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000001 | \subsection{M\"obius Matching and the Minimal Gluing Class}
000002 | For the M\"obius $\mathbb{Z}_2$ identification, the matching satisfies $U_\Gamma^2=\mathbf{1}$.
000003 | The minimal nontrivial class is modeled by the unitary phase
000004 | \begin{equation}
000005 |     U_\Gamma(\theta)=e^{i\theta}.
000006 | \end{equation}
000007 | 
000008 | \subsection{Spectral Flow Computation}
000009 | Interpolate boundary conditions by a phase $\varphi$,
000010 | \begin{equation}
000011 |     \psi(\theta+2\pi)=e^{i\varphi}\psi(\theta), \qquad \varphi\in[0,2\pi],
000012 | \end{equation}
000013 | which gives the eigenvalues
000014 | \begin{equation}
000015 |     \lambda_n(\varphi)=n+\frac{\varphi}{2\pi}, \qquad n\in\mathbb{Z}.
000016 | \end{equation}
000017 | As $\varphi$ runs from $0$ to $2\pi$, exactly one eigenvalue crosses zero, so
000018 | $\mathrm{SF}(U_\Gamma)=1$ and $\Delta_\Gamma=2\pi$.
000019 | 
000020 | \begin{remark}
000021 | For unitary matching maps $U_\Gamma:S^1\to U(1)$, the spectral flow equals the winding
000022 | number of $\det U_\Gamma$, consistent with the $K^1(S^1)\cong\mathbb{Z}$ classification.
000023 | \end{remark}