TFPT scientific source excerpt
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000013 | \begin{proposition}[Carrier susceptibility exponent]
000014 | \label{prop:carrier-susceptibility-exponent}
000015 | On the minimal carrier branch,
000016 | \[
000017 | B\gamma=\frac32\cdot\frac56=\frac54.
000018 | \]
000019 | \end{proposition}
000020 | \begin{definition}[Minimal primitive seam transfer]
000021 | \label{def:exact-seam-generating-function}
000022 | Let $\psi_{\Sigma}$ be the normalized primitive seam mode singled out by
000023 | \Cref{thm:primitive-seam-generator}. Define the minimal seam transfer operator by the rank-one
000024 | projector
000025 | \[
000026 | U_{\Sigma}^{\min}:=|\psi_{\Sigma}\rangle\langle\psi_{\Sigma}|.
000027 | \]
000028 | With
000029 | \[
000030 | q(\alpha):=\deltatop e^{-2\alpha},
000031 | \qquad
000032 | B\gamma=\frac54,
000033 | \]
000034 | one has
000035 | \[
000036 | \mathcal W_{\Sigma}(\alpha)
000037 | :=
000038 | -B\gamma\,\tr_{\mathrm{adm}}\log\!\bigl(1-q(\alpha)U_{\Sigma}^{\min}\bigr)
000039 | =-\frac54\log(1-q(\alpha)).
000040 | \]
000041 | \[
000042 | \mathcal T_{\Sigma}(\alpha)=e^{\mathcal W_{\Sigma}(\alpha)}=(1-q(\alpha))^{-5/4}.
000043 | \]
000044 | \begin{equation}
000045 | \label{eq:phi-opening}
000046 | \phiseam(\alpha)=\phibase+q(\alpha)(1-q(\alpha))^{-5/4}.
000047 | \end{equation}
000048 | \end{definition}
000049 | 
000050 | \begin{proposition}[Normalized scalar closure potential]
000051 | \label{prop:log-det-variation-u1}
000052 | Let
000053 | \[
000054 | F_{U(1)}(\alpha):=\alpha^3-2\cthree^3\alpha^2-8\cthree^6 b_1\log\bigl(\phiseam(\alpha)^{-1}\bigr).
000055 | \]
000056 | There exists a unique quartic local counterterm $P_{\mathrm{loc}}$ satisfying
000057 | \[
000058 | P_{\mathrm{loc}}(0)=P_{\mathrm{loc}}'(0)=0,
000059 | \qquad
000060 | P_{\mathrm{loc}}'(\alpha)=\alpha^3-2\cthree^3\alpha^2,
000061 | \]
000062 | namely
000063 | \[
000064 | P_{\mathrm{loc}}(\alpha)=\frac{\alpha^4}{4}-\frac23\cthree^3\alpha^3.
000065 | \]
000066 | Hence the normalized scalar closure potential
000067 | \[
000068 | \Gamma_{U(1)}(\alpha)
000069 | :=
000070 | \frac{\alpha^4}{4}-\frac23\cthree^3\alpha^3
000071 | -8\cthree^6 b_1\int_0^\alpha \log\bigl(\phiseam(a)^{-1}\bigr)\,da
000072 | \]
000073 | satisfies
000074 | \[
000075 | \partial_{\alpha}\Gamma_{U(1)}(\alpha)=F_{U(1)}(\alpha).
000076 | \]
000077 | \end{proposition}