The statement

Let \(V\) be a complex vertex operator algebra, with vacuum \(\Omega\), conformal vector \(\nu\) and conformal energy \(L_0\). It is of CFT type if \[V=\bigoplus_{n\geq0}V_n,\qquad V_0=\mathbb C\Omega.\] Write \(C_2(V)=\operatorname{span}\{a_{(-2)}b:a,b\in V\}\). We call \(V\) strongly rational if it is of CFT type, is isomorphic to its graded contragredient, is rational, and has finite-dimensional quotient \(V/C_2(V)\). Here rationality means complete reducibility of admissible modules. Unitarity means that \(V\) has a normalized positive invariant Hermitian form and its corresponding antilinear PCT involution \(\Theta\).

We use grading-restricted finite-length modules: their conformal-energy spaces are finite dimensional, and their weights are bounded below. Their algebraic braided tensor category is denoted by \(\mathop{\mathrm{Rep}}(V)\). Its tensor product is the Huang–Lepowsky fusion product. The category \(\mathop{\mathrm{Rep}}^u(V)\) consists of such modules with positive invariant Hermitian forms, and module homomorphisms. We call \(V\) strongly unitary when every simple module is unitarizable, and completely unitary when, in addition, the canonical invariant Hermitian fusion forms are positive definite. In the latter case the module category is a unitary modular tensor category.

For a homogeneous \(a\in V\) of weight \(d_a\), write \[Y(a,z)=\sum_{n\in\mathbb Z}a_{(n)}z^{-n-1}.\] Polynomial energy bounds mean that, for each \(a\), there are \(C>0\) and nonnegative integers \(p,k\) such that \[ \|a_{(n)}b\|\leq C(1+|n|)^p\|(1+L_0)^k b\|, \qquad n\in\mathbb Z,\quad b\in V. \tag{1}\] These bounds define closable smeared fields \[ Y(a,f)=\oint_{S^1}Y(a,z)f(z)z^{d_a}\frac{dz}{2\pi i z}, \qquad f\in C^\infty(S^1), \tag{2}\] initially on \(V\). For a nonempty nondense open interval \(I\subset S^1\), set \[ A_V(I)=W^*\!\left\{\overline{Y(a,f)}: a\in V\text{ homogeneous},\ \mathop{\mathrm{supp}}f\subset I\right\}. \tag{3}\] Here the von Neumann algebra generated by closed operators is the algebra generated by their bounded graph or polar data. The VOA is CKLW-strongly local if it is energy-bounded and \(A_V(I)\subset A_V(I')'\), where \(I'\) is the interior of the complementary arc.

A conformal net \(A\) is completely rational if it is irreducible, split, strongly additive and has finite \(\mu\)-index. Split means that separated interval algebras generate their spatial tensor product. Strong additivity means that removing one interior point from an interval does not change the algebra generated by its two components. For two disjoint intervals \(E=I_1\cup I_3\), the \(\mu\)-index is the index of \(A(E)\subset A(E')'\). We write \(\mathop{\mathrm{Rep}}^f(A)\) for the additive category of finite-index, equivalently dualizable, locally normal DHR representations, including finite sums of the tensor unit. The scalar case is described in Section 2.

A unitary \(V\)-module \(M\) is strongly integrable if there are compatible normal representations \(\pi_I^M\) of \(A_V(I)\) on its Hilbert completion \(\mathcal H_M\) satisfying \[ \pi_I^M\!\left(\overline{Y^V(a,f)}\right) =\overline{Y^M(a,f)},\qquad \mathop{\mathrm{supp}}f\subset I. \tag{4}\] The equality includes the domains of the closed operators. Normal representations act on affiliated operators through their graph projections. The resulting Hilbert-completion functor is denoted by \(F_{\mathrm{CWX}}^V\).

The equivalence also compares extensions of the two theories. A conformal VOA extension enlarges \(V\) while preserving its vacuum and conformal vector; a local net extension enlarges the interval algebras compatibly with their conformal symmetries. Gui’s comparison theorems identify their finite-index counterparts through the corresponding commutative algebra objects. Corollary 22 gives the precise normalized correspondence and identifies the representation functors of the extended theories.

The correspondence problem and earlier work

Carpi, Kawahigashi, Longo and Weiner made strong locality the analytic condition for constructing a conformal net from a simple unitary VOA [5]. Their examples include unitary affine and Virasoro VOAs and the moonshine VOA, and their construction is compatible with unitary subalgebras. They conjectured strong locality for every simple unitary VOA [5]. Theorem 1 establishes this assertion under strong rationality and then compares the full representation theories in that setting. The locality issue is substantial: commuting unbounded fields on a common domain need not generate commuting von Neumann algebras.

The two representation theories already possess rich tensor structures. Huang’s results give rigidity and modularity for strongly rational VOAs [22, 23], while Kawahigashi–Longo–Müger establish modularity for completely rational conformal nets [25]. Identifying these structures requires more than constructing the vacuum net. The Carpi–Weiner–Xu functor integrates unitary modules through their closed fields and is fully faithful on strongly integrable modules [18], [14]. Gui’s tensor comparison then identifies VOA fusion and braiding with Connes fusion and DHR braiding under charged-field hypotheses [17].

Complete unitarity, full braided equivalence and complete rationality are already known for affine VOAs at positive integral levels, even positive-definite lattice VOAs, parafermion VOAs at positive integral levels, and type \(ADE\) discrete-series \(W\)-algebras. Gui’s results also cover finite tensor products of these classes and regular cosets of unitary inclusions between such tensor products [17]. His extension results propagate the comparison through further extensions and coset constructions under their stated hypotheses [14]. Here the analytic and positivity hypotheses for the tensor embedding are established from strong rationality and unitarity, and a separate reconstruction argument proves that every finite-index sector is in its image.

In the converse direction, Henriques and Tener construct a unitary VOA from a conformal net with finite-dimensional vacuum energy spaces, recover a VOA from its associated net under their AQFT-locality condition, and reconstruct modules from net representations [20]. Their AQFT-locality condition does not include polynomial energy bounds. Their annulus representation theory and bounded vacuum operations for multidisc embeddings provide the geometric tools used here [21, 19]. Those constructions start from an existing net. In this paper they enter after the independent proof of strong locality, first for creation fields and ultimately for the reconstruction of arbitrary irreducible sectors.

Two earlier analytic methods are especially important to the proof. Carpi, Tanimoto and Weiner showed that optimal energy order for Hermitian primary generators implies strong locality [7], with the correction [6]. Carpi and Tomassini obtained a \(C_2\) exponential bound and used a contracting estimate to improve exponential growth to polynomial growth [8]. Here a finite-\(C_2\) filtration supplies the first contraction for a generating family. A sharper normal-product count and a second contraction then yield the optimal order required for locality.

For loop-group models based on \(SU(N)\), Wassermann computed Connes fusion using bounded primary fields and their transport relations [32]. Gui’s transport identities and Tener’s transport-form framework express candidate VOA fusion inner products through ordinary vertex operators [18, 30]; Gui further developed their canonical normalization and geometric positivity [16]. Tener’s positivity theorem for regular VOAs assumes a unitary structure on the algebraic fusion product and a corresponding net representation of the source module [30]. Our argument constructs positivity on the full algebraic fusion object before presuming that it is unitarizable. To reach the remaining simple modules, we use Carnahan–Miyamoto’s fixed-point regularity and the Barron–Dong–Mason permutation construction [3, 2]. The required permutation product rests on the extension tensor structures of Creutzig–Kanade–McRae and McRae [9, 28] and the one-slot intertwiner correspondence of Dong–Li–Xu–Yu [11].

Proof strategy

The central ordering issue is that a positive form on \(V\) does not initially give positive forms on its modules or fusion products. We first work with modules already carrying such forms. An intertwining operator is viewed as a charged field, with a charge module, a source module and a target module. Its creation case has the vacuum \(V\) as source. This supplies a starting point for the following sequence.

1. From \(C_2\)-cofiniteness to a net. In Section 3, a finite generating family controls all normal products, and positivity of certain zero modes gives an exponential starting bound. Two contraction inequalities first give polynomial growth and then, for Hermitian quasi-primary fields of weight \(d\geq2\), the energy order \(d-1\) needed for locality. A sharper normal-product count and a Möbius estimate supply the second contraction.

2. Integrating modules with given positive forms. In Section 4, a fixed doubled annulus bounds local insertions by the norm of the inserted state. A Möbius factorization makes a round-annulus estimate uniform as the insertion approaches the circle. These estimates give polynomial bounds and bounded local regularizations for creation fields. Section 5 uses their total ranges to construct normal local-algebra actions on every already unitary module, with equality of the actual closed smeared fields.

3. Positive fusion for unitary inputs. Creation operators give positive Gram matrices in the vacuum local algebra. The normal representations just constructed transfer those matrices to any unitary source module. In Section 6, rigidity supplies a nondegenerate invariant form on the algebraic fusion product, and positivity of polynomial moments separates its conformal-energy spaces. This proves positivity of the canonical fusion form. The resulting charged-field bounds then apply to every intertwining operator between unitary modules.

4. Unitarizing all simple modules. Fix a simple module \(P\), with graded contragredient \(P^\vee\), set \(D=V\otimes V\), and let \(U\) be its flip-fixed subalgebra. The transposition-twisted \(D\)-module \(T\) obtained from the vacuum \(V\) is unitary over \(U\). In Section 7, the permutation product detects every simple constituent of \(R=P\otimes P^\vee\), restricted to \(U\), inside the positive fusion product \(T\boxtimes_U T^\vee\). Thus \(R\) has a positive \(U\)-invariant form. The action of the anti-invariant part of \(D\) produces a second such form; their sum is invariant under all of \(D\). Restriction to a copy of \(P\) gives the required positive \(V\)-invariant form.

5. From a tensor embedding to all sectors. Section 8 now verifies Gui’s hypotheses for the CWX braided unitary tensor embedding, so every module gives a finite-index sector. Section 9 starts with an arbitrary irreducible locally normal net representation, without an index or finite-energy-multiplicity assumption. Its rotation spectrum is discrete. Henriques–Tener reconstruction, including [20], and algebraic regularity produce an ordinary simple module inside it; a reducing-projection argument identifies the whole sector with that module. The finite list of sectors, the split property and the Longo–Xu dichotomy give complete rationality. Decomposition into irreducibles then proves the equivalence for all additive finite-index objects.

Optimal energy order is needed for ordinary fields to generate a local net. For charged fields, polynomial bounds combined with bounded annulus regularization suffice.

Modules, duality and fusion

By the regularity theorem of Abe–Buhl–Dong [1], every weak \(V\)-module is an algebraic direct sum of ordinary simple modules. There are finitely many simple isomorphism classes. Their lowest conformal weights are rational [10]. These weights are not assumed nonnegative before a positive invariant form is known. In the grading-restricted category, each simple occurs with finite multiplicity: infinitely many copies would give an infinite-dimensional space at its lowest weight. Thus the categories used here have finite length.

The Huang–Lepowsky tensor theory supplies the fusion product \(M\boxtimes N\) and its universal intertwining operator. For a target \(K\), postcomposition identifies \[\mathop{\mathrm{Hom}}_V(M\boxtimes N,K) \quad\text{with intertwining operators of type}\quad \binom{K}{M\ N}.\] Products and iterates of these operators have absolutely convergent matrix coefficients in their respective regions, and the fusing isomorphism identifies them. We use the positive-real convention \[x_1>x_2>x_1-x_2>0\] for this identification, with real logarithms. The convergence and associativity statements include sums over intermediate conformal-energy projections [22]. Huang’s rigidity and modularity theorems apply under precisely the present strong rationality hypotheses [23]. In particular the graded contragredient \(M^\vee\) is a categorical dual. No positive form on \(M\) is needed for these algebraic assertions.

We use ordinary, nonlogarithmic intertwining operators; the modules under consideration are semisimple with diagonalizable \(L_0\). Their skew-symmetry and contragredient transformations are those of Frenkel–Huang–Lepowsky [13]. The conjugate module \(\bar M\) is the conjugate vector space with the action conjugated using \(\Theta\). For an arbitrary module its Hermitian dual is \[M^{\mathrm h}=(\bar M)^\vee.\] The natural pairing is \[ [x,\lambda]_M=\overline{\lambda(\bar x)}, \qquad x\in M,\quad\lambda\in M^{\mathrm h}. \tag{5}\] It is linear in \(x\) and conjugate-linear in \(\lambda\). This pairing has no positivity assertion. An invariant sesquilinear form on \(M\) corresponds to a module map \(M\to M^{\mathrm h}\).

All Hermitian inner products in the paper are linear in their first variable. A unitary module is a finite orthogonal sum of unitary simples: the grade-wise orthogonal complement of a submodule is invariant by the mode adjoint identities. A homomorphism between finite-length unitary modules extends to a bounded operator on their completions, and its adjoint is again a module homomorphism. To see the boundedness, decompose into simple types. On an irreducible type invariant positive forms differ by a positive scalar, by Schur’s lemma; the remaining maps act on finite-dimensional multiplicity spaces.

The spectrum of \(L_0\) on a unitary module is nonnegative. At a lowest vector of weight \(h\), the relation \([L_1,L_{-1}]=2L_0\) gives \(\|L_{-1}x\|^2=2h\|x\|^2\). Every weight in that simple module is \(h+n\) for an integer \(n\geq0\); finite orthogonal sums give the general case. Constants in our estimates may depend on the fixed module; uniformity over all modules is unnecessary.

Modes, smooth vectors and closed operators

For a homogeneous state \(a\) of weight \(d\), the physical modes are \[ a[m]=a_{(m+d-1)},\qquad [L_0,a[m]]=-m\,a[m]. \tag{6}\] We use the same convention for intertwining fields, whose physical indices may lie in nonintegral translates of \(\mathbb Z\). On an algebraic module, let \(\Pi_\lambda\) denote the projection onto the \(L_0\)-eigenspace of energy \(\lambda\) along the other eigenspaces. This definition does not require a Hermitian form. On a unitary completion, the same symbol denotes its orthogonal extension, and \(P_N\) denotes the spectral projection onto energies at most \(N\). We write \[\mathcal H_M^\infty =\bigcap_{k\geq0}\mathop{\mathrm{Dom}}(1+L_0)^k\] for the smooth vectors, with their energy Sobolev seminorms. Their finite-energy subspace is the algebraic module \(M\).

A state is quasi-primary if \(L_1a=0\), and primary if \(L_na=0\) for every \(n>0\). A quasi-primary state of integral weight \(d\) is called Hermitian when \((-1)^d\Theta a=a\). The invariant form then gives the level-wise adjoint identity \[ a[m]^*=a[-m]. \tag{7}\] For a general state, the adjoint formula has the additional, finitely many \(L_1\)-descendant terms. We use (7) only with its stated hypothesis.

Polynomial mode bounds imply that smooth smearings and their formal adjoints map smooth vectors continuously to smooth vectors. Indeed Fourier coefficients of a smooth test decay faster than every power, and commuting \((1+L_0)^k\) across a mode costs only a polynomial in its energy shift. Finite-energy truncations converge in all these seminorms. Consequently the closures of a polynomially bounded smear from finite-energy vectors and from smooth vectors agree. We will repeatedly use this explicit core.

For a charged field, smearing is performed on an interval with a chosen angular lift; its physical normalization is \(z^d\mathcal Y(a,z)\). Fractional powers use the branch specified by that lift. Changing the normalization by a smooth nowhere-zero phase merely changes the test function. At a reflected coordinate the conjugate branch is always used.

Positive-energy unitary Virasoro representations integrate to the centrally extended covering group of circle diffeomorphisms and satisfy the usual linear energy bounds [5]; see also [31]. The resulting Möbius and diffeomorphism operators preserve the smooth space. We use their fixed conformal normalization; in particular the Virasoro relation \([L_1,L_{-1}]=2L_0\) fixes the additive constant in \(L_0\). Annulus actions and their Sobolev properties will be introduced only after the local net has been constructed.

For a closed densely defined operator \(T\), its graph is a closed subspace of the direct sum of its source and target Hilbert spaces. The entries of the orthogonal graph projection are bounded operators. Graph projections allow us to state precisely that a normal representation carries one closed field to another, or that a bounded projection reduces a closed field. Agreement on a dense set alone will not be used as a substitute for agreement of closed graphs.

Creation and the adjoint convention

For a unitary module \(M\), the creation intertwining operator is \[ \mathcal Y_M^+(a,z)v=e^{zL_{-1}}Y^M(v,-z)a, \qquad a\in M,\quad v\in V. \tag{8}\] Its type is \(\binom{M}{M\ V}\). We use Tener’s adjoint convention [30]; its adjoint, the annihilation operator, is \[\mathcal Y_M^-=(\mathcal Y_M^+)^\dagger, \qquad\text{of type }\binom{V}{\bar M\ M}.\] For positive \(r\), put \[ \theta_r a=e^{rL_1}e^{i\pi L_0}r^{-2L_0}\bar a\in\bar M. \tag{9}\] For complex coordinates the logarithms follow the chosen angular lift. The reflection formula uses \(\theta_{\bar w}\) and the coordinate \(\bar w^{-1}\). These conventions fix the phase of every transport form below, including when the lowest weight of \(M\) is nonintegral. The comparison with Gui’s canonical fusion normalization is made explicitly in the positivity argument.

Tener’s Propositions 3.10–3.11 [30] give two inputs we will need before general fusion positivity. First, creation satisfies the vacuum-source transport identity. Second, for unitary charge, source and target, an intertwining operator applied to finite-energy vectors has Hilbert-space values on \(0<|z|<1\), holomorphic on each branch. The latter conclusion uses regularity and convergence of products with the adjoint intertwiner. It does not require an energy bound or a net representation on the source. We will distinguish this Hilbert-valued convergence from the operator bounds that we prove later.

The scalar case

If \(V=\mathbb C\Omega\), the conformal vector is zero, every grading-restricted module is finite dimensional in degree zero, and all fields are constant scalar fields. The net on the one-dimensional vacuum space is the scalar net. It is irreducible, split and strongly additive, and its two-interval index is one. Its additive dualizable representation category is the category of finite-dimensional Hilbert spaces. Creation, fusion and CWX integration are the usual tensor product over \(\mathbb C\), with the flip braiding. Thus Theorem 1 holds in this case.

We assume \(V\neq\mathbb C\Omega\) henceforth. Its central charge is positive: \(\|\nu\|^2=c/2\), so \(c=0\) would force \(\nu=0\), and then CFT type would force \(V=V_0=\mathbb C\Omega\). The nontriviality assumption is used in the flip-orbifold argument and in the usual conformal-net sector theory.

A finite generating filtration

Choose an integer \(D\geq2\) such that \(V_w=C_2(V)_w\) for all \(w>D\). In each weight \(1,\ldots,D\), choose a finite set of Hermitian vectors spanning the quasi-primary space, and let \(S\) be their union. Such a choice is possible because \((-1)^w\Theta\) is an antilinear involution of that space. We distinguish the conformal weight of a monomial from the sum of the weights of its generators.

We shall use only finitely many expansions from Lemma 3 at a time. Constants may depend on these fixed expansions. In particular, no uniformity over all monomials is asserted.

Here is the mode expansion that controls the intermediate energies. Iterating the negative-product formula from the outside of (12) places each generator either on the left, with physical index \[ -d_{a^i}-n_i-j_i, \qquad\text{or on the right, with index}\qquad 1-d_{a^i}+j_i,\qquad j_i\geq0. \tag{14}\] The coefficients are products of signed binomial coefficients, bounded by \(C_b\prod_i(1+j_i)^{n_i}\). There are finitely many order patterns; in each pattern every left factor precedes every right factor. The vacuum mode imposes the constraint that the physical indices sum to the desired mode of \(b\).

In a matrix element with both endpoint energies at most \(N\), move the left factors to the testing vector by adjunction. They then all lower energy. Each right factor can raise energy by at most \(d_{a^i}-1\). Every intermediate vector therefore lies in \(P_{N+c_b}W\) for a fixed integer \(c_b\), and every relevant \(j_i\) is \(O_b(1+N)\). This proves finiteness of the sums used below and permits the insertion of those fixed intermediate cutoffs.

Polynomial energy bounds

For integer \(N\geq1\), define \[B_N=\max\left(1, \max_{a\in S,\,m\in\mathbb Z} \|P_Na[m]P_N\|^{1/d_a}\right).\] The \(d_a\)th root measures growth per unit of generator weight, so a monomial of generator degree \(D_b\) contributes the power \(B_{N+c_b}^{D_b}\). Only \(|m|\leq N\) can contribute. The preceding mode expansion gives, for each fixed homogeneous \(u\in V_w\), \[ \|P_Nu[\mu]P_N\| \leq C_u(1+N)^{p_u}B_{N+c_u}^{w}. \tag{15}\] All cutoffs and exponents in this preliminary estimate are fixed once the expression for \(u\) is fixed.

We need an initial growth bound before improving (15). For \(a\in S\) of weight \(d\), set \(T_a=(a_{(-1)}a)[0]\). The normal-product formula reads \[ T_a=\sum_{j\geq0} \bigl(a[-d-j]a[d+j]+a[d-1-j]a[j+1-d]\bigr). \tag{16}\] Every summand is a positive square as a finite-energy quadratic form. The second family contains \(a[m]^*a[m]\) for every \(m\geq0\). Consequently \[ \sum_{m\geq0}\|a[m]\xi\|^2 \leq \langle T_a\xi,\xi\rangle. \tag{17}\] Zhu’s ordinary-module one-point convergence theorem applies to \(a_{(-1)}a\) [34]. Its finiteness Condition C holds here: besides \(C_2\)-cofiniteness and finite-dimensional grades, the required spanning by Virasoro descendants of primary vectors follows from unitarity. For fixed \(0<q<1\) we therefore have \[Z_a(q):=\operatorname{Tr}_W(T_aq^{L_0})<\infty.\] Here the trace is the convergent graded sum \(\sum_h q^h\operatorname{Tr}_{W(h)}(T_a|_{W(h)})\); no bounded operator extension of \(T_a\) is presumed. Positivity on each energy level and (17) give \[\sup_{\substack{\xi\in W(h)\\\|\xi\|=1}} \sum_{m\geq0}\|a[m]\xi\|^2 \leq q^{-h}Z_a(q).\] Negative compressed modes are controlled by adjunction. It follows that \(\log B_N=O(N+1)\). This is the positive-trace method used for the vacuum exponential bound in [8]; the ordinary-module trace theorem gives the version needed here.

For \(a\in S\) of weight \(d\) and \(m>0\), the mode commutator is \[ C_{a,m}:=[a[m],a[-m]] =\sum_{j\geq0}\binom{d-1+m}{j}(a_{(j)}a)[0]. \tag{18}\] The nonzero states on the right have weights at most \(2d-1\). Thus (15), uniformly for \(1\leq m\leq N\), implies \[ \|C_{a,m}P_N\| \leq C(1+N)^p B_{N+c}^{2d-1}. \tag{19}\]

The descent from a commutator estimate to a mode estimate is the mechanism used in [7]. We record it explicitly with its input-energy cutoff, which enters the subsequent iteration. Put \(A_h=\|a[-m]:W(h)\longrightarrow W(h+m)\|\), interpreting absent levels as zero. Adjunction and (18) give \[ A_h^2\leq A_{h-m}^2+\|C_{a,m}|_{W(h)}\|. \tag{20}\] Indeed, apply the identity to a unit vector in \(W(h)\), and note that \(\|a[m]|_{W(h)}\|=A_{h-m}\). Iterating descends to negative energy in at most \(1+N/m\) steps when \(h\leq N\). There is no factor depending on the dimensions of the levels. The raising mode may have output above \(N\); only the input level enters the commutator estimate.

The same argument applied to \([L_m,L_{-m}]=2mL_0+\frac c{12}(m^3-m)\) gives \[ \|L_m\xi\|\leq C(1+|m|)^2\|H\xi\|. \tag{21}\] In particular \(L_{\pm1}\) have linear energy bounds. For the zero mode of \(a\), use \[ d\,a[0]=[L_1,a[-1]]. \tag{22}\] The two products involve only a fixed shift of input energy. Combining (19)–(22), and enlarging a common fixed cutoff, yields \[ B_N\leq C(1+N)^p B_{N+c}^{\rho}, \qquad \rho=1-\frac1{2D}<1. \tag{23}\]

The use of an exponential starting bound to close a contraction with an additive cutoff also appears in the proof of [8]. Here the finite-\(C_2\) filtration supplies the contracting estimate. The contraction remains effective despite the increasing cutoff: after \(r\) iterations, \[\log B_N\leq \sum_{j=0}^{r-1}\rho^j \bigl(\log C+p\log(1+N+jc)\bigr) +\rho^r\log B_{N+rc}.\] The final term tends to zero by the exponential starting bound. The inequality \(1+N+jc\leq(1+N)(1+jc)\) bounds the sum by \(C'+\frac p{1-\rho}\log(1+N)\). Hence \(B_N\) grows polynomially.

For an arbitrary mode \(u[\mu]\) on \(W(h)\), apply (15) with an endpoint cutoff of size \(O(1+h+|\mu|)\). This bounds its norm by a polynomial in \(1+h+|\mu|\). Distinct input energy levels have distinct output levels under a fixed mode, so the squared norms add. We have proved polynomial energy bounds for every vertex operator on \(W\).

In particular, smooth smears and their formal adjoints map \(\mathcal H_W^\infty=\bigcap_{r\geq0}\mathcal D(H^r)\) continuously into itself: the mode shifts cost only polynomial factors, while smooth Fourier coefficients decrease faster than any power. Spectral truncation shows that closing a smear from \(W\) or from \(\mathcal H_W^\infty\) gives the same operator. These facts will justify the covariance calculation below.

A sharper normal-product estimate

Normal-product estimates also prove the closure of polynomial energy bounds under homogeneous generation [5]. The estimate below keeps the specific energy power needed for the second contraction and locality. We divide the zero-mode norm by its target power \(N^{d-1}\) for \(d\geq2\). The column normalization is chosen so that its square has the desired energy power \(N^{2d-1}\) for the positive zero mode \(T_a\). Define \[R_a(N)=\sup_{\substack{\xi\in P_NW\\\|\xi\|\leq1}} \left(\sum_{m\geq0}\|a[m]\xi\|^2\right)^{1/2}\] and, for integer \(N\geq1\), \[ X_N=\max\left( 1, \max_{a\in S}\left(\frac{R_a(N)}{N^{d_a-1/2}}\right)^{1/d_a}, \max_{\substack{a\in S\\d_a\geq2}} \left(\frac{\|a[0]P_N\|}{N^{d_a-1}}\right)^{1/d_a} \right). \tag{24}\] The preceding bounds show that \(X_N\) grows at most polynomially. The column contains at most \(1+N\) nonzero modes.

For \(a\in S\) of weight \(d\geq2\), subtract the identity (18) at \(m=0\) from that at \(m>0\). The former is the zero operator. This removes the \(j=0\) term without requiring the state \(a_{(0)}a\) to vanish. Only fields of weights at most \(2d-2\) remain, with coefficients polynomial in \(m\). Lemma 4 now gives \[ \|C_{a,m}P_N\| \leq C(1+m)^p N^{2d-3}X_{N+c}^{2d-2}. \tag{28}\] The weight-one estimate also fits this bound because \(N^{1/2}\leq N^{2d-3}\); the weight-zero term is bounded. For all other terms the generator degree is at most their conformal weight. The descent (20), with at most \(1+N/m\) steps, consequently proves \[ \|a[m]P_N\| \leq C(1+|m|)^pN^{d-1}X_{N+c}^{d-1}, \qquad m\ne0. \tag{29}\] For a negative mode this follows directly by applying the raising-mode recurrence at input energy at most \(N\). Thus the cutoff in \(X\) does not grow with \(|m|\). Positive modes of index greater than \(N\) vanish.

Transferring the bound to the zero mode

The commutator (22) would lose one energy power. Möbius covariance transfers the estimate without that loss. For Hermitian primary fields, [7] transfers an energy bound from one Fourier mode to all smooth smearings without increasing its energy order. We give the quantitative cutoff transfer for quasi-primary generators because its uniformity is needed in the last iteration.

Let \(a\in S\) have weight \(d\geq2\). Positive-energy unitary Möbius integration acts on \(\mathcal H_W\); it does not require a net representation. With the physical smearing convention, the test \(f(e^{i\vartheta})=e^{i\vartheta}\) selects \(a[1]\). Write \(a(f)=Y^W(a,f)\) for this module smear. The quasi-primary commutators (11) integrate on the smooth domain to \[U_g a(f)U_g^*=a(f_g),\qquad f_g(g(\vartheta))=g'(\vartheta)^{d-1}f(e^{i\vartheta}).\] One obtains this identity by differentiating the conjugation with the inversely evolving test; the preliminary polynomial bounds justify the differentiation. The positive-energy Möbius representation and these covariance facts are recalled in [7].

Take the flow generated by \((L_1-L_{-1})/2\). Its infinitesimal action on \(a[1]\) is \[\left[\frac{L_1-L_{-1}}2,a[1]\right] =\frac{d-2}{2}a[2]+\frac d2a[0].\] The mean of \(f_g\) therefore has nonzero derivative at the identity. Fix a small \(g\) for which that mean, denoted \(c_g\), is nonzero. Compression to one energy level gives \[ \Pi_hU_ga[1]U_g^*\Pi_h=c_g a[0]\Pi_h. \tag{30}\]

The conjugate \(H'=U_gHU_g^*\) is a fixed linear combination of \(1,L_0,L_1,L_{-1}\). By (21), \[H'P_MW\subset P_{M+1}W,\qquad \|H'P_M\|\leq C_g(1+M).\] Consequently, for every integer \(j\geq1\), \[ \|H^jU_g^*P_N\| =\|(H')^jP_N\| \leq C_g^j\prod_{i=0}^{j-1}(1+N+i) \leq C_g^j(1+N+j)^j. \tag{31}\] For fixed powers, the same product estimate gives the usual Sobolev continuity of \(U_g\) and \(U_g^*\), and hence preservation of \(\mathcal H_W^\infty\). For a growing power it controls a spectral tail. Choose an integer \(K>4C_g\). If \(r\) is fixed and \(N>r\), then \[\begin{align*} \|H^r(1-P_{KN})U_g^*P_N\| &\leq (KN)^{-(N-r)}\|H^NU_g^*P_N\|\\ &\leq C_rN^r(3C_g/K)^N. \tag{32}\end{align*}\] The finitely many remaining \(N\) are absorbed into the constant. In particular, the tail is bounded in every fixed graph norm, with \(K\) independent of \(N\) and of the dimension of \(P_NW\).

Choose \(r\) at least the preliminary polynomial energy order of \(a[1]\). Equations (29) and (32) give \[\begin{align*} \|a[1]U_g^*P_N\| &\leq \|a[1]P_{KN}\| +\|a[1]H^{-r}\|\, \|H^r(1-P_{KN})U_g^*P_N\|\\ &\leq C N^{d-1}X_{KN+c}^{d-1}+O(1). \end{align*}\] Compress using (30). Since \(a[0]\) preserves energy, taking the supremum of its level norms proves \[ \|a[0]P_N\| \leq C N^{d-1}X_{KN+c}^{d-1}. \tag{33}\] The bounded error is absorbed because \(X\geq1\) and \(N\geq1\).

Closing the optimal estimate

Return to the positive operator \(T_a\) in (16), for any generator of weight \(d\), including \(d=1\). If \(2d>D\), its defining state lies in \(C_2(V)_{2d}\), so Lemma 3 supplies an expansion with generator degree at most \(2d-1\). If \(2d\leq D\), decompose that state into a quasi-primary part and an element of \(L_{-1}V_{2d-1}\). The latter is in \(C_2(V)\) and has the same strict degree saving. The former is a fixed combination of generators of weight \(2d\) and is controlled by (33). In both cases, \[ \|T_aP_N\| \leq C N^{2d-1}X_{K_0N}^{2d-1} \tag{34}\] for a sufficiently large fixed integer \(K_0\).

Here and below the different enlarged cutoffs can indeed be replaced by a common one. Although \(X_N\) itself need not be monotone, all its unnormalized column and zero-mode numerators are monotone. If \(N\leq M\leq cN\), its definition gives \(X_N\leq cX_M\). The finitely many fixed cutoff enlargements therefore change only the constant in (34).

Take the square root in (17) and (34), and then the \(d\)th root in the definition of \(X_N\). The column term contracts with exponent \((2d-1)/(2d)\). Equation (33) contracts each zero-mode term with exponent \((d-1)/d\). Thus \[ X_N\leq C X_{K_0N}^{\sigma},\qquad \sigma=1-\frac1{2D}<1. \tag{35}\] The first contraction allowed an additive cutoff enlargement and used an exponential starting bound. Here the cutoff grows multiplicatively; the polynomial bound already proved for \(X_N\) suffices. Namely, \[\log X_N\leq\frac{\log C}{1-\sigma} +\sigma^j\log X_{K_0^jN},\] and the last term is bounded by \(\sigma^j(C'+s\log N+sj\log K_0)\), which tends to zero. We conclude that \(X_N\) is uniformly bounded.

Equations (29) and (33) now give (10) for every \(a\in S\) of weight at least two. To pass from cutoff estimates to a general finite-energy vector, use \(N=\max(1,\lceil h\rceil)\) on each input level \(W(h)\) and add squared norms: a fixed mode has different output energies for different input energies. Any specified Hermitian quasi-primary can be included in such an \(S\) by enlarging the fixed integer \(D\).

For a weight-one generator, boundedness of \(X_N\) gives \(R_a(N)\leq CN^{1/2}\). For positive modes this is already a half-order energy estimate. For a negative mode \(-m\), adjunction on the output level gives a bound by \(C(1+h+m)^{1/2}\) on \(W(h)\), and \[(1+h+m)^{1/2}\leq(1+m)^{1/2}(1+h)^{1/2}.\] Thus weight-one fields satisfy, in particular, at most linear energy bounds. Linear combinations of the Hermitian generators give the same conclusion for all of \(V_1\). The Virasoro estimate was proved in (21). This completes the proof of Theorem 2.

We have constructed \(A_V\) independently of any fusion positivity or module integration. In the following section it is this vacuum net that supplies the annulus operators.

Localized annuli and their ranges

We use the annulus constructions of Henriques–Tener [21, 20, 19]. An annulus may have incoming and outgoing boundaries that touch, but do not cross. A lift of an annulus means an element of the centrally extended covering semigroup \(\widetilde{\operatorname{Ann}}_c\), where \(c\) is the central charge. Lifts act by bounded operators in positive-energy unitary Virasoro representations. Gluing is multiplication, with the right factor acting first, and reflection is denoted by \(\dagger\) and acts by the Hilbert-space adjoint. On the grading-restricted unitary modules used below, fixed lifted annuli and their adjoints preserve every energy Sobolev space and hence the smooth domain continuously [21].

The VOA reconstructed from \(A_V\) is \(V\) itself [20]. Thus the point-insertion labels in that construction are the vectors of \(V\). We write \(\underline A[v(z)]\) for an insertion with the standard local coordinate \(\zeta\mapsto z+\zeta\). More generally, \(p_*v(0)\) denotes the label obtained by a change of local coordinate \(p\); for \(p(\zeta)=z+q\zeta\), \(q>0\), it is \((q^{L_0}v)(z)\). Finite-energy labels are transported by the usual formal coordinate action. Insertions compose under gluing and depend holomorphically on their positions. They also commute with reflection [20].

For \(0<\rho\leq1\) a smooth function on \(S^1\), consider the radial annulus whose incoming and outgoing parametrizations are \[e^{i\vartheta}\longmapsto\rho(e^{i\vartheta})e^{i\vartheta}, \qquad e^{i\vartheta}\longmapsto e^{i\vartheta}.\] If \(1-\rho\) is compactly supported in an interval \(I\), radial interpolation gives a lift \(\underline A\) localized in \(I\). Its vacuum operator belongs to \(A_V(I)\). The same holds for its point insertions, and their actions in a locally normal representation \(\pi\) are their images under \(\pi_I\) [20]. The reflected double \[\widehat A=\underline A^\dagger\underline A\] can be drawn in the plane between the radial curves \(\rho\) and \(\rho^{-1}\). The two annuli meet along the unit circle.

The complementary-annulus argument below is the exact-preimage argument of [20]. We record its range and density consequences in the form used for charged fields.

Let \(N\) be an already strongly integrable unitary \(V\)-module, with local representations \(\pi_I^N\). Subscripts \(N\) on annulus operators refer to the resulting net representation. The Virasoro action used here is the module Virasoro action: the closed stress-energy smears agree by strong integration, and a partition of unity identifies their modes, including \(L_0\). The reconstructed module is consequently \(N\), by the uniqueness statement in [20]. In particular, for the standard lifted round annulus with radii \(s<R\), \[ A_{R,s,N}[v(z)]=R^{-L_0}Y^N(v,z)s^{L_0} \tag{36}\] in finite-energy matrix coefficients. For a radial annulus \(A\) and \(z\) in its interior, gluing with the complementary factor in Lemma 6 gives \[ \begin{split} \underline A_N[v(z)]\xi &=Y^N(v,z)\underline A_N\xi,\\ \langle\widehat A_N[v(z)]\xi,\eta\rangle &=\langle Y^N(v,z)\underline A_N\xi, \underline A_N\eta\rangle, \qquad \xi,\eta\in\underline A_N^{-1}N. \end{split} \tag{37}\] These are identities on the indicated exact dense preimages, rather than identities of arbitrary products of unbounded operators.

A fixed-disc estimate and a uniform round estimate

In particular, over a compact subarc on which \(\rho<1\), put \[\delta=|z|^{-1}-|z|,\qquad q=2\delta.\] For \(|z|<1\) sufficiently close to one, the small disc \(z+q\mathbb D\) lies in one of finitely many fixed discs strictly inside the double. Lemma 7 is thus uniform for the labels \((q^{L_0}v)(z)\) throughout that collar. The next estimate supplies the corresponding uniform control when both boundaries are round and shrink towards the insertion. A bound for each fixed annulus would not control this degeneration. We instead factor the round cutoff through one fixed dented annulus: the moving factors will be a Möbius unitary and a contraction, while the pulled-back insertion stays in one fixed disc.

Transport forms and infinite insertion states

Recall from (8) that, for a unitary module \(M\), the creation and annihilation intertwiners are \[\mathcal Y_M^+(a,z)v=e^{zL_{-1}}Y^M(v,-z)a, \qquad \mathcal Y_M^-=(\mathcal Y_M^+)^\dagger.\] Their types are respectively \(\binom{M}{M\,V}\) and \(\binom{V}{\bar M\,M}\). The charge transform (9), for a complex coordinate with a chosen logarithm, is \[\theta_{\bar z}a =e^{\bar zL_1}e^{i\pi L_0}\bar z^{-2L_0}\bar a,\] using the conjugate logarithm at \(\bar z\). For unitary modules \(M,N\), define the diagonal transport form by \[ \mathfrak t_z^{M,N}(a\otimes b,a'\otimes b') =\big\langle Y^N\big(\mathcal Y_M^-(\theta_{\bar z}a', \bar z^{-1}-z)a,z\big)b,b'\big\rangle. \tag{43}\] We use this for \(0<|z|<1\) and \(|z|>|z|^{-1}-|z|\). At positive real \(z\) all positive-coordinate logarithms are real; on a rotated diagonal the two insertion coordinates have the same argument and the reflected logarithm is the conjugate one. The form is linear in its first entry.

Regularity gives absolute convergence of these matrix coefficients. More precisely, the ordinary intertwiners between unitary modules have Hilbert-space values at \(0<|z|<1\), with branchwise norm holomorphy [30]. Creation satisfies the identity \[ \mathfrak t_z^{M,V}(a\otimes b,a'\otimes b') =\langle\mathcal Y_M^+(a,z)b, \mathcal Y_M^+(a',z)b'\rangle \tag{44}\] without assuming energy bounds, integration of \(M\), or positivity of general fusion forms [30].

Charged modes and bounded boundary smears

A positive matrix in the local algebra

We begin with positivity of the diagonal form. Fix an interval \(I\) near \(1\in S^1\), a localized dented annulus \(\underline A\) as in Section 4, and its double \(\widehat A=\underline A^\dagger\underline A\). Choose a real \(r<1\) sufficiently close to \(1\) in its inner collar. This choice works for every finite list of charge vectors; constants in estimates may depend on that list. Write \(\delta=r^{-1}-r\) and \(q=2\delta\).

For \(a_1,\ldots,a_m\in M\), set \[u_{ij}=\mathcal Y_M^-(\theta_r a_i,\delta)a_j \quad\text{in the grade completion of }V.\] By linearity and (45), \(q^{-L_0}u_{ij}\in\mathcal H_V\). The fixed-disc estimate (38) therefore defines bounded operators \[ K_{ij}=\widehat A_V[u_{ij}(r)]\in A_V(I) \tag{59}\] as operator-norm limits of the insertions of the finite level truncations of \(u_{ij}\). In particular these limits remain in the same local von Neumann algebra.

The vacuum transport identity for creation, namely Proposition 3.10 of [30], and the insertion identity (37) give \[ K_{ij}=T_i(r)^*T_j(r),\qquad T_j(r)=\mathcal Y_M^+(a_j,r)\underline A_V. \tag{60}\] Here the creation operators with the displayed cutoff are bounded by Lemma 10. The equality is first an equality of matrix elements on \(\underline A_V^{-1}V\), which is dense by Lemma 6; boundedness extends it to the Hilbert space. Thus \((K_{ij})_{i,j=1}^m\) is a positive element of \(M_m(A_V(I))\).

By Theorem 13, \(N\) has a normal local representation \(\pi_I^N\). Applying its matrix amplification preserves this positivity. For arbitrary \(b_1,\ldots,b_m\in N\), choose exact preimages \(\xi_i\) under \(\underline A_N\) as follows. If the complementary annulus of Lemma 6 satisfies \(\underline A_N\underline B_N=s^{L_0}\), put \[\xi_i=\underline B_Ns^{-L_0}b_i.\] The inverse power acts on a finite-energy vector, so this expression is defined and \(\underline A_N\xi_i=b_i\). The norm-convergent insertion identity, followed by absolute convergence of the transport series, now gives \[ \sum_{i,j=1}^m \mathfrak t^{M,N}_{r,r}(a_j\otimes b_j,a_i\otimes b_i) =\sum_{i,j=1}^m \langle\pi_I^N(K_{ij})\xi_j,\xi_i\rangle_N \geq0. \tag{61}\] Only the creation case was used to obtain the vacuum matrix. General fusion positivity has not entered this argument.

The invariant form on the algebraic fusion object

Assume first that \(M,N\) are simple. Associativity in the positive-real region produces an intertwining operator \(B\) of type \(\binom{N}{\bar M\ X}\) such that \[ Y^N\big(\mathcal Y_M^-(c,w^{-1}-z)a,z\big)b =B(c,w^{-1})\mathcal Y_X(a,z)b \tag{62}\] in convergent matrix elements, for \(c\in\bar M\), \(a\in M\) and \(b\in N\). Its morphism is the associator followed by the annihilation morphism \(\bar M\boxtimes M\longrightarrow V\) and the left unit action on \(N\). The annihilation is nonzero: its adjoint is creation, which sends \(a\otimes\Omega\) to \(e^{zL_{-1}}a\). Since \(M\) is simple, unitarity identifies \(\bar M\) with \(M^\vee\), and the annihilation morphism is a nonzero multiple of rigid evaluation. We also write \(B\) for its corresponding morphism. Consequently rigidity gives an isomorphism \[ \operatorname{Hom}_V(E,X)\longrightarrow \operatorname{Hom}_V(\bar M\boxtimes E,N), \qquad f\longmapsto B(1_{\bar M}\boxtimes f) \tag{63}\] for every grading-restricted finite-length module \(E\).

To turn (62) into a form, recall the Hermitian duality from Section 2, which does not require positive forms. For a module \(E\), its conjugate \(\bar E\) has mode action \[v_{(n)}^{\bar E}\bar x =\overline{(\Theta v)_{(n)}^E x}.\] Its Hermitian dual is \(E^{\mathrm h}=(\bar E)^\vee\), with pairing \[ [x,\lambda]_E=\overline{\lambda(\bar x)}. \tag{64}\] The pairing is linear in \(x\) and conjugate-linear in \(\lambda\). Restricted contragredience gives its invariant adjoint relation. The given inner product of \(N\) is represented by a linear module isomorphism \(H_N:N\longrightarrow N^{\mathrm h}\).

Conjugate \(B\) and apply the FHL contragredient transformation [13], in the explicit algebraic form [24], taking the integer branch index in that formula to be \(-1\). Thus we use \[e^{wL_1}e^{-i\pi L_0}w^{-2L_0}\] on the conjugated charge, and precompose the second input with \(H_N\). This gives an intertwining operator \(B^\sharp\) of type \(\binom{X^{\mathrm h}}{M\ N}\) with \[ [x,B^\sharp(a,w)b]_X =\langle B(\theta_w a,w^{-1})x,b\rangle_N \qquad(x\in X,\ a\in M,\ b\in N, w>0). \tag{65}\] The minus phase before conjugation gives the plus phase in \(\theta_w\). All module weights are real, so these transformations also preserve the prescribed positive-real branches. Universality of \(\mathcal Y_X\) gives a module map \(H:X\longrightarrow X^{\mathrm h}\) such that \(B^\sharp=H\mathcal Y_X\).

The map \(H\) is surjective. To see the role of rigidity, suppose that the quotient map \(q:X^{\mathrm h}\longrightarrow C=\operatorname{coker}H\) has nonzero target. Hermitian duality identifies \((X^{\mathrm h})^{\mathrm h}\) with \(X\), so dualizing \(q\) gives a nonzero injection \[f=q^{\mathrm h}:E=C^{\mathrm h}\longrightarrow X, \qquad f^{\mathrm h}H=qH=0.\] Pairing (65) with \(f(E)\) would imply \(B(1_{\bar M}\boxtimes f)=0\): the charge transformation \(\theta_w\) is invertible, and the pairing with \(N\) is nondegenerate. This contradicts (63). Hermitian duality preserves finite length, so the surjective map \(H\) is an isomorphism.

We have therefore constructed the nondegenerate invariant sesquilinear form \[ h_X(x,y)=[x,Hy]_X. \tag{66}\] It pairs distinct energy spaces to zero. At this stage neither Hermitian symmetry nor positivity has been assumed. Combining (62) and (65) proves (58) for this form. Absolute convergence is the ordinary convergence of the product of \(B\) and \(\mathcal Y_X\), applied to the restricted-dual functional supplied by a finite-energy vector of \(N\).

Extracting positivity at one energy

The positivity in (61) concerns a sum over all output energies. The following lemma separates those energies without first constructing a norm on \(X\).

Fix the radius \(r\) used in (61). Consider a finite sum in the grade completion, \[x=\sum_i\mathcal Y_X(a_i,r)b_i, \qquad x_\lambda=\Pi_\lambda x, \qquad c_\lambda=h_X(x_\lambda,x_\lambda).\] After decomposing the inputs, we may take all \(a_i,b_i\) homogeneous. The numbers \(c_\lambda\) have an exponential moment in absolute value. Indeed, choose \(z>r\) sufficiently close to \(r\) that \(z<r^{-1}\). In the absolutely convergent product with the outer insertion fixed at \(r^{-1}\), replacing the inner radius \(r\) by \(z\) multiplies its component of weight \(\lambda\) by \[(z/r)^{\lambda-\operatorname{wt}(a_i)-\operatorname{wt}(b_i)}.\] Absolute product convergence for each of the finitely many pairs of inputs gives \(\sum_\lambda e^{\epsilon\lambda}|c_\lambda|<\infty\) for a common \(\epsilon>0\). This uses only the sesquilinear form (66) and algebraic convergence.

The space of such finite sums \(x\) is stable under \(L_0\), since \[ L_0\mathcal Y_X(a,r)b =\mathcal Y_X(L_0a,r)b+\mathcal Y_X(a,r)L_0b +r\mathcal Y_X(L_{-1}a,r)b. \tag{67}\] The radius \(r\) stays fixed in this step. Each polynomial in \(L_0\) introduces only finitely many further charge and source vectors, and the fixed-disc construction of (61) applies at this same \(r\) to every such finite list. Its constants may change; its positivity conclusion does not. Apply (61) to \(p(L_0)x\). Grade orthogonality gives \[\sum_\lambda |p(\lambda)|^2h_X(x_\lambda,x_\lambda)\geq0.\] Lemma 15 proves \(h_X(x_\lambda,x_\lambda)\geq0\) at each energy.

These \(x_\lambda\) run through the whole space \(X_\lambda\). In fact the span of all coefficients of \(\mathcal Y_X(a,z)b\) is a graded submodule by the intertwiner commutator formula. The quotient map by that submodule has zero postcomposition with the universal intertwiner, and is therefore zero. For homogeneous inputs each projection at the fixed nonzero radius \(r\) is a nonzero scalar multiple of its coefficient. This proves the asserted spanning at each energy.

It follows by polarization that \(h_X\) is Hermitian and positive semidefinite on every energy space, and hence on \(X\). A zero-norm vector for a positive semidefinite Hermitian form belongs to its radical. Nondegeneracy of (66) therefore makes \(h_X\) positive definite. In particular, \(X\) is now a unitary module. Finite orthogonal decompositions of \(M,N\) extend the construction to arbitrary unitary finite-length modules: creation, annihilation and the transport identity split in exactly these orthogonal blocks.

Canonical normalization and general charged fields

With \(h_X\) now positive, (65) says that \(B\) is the adjoint of \(\mathcal Y_X\) in Tener’s convention. Thus (62) is the exact normalized fusion identity for that convention.

We check the normalization, since fractional charge weights make the adjoint phase significant. Put, for \(x>0\), \[U_\pm(x)=e^{xL_1}e^{\pm i\pi L_0}x^{-2L_0}.\] Gui’s contragredient convention is \(U_+(x)\) in [16]; solving for the adjoint uses \(U_-(x)=U_+(x^{-1})^{-1}\). On a simple charge module of lowest weight \(h\), \(U_-=e^{-2\pi i h}U_+\). With inner products linear in the first variable, this means that Gui’s adjoint, after the unitary identification of the conjugate charge with the contragredient charge, is \(e^{2\pi i h}\) times Tener’s adjoint. Gui’s annihilation operator has the same factor relative to \(\mathcal Y_M^-\): its vacuum coefficient on a top vector \(a\) is \(e^{i\pi h}x^{-2h}\|a\|^2\), whereas the coefficient for \(\mathcal Y_M^-\) is \(e^{-i\pi h}x^{-2h}\|a\|^2\).

The two factors cancel in the fusion identity [16]. Thus the canonical operator called \(A\) in that identity is \(1\) for the form just constructed. The identity leaves no scalar freedom: the coefficients of the universal intertwiner span \(X\), and their pairings determine its form. This proves the canonical-normalization assertion in Theorem 14. The comparison applies on each orthogonal simple charge summand, so also covers finite sums. Common-angle versions follow by simultaneous \(L_0\) rotations, using conjugate logarithms at reflected coordinates. In particular (58) at a rotated diagonal is the squared norm of the universal intertwiner there.

We have proved closure of the unitary modules under the full algebraic fusion product, without yet knowing that every simple module is unitary. The following consequence supplies the charged fields needed for the comparison with conformal nets.

The algebraic permutation product

From detection to a positive invariant form

Finite-index local extensions

Gui’s comparison theorems now give an extension-level application of the base equivalence. We use the separable-representation and normalized extension conventions of [14]: the VOA embedding is isometric and preserves the vacuum and conformal vector, while the net extension includes the specified isometric vacuum embedding and compatibility of the projective diffeomorphism actions.