• lattice and B-field

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 $T^{b_1}$, $T=S^1$. The spectrum depends on the parameters.

Can one find some special solvable model?

Let $Q$ be a graph on a surface $S$ of genus $g$.

To each node, we associate a complex hermitian vector space. Let $V = \oplus_i V_i$, then $V$ is a hermitian vector space. Consider a hermitian form $$ B: V \times V \to \C $$ such that $$ B(v, w) = \overline{B(w,v)}. $$

If we equip $S$ with a flat $U(1)$ connection on a trivial line bundle, then we can twist the hermitian form. And, we can ask for the spectrum of the twisted matrix. It is a function on $T^{2g}$, but maybe we can extend it to a function on $(\C^*)^{2g}$. This is the generalization of the Fourier transformation.

Now, we can turn on an 'integral' B-field. This means introduce a complex line bundle with some given first Chern class. But this is not enough to specify the effect. We may need to specify some (unitary?) parallel connection data to each edge. Fix an orientation of the edge, then we assign $E \to \Q/\Z$. This defines a $1$-cochain on $Q$. You can trivialize along a (max) spanning tree.

How to solve the 1D spin chain cycle.