TFPT scientific source excerpt
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000220 | spectral-flow term determined by the matching operator $U_\Gamma$. This makes the seam
000221 | contribution quantized and computable, rather than a geometric guess.
000222 | 
000223 | \begin{tcolorbox}[colback=green!3, colframe=green!50!black, 
000224 |     title={\textbf{Proof Sketch for $\phitree = 1/(6\pi)$ (via $\eta$-gluing)}}, fonttitle=\bfseries\small]
000225 | \textbf{Setup:} Cut the orientable double cover along $\Gamma$ to obtain
000226 | $\widetilde{M}_\Gamma$ with boundary components $(C_1,C_2,\Gamma_+,\Gamma_-)$.
000227 | Impose APS boundary conditions on $C_{1,2}$ and gluing by a unitary operator
000228 | $U_\Gamma$ between $\Gamma_+$ and $\Gamma_-$.
000229 | 
000230 | \textbf{Step 1 (Boundary operator):} On the seam, take the tangential Dirac operator
000231 | \begin{equation}
000232 |     D_\Gamma = i\,\frac{d}{d\theta}, \qquad \theta\in[0,2\pi).
000233 | \end{equation}
000234 | 
000235 | \textbf{Step 2 (Minimal gluing class):} The minimal M\"obius class is modeled by the
000236 | unitary phase
000237 | \begin{equation}
000238 |     U_\Gamma(\theta)=e^{i\theta}.
000239 | \end{equation}
000240 | 
000241 | \textbf{Step 3 (Spectral flow):} Interpolate the boundary condition
000242 | $\psi(\theta+2\pi)=e^{i\varphi}\psi(\theta)$ for $\varphi\in[0,2\pi]$. The eigenvalues are
000243 | \begin{equation}
000244 |     \lambda_n(\varphi)=n+\frac{\varphi}{2\pi}, \qquad n\in\mathbb{Z},
000245 | \end{equation}
000246 | so exactly one eigenvalue crosses zero as $\varphi$ runs from $0$ to $2\pi$:
000247 | $\mathrm{SF}(U_\Gamma)=1$.
000248 | 
000249 | \textbf{Step 4 (Gluing term):} The APS gluing theorem yields
000250 | $\eta(\widetilde{M}) = \eta(\widetilde{M}_\Gamma) + 2\,\mathrm{SF}(U_\Gamma)$,
000251 | so the seam contribution is $\Delta_\Gamma=2\pi$.
000252 | 
000253 | \textbf{Step 5 (Normalization):} The two physical boundary cycles contribute $2\pi+2\pi$,
000254 | and the seam adds $2\pi$, so $R_{\text{total}}=6\pi$. The stationarity condition
000255 | $\varphi\cdot R_{\text{total}}=1$ then gives $\phitree=1/(6\pi)$.
000256 | \end{tcolorbox}
000257 | 
000258 | \begin{axiom}{Discrete Topological Corrections}{TC}