TFPT scientific source excerpt
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000165 | Let $N_\Seam$ be the oriented two-dimensional normal slice to the seam. For $a\in\{2,3\}$ define
000166 | \[
000167 | \mathcal B_{E_a}:=\sigma^\mu\nabla^{(a)}_\mu+\Phi_a
000168 | \]
000169 | on $N_\Seam$, where $\Phi_a$ is asymptotically invertible on the boundary at infinity. Under the determinant-class data of \Cref{thm:determinant-class-higgs-selection}, assume
000170 | \[
000171 | (\mathcal B_{E_2}^-)^\dagger\mathcal B_{E_2}^-\ge m_-^2>0,
000172 | \qquad
000173 | (\mathcal B_{E_3}^{\pm})^\dagger\mathcal B_{E_3}^{\pm}\ge m_3^2>0.
000174 | \]
000175 | Then the bosonic relative index satisfies
000176 | \[
000177 | \Ind_{\mathrm{rel}}^{\Padm}(\mathcal B_{E_2}^+)=1,
000178 | \qquad
000179 | \Ind_{\mathrm{rel}}^{\Padm}(\mathcal B_{E_3}^+)=0.
000180 | \]
000181 | Moreover,
000182 | \[
000183 | \dim\ker\mathcal B_{E_2}^+=1,
000184 | \qquad
000185 | \ker\mathcal B_{E_2}^-=\ker\mathcal B_{E_3}^{\pm}=0.
000186 | \]
000187 | Consequently the unique light seam-even bosonic zero mode lies in
000188 | \[
000189 | \Phi\in\Gamma(E_2)\cong (1,2)_{1/2},
000190 | \qquad
000191 | \inv\Phi=+\Phi.
000192 | \]
000193 | \end{theorem}
000194 | 
000195 | \begin{proof}[Proof sketch]
000196 | By \Cref{thm:determinant-class-higgs-selection}, the retained branch fixes the asymptotic winding data to
000197 | \[
000198 | \mathrm{wind}(\det\Phi_2)=1,
000199 | \qquad
000200 | \mathrm{wind}(\det\Phi_3)=0.
000201 | \]
000202 | The Callias index theorem then gives
000203 | \[
000204 | \Ind_{\mathrm{rel}}^{\Padm}(\mathcal B_{E_2}^+)=1,
000205 | \qquad
000206 | \Ind_{\mathrm{rel}}^{\Padm}(\mathcal B_{E_3}^+)=0.
000207 | \]
000208 | The positivity bounds remove the opposite chirality and all triplet zero modes, so the index lifts to an actual uniqueness statement for one seam-even Higgs doublet.
000209 | \end{proof}
000210 | 
000211 | This is the sharp form of the Higgs claim used here. The issue is not whether a Higgs doublet can be written down, but whether the relative bosonic index can be shown to select it uniquely.
000212 | Without a genuine Callias / Dirac / Dolbeault-type index problem there would be no mathematical reason for one light doublet and no light triplet to emerge from the bosonic sector. Once that uniqueness statement is fixed, the renormalizable carrier-compatible Yukawa couplings are the standard ones,