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000095 | 
000096 | \begin{theorem}[Bosonic index and unique Higgs doublet from the minimal determinant class]
000097 | \label{thm:callias-higgs-selection}
000098 | \label{thm:compact-higgs-unique}
000099 | \label{thm:compact-determinant-higgs-index}
000100 | Compactify the oriented two-dimensional normal slice $N_\Seam$ to $S^2$. For $a\in\{2,3\}$
000101 | define
000102 | \[
000103 | \mathcal B_{E_a}:=\sigma^\mu\nabla^{(a)}_\mu+\Phi_a
000104 | \]
000105 | on the compactified normal sphere. Under the determinant-class data of
000106 | \Cref{thm:determinant-class-higgs-selection}, one has
000107 | \[
000108 | L_2\cong\mathcal O(1),
000109 | \qquad
000110 | L_3\cong\mathcal O,
000111 | \]
000112 | and hence
000113 | \[
000114 | H^0(S^2,\mathcal O(1))\cong\mathbb C^2,
000115 | \qquad
000116 | H^1(S^2,\mathcal O(1))=0,
000117 | \]
000118 | and the compact bosonic index satisfies
000119 | \[
000120 | \Ind(\mathcal B_{E_2}^+)=1,
000121 | \qquad
000122 | \Ind(\mathcal B_{E_3}^+)=0.
000123 | \]
000124 | Consequently the closed branch contains exactly one complex weak doublet and no light color triplet,
000125 | and the unique light seam-even bosonic zero mode lies in
000126 | \[
000127 | \Phi\in\Gamma(E_2)\cong (1,2)_{1/2},
000128 | \qquad
000129 | \taudbl^{\ast}\Phi=+\Phi.
000130 | \]
000131 | \end{theorem}
000132 | 
000133 | \begin{proof}
000134 | By \Cref{thm:determinant-class-higgs-selection}, the weak determinant line is the degree-one line
000135 | bundle
000136 | \[
000137 | L_2\cong\mathcal O(1)
000138 | \]
000139 | on
000140 | \[
000141 | S^2\cong\mathbb CP^1,
000142 | \]
000143 | whereas the color determinant class is trivial and
000144 | \[
000145 | L_3\cong\mathcal O.
000146 | \]
000147 | For $\mathbb CP^1$, Birkhoff--Grothendieck identifies every holomorphic line bundle by its degree,
000148 | and Riemann--Roch gives
000149 | \[
000150 | \chi(\mathcal O(1))=1+1=2.
000151 | \]
000152 | Serre duality yields
000153 | \[
000154 | H^1(S^2,\mathcal O(1))
000155 | \cong
000156 | H^0(S^2,\mathcal O(-3))^\vee
000157 | =0,
000158 | \]
000159 | so
000160 | \[
000161 | H^0(S^2,\mathcal O(1))\cong\mathbb C^2,
000162 | \qquad
000163 | H^1(S^2,\mathcal O(1))=0.
000164 | \]
000165 | 
000166 | The resulting two-dimensional zero-mode space is exactly one complex weak doublet, so the compact
000167 | bosonic index records it as
000168 | \[
000169 | \Ind(\mathcal B_{E_2}^+)=1.
000170 | \]
000171 | 
000172 | In the color sector one has $L_3\cong\mathcal O$, so no positive determinant degree is available.
000173 | Center neutrality forbids an unpaired seam-even color contribution, hence the positive and negative
000174 | triplet modes pair with zero net index:
000175 | \[
000176 | \Ind(\mathcal B_{E_3}^+)=0.
000177 | \]
000178 | Therefore the closed branch contains exactly one seam-even Higgs doublet and no light color
000179 | triplet, and the unique light bosonic zero mode lies in the weak block.
000180 | \end{proof}