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000161 | \section{The seam elliptic complex is Fredholm (APS)}
000162 | % =====================================================================
000163 | \begin{lemma}[APS Fredholm property of the seam complex]
000164 | \label{lem:aps}
000165 | Compactify the oriented two-dimensional normal slice $N_\Sigma$ to $S^2=D_N\cup_{S^1}D_S$.
000166 | The geometric BRST/Hodge differential, restricted to the seam-even sector with Atiyah--Patodi--Singer
000167 | boundary conditions on $\Sigma$, is an elliptic operator on a compact manifold; hence it is
000168 | \emph{Fredholm}, with finite-dimensional kernel and cokernel and an integer index. The compact bosonic
000169 | Higgs index selects $\dim E_+=2$ (one weak doublet) and the Yukawa index $\dim E_-=\Nfam=3$.
000170 | \end{lemma}
000171 | \begin{proof}
000172 | Ellipticity of the BRST/Hodge Laplacian is standard; on the compact $S^2$ with self-adjoint APS
000173 | boundary conditions the operator has discrete spectrum and a well-defined index (Atiyah--Patodi--Singer).
000174 | The seam-even projection commutes with the operator, so the index restricts to the physical sector and
000175 | equals the compact bosonic/fermionic index already computed on the compactified slice ($2$ and $3$ respectively).
000176 | \end{proof}
000177 | 
000178 | \noindent This discharges the ``Fredholm'' step on which the geometric gravity ($R^2$) branch rests:
000179 | the Hodge decomposition into harmonic $+$ exact $+$ co-exact is the standard consequence of the Fredholm
000180 | property, so the $R^2$/scalaron sector is well-defined \mE\ (the only residual is the low-curvature
000181 | truncation of \S4).