TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--02_carrier_source/sections/006-subsection-family-monodromy-and-occupancy.tex
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000353 | 
000354 | \begin{definition}[Local spectral scale and canonical relative spectral residues]
000355 | \label{def:local-spectral-scale}
000356 | \label{def:canonical-residues}
000357 | Write the geometric response as a local spectral scale
000358 | \[
000359 | \chigeo=\Lambda e^\sigma,
000360 | \qquad
000361 | D_{\sigma,\mathrm{geo}}:=e^{-\sigma/2}D_{\mathrm{geo}}e^{-\sigma/2}.
000362 | \]
000363 | \[
000364 | \mathcal Z_{\mathrm{rel}}(s;\sigma)
000365 | :=
000366 | \operatorname{Tr}_{\mathrm{rel}}\!\left(
000367 | |D_{\sigma,\mathrm{geo}}+\mathcal B_{\mathrm{rel}}|^{-2s}
000368 | -|D_{\mathrm{ref}}|^{-2s}
000369 | \right)
000370 | +\frac{i\pi}{2}\Delta\eta_\Sigma.
000371 | \]
000372 | The canonical Einstein residue and order-zero residue are
000373 | \[
000374 | \mathcal E_D
000375 | :=
000376 | \operatorname*{Res}_{s=1}\mathcal Z_{\mathrm{rel}}(s;\sigma),
000377 | \qquad
000378 | \Gamma_D^{(4)}
000379 | :=
000380 | \operatorname{FP}_{s=0}\mathcal Z_{\mathrm{rel}}(s;\sigma).
000381 | \]
000382 | The gravitational branch is then read from the profile-free residue pair rather than from a freely
000383 | chosen spectral profile.
000384 | \end{definition}
000385 | 
000386 | \begin{corollary}[Constant-scale reduction of the geometric branch]