Strongly local unitary VOA data

For this page, a VOA means a conformal vertex algebra in the CFT-type convention fixed by the library: it has the state–field map, vacuum, translation, conformal vector, Virasoro action, nonnegative integer grading, finite-dimensional weight spaces, and $V_0=\mathbb{C}\mathbf{1}$ stated there. It is simple when its only ideals are $0$ and $V$. The PCT map $\theta$ is an antilinear bijection preserving the vacuum, translation, conformal vector, and state-field products, and satisfies ${\theta }^2=1$. The energy operator $L_0$ acts by $d$ on each weight space $V_d$. In the state-field series $Y(a,z)={\sum }_{n\in \mathbb{Z}}{a}_{(n)}{z}^{-n-1}$, $a_{(n)}$ is the $n$th mode from the fixed state-field convention.

A normalized invariant unitary structure consists of a positive-definite Hermitian form and an antilinear involutive VOA automorphism $\theta$ (the PCT involution) such that $\mathrm{\parallel }\mathbf{1}\mathrm{\parallel }=1$ and the induced invariant bilinear form satisfies, for all $a,b,c\in V$,

$$(Y(a,z)b,c{)}_{\mathrm{b}\mathrm{i}\mathrm{l}}=(b,Y(e^{z{L}_1}(-z^{-2}{)}^{L_0}a,z^{-1})c{)}_{\mathrm{b}\mathrm{i}\mathrm{l}}\mathrm{.}$$

The library inner product convention is linear in its first argument. To match the source's linear-in-the-second convention, write $(u\mathrm{\mid }v{)}_{\mathrm{s}\mathrm{r}\mathrm{c}}=\overline{⟨u,v⟩}$ and $(a,b{)}_{\mathrm{b}\mathrm{i}\mathrm{l}}=(\theta a\mathrm{\mid }b{)}_{\mathrm{s}\mathrm{r}\mathrm{c}}$; this makes the displayed form bilinear and preserves positivity of $(u\mathrm{\mid }u{)}_{\mathrm{s}\mathrm{r}\mathrm{c}}=⟨u,u⟩$. The PCT operator also fixes the conformal vector.

For homogeneous $a\in V_d$, polynomial energy bounds mean that there are constants $M_a>0$ and integers $s_a,k_a\ge 0$ such that

$$\mathrm{\parallel }{a}_{(n)}b\mathrm{\parallel }\le M_a(1+\mathrm{\mid }n\mathrm{\mid }{)}^{s_a}\mathrm{\parallel }(1+L_0{)}^{k_a}b\mathrm{\parallel }(n\in \mathbb{Z},b\in V)\mathrm{.}$$

For homogeneous $a\in V_d$, use the energy modes $a_n=a_{(n+d-1)}$, so $[L_0,a_n]=-n{a}_n$. For $f\in C^{\mathrm{\infty }}(S^1)$ define ${\widehat{f}}_n={\int }_{-\pi }^{\pi }f(e^{i\vartheta })e^{-in\vartheta }d\vartheta \mathrm{/}(2\pi )$ and on the algebraic finite-energy domain $V$ set $Y_0(a,f)b={\sum }_{n\in \mathbb{Z}}{\widehat{f}}_n{a}_{n}b$. The Fourier coefficients decrease faster than every power; shifting the index in the energy bound preserves a polynomial bound, so this series converges in norm for every $b\in V$. Extend linearly to inhomogeneous states. The Hilbert space $H_V$ is the completion of $V$ in its positive Hermitian norm.

The invariant form gives a densely defined adjoint on $V$, so $Y_0(a,f)$ is closable. Write $Y(a,f)$ for its closure. For a closed densely defined operator $T$, its polar decomposition is $T=V_T\mathrm{\mid }T\mathrm{\mid }$, where $V_T$ is the partial isometry and $\mathrm{\mid }T\mathrm{\mid }=(T^{\ast }T{)}^{1\mathrm{/}2}$ is positive self-adjoint with $D(\mathrm{\mid }T\mathrm{\mid })=D(T)$. Let $E_{\mathrm{\mid }T\mathrm{\mid }}(B)$ denote the spectral projection of $\mathrm{\mid }T\mathrm{\mid }$ for a Borel set $B\subseteq [0,\mathrm{\infty })$. For each interval $I$, define ${\mathcal{A}}_V(I)=W^{\ast }({\mathcal{G}}_I)$, where ${\mathcal{G}}_I$ consists of the $V_T$ and all $E_{\mathrm{\mid }T\mathrm{\mid }}(B)$ for $T=Y(a,f)$, $a\in V$, $f\in C_c^{\mathrm{\infty }}(I)$, and Borel $B\subseteq [0,\mathrm{\infty })$. Here $W^{\ast }$ is the weak-operator closure of the unital $\ast$-algebra generated by bounded operators, as in the concrete von Neumann-algebra definition. A closed densely defined operator $T$ is affiliated with a von Neumann algebra $M$ if every unitary $u\in M^{\mathrm{\prime }}$ maps $D(T)$ onto itself and satisfies $uT\xi =Tu\xi$ on that domain. Every $T=Y(a,f)$ is affiliated with ${\mathcal{A}}_V(I)$: unitaries in ${\mathcal{A}}_V(I{)}^{\mathrm{\prime }}$ commute with $V_T$ and the spectral projections of $\mathrm{\mid }T\mathrm{\mid }$, hence preserve $D(\mathrm{\mid }T\mathrm{\mid })$ and commute with $\mathrm{\mid }T\mathrm{\mid }$ by the spectral calculus, and then preserve $D(T)$ and commute with $T=V_T\mathrm{\mid }T\mathrm{\mid }$. This uses closed affiliated fields and does not assume essential self-adjointness from polynomial energy bounds alone.

The grades $V_d$ are mutually orthogonal by invariance of the form. On $H_V$, use the self-adjoint extension of $L_0$ given by $L_0\xi ={\sum }_{d\ge 0}d{\xi }_d$, with domain $\{\xi ={\sum }_{d\ge 0}{\xi }_d:{\xi }_d\in V_d,{\sum }_{d\ge 0}{d}^2\mathrm{\parallel }{\xi }_d{\mathrm{\parallel }}^2<\mathrm{\infty }\}$. Its smooth-energy domain is $H_V^{\mathrm{\infty }}={\bigcap }_{r\in {\mathbb{Z}}_{\ge 0}}D((1+L_0{)}^r)$. This is a common core for all $Y(a,f)$: a core for a closed operator $T$ means a subspace $D\subseteq D(T)$ whose restriction $T{\mathrm{\mid }}_D$ has closure $T$. Indeed, the energy bound and $[L_0,a_n]=-n{a}_n$ give, for homogeneous $a$, every integer $r\ge 0$ and $b\in V$, $\mathrm{\parallel }(1+L_0{)}^r{Y}_0(a,f)b\mathrm{\parallel }\le C_{a,r}({\sum }_{n\in \mathbb{Z}}\mathrm{\mid }{\widehat{f}}_n\mathrm{\mid }(1+\mathrm{\mid }n\mathrm{\mid }{)}^{s_a+r})\mathrm{\parallel }(1+L_0{)}^{k_a+r}b\mathrm{\parallel }\mathrm{.}$ The sum is finite. Finite-energy truncations converge in every indicated weighted norm, so the closed fields extend this action to $H_V^{\mathrm{\infty }}$, preserve it, and have it as a core since it contains their defining core $V$.

A homogeneous vector $a\in V_d$ is quasi-primary when $L_{1}a=0$. For such $a$, invariance gives the mode adjoint $⟨a_{n}b,c⟩=⟨b,(-1{)}^d(\theta a{)}_{-n}c⟩$ for $b,c\in V$. Summing against the Fourier coefficients and taking graph closures yields $Y(a,f{)}^{\ast }\supseteq (-1{)}^{d}Y(\theta a,\bar{f})$, where $\bar{f}(z)=\overline{f(z)}$. In particular both sides are defined and agree on $H_V^{\mathrm{\infty }}$; the inclusion also covers the entire domain of the closed operator on its right. Thus Hermitian combinations satisfy $a=(-1{)}^d\theta a$; theta-fixed odd-weight states alone are not asserted to give symmetric real smearings. For descendants the complete adjoint is determined by the invariant-form identity above. The algebra definition includes all closed fields, so no restricted Hermitian generating set is silently substituted.

The strong-locality condition is the operator inclusion ${\mathcal{A}}_V(I)\subseteq {\mathcal{A}}_V(I^{\mathrm{\prime }}{)}^{\mathrm{\prime }}$ for every $I\in \mathcal{I}$. The inclusion is not replaced by equality, and this definition claims no theorem that every unitary VOA is strongly local. A strongly local unitary VOA datum is a simple VOA with the normalized invariant unitary structure, polynomial energy bounds, and this local-algebra inclusion.

Möbius and conformal nets on the circle

Conformal vertex algebra of CFT type

Real and complex inner-product spaces and their induced length

Hilbert space

Von Neumann algebras and commutants

Strong and weak operator topologies

The Axiom of Choice

Polar decomposition of a closed densely defined operator

Unbounded Borel functional calculus: domains, products, spectral mapping