TFPT scientific source excerpt
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000229 | \begin{definition}[Projected modular free energy and stratified master functional]
000230 | \label{def:projected-modular-free-energy}
000231 | Fix the unique joint discrete datum $d^\star_{\mathrm{disc}}$ of
000232 | \Cref{thm:joint-discrete-admissible-sector}, and let $\Padm$ be the derived full selector of
000233 | \Cref{cor:derived-post-carrier-kernel}. For
000234 | \[
000235 | \xi=(\alpha,\chigeo,\delta_{\mathrm{ph}},\rho_{\mathrm{vac}})
000236 | \]
000237 | on the retained admissible branch, define
000238 | \[
000239 | \begin{aligned}
000240 | \mathcal M_{d^\star_{\mathrm{disc}}}(\xi)
000241 | :={}&-\log Z^{\Padm}_{\mathrm{rel}}[\xi]
000242 | +\Re\log\det_{\zeta,\Padm}(D_\xi^2+\mu^2)\\
000243 | &+\frac12\eta_{\Sigma,\Padm}(D_\xi)
000244 | -\log\det(1-U_{6,\xi})
000245 | +\Gamma_{\mathrm{vac}}(\rho_\xi),
000246 | \end{aligned}
000247 | \]
000248 | where every term is evaluated on the retained admissible sector $\operatorname{Ran}(\Padm)$.
000249 | For a general admissible spectral-bordism state
000250 | \[
000251 | \mathfrak s=(\mathcal B,\xi),
000252 | \qquad
000253 | \mathcal B\in\mathbf{SBord}^{\mathrm{adm}},
000254 | \]
000255 | define the lexicographic barrier
000256 | \[
000257 | \mathbb B_{\mathrm{lex}}(\mathfrak s)
000258 | :=
000259 | \begin{cases}
000260 | 0,& \mathcal B\cong \mathcal B_{\min},\\
000261 | +\infty,& \text{otherwise},
000262 | \end{cases}
000263 | \]
000264 | and the stratified master functional
000265 | \[
000266 | \mathbb M(\mathfrak s)
000267 | :=
000268 | \mathbb B_{\mathrm{lex}}(\mathfrak s)+\mathcal M_{d^\star_{\mathrm{disc}}}(\xi).
000269 | \]
000270 | Thus the continuous retained-branch functional $\mathcal M_{d^\star_{\mathrm{disc}}}$ is the
000271 | restriction of one stratified functional to the unique minimal admissible stratum.
000272 | \end{definition}