TFPT scientific source excerpt
Original repository path: documents/archive--tfpt-45--source_extracts--03_em_flavor_source/sections/003-subsection-discrete-phase-operator-and-transport-grammar.tex
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000090 | \[
000091 | P_{\mathrm{str}}:=P_{\Seam,+}P_{\mathrm{sing}}P_\Theta,
000092 | \qquad
000093 | P_{\Seam,+}:=\frac{\mathbf 1+\iC}{2},
000094 | \qquad
000095 | P_\Theta:=\frac12(\mathbf 1+Q_\Theta),
000096 | \qquad
000097 | P_{\mathrm{sing}}:=\frac13(\mathbf 1+Z_c+Z_c^2).
000098 | \]
000099 | 
000100 | \begin{definition}[Structural transport kernel]
000101 | \label{def:structural-transport-kernel}
000102 | For each fermion sector define the structural positive transport kernel by
000103 | \[
000104 | K_f^{\mathrm{str}}:=\bigl(P_{\mathrm{str}} D_f^\dagger D_f P_{\mathrm{str}}\bigr)^{-1/2},
000105 | \]
000106 | whenever the inverse exists on $\mathrm{Ran}(P_{\mathrm{str}})$. The final admissible kernel is
000107 | the restriction of this operator to the gap-stable branch selected below. The transport pole
000108 | $\delta_{\mathrm{ph}}$ is fixed by the structural stationarity of the exact determinant functional
000109 | \[
000110 | \mathcal F_{\mathrm{tr}}(\delta)
000111 | :=
000112 | \sum_{y\in\{1,1/3,2/3\}}
000113 | \Bigl[-\log\det\!\bigl(D_{y,\delta}^\dagger D_{y,\delta}+\varepsilon^2\bigr)\Bigr],
000114 | \]
000115 | with
000116 | \[
000117 | D_{y,\delta}:=y\,\mathbf 1-\delta U_6.
000118 | \]
000119 | On the closed branch one has
000120 | \begin{equation}
000121 | \label{eq:transport-bridge-offset}
000122 | \partial_\delta \mathcal F_{\mathrm{tr}}(\delta_{\mathrm{ph}})=0,
000123 | \qquad
000124 | \delta_{\mathrm{ph}}\in\left(\frac13,\frac23\right).
000125 | \end{equation}