Unitary VOAs, Energy Bounds, and Strong Locality

A unitary VOA becomes an operator-algebraic chiral theory only after its formal fields act on a Hilbert completion with controlled domains. A positive invariant inner product supplies adjoints; polynomial energy bounds make smooth smearing convergent and closable; strong locality then requires the von Neumann algebras generated in disjoint intervals to commute. Formal locality on the algebraic state space is necessary but does not prove this last operator statement.

Required background. Vertex Operator Algebras: Axioms, Grading, and Locality supplies the formal fields. Self-Adjointness, Extensions, and Unitary Evolution supplies closure and strong-commutativity language.

Helpful background. Banach and Hilbert Spaces, Completion, and Riesz Representation reviews the Hilbert completion used here.

Unitary structure and polynomial energy control

Let $V$ carry a positive-definite Hermitian form $(\cdot\mid\cdot)$ with $(\mathbf1\mid\mathbf1)=1$. Invariance is most compactly expressed using an antilinear involutive VOA automorphism $\theta$, the PCT operator. For homogeneous $v$, the adjoint field is obtained from the coordinate inversion $z\mapsto z^{-1}$ together with $e^{zL_1}(-z^{-2})^{L_0}\theta v$. In particular,

$$L_n^*=L_{-n},$$

and for a quasi-primary $\theta$-real state $v$ of integral weight $d$, its modes obey $v_n^*=(-1)^d v_{-n}$ in the circle convention. Carpi, Kawahigashi, Longo, and Weiner give the invariant-inner-product definition and its PCT characterization in Carpi et al. 2018, §5.1–5.2, PDF pp. 31–39.

Complete $V$ to a Hilbert space $\mathcal H$. The finite-energy vectors remain the algebraic sum of $L_0$ eigenspaces. A homogeneous state $v$ satisfies a polynomial energy bound if constants $M>0$ and integers $s,k\geq0$ exist such that

$$\lVert v_n\psi\rVert \leq M(1+|n|)^s\lVert(1+L_0)^k\psi\rVert$$

for every mode $n$ and finite-energy vector $\psi$. If a homogeneous generating set satisfies such estimates, the whole VOA does: derivatives multiply a mode by a polynomial in $n$, nonnegative products are finite sums, and normal-ordered products are controlled by the $L_0$ grading. This is Carpi et al. 2018, Proposition 6.1, PDF pp. 46–48.

For $f\in C^\infty(S^1)$ with Fourier coefficients $\widehat f_n$, define on finite-energy vectors

$$Y_0(v,f)\psi=\sum_{n\in\mathbb Z}\widehat f_n v_n\psi.$$

Rapid Fourier decay and the mode bound give

$$\lVert Y_0(v,f)\psi\rVert \leq M\!\left(\sum_n(1+|n|)^s|\widehat f_n|\right) \lVert(1+L_0)^k\psi\rVert.$$

The adjoint relation makes $Y_0(v,f)$ closable; write $Y(v,f)$ for its closure. The smooth domain $\mathcal H^\infty=\bigcap_{r\geq0}\operatorname{Dom}(1+L_0)^r$ is a common invariant core Carpi et al. 2018, equations (105)–(113), PDF pp. 48–50.

Strong locality

For an interval $I\subset S^1$, let

$$\mathcal A_V(I)=W^*\{Y(v,f):v\in V,\ \operatorname{supp}f\subset I\},$$

where $W^*$ denotes the von Neumann algebra generated by the spectral data of the affiliated closed operators. The unitary VOA is strongly local when it is energy-bounded and

$$\mathcal A_V(I)\subset \mathcal A_V(I')'$$

for every interval, with $I'$ the interior of the complement. This is a property of closures and their spectral projections. By contrast, formal locality yields only $[Y(u,f),Y(v,g)]\psi=0$ on a common core for disjoint supports. Two symmetric unbounded operators can commute on a dense domain while their self-adjoint closures fail to commute strongly, so the implication is invalid without an additional theorem.

The distinction is also visible in point-field reconstruction. A local net plus suitable energy regularity can produce closed smeared fields affiliated with interval algebras, but affiliation, domains, and covariance must be proved before one recovers a pointlike field system Fredenhagen and Jörß 1996, §§2–3, pp. 545–550. Strong locality supplies the bounded-algebra side of that bridge. It does not follow merely because the formal series has vanishing coefficients away from the diagonal.

Heisenberg current as the first application

The model is the current theory of Free Bosons and Vertex Operators. On the rank-one Fock space, $a_n^*=a_{-n}$ and

$$\lVert a_n\psi\rVert\leq C(1+|n|)^{1/2}\lVert(1+L_0)^{1/2}\psi\rVert$$

on finite-particle vectors. Thus $J(f)=\sum_n\widehat f_na_n$ is closable; for real $f$ it is essentially self-adjoint on the finite-energy core. The commutator is the scalar

$$[J(f),J(g)]=i\,\sigma(f,g)\mathbf1, \qquad \sigma(f,g)=\frac1{2\pi}\int_0^{2\pi}f(\theta)g'(\theta)\,\mathrm d\theta,$$

up to the declared Fourier orientation. If $f$ and $g$ have disjoint supports, $\sigma(f,g)=0$. The Weyl relations then give

$$e^{i\overline{J(f)}}e^{i\overline{J(g)}} =e^{i\overline{J(g)}}e^{i\overline{J(f)}},$$

which is strong commutativity, not just a core commutator. This realizes the exact first application: the Heisenberg adjoint relations and energy growth lead to commuting interval algebras.

An independent normalization check chooses $f(\theta)=e^{im\theta}$ and $g(\theta)=e^{in\theta}$. The symplectic form becomes proportional to $m\delta_{m+n,0}$, reproducing the oscillator commutator and fixing the derivative sign.

Failure boundary: a common core is not enough

Suppose formal locality is verified and each $Y(v,f)$ is closable, but no common invariant core is controlled after products or adjoints. Then the equality of weak commutators on the original finite-energy vectors does not determine the spectral projections of the closures. The local-net conclusion must be withheld. The surviving statement is algebraic locality of smeared forms on the tested domain; missing are the energy estimate, essential self-adjointness or affiliation control, and a strong-commutativity argument.

Exercises

1. Show that smoothness of $f$ makes $\sum_n(1+|n|)^s|\widehat f_n|$ finite.

Solution

Repeated integration by parts gives $|\widehat f_n|\leq C_N(1+|n|)^{-N}$ for every $N$. Choose $N>s+2$; comparison with a convergent $p$-series proves the claim.

2. Why does $\sigma(f,g)$ vanish for disjoint supports?

Solution

The support of $g'$ is contained in the support of $g$. Hence $f(\theta)g'(\theta)=0$ pointwise when the supports are disjoint, so the integral vanishes.

References

• Carpi, Sebastiano, Yasuyuki Kawahigashi, Roberto Longo, and Mihály Weiner. “From Vertex Operator Algebras to Conformal Nets and Back.” Communications in Mathematical Physics 364 (2018), 101–145. DOI. Open PDF.
• Fredenhagen, Klaus, and Martin Jörß. “Conformal Haag–Kastler Nets, Pointlike Localized Fields and the Existence of Operator Product Expansions.” Communications in Mathematical Physics 176 (1996), 541–554. DOI.