TFPT scientific source excerpt
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000334 | \begin{definition}[Projected modular free energy and stratified master functional]
000335 | \label{def:projected-modular-free-energy}
000336 | Fix the unique joint discrete datum $d^\star_{\mathrm{disc}}$ of
000337 | \Cref{thm:joint-discrete-admissible-sector}, and let $\Padm$ be the derived full selector of
000338 | \Cref{cor:derived-post-carrier-kernel}. For
000339 | \[
000340 | \xi=(\alpha,\chigeo,\delta_{\mathrm{ph}},\rho_{\mathrm{vac}})
000341 | \]
000342 | on the retained admissible branch, define
000343 | \[
000344 | \begin{aligned}
000345 | \mathcal M_{d^\star_{\mathrm{disc}}}(\xi)
000346 | :={}&-\log Z^{\Padm}_{\mathrm{rel}}[\xi]
000347 | +\Re\log\det_{\zeta,\Padm}(D_\xi^2+\mu^2)\\
000348 | &+\frac12\eta_{\Sigma,\Padm}(D_\xi)
000349 | -\log\det(1-U_{6,\xi})
000350 | +\Gamma_{\mathrm{vac}}(\rho_\xi),
000351 | \end{aligned}
000352 | \]
000353 | where every term is evaluated on the retained admissible sector $\operatorname{Ran}(\Padm)$.
000354 | For a general admissible spectral-bordism state
000355 | \[
000356 | \mathfrak s=(\mathcal B,\xi),
000357 | \qquad
000358 | \mathcal B\in\mathbf{SBord}^{\mathrm{adm}},
000359 | \]
000360 | define the lexicographic barrier
000361 | \[
000362 | \mathbb B_{\mathrm{lex}}(\mathfrak s)
000363 | :=
000364 | \begin{cases}
000365 | 0,& \mathcal B\cong \mathcal B_{\min},\\
000366 | +\infty,& \text{otherwise},
000367 | \end{cases}
000368 | \]
000369 | and the stratified master functional
000370 | \[
000371 | \mathbb M(\mathfrak s)
000372 | :=
000373 | \mathbb B_{\mathrm{lex}}(\mathfrak s)+\mathcal M_{d^\star_{\mathrm{disc}}}(\xi).
000374 | \]
000375 | Thus the continuous retained-branch functional $\mathcal M_{d^\star_{\mathrm{disc}}}$ is the
000376 | restriction of one stratified functional to the unique minimal admissible stratum.
000377 | \end{definition}