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 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 $$H(s_1, s_2) = \la s_1, s_2 \ra + \la \Phi(s_1), s_2 \ra $$