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000044 | 
000045 | Let $I\mapsto\B(I)$ be an irreducible local conformal net on $S^1$ in
000046 | its vacuum representation $(\Hh_{\B},U_{\B},\Omega_{\B})$.  Let
000047 | $G=\Z_4$ act properly by vacuum-preserving internal symmetries
000048 | $\alpha_r$, meaning
000049 | \[
000050 |  \alpha_r(\B(I))=\B(I),\qquad
000051 |  \alpha_r\Ad U_{\B}(g)=\Ad U_{\B}(g)\alpha_r
000052 | \]
000053 | for every interval $I$, every conformal transformation $g$, and
000054 | $r\in\Z_4$.  Put
000055 | \[
000056 |  \A(I)=\B(I)^G.
000057 | \]
000058 | We assume that the local inclusion is irreducible and has Jones index
000059 | four:
000060 | \[
000061 |  [\B(I):\A(I)]=4,\qquad
000062 |  \A(I)'\cap\B(I)=\C\id.
000063 | \tag{1.1}\label{eq:local-inclusion}
000064 | \]
000065 | For a proper outer action of a finite group on a factor, the first
000066 | identity follows from the fixed-point theorem; it is also included
000067 | explicitly among our hypotheses so no outerness convention is hidden.
000068 | 
000069 | The net $\B$ is assumed holomorphic in the operator-algebraic sense:
000070 | it is completely rational and
000071 | \[
000072 |  \mu_{\B}=1.
000073 | \tag{1.2}\label{eq:holomorphic}
000074 | \]
000075 | The central charge $c=8$ identifies the intended endpoint as the
000076 | $(E_8)_1$ net, but is not needed for KMS uniqueness once
000077 | \eqref{eq:holomorphic} is known.
000078 | 
000079 | \subsection{The rotational KMS state}
000080 | 
000081 | Let $L_0^{\B}\geq0$ be the conformal Hamiltonian and let
000082 | $\tau_t=\Ad e^{itL_0^{\B}}$ be the rotation dynamics on the locally
000083 | normal universal algebra $C^*_{\mathrm{ln}}(\B)$.  For $\beta>0$ put
000084 | \[
000085 |  \Phi_\beta^{\B}(x)
000086 |  =\frac{\Tr_{\Hh_{\B}}(e^{-\beta L_0^{\B}}x)}
000087 |  {\Tr_{\Hh_{\B}}(e^{-\beta L_0^{\B}})}.
000088 | \tag{1.3}\label{eq:gibbs}
000089 | \]
000090 | For a completely rational conformal net the character is finite, and
000091 | the rotational KMS classification quoted below in fact recovers the
000092 | trace-class property from any extremal KMS state.  In the concrete
000093 | $(E_8)_1$ case it is also immediate from the character
000094 | $E_4/\eta^8$.
000095 | 
000096 | \begin{definition}[Geometric rotational state used here]
000097 | \label{def:geometric}
000098 | The state on the subnet is
000099 | \[
000100 |  \omega_\beta:=\Phi_\beta^{\B}|_{C^*_{\mathrm{ln}}(\A)}.
000101 | \tag{1.4}\label{eq:omega-def}
000102 | \]
000103 | At the canonical angular normalization we set
000104 | $\beta=\beta_{\mathrm{angle}}=2\pi$.
000105 | \end{definition}
000106 | 
000107 | \begin{remark}[Ground state versus rotational KMS state]
000108 | The vector functional
000109 | $\langle\Omega_{\B},\,\cdot\,\Omega_{\B}\rangle$ is a ground state for
000110 | rotations.  It is not the finite-temperature state
000111 | \eqref{eq:gibbs}.  The geometric KMS state on the real line for
000112 | translations/dilations and the rotational Gibbs state on the circle
000113 | are likewise different constructions.  The theorem below concerns
000114 | the latter, because its unconditional classification by
000115 | superselection sectors is available in exactly this setting.
000116 | \end{remark}
000117 | 
000118 | \section{Classical theorems used, stated in full}