Möbius and conformal nets on the circle

Set $S^1=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ with counterclockwise orientation. Define $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})=\mathrm{S}\mathrm{L}(2,\mathbb{R})\mathrm{/}\{\pm I\}$. With $C(t)=(t-i)\mathrm{/}(t+i)$ for $t\in \mathbb{R}$ and $C(\mathrm{\infty })=1$, its action on $S^1$ is $g\cdot C(t)=C((at+b)\mathrm{/}(ct+d))$ when $g$ is represented by $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$; the fractional-linear formula is interpreted on $\mathbb{R}\cup \{\mathrm{\infty }\}$ and is independent of the representative. Write $r_{\alpha }(z)=e^{i\alpha }z$ for rotations. The group ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{+}(S^1)$ consists of smooth bijections $S^1\to S^1$ with smooth inverses that preserve the orientation, with composition as group operation; use its $C^{\mathrm{\infty }}$ topology, where convergence means uniform convergence of each derivative in local smooth charts. For a Hilbert space $\mathcal{H}$, let $\mathrm{U}(\mathcal{H})$ be its group of surjective linear isometries and $\mathrm{P}\mathrm{U}(\mathcal{H})=\mathrm{U}(\mathcal{H})\mathrm{/}\mathbb{T}$, where $\mathbb{T}=\{\lambda \in \mathbb{C}:\mathrm{\mid }\lambda \mathrm{\mid }=1\}$ acts by scalar multiples; give $\mathrm{P}\mathrm{U}(\mathcal{H})$ the quotient of the strong operator topology.

Let $\mathcal{I}$ be the set of nonempty connected open subsets $I\subset S^1$ with $\bar{I}\mathrm{\ne }{S}^1$ (the interval convention used here), and let $I^{\mathrm{\prime }}$ be the interior of $S^1\setminus \bar{I}$. A Möbius-covariant net on a complex Hilbert space $\mathcal{H}$ assigns a concrete von Neumann algebra $\mathcal{A}(I)\subseteq \mathcal{B}(\mathcal{H})$ to each $I\in \mathcal{I}$. Here a von Neumann algebra is a unital, self-adjoint operator algebra closed in the weak operator topology. For $\mathcal{S}\subseteq \mathcal{B}(\mathcal{H})$, define ${\mathcal{S}}^{\mathrm{\prime }}=\{T\in \mathcal{B}(\mathcal{H}):TS=ST\text{ for every }S\in \mathcal{S}\}$ and ${\mathcal{S}}^{\mathrm{\prime }\mathrm{\prime }}=({\mathcal{S}}^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$. The cited double-commutant theorem gives $\mathcal{M}={\mathcal{M}}^{\mathrm{\prime }\mathrm{\prime }}$ for every such von Neumann algebra $\mathcal{M}$. Here $\mathcal{B}(\mathcal{H})$ is the bounded-operator algebra. These use a complete Hilbert norm, weak operator closure, and the concrete commutant convention.

The assignment satisfies isotony, $\mathcal{A}(I_1)\subseteq \mathcal{A}(I_2)$ whenever $I_1\subseteq I_2$, and locality, $[\mathcal{A}(I_1),\mathcal{A}(I_2)]=0$ whenever $I_1\cap I_2=\mathrm{\varnothing }$. It has a strongly continuous unitary representation $U$ of $\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})$, the Möbius group acting on $S^1$. Positive energy and a specified vacuum vector $\mathrm{\Omega }$, as defined below, are required parts of the net data. Here a unitary representation is a group homomorphism into the surjective linear isometries of $\mathcal{H}$, and strong continuity means $\mathrm{\parallel }U(g)\xi -U(g_0)\xi \mathrm{\parallel }\to 0$ as $g\to g_0$ for each $\xi \in \mathcal{H}$. Its smooth-vector domain is ${\mathcal{H}}_U^{\mathrm{\infty }}=\{\xi \in \mathcal{H}:g\mapsto U(g)\xi \text{ is a norm-}{C}^{\mathrm{\infty }}\text{ map on }\mathrm{P}\mathrm{S}\mathrm{L}(2,\mathbb{R})\}\mathrm{.}$ For a real Lie algebra element $X\in {\mathfrak{p}\mathfrak{s}\mathfrak{l}}_2(\mathbb{R})$, the infinitesimal generator on this domain is $dU(X)\xi ={\frac{d}{dt}\mid }_{t=0}U(\exp (tX))\xi$; extend $dU$ complex-linearly to ${\mathfrak{p}\mathfrak{s}\mathfrak{l}}_2(\mathbb{C})$. The operators $L_{-1},L_0,L_1$ below are the complexified infinitesimal generators in the Möbius basis, with their normalization fixed by the displayed brackets, adjoint relations, and rotation convention. In this convention, $U(g)\mathcal{A}(I)U(g{)}^{\ast }=\mathcal{A}(gI)\mathrm{.}$ On ${\mathcal{H}}_U^{\mathrm{\infty }}$, the generators satisfy $[L_m,L_n]=(m-n)L_{m+n}$ for $m,n\in \{-1,0,1\}$. Their adjoint convention on this domain is the form identity $⟨L_{-1}\xi ,\eta ⟩=⟨\xi ,L_1\eta ⟩$ and $⟨L_0\xi ,\eta ⟩=⟨\xi ,L_0\eta ⟩$ for all $\xi ,\eta \in {\mathcal{H}}_U^{\mathrm{\infty }}$. The self-adjoint generator of the rotation subgroup is normalized by $U(r_{\alpha })=e^{i\alpha L_0}$, and positive energy means its spectrum is contained in $[0,\mathrm{\infty })$. Since $r_{2\pi }$ is the identity, $e^{2\pi i{L}_0}=I$; the unbounded Borel functional calculus then gives $E_{L_0}(\mathbb{R}\setminus \mathbb{Z})=0$. Positivity restricts this support to ${\mathbb{Z}}_{\ge 0}$. Define ${\mathcal{H}}_{\mathrm{f}\mathrm{i}\mathrm{n}}={\mathrm{span}}_{\mathrm{a}\mathrm{l}\mathrm{g}}\{\ker (L_0-nI):n\in {\mathbb{Z}}_{\ge 0}\}$. The increasing projections $E_{L_0}([0,N])={\sum }_{n=0}^N{E}_{L_0}(\{n\})$ converge strongly to $I$, so ${\mathcal{H}}_{\mathrm{f}\mathrm{i}\mathrm{n}}$ is dense in $\mathcal{H}$. A vacuum vector is a unit vector $\mathrm{\Omega }\in \mathcal{H}$ with $U(g)\mathrm{\Omega }=\mathrm{\Omega }$ for all $g$ and with $\mathrm{\Omega }$ cyclic for the global algebra ${\bigvee }_{I\in \mathcal{I}}\mathcal{A}(I)$; cyclic means that the span of $\{A\mathrm{\Omega }:A\in {\bigvee }_I\mathcal{A}(I)\}$ is dense. The symbol $\bigvee$ denotes the smallest von Neumann algebra containing the listed local algebras.

A conformal net additionally has a continuous group homomorphism ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{+}(S^1)\to \mathrm{P}\mathrm{U}(\mathcal{H})$, called a strongly continuous projective unitary representation, extending the Möbius representation and satisfying the same interval covariance. Equivalently, unitary lifts represent multiplication up to scalar phases, or define a genuine representation of a specified central extension. It also satisfies local diffeomorphism implementation: if $\gamma \in {\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{+}(S^1)$ equals the identity on an interval $I$, then $U(\gamma )AU(\gamma {)}^{\ast }=A$ for every $A\in \mathcal{A}(I)$; this is independent of the lift's phase. Equivalently, a diffeomorphism supported in $I^{\mathrm{\prime }}$ acts trivially on $\mathcal{A}(I)$. No Haag-duality or geometric modular-action property is assumed by this axiom. A net is irreducible when ${\bigvee }_{I\in \mathcal{I}}\mathcal{A}(I)=\mathcal{B}(\mathcal{H})$. Irreducibility is a separate condition, not part of the two covariance definitions.

The selected concrete von Neumann-algebra supplier assumes full Axiom of Choice for its bounded-adjoint interface; this item carries that exact assumption through the selected concrete von Neumann-algebra interface. No claim is made that every formulation of a conformal net uses choice.

Hilbert space

Von Neumann algebras and commutants

The double commutant theorem for concrete von Neumann algebras

Strong and weak operator topologies

The Axiom of Choice

Unbounded Borel functional calculus: domains, products, spectral mapping