrix product, so sometimes you will see $F(A) = dA + A \wedge A$ in the context of matrix Lie groups.

2. Simply use the definition of the derivative in an affine space (since $\mathcal{A}$ is affine):

\begin{align} (T_A F)(\phi) & = \lim_{t \to 0} \frac{1}{t} (F(A + t\phi) - F(A))\\ & = \lim_{t \to 0} \frac{1}{t}(d_{A + t\phi} (A + t\phi) - d_A A)\\ & = \lim_{t \to 0} \frac{1}{t}(d(A + t\phi) + \tfrac{1}{2}[A, A + t\phi] - dA - \tfrac{1}{2}[A,A])\\ & = \lim_{t \to 0} \frac{1}{t}(dA + td\phi + \tfrac{1}{2}[A,A] + \tfrac{1}{2}t[A, \phi] - dA - \tfrac{1}{2}[A,A])\\ & = d\phi + \tfrac{1}{2}[A, \phi]\\ & = d_A\phi. \end{align}