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000068 | \section{The holomorphic KMS extension theorem}
000069 | 
000070 | \begin{theorem}[Holomorphic rotational KMS extension]
000071 | \label{thm:main}
000072 | Let $\A\subset\B$ satisfy the hypotheses of Section~1: for every
000073 | interval $I$, $\A(I)\subset\B(I)$ is an irreducible index-four
000074 | inclusion, $\A=\B^{\Z_4}$ for a proper internal gauge action
000075 | commuting with rotations, and $\B$ is a holomorphic conformal net
000076 | with $\mu_{\B}=1$ and $c=8$.  Let $\beta>0$ and let
000077 | $\omega_\beta$ be the restriction \eqref{eq:omega-def} of the
000078 | vacuum-sector rotational Gibbs state.  Then:
000079 | \begin{enumerate}[label=\textup{(\alph*)}]
000080 | \item $\omega_\beta$ has a rotational $\beta$-KMS extension to
000081 |   $\B$.
000082 | \item The maps
000083 |   \[
000084 |     E_I=\frac14\sum_{r=0}^{3}\alpha_r:
000085 |     \B(I)\to\A(I)
000086 |   \]
000087 |   are the unique normal faithful conditional expectations.  They
000088 |   form a net expectation $E$, and
000089 |   \[
000090 |     \widetilde\omega_\beta
000091 |     =\omega_\beta\circ E
000092 |     =\Phi_\beta^{\B}.
000093 |   \tag{4.1}\label{eq:main-extension}
000094 |   \]
000095 |   This is the unique $\Z_4$-invariant state extension of
000096 |   $\omega_\beta$, hence in particular the unique
000097 |   $\Z_4$-invariant rotational KMS extension.
000098 | \item Every rotational $\beta$-KMS extension of $\omega_\beta$ is
000099 |   $\Z_4$-invariant.  Consequently
000100 |   $\widetilde\omega_\beta$ is the unique rotational KMS extension
000101 |   without an invariance qualifier.
000102 | \end{enumerate}
000103 | \end{theorem}
000104 | 
000105 | \begin{proof}