Let $\Sigma$ be a Riemann surface. Let $Q$ be a quiver on $\Sigma$, consist of vertices and directed edges. (No edge is contractible to a point, and no two edges are homotopic. )

Fix a hermitian vector bundle $E$ over $\Sigma$ with unitary connection $\nabla$.

We define the Hilbert space to be $\gdef\hcal{\mathcal H}$ $$\hcal = \Gamma(Q_0, E). $$ with obvious inner product. There is an endomorphism $$ \Phi: \hcal \to \hcal, \quad \Phi(s)_i = \sum_{j \to i} P_{j \to i}(s_j) $$ where $P_{j \to i}$ is the parallel transport along the edge from $j$ to $i$.

We define a sesquilinear form $$H(s_1, s_2) = \langle s_1, s_2 \rangle + \langle \Phi(s_1), s_2 \rangle $$ Since $ \langle (s_1)_{i \to j}, (s_2)_j \rangle = \langle (s_1)_{i}, (s_2)_{j \to i} \rangle$.

Well, this only says, we have a graph. We label the vertices with hermitian vector spaces and label the edges with isometries. We can ask for the spectrum of this system.

Given two such system, we can form a tensor product. Hence, we can tensor by another line bundle, or even vector bundle.

Let $Loc_{U(1)}(\Sigma)$ be the space of $U(1)$ local system on $\Sigma$, which is a real $2b_1$ torus $T^{b_1}$, $T=S^1$. We can choose a fundamental domain, cut out by some circles, and