Let $L \to M$ be a complex line bundle over a smooth manifold $M$. Let $\{U_\alpha\}$ be a trivializing open cover with transition maps $z_{\alpha\beta}:U_\alpha \cap U_\beta \to \Bbb C^* = \text{GL}(1,\Bbb C)$. The bundle of endomorphisms of $L$, $\text{End}(L) \cong L^* \otimes L$ is trivial since it can be defined by transition maps $(z_{\alpha\beta})^{-1} \otimes z_{\alpha\beta} = 1$. This, the space of connections on $L$, $\mathcal{A}(L)$ is an affine space modeled by the linear space of complex valued $1$-forms. A connection on $L$ is simply a collection of $\Bbb C$-valued $1$-forms $\omega^\alpha$ on $U_\alpha$ related on overlaps by $$\omega^\beta = \frac{dz_{\alpha\beta}}{z_{\alpha\beta}} + \omega^\alpha = d\log z_{\alpha\beta} + \omega^\alpha.$$
Could someone explain to me what is going on here? I'm familiar with line bundles, transition maps, connections and bunch of other stuff, but this seems very weird to me.
I have never seen for example this affine space $\mathcal{A}(L)$ which I believe is defined to be somehow realted to $\Omega^1(\text{End}(L))$. This expression involving the logarithm is also pulled out of thin air, how does one go about deriving such a thing?
Best Answer
Let $\nabla_0\colon C^\infty(M,L)\to \Omega^1(M,L)$ be a reference connection. Then any other connection on $L$ can be written as $\nabla_A = \nabla_0+A$, where $A\in \Omega^1(M,\mathbb C)\cong \Omega^1(M,\mathrm{End}(L))$. This is what makes $\mathcal{A}(L)$ into an affine space.
Let now $(U_\alpha)$ be an open cover of $M$ and $s_\alpha\in C^\infty(U_\alpha,L|_{U_\alpha})$ be collection of non-vanishing sections. These automatically trivialise the bundle and thus there exist transition functions $z_{\alpha\beta}\in C^\infty(U_{\alpha\beta},\mathbb C)$ on $U_{\alpha\beta} =U_\alpha\cap U_\beta$ such that $$s_\alpha = z_{\alpha\beta}s_\beta \quad \text{ on } U_{\alpha\beta}. \tag{1}$$ Moreover, we can express $\nabla_A s_\alpha$ via some $\omega_{\alpha}\in \Omega^1(U_\alpha,\mathbb C)$ as follows: $$ \nabla_A s_\alpha = \omega_{\alpha} s_\alpha \tag{2} $$ Now apply $\nabla_A$ to $(1)$ and use the Leibniz rule: $$ \omega_\alpha s_\alpha = \nabla_A(s_\alpha) = dz_{\alpha\beta} s_\beta + z_{\alpha\beta} \nabla_As_{\beta} = dz_{\alpha\beta} s_\beta + z_{\alpha\beta} \omega_\beta s_\beta = \frac{dz_{\alpha\beta}}{z_{\alpha\beta}} s_\alpha + \omega_\beta s_\alpha $$ Since the $s_\alpha'$s don't vanish we get the following relation you encountered: $$ \omega_\alpha = \frac{dz_{\alpha\beta}}{z_{\alpha\beta}} + \omega_\beta \tag{3} $$ What I've explained is how to associate to a connection $\nabla_A$ a collection of $1$-forms $(\omega_\alpha)$ satisfying relation $(3)$. Now try to reverse engineer a connection $\nabla_A$ out of the $1$-forms!