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000039 | \subsection{Canonical spectral Einstein functional and geometric Hodge closure}
000040 | 
000041 | The local target action is written as the low-curvature expansion of a canonical spectral
000042 | Einstein functional:
000043 | \begin{equation}
000044 | \label{eq:chi-effective-action}
000045 | \begin{aligned}
000046 | \Gammaeff
000047 | =
000048 | \int d^4x\sqrt{-g}\,
000049 | \Big[
000050 | c_R\chigeo^2R
000051 | +Z_\chi(\partial\chigeo)^2
000052 | -\mu_\Phi^2(\chigeo)\,\Phi^\dagger\Phi
000053 | +\lambda_\Phi(\Phi^\dagger\Phi)^2
000054 | +\frac{1}{6M^2}R^2
000055 | +\cdots
000056 | \Big].
000057 | \end{aligned}
000058 | \end{equation}
000059 | The relative bosonic endomorphism used below is
000060 | \begin{equation}
000061 | \label{eq:relative-bosonic-endomorphism}