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000003 | \begin{theorem}[Minimal determinant classes from the primitive charge generator]
000004 | \label{thm:determinant-class-higgs-selection}
000005 | \label{thm:determinant-clutching}
000006 | Let
000007 | \[
000008 | L_2:=\det E_2,
000009 | \qquad
000010 | L_3:=\det E_3
000011 | \]
000012 | over the compactified oriented normal sphere $S^2=D_N\cup_{S^1}D_S$. Then the primitive charge
000013 | generator
000014 | \[
000015 | X=-2P_3+3P_2
000016 | \]
000017 | together with center neutrality and seam-evenness forces the clutching map
000018 | \[
000019 | g_2(\varphi)=e^{i\varphi}
000020 | \]
000021 | for $L_2$, while center neutrality forces
000022 | \[
000023 | g_3(\varphi)=1
000024 | \]
000025 | for $L_3$. Therefore
000026 | \[
000027 | c_1(L_2)=\deg g_2=1,
000028 | \qquad
000029 | c_1(L_3)=\deg g_3=0.
000030 | \]
000031 | This pair is the unique minimal nonnegative admissible determinant class.
000032 | \end{theorem}
000033 | 
000034 | \begin{proof}
000035 | Complex line bundles on
000036 | \[
000037 | S^2=D_N\cup_{S^1}D_S
000038 | \]
000039 | are classified by their clutching maps along the equator:
000040 | \[
000041 | \pi_1(U(1))\cong \mathbb Z.
000042 | \]
000043 | Hence the first Chern class is exactly the winding degree of the transition function.
000044 | 
000045 | In the weak rank-two sector the primitive charge generator
000046 | \[
000047 | X=-2P_3+3P_2
000048 | \]
000049 | assigns the unique positive determinant winding to the seam-even weak block. The spin lift on the
000050 | oriented normal sphere acts there by the half-angle representation, so on the determinant line the
000051 | transition factor is
000052 | \[
000053 | g_2(\varphi)=e^{i\varphi},
000054 | \]
000055 | hence
000056 | \[
000057 | \deg g_2=1,
000058 | \qquad
000059 | c_1(L_2)=1.
000060 | \]
000061 | 
000062 | In the color sector admissibility imposes center neutrality. Therefore the determinant transition
000063 | function of the color block is null-homotopic and may be chosen constant:
000064 | \[
000065 | g_3(\varphi)=1.
000066 | \]
000067 | Consequently
000068 | \[
000069 | \deg g_3=0,
000070 | \qquad
000071 | c_1(L_3)=0.
000072 | \]
000073 | 
000074 | The full primitive seam twist is therefore exhausted in the weak determinant class. Any nonnegative
000075 | pair strictly smaller than $(1,0)$ would force vanishing weak determinant winding and hence remove
000076 | the seam-even weak line required by the primitive charge generator. Any larger nonnegative pair
000077 | would insert extra clutching not carried by the primitive generator and therefore violate
000078 | minimality. Thus the unique minimal nonnegative admissible determinant class is
000079 | \[
000080 | (c_1(L_2),c_1(L_3))=(1,0).
000081 | \]
000082 | \end{proof}
000083 | 
000084 | \begin{remark}[Energy interpretation on the retained branch]
000085 | The older asymptotic bosonic functional
000086 | \[
000087 | \mathcal E_{\mathrm{asym}}(\Phi_2,\Phi_3)
000088 | :=
000089 | \tau_2\bigl|\mathrm{wind}(\det\Phi_2)\bigr|
000090 | +\tau_3\bigl|\mathrm{wind}(\det\Phi_3)\bigr|
000091 | +m_3^2\dim\ker\mathcal B_{E_3}^+,
000092 | \]
000093 | with $\tau_3>\tau_2>0$, remains a useful physical intuition for why the retained branch is energetically preferred. In the present rewrite, however, the structural weight is carried by the determinant-class theorem above rather than by the cost functional itself.
000094 | \end{remark}
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.