A strongly local unitary VOA generates a conformal net

Isotony follows by inclusion of the supported generators. If $I\cap J=\mathrm{\varnothing }$, then $J\subset I^{\mathrm{\prime }}$; isotony and strong locality give ${\mathcal{A}}_V(J)\subset {\mathcal{A}}_V(I^{\mathrm{\prime }})\subset {\mathcal{A}}_V(I{)}^{\mathrm{\prime }}$. Thus the interval assignment is local. The vacuum norm is one. Global smeared fields recover every mode by Fourier tests and a finite interval partition, so their action on the vacuum contains $a=a_{-d}\mathbf{1}$ for every homogeneous $a\in V_d$. Their global algebra therefore has cyclic vacuum.

The stress modes satisfy a linear energy bound with a polynomial Fourier weight. Indeed $c=2\mathrm{\parallel }{L}_{-2}\mathbf{1}{\mathrm{\parallel }}^2\ge 0$. For $n\ge 1$, put $K_{-}=L_n\mathrm{/}n$, $K_{+}=L_{-n}\mathrm{/}n$ and $K_0=L_0\mathrm{/}n+c(n-1\mathrm{/}n)\mathrm{/}24$. Then $[K_{-},K_{+}]=2K_0$, $[K_0,K_{-}]=-K_{-}$, $[K_0,K_{+}]=K_{+}$, and $K_{+}^{\ast }=K_{-}$. In each residue class of the integer energy modulo $n$, decompose inductively into lowest-weight chains using the orthogonal decomposition into $\ker K_{-}$ and $\mathrm{ran}{K}_{+}$ at each finite-dimensional energy level. A lowest-weight vector of $K_0$ weight $h\ge 0$ gives normalized chain coefficients $\mathrm{\parallel }{K}_{-}{e}_j{\mathrm{\parallel }}^2=j(2h+j-1)\le (h+j{)}^2$. Hence $\mathrm{\parallel }{L}_{n}v\mathrm{\parallel }\le \mathrm{\parallel }(L_0+c(n^2-1)\mathrm{/}24)v\mathrm{\parallel }$. The commutator $[L_n,L_{-n}]=2n{L}_0+c(n^3-n)\mathrm{/}12$ gives the corresponding bound for $L_{-n}$. Summing orthogonal chains yields $\mathrm{\parallel }{L}_{n}v\mathrm{\parallel }\le C_c(1+\mathrm{\mid }n\mathrm{\mid }{)}^2\mathrm{\parallel }Av\mathrm{\parallel }$ for all $v\in V$, with one constant independent of $n$.

The global commutant is scalar. Let $B$ be any bounded operator in it. For a self-adjoint contraction $X$ in a unital von Neumann algebra, bounded functional calculus makes $u=X+i(I-X^2{)}^{1\mathrm{/}2}$ unitary and $X=(u+u^{\ast })\mathrm{/}2$; decomposing a general operator into scaled real and imaginary self-adjoint parts shows every element is a finite linear combination of unitaries. Thus the unitary commutant condition in the affiliation definition extends to $B$ and $B^{\ast }$, so they commute with each closed local smearing and its adjoint, and with finite sums on the common core. In particular, partitioning the constant stress test shows that $B$ commutes with the self-adjoint $L_0$ and preserves its finite-dimensional eigenspaces, hence $V$. Since $V_0=\mathbb{C}\mathbf{1}$, $B\mathbf{1}=\lambda \mathbf{1}$. Global Fourier stress/field tests, again obtained by finite interval partitions, give $Ba=a_{-d}B\mathbf{1}=\lambda a$ for every homogeneous $a$. Thus $B=\lambda 1$ by density. The concrete double-commutant theorem [F4] now gives that the global von Neumann algebra is $\mathcal{B}(H)$.

Put $T(f)={\sum }_n{\widehat{f}}_n{L}_n$ on $H^{\mathrm{\infty }}$. The estimate in 1.2 proves $\mathrm{\parallel }T(f)v\mathrm{\parallel }\le C_c\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }Av\mathrm{\parallel }$, where $\mathrm{\parallel }f{\mathrm{\parallel }}_q={\sum }_n(1+\mathrm{\mid }n\mathrm{\mid }{)}^q\mathrm{\mid }{\widehat{f}}_n\mathrm{\mid }$. Energy shifts and the identity $[A,L_n]=-n{L}_n$ give, by expanding $[A^k,L_n]$, $\mathrm{\parallel }[A^k,T(f)]v\mathrm{\parallel }\le C_k\mathrm{\parallel }f{\mathrm{\parallel }}_{k+2}\mathrm{\parallel }{A}^{k}v\mathrm{\parallel }$. Thus $T(f)$ maps $H^{\mathrm{\infty }}$ continuously into itself and is symmetric there for real $f$. These estimates hold uniformly for a smooth time-dependent test function on a compact time interval.

Here is an explicit integration argument, rather than an appeal to Virasoro integration. For real $f_t$ set $J_{ϵ}=(1+ϵA{)}^{-1}$ and $T_{ϵ}(t)=J_{ϵ}T(f_t)J_{ϵ}$, a bounded self-adjoint operator. The norm-convergent time-ordered series defines its unitary propagator $U_{ϵ}(t,s)$. Since $J_{ϵ}$ commutes with $A$, the commutator estimate in 2.2 and differentiation of $\mathrm{\parallel }{A}^k{U}_{ϵ}(t,s)v{\mathrm{\parallel }}^2$ give $\mathrm{\parallel }{A}^k{U}_{ϵ}(t,s)v\mathrm{\parallel }\le e^{C_k\mathrm{\mid }t-s\mathrm{\mid }}\mathrm{\parallel }{A}^{k}v\mathrm{\parallel }$, uniformly in $ϵ$. Also $\mathrm{\parallel }(T_{ϵ}(t)-T_{\delta }(t))v\mathrm{\parallel }\le C(ϵ+\delta )\mathrm{\parallel }{A}^{2}v\mathrm{\parallel }$, by $\mathrm{\parallel }(1-J_{ϵ})v\mathrm{\parallel }\le ϵ\mathrm{\parallel }Av\mathrm{\parallel }$. Duhamel's identity and the uniform $k=2$ estimate show $U_{ϵ}$ is strongly Cauchy on $D(A^2)$, hence on $H$. Applying the same argument to inverse propagators makes its limit unitary. Higher graph-norm estimates and the same Cauchy argument with $A^k$ give invariance of $H^{\mathrm{\infty }}$. The limit solves ${\mathrm{\partial }}_{t}U(t,s)v=iT(f_t)U(t,s)v$ there, and differentiation of the norm of the difference proves uniqueness. For constant $f$ it is a unitary group whose generator is the closure of $T(f)$: $H^{\mathrm{\infty }}$ is an invariant dense differentiability domain, and it has $H^{\mathrm{\infty }}$ as a graph core. Explicitly, convolve the group with a compactly supported smooth time function. Applied to finite-energy vectors these convolutions lie in $H^{\mathrm{\infty }}$ by the uniform graph estimates. Applied to arbitrary vectors they lie in the generator domain, and their generator is the bounded convolution with the derivative of the time function. First approximate any vector by finite-energy vectors at a fixed time smoothing, obtaining graph convergence, and then let the smoothing approximate the identity on a vector in the generator domain. This proves the claimed core assertion.

For a smooth path ${\gamma }_t$ of circle diffeomorphisms starting at the identity, take its right logarithmic velocity $f_t=({\mathrm{\partial }}_t{\gamma }_t)\circ {\gamma }_t^{-1}$ and use 3.1. The Fourier Virasoro commutators give an anti-representation of real vector fields up to a scalar cocycle; the scalar series is absolutely convergent since it is a cubic polynomial in the Fourier indices. For a two-parameter path, its logarithmic velocities obey ${\mathrm{\partial }}_{s}f-{\mathrm{\partial }}_{t}g=[f,g]$. Differentiating the propagator on $H^{\mathrm{\infty }}$ shows that its curvature is exactly the scalar Virasoro cocycle: the noncentral terms cancel by this identity. Consequently varying the path with endpoints fixed changes its propagator only by a scalar phase. This follows directly by solving the resulting scalar first-order equation in the homotopy parameter; it is not a claim of vanishing scalar curvature. Thus the projective propagator defines a representation of the universal covering diffeomorphism group. Circle diffeomorphisms deformation retract to rotations: for increasing lifts, subtract the value at zero and interpolate linearly to the identity lift, whose derivative stays positive. Hence the only covering generator is a full rotation; it acts as $e^{2\pi i{L}_0}=1$. The projective representation descends to ${\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}}^{+}(S^1)$. Its strong continuity follows from the estimate $\mathrm{\parallel }(U-1)v\mathrm{\parallel }\le \int C\mathrm{\parallel }{f}_t{\mathrm{\parallel }}_2\mathrm{\parallel }AU(t,0)v\mathrm{\parallel }dt$ on $D(A)$ and density.

This projective representation transforms supported fields locally. The vertex-algebra residue identity with the conformal vector gives $[L_m,Y(a,z)]={\sum }_{j\ge -1}\left(\genfrac{}{}{0px}{}{m+1}{j+1}\right)z^{m-j}Y(L_{j}a,z)\mathrm{.}$ For a weight-d state the sum has only finitely many nonzero positive-j terms. The $j=-1$ term is ${\mathrm{\partial }}_{z}Y(a,z)$ and is transferred to the test function by integration by parts. Thus, after circle smearing, commutation by $T(f)$ gives a finite linear combination of fields with states in ${⨁}_{e\le d}{V}_e$, with test functions obtained by differential operators with smooth coefficients from $f$. There is no infinite state sum. Along a smooth diffeomorphism path these equations are a finite-dimensional matrix transport equation for test functions. Solve its first-order transport along the circle flow; the remaining matrix equation has its uniformly convergent iterated-integral solution on compact time intervals. Zero initial data off the transported support remain zero by uniqueness. Differentiating the conjugated fields on $H^{\mathrm{\infty }}$, using the graph-norm continuity of their energy bounds and 3.1, proves that $U(\gamma )Y(a,h)U(\gamma {)}^{\ast }$ is a finite sum of these smearings with support in $\gamma (\mathrm{supp}h)$. Because $U(\gamma )H^{\mathrm{\infty }}=H^{\mathrm{\infty }}$ and this is a core for every smearing, the identity holds for their closures. Therefore $U(\gamma ){\mathcal{A}}_V(I)U(\gamma {)}^{\ast }\subseteq {\mathcal{A}}_V(\gamma I)$, and applying ${\gamma }^{-1}$ gives equality.

The standard local diffeomorphism axiom also holds. For real $f$ supported in $I$, the stress field is one of the closed affiliated generators of ${\mathcal{A}}_V(I)$ by F2. Its self-adjoint closure from step 1.2 has all exponentials in this algebra. A smooth isotopy of diffeomorphisms compactly supported in $I$ has real logarithmic velocities supported there; its propagator is the strong limit of time-ordered products of these exponentials, so it lies in ${\mathcal{A}}_V(I)$ up to its immaterial scalar phase. If a smooth diffeomorphism is supported only in $\bar{I}$, choose its lift equal to the identity on $I^{\mathrm{\prime }}$ and interpolate its displacement linearly to zero. The displacement and all derivatives vanish at the endpoints of $I$. Multiplication by cutoffs supported inside $I$ therefore approximates the displacement in every smooth seminorm (flatness absorbs every cutoff derivative); for sufficiently close approximants the derivative remains positive. Their isotopies are supported compactly in $I$, and the propagator estimates of 3.1--4.1 give strong convergence. Strong closure yields the same algebra inclusion for the original diffeomorphism. Consequently if $\gamma$ equals the identity on $I$, its implementer belongs to ${\mathcal{A}}_V(I^{\mathrm{\prime }})$, commutes with ${\mathcal{A}}_V(I)$ by locality, and fixes its elements pointwise.

For the Möbius subalgebra the central cocycle vanishes, and $L_{-1},L_0,L_1$ annihilate the vacuum. Thus the Möbius restriction is genuine, fixes the vacuum, and has rotation generator $L_0\ge 0$. The already proved strong continuity, interval covariance, locality, isotony, cyclicity and irreducibility give exactly the conformal-net axioms of Möbius and conformal nets on the circle.