From the paper "Gapped Boundary Phases of Topological Insulators via Weak Coupling" on page 11,

https://arxiv.org/abs/1602.04251

the authors states that on a curved manifold with a $spin_{\mathbb{C}}$ structure, the $U(1)$-connection

$$D_{\mu}^{(n)}=D_{\mu}^{(0)}+inA_{\mu}$$ is well-defined only for odd electric charge $n$, where $D_{\mu}^{(0)}$, I suppose, is the spin connection.

Why is the definition of $spin_{\mathbb{C}}$ connection related with the parity of electric charge that the spinor couples?