I have troubles to digest the following messages/discussions in the following work in p.10-12; Which construct a map from the moduli space of flat connections $M_{\rm flat}=\mathbb{E} / {\mathfrak S}_N \to CP^{N-1}$ to the projective space explicitly.
My questions:
What are the purposes of using the root and weight lattices of SU(N) here?
$\theta_k$ are usually theta fuctions. But they emphasize $\theta_k$ are not functions, but sections of a line bundle $L$. Why is that?
Their discussions are detailed below:
We now make the map $M_{\rm flat}=\mathbb{E} / {\mathfrak S}_N \to CP^{N-1}$ more explicit.
We denote the root lattice of $SU(N)$ by ${\mathbb L}$ $$ {\mathbb L}=\left\{ \vec{\ell} =(\ell_1, \cdots, \ell_N) \in \mathbb Z^N; \sum_i \ell_i=0 \right\}\;. $$ The weight lattice is spanned by the fundamental weights $$ \vec{e}_k=(\overset{1}{1},\cdots,\overset{k}{1},0,\cdots,0) - \frac{k}{N}(1,\cdots,1)\;. $$
We define theta functions as $$ \theta_k(\vec{\phi}) := \sum_{ \vec{\ell} \in {\mathbb L}} e^{ \pi i \tau (\vec{\ell}+\vec{e}_k)^2+ 2\pi i (\vec{\ell}+\vec{e}_k) \cdot \vec{\phi} } ~~~~~(k=1,\cdots,N)\;, $$ where $\vec{\phi}=(\phi_1,\cdots,\phi_N)$ and the inner product between vectors is defined as $\vec{\phi}\cdot \vec{\ell}=\sum_i \phi_i \ell_i$. These theta functions are invariant under the Weyl symmetry ${\mathfrak S}_N$ acting on $\vec{\phi}$, because each set $$ \vec{e}_k+{\mathbb L}=\{ \vec{e}_k+\vec{\ell} ; \vec{\ell} \in {\mathbb L} \} $$ is Weyl invariant, and the Weyl symmetry preserves the inner product. Furthermore, under the shift $$ \vec{\phi} \to \vec{\phi} +\tau \vec{m} - \vec{n}~~~~~(\vec{m}, \vec{n} \in {\mathbb L}) \;, $$ they transform as $$ \theta_k(\vec{\phi} + \tau \vec{m} - \vec{n}) =e^{ -\pi i \tau \vec{m}^2-2\pi i \vec{m} \cdot \vec{\phi}}\theta_k(\vec{\phi}) \;. $$ Note that the factor $e^{ -\pi i \tau \vec{m}^2-2\pi i \vec{m} \cdot \vec{\phi}}$ is independent of $k$.
We denote points of $CP^{N-1}$ by using homogeneous coordinates as $[Z_1,\cdots,Z_N]$. Then, if we define $$ \varphi (\vec{\phi}):=[\theta_1(\vec{\phi}), \cdots, \theta_N(\vec{\phi})] \;, $$ then the above properties imply that this is a well-defined map from $M_{\rm flat}$ to $CP^{N-1}$.
We claim that this is an isomorphism between $M_{\rm flat}$ and $CP^{N-1}$.