TFPT scientific source excerpt
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000095 | \label{def:exact-seam-generating-function}
000096 | Let $\psi_{\Sigma}$ be the normalized primitive seam mode singled out by
000097 | \TFPTxref{thm:primitive-seam-generator}. Define the minimal seam transfer operator by the rank-one
000098 | projector
000099 | \[
000100 | U_{\Sigma}^{\min}:=|\psi_{\Sigma}\rangle\langle\psi_{\Sigma}|.
000101 | \]
000102 | With
000103 | \[
000104 | q(\alpha):=\deltatop e^{-2\alpha},
000105 | \qquad
000106 | B\gamma=\frac54,
000107 | \]
000108 | one has
000109 | \[
000110 | \mathcal W_{\Sigma}(\alpha)
000111 | :=
000112 | -B\gamma\,\tr_{\mathrm{adm}}\log\!\bigl(1-q(\alpha)U_{\Sigma}^{\min}\bigr)
000113 | =-\frac54\log(1-q(\alpha)).
000114 | \]
000115 | \[
000116 | \mathcal T_{\Sigma}(\alpha)=e^{\mathcal W_{\Sigma}(\alpha)}=(1-q(\alpha))^{-5/4}.
000117 | \]
000118 | \begin{equation}
000119 | \label{eq:phi-opening}
000120 | \phiseam(\alpha)=\phibase+q(\alpha)(1-q(\alpha))^{-5/4}.
000121 | \end{equation}
000122 | \end{definition}
000123 | 
000124 | \begin{proposition}[Normalized scalar closure potential]
000125 | \label{prop:log-det-variation-u1}
000126 | Let
000127 | \[
000128 | F_{U(1)}(\alpha):=\alpha^3-2\cthree^3\alpha^2-8\cthree^6 b_1\log\bigl(\phiseam(\alpha)^{-1}\bigr).
000129 | \]
000130 | There exists a unique quartic local counterterm $P_{\mathrm{loc}}$ satisfying
000131 | \[
000132 | P_{\mathrm{loc}}(0)=P_{\mathrm{loc}}'(0)=0,
000133 | \qquad
000134 | P_{\mathrm{loc}}'(\alpha)=\alpha^3-2\cthree^3\alpha^2,
000135 | \]
000136 | namely
000137 | \[
000138 | P_{\mathrm{loc}}(\alpha)=\frac{\alpha^4}{4}-\frac23\cthree^3\alpha^3.
000139 | \]
000140 | Hence the normalized scalar closure potential
000141 | \[
000142 | \Gamma_{U(1)}(\alpha)
000143 | :=
000144 | \frac{\alpha^4}{4}-\frac23\cthree^3\alpha^3
000145 | -8\cthree^6 b_1\int_0^\alpha \log\bigl(\phiseam(a)^{-1}\bigr)\,da
000146 | \]
000147 | satisfies
000148 | \[
000149 | \partial_{\alpha}\Gamma_{U(1)}(\alpha)=F_{U(1)}(\alpha).
000150 | \]