Strongly rational unitary vertex operator algebras and conformal nets

Proof. Let E_t be the span of arbitrary-mode strings of elements of S on \Omega whose total generator weight is at most t. Induction on string length, using the iterate identity, gives

For the induction step, write a chosen string as u=a_{(m)}u', with a\in S and u' of generator degree at most t-d_a. The iterate formula expresses u_{(r)}v, v\in E_s, as a sum of terms of the two forms

For the first, apply induction to u' and then prepend the generator mode. For the second, a_{(j)}v\in E_{s+d_a}, so induction on u' again gives a vector in E_{t+s}. Each of the two sums truncates on the fixed vector v.

We first prove V_w\subset E_w by induction on w. At weights at most D, use the orthogonal decomposition

The first summand is spanned by our chosen generators, and differentiation does not increase generator weight. A homogeneous term u_{(-2)}v of weight w has \mathrm{wt}(u)+\mathrm{wt}(v)=w-1. If both factors have positive weight, induction and (3.4) put it in E_{w-1}. If v=\Omega, it is L_{-1}u and has the same property; terms with u\in\mathbb C\Omega vanish. This proves C_2(V)_w\subset E_{w-1} and completes the induction above D as well.

Now order an arbitrary-mode string by moving nonnegative ordinary modes to the right. A commutator of modes of generators a,b involves a_{(j)}b, j\geq0, of weight at most d_a+d_b-1. The inclusion just proved places this state in E_{d_a+d_b-1}. Thus every commutator term lowers the total generator weight, whereas the uncommuted term moves the chosen nonnegative mode one place to the right. Induction on total generator weight, and then on the number of such moves, puts the string into negative ordinary-mode form. A nonnegative mode reaching \Omega vanishes. Taking homogeneous components proves both assertions.□

Proof. Use the left/right expansion (3.5) for the zero mode, between unit vectors \eta,\xi\in P_NW. Consider first an order pattern having p\geq1 left and q\geq1 right factors. Let D_L,D_R be the sums of generator weights on those sides, and let L(\boldsymbol j_L),R(\boldsymbol j_R) denote their ordered products of modes. Write

Repeated use of the column bound defining X, with the fixed intermediate cutoff already established, gives

To include the finitely many negative right indices, observe that for a fixed positive integer s,

Thus adding those finitely many indices to the column costs only a constant and a fixed cutoff enlargement. Equations (3.17)–(3.18) follow by applying this one-factor estimate successively to the vectors on each path and then summing. Orthogonality between different paths is not needed.

Let \mathcal L,\mathcal R be the sets of left and right factors. By (3.5), the zero-mode constraint is

Thus one index is determined by the others. Given a left tuple, at most CN^{q-1} right tuples complete it: the last right index is determined by the other q-1 indices. Conversely, there are at most CN^{p-1} left completions of a right tuple. Cauchy–Schwarz over the constrained pairs therefore gives

The constraint has saved one power of N from the unconstrained count. The binomial coefficients cost at most N^{\sum_i n_i}, producing exactly the power w-1 in (3.16).

An all-left pattern cannot have total physical index zero. In an all-right pattern, the indices have fixed lower bounds and sum to zero, so every index and the number of terms are bounded independently of N. If \ell\geq2, individual column bounds give a power D_b-\ell/2\leq D_b-1, which suffices. If \ell=1 and n_1=0, use the zero-mode term in X_N. If \ell=1 and n_1>0, the weaker power d_{a^1}-1/2 is bounded by w-1; differentiation at the zero physical mode has a fixed scalar coefficient. The vacuum and weight-one cases follow directly from their defining bounds.□

Proof. The unitary Virasoro decomposition expresses V as a sum of highest-weight representations. Equivalently, at each energy the orthogonal complement of the negative-Virasoro-mode descendants consists of primary vectors. These primary spaces are spanned by Hermitian vectors. Hence the conformal vector and Hermitian primary vectors generate V; see [7, Remark 3.9]. A finite family suffices: decompose the vectors in S into Virasoro descendants and take Hermitian bases of the finitely many primary spaces that occur. Together with \nu, these vectors generate S and hence V.

Theorem 3.1 gives the optimal order d-1 for every such primary of weight d\geq2, and supplies the required current and Virasoro bounds. The corrected strong-locality criterion [7, Theorem 3.7], with its Hermitian hypothesis [6], therefore applies. Finally [5, Theorem 6.8] constructs the irreducible conformal net from this simple unitary strongly local VOA.□

Proof. The geometric complement between the circle of radius s_0 and the incoming boundary gives B. Choose its lift so that the product is the standard lift of the round annulus; any necessary scalar and winding are absorbed in this fixed choice. Annulus operators are injective with dense range [20, Lemma 4.15]. Since s_0^{L_0} is invertible on E, its range contains E, and \underline B E\subset \underline A^{-1}E. Conversely, if \underline A\xi\in E, injectivity of \underline A gives

Finally, \underline B is bounded with dense range and E is dense. These observations also prove the assertion when the energy eigenspaces are not finite dimensional; compare [20, Lemma 7.9].□

Proof. Write the underlying double as D_{\rm out}\setminus \mathring D_{\rm in}. The inclusion S_0\sqcup D_{\rm in}\to D_{\rm out} is a multidisc embedding: the incoming closed discs are disjoint, although D_{\rm in} may meet the outgoing boundary. The bounded map of [19, Theorem A, or Theorem 3.6] yields

Evaluating its first slot at an inserted vector, and using [19, Corollary A.1], gives the corresponding inserted annulus operator. The boundary identification operators are unitary; the scalar comparing the fixed lift with this vacuum map is independent of the insertion [20, Lemma 6.10].

Identify S_0 conformally with \mathbb D. The resulting Möbius embedding factors into disc automorphisms and a dilation of radius less than one. Its covered univalent lift acts by unitaries and a positive-energy contraction. The state associated to p_*v(0) in H_0(S_0) therefore has norm at most \|v\|. This proves the vacuum bound. The inserted vacuum operator belongs to A_V(I) for the localizing interval of the double. Its image under \pi_I^N is the inserted operator on N; contractivity proves the same bound there.□

Proof. The fixed geometry. Fix p_0(\zeta)=-0.2+0.8\zeta, whose image disc has diameter [-1,0.6]. Choose a radial annulus A_* with inner radius 0.85 for |\vartheta|\leq0.1, inner radius between 0.85 and 1 everywhere, and inner radius 1 for |\vartheta|\geq0.2. Its incoming disc contains p_0(\mathbb D): over |\vartheta|<0.2 the latter disc has radial boundary less than 0.61, and elsewhere it lies in \mathbb D. The double of A_* contains

strictly in its interior, since this disc has argument of absolute value less than 0.051 and radii between 0.95 and 1.05. Factor the univalent annulus p_0 as A_*C, and fix lifts with \underline A_*\underline C equal to the covered univalent lift of p_0. The factor C is fixed; Figure 1 shows the geometry of its outer factor.

Factoring the shrinking annulus. The map \gamma_t moves the fixed point a to r. We arrange the remaining incoming boundary by writing s\operatorname{id}=\gamma_t\circ p_0\circ b_t, namely by defining

After doubling, the factor \gamma_t will cancel against its adjoint. The insertion will then lie between the same fixed factors \underline C^* and \underline C, with only the contractions \underline b_t^* and \underline b_t outside them.

The disc \gamma_t^{-1}(s\mathbb D) has real diameter with endpoints -(s+t)/(1+st)>-1 and 1/2. It lies strictly in p_0(\mathbb D), so b_t is a strict Möbius disc embedding. Its covered univalent lift is a contraction, whereas the lift of \gamma_t is unitary. Choose these lifts so that

is the standard round-annulus lift. This is an equality in the covered univalent semigroup, whose inclusion lifts to \widetilde{\operatorname{Ann}}_c [20, Section 4.2]: the Virasoro cocycle vanishes on the univalent subalgebra, so this lift introduces no scalar ambiguity. Any winding adjustment acts unitarily.

Consider the insertion coordinate p_t(\zeta)=\gamma_t^{-1}(r+q\zeta). The identities

and

show, for |h|\leq q, that

Consequently p_t(\mathbb D)\subset\mathring S_0 for every t. The map has no pole there. Also s>t, so the pole 1/t of \gamma_t^{-1} is outside the round annulus with outer radius 1/s. Since \gamma_t commutes with reflection in the unit circle, we may double (4.6), carrying along the insertion. The unitary factors cancel, and gluing gives the exact operator identity

The uniform bound. Lemma 4.2 bounds the middle factor by a fixed constant times \|v\|. Thus the right side has norm at most C_{\widehat A_*,S_0}\|\underline C_N\|^2\|v\|. In particular, no radius-dependent comparison scalar is introduced.

Finally,

which proves (4.5).□

Proof. Expand \theta_{\bar z}a' into finitely many vectors of weights d_j\leq d'. Their coefficients are bounded as |z| tends to one. By L_0-covariance, applying q^{-L_0} to each annihilation value produces the factor q^{-d_j-d} and an annihilation value at modulus \delta/q=1/2. The latter is a Hilbert-space vector by [30, Proposition 3.11]; rotations preserve its norm. This proves (4.10).

Let P_n be the vacuum energy cutoff. Apply the uniform collar version of (4.3) to q^{-L_0}(P_n-P_m)u_z. It gives

Thus these operators converge in norm, with the asserted bound. Locality and representation transfer persist under that norm limit. For finite truncations, (4.2) applies. Absolute convergence of the transport iterate identifies its limiting matrix coefficients on the exact dense preimages, proving (4.11). The identical argument using (4.4) and (4.1) gives the round-annulus assertion.□

Proof. Lemma 4.4 and Proposition 4.3 give

Write the physical expansion as \Phi(a,z)=\sum_\mu\Phi(a)[\mu]z^{-\mu-d}, so that a coefficient maps energy h to h-\mu. For b of energy h, project (4.14) to that output energy. Fix 0<\varepsilon_a\leq10^{-4} small enough that the transport domination and the cutoff estimate hold whenever 1-\varepsilon_a\leq r<1, and choose

The corresponding t lies in [0,1). By (4.5), s\geq1/2 and 1-s\leq(6667)10^{-4}/(1+h+|\mu|). Since -\log s\leq2(1-s), the factor s^{-h} is uniformly bounded. The factor r^{\mu+d} is uniformly bounded as well, using d\geq0 and r\geq1-10^{-4}. Hence

Distinct input energies have distinct output energies for a fixed coefficient. Orthogonality therefore turns this into a polynomial energy bound on N. Taking adjoints between individual energy spaces gives the same kind of bound for the adjoint coefficients. Choose any integer energy order at least d when an integer order is desired.

On a fixed argument interval, define \Phi(a,f) by smearing its physical field z^d\Phi(a,z) against f(e^{i\vartheta})\,d\vartheta/(2\pi). Other smooth weight factors can be absorbed into f. Rapid decay of Fourier coefficients, (4.15), and the energy shifts show that this smear and its formal adjoint map smooth vectors continuously to smooth vectors. For clarity, a target Sobolev seminorm of order j is bounded by an input seminorm of a fixed higher order, times a summable series of polynomially weighted Fourier coefficients. This also shows that finite-energy truncations converge in the graph norm of the smear. The closures of the smears defined on finite-energy and on smooth vectors consequently agree.

Choose \underline A with \rho<1 on a neighborhood of \operatorname{supp}f and with 1-\rho compactly supported in I. On the fixed dense subspace \underline A_N^{-1}N, consider

The localized part of Lemma 4.4, together with (4.12), gives a bounded extension satisfying

throughout an inner collar over this neighborhood. Its values on the dense initial domain are Hilbert-valued holomorphic functions, by [30, Lemma 2.9 and Proposition 3.11]. Local uniform boundedness therefore gives operator-norm holomorphy.

For neutral fields, annulus regularization and the Banach-valued boundary argument appear in [20, Proposition 8.4 and Lemma 8.5]. The charged collar estimate (4.16) supplies the hypothesis needed here. We spell out the boundary argument. Put

All integrals here use the chosen argument and the compact support of f. Holomorphy gives t\partial_tT=-i\partial_\vartheta T. After integrating by parts, every radial derivative of order p of F_f is a finite sum of integrals of T against derivatives of the fixed smooth test, multiplied by bounded powers of t^{-1}. Thus

Choose an integer p>d. Taylor’s integral formula from a fixed t_0<1 expresses F_f(t) as a polynomial plus the integral of (t-u)^{p-1}F_f^{(p)}(u)/(p-1)!. The norm of the latter integrand is bounded, uniformly for u\leq t, by a constant times (1-u)^{p-1-d}, which is integrable at one. Dominated convergence proves that F_f(t) has an operator-norm limit T_f as t tends to one.

On \underline A_N^{-1}N, the mode expansion and its polynomial bounds identify this limit with \Phi(a,f)\underline A_N. For arbitrary \xi\in\mathcal H_N, approximate \xi by \xi_j\in\underline A_N^{-1}N. Then \underline A_N\xi_j\to\underline A_N\xi and T_f\xi_j\to T_f\xi. Closedness of the smeared field proves (4.13) on all of \mathcal H_N. Lemma 4.1 supplies the asserted range and density.

The product in (4.13) preserves smooth vectors, since both its factors do. If \Phi(a,f)^\dagger denotes the formal adjoint smear, then every smooth target vector \eta belongs to the adjoint domain of \overline{\Phi(a,f)}, and

Indeed this equality follows by pairing with the dense finite-energy core and using the adjoint coefficient bounds. The right side is smooth: the formal adjoint smear preserves smooth vectors and the fixed annulus adjoint preserves every Sobolev space. This proves the remaining domain assertion. Finally, (4.9) supplies (4.12) with constant one for creation, with the vacuum as its integrated source.□

Proof. Write \mathcal D_N=\mathcal H_N^\infty and \mathcal D_K=\mathcal H_K^\infty. The polynomial mode estimates imply that smeared fields and their formal adjoints map the corresponding smooth spaces continuously to smooth spaces. Spectral truncations converge in all the smooth seminorms. Consequently a smear has the same closure whether it is initially defined on finite-energy vectors or on smooth vectors. These facts apply both to the observables and, by Lemma 4.5, to the charged field.

Choose the localized annulus \underline A in that lemma with its dent containing \operatorname{supp}f and localized in I. Its action \underline A_N is a bounded injective operator with dense range, and

is bounded on all of \mathcal H_N. Both T_f and T_f^* preserve the respective smooth spaces. On \mathcal D_K the adjoint is \underline A_N^*\Phi(a,f)^\dagger, where the dagger denotes the formal smeared adjoint. These are assertions about the full bounded regularization, not merely about its restriction to finite energies.

For homogeneous v\in V and a smooth g supported in J, put

On the smooth spaces, the intertwining-operator Jacobi identity gives

To pass from the formal identity to this equality, multiply the two field orderings by a sufficiently high integral power of the coordinate difference. Polynomial energy bounds make the resulting series smooth-domain distributions. Since the two test supports are disjoint, one can divide the product test by that power. The same argument applies to the adjoint relation, whose observable test has the same support. This is the weak intertwining property in [17, Theorem 3.5].

The operator \underline A_N and its adjoint commute strongly with these observables: \underline A_N is localized in I, whereas N is already integrated and the observables are localized in J. Combining this with the weak identities gives

For the second equality one can pair the first weak intertwining identity for the formal adjoint observable. More explicitly, let S_N^\dagger and S_K^\dagger denote its actions on the smooth spaces. The invariant adjoint formula expresses it as a finite sum of smears of PCT-transformed states and their L_1-descendants, with tests still supported in J. Thus the first equality also gives T_fS_N^\dagger=S_K^\dagger T_f on \mathcal D_N. For \xi\in\mathcal D_N and \eta\in\mathcal D_K, using our linear-first inner product convention, we obtain

All vectors in these pairings are smooth. This proves the second equality in (5.3) without identifying a formal-adjoint closure with the full Hilbert-space adjoint.

We spell out the passage to closed operators. On \mathcal H_N\oplus\mathcal H_K, let

Approximation in the graph norms, using the smooth cores, shows that both Q_f and Q_f^* preserve \operatorname{Dom}D and commute with D there. Hence Q_f\oplus Q_f and its adjoint preserve the closed graph of D. They therefore commute with its orthogonal projection. The matrix entries of this graph projection determine the usual bounded transform and polar data of D. Thus Q_f commutes with the von Neumann algebra generated by D.

As v and g vary, the preceding argument proves Q_f\in\mathcal B_{N,K}(J)'.

It remains to remove the annulus. The first-summand compression of \mathcal B_{N,K}(J) is \pi_J^N(A_V(J)): its normal range is generated by the module smears, which are the images of the vacuum generators under strong integration. Since \underline A_N is localized in I, both \operatorname{diag}(\underline A_N,1) and its adjoint commute with \mathcal B_{N,K}(J). The closed inverse \underline A_N^{-1} is densely defined on \operatorname{ran}\underline A_N, and inversion preserves the resulting affiliation with the commutant. Therefore

is densely defined, and its domain and action commute with every unitary in \mathcal B_{N,K}(J). It is closable: by (5.2) it is a restriction of the closed operator with lower corner \overline{\Phi(a,f)} and all other corners zero. Its domain contains the finite-energy core in the first summand, since N\subset\operatorname{ran}\underline A_N by Lemma 4.1. Its closure is consequently exactly that closed lower-corner operator. This proves its affiliation with \mathcal B_{N,K}(J)'.

This affiliation is the strong intertwining assertion, including the diagram convention used in [17, Section 3.3]. Explicitly, on two copies of \mathcal H_N\oplus\mathcal H_K, repeat the charged corner diagonally and place a diagonal observable in a matrix corner between the copies. The latter is affiliated with M_2(\mathcal B_{N,K}(J)), and the repeated charged corner with its commutant. Hence the diagram commutes strongly. All operations above are valid for each fixed branch on I.□

Proof. The transport domination needed here holds with equality for creation and vacuum source. It uses the given form on M, and assumes neither integration of M nor positivity of a general fusion product. Since \Omega\in\operatorname{ran}\underline A_V, each range contains \mathcal Y_M^+(a,f)\Omega.

Suppose \eta\in\mathcal H_M is orthogonal to all these vectors. For each homogeneous a, consider

Creation gives \mathcal Y_M^+(a,z)\Omega=e^{zL_{-1}}a. The charged energy estimate bounds the Taylor coefficients of F_a by a polynomial in their index. Its boundary value is therefore a distribution with only nonnegative Fourier frequencies. Orthogonality to all the creation smears says that this distribution vanishes on I. The smooth weight or branch factor in the smearing convention can be absorbed into the test on this interval.

For completeness, such a one-sided distribution is zero if it vanishes on a nonempty arc. Convolve it with smooth approximate identities supported in increasingly small arcs. Each convolution still has only nonnegative frequencies and vanishes on a smaller nonempty arc. It is the smooth boundary value of a holomorphic function in the disc. The Schwarz reflection principle across an arc on which that function is zero, followed by analytic uniqueness, shows that the convolution is identically zero. Passing to the distributional limit proves the claim. Thus F_a=0, and its constant coefficient gives \langle a,\eta\rangle=0. Homogeneous vectors span M, which is dense in \mathcal H_M; hence \eta=0.□

Proof. Module energy bounds follow from Theorem 3.1, and the vacuum net exists by Corollary 3.4. Fix J and form \mathcal B(J)=\mathcal B_{V,M}(J) as in (5.1), including every homogeneous observable. Its two Hilbert summands reduce all its generators. Vacuum restriction is therefore a normal unital homomorphism

It is onto. Indeed, compression of the graph data of a direct sum gives the graph data of its first summand, and these compressed data generate A_V(J). The range of a normal homomorphism of von Neumann algebras is a von Neumann algebra, so it contains that generated algebra and is equal to it.

The kernel of r_0 is \mathcal B(J)q for a central projection q. Choose any interval I disjoint from J. Apply Proposition 5.1 to creation, whose source is the integrated vacuum. Its bounded regularizations commute, as lower corners, with \mathcal B(J). Since q annihilates the vacuum summand, its restriction q_M to the other summand satisfies

Lemma 5.2 gives q_M=0, hence q=0. Thus r_0 is a normal isomorphism. Its inverse is normal as well: an order isomorphism preserves suprema of bounded increasing positive nets. If r_M denotes restriction to the module summand, define

For J_1\subset J_2 the diagonal algebras are nested, and both compression maps respect the inclusions. It follows that \pi_{J_2}^M|_{A_V(J_1)}=\pi_{J_1}^M.

We verify the exact closed-operator identity. Let S_V=\overline{Y^V(v,g)} and S_M=\overline{Y^M(v,g)}. The graph projection of S_V\oplus S_M has its matrix entries in \mathcal B(J). Compressing its entries to the two summands yields the graph projections of S_V and S_M, respectively. Consequently the matrix amplification of \pi_J^M sends the entire graph projection of S_V to that of S_M. This is precisely the normal image of an affiliated closed operator, and proves (5.4). In particular the argument identifies closed graphs, rather than merely operators on a dense common domain.

The graph data of the vacuum smears generate each A_V(J), so the same identity also proves uniqueness. With the module energy bounds, it is the defining property of strong integrability and of the CWX action; see [14, Definition 4.22]. Applying the identity to the conformal vector identifies all localized stress-energy fields. Their canonical centrally extended Virasoro actions agree by [17, Proposition 3.10], equivalently [14, Remark 4.23]. This gives exactly e^{itL_0} for rotations, not an arbitrarily shifted positive-energy generator. The normalization is important when using r^{L_0} in annulus factorizations.□

Proof. Set \mu=\sum_\lambda c_\lambda\delta_\lambda. Polynomials are dense in L^2(|\mu|). To see this, let g be orthogonal to them. Cauchy–Schwarz and the exponential moment show that the finite complex measure g\,|\mu| has an exponential moment of smaller positive order. Its bilateral Laplace transform is holomorphic in a strip containing the imaginary axis. Orthogonality makes every derivative at zero vanish, so the identity theorem makes the transform zero on this strip. Uniqueness of the Fourier transform of a finite measure then gives g\,|\mu|=0. For completeness, this last uniqueness follows by convolving the measure with Gaussian approximate identities: every convolution has zero Fourier transform and is zero, and the convolutions converge weakly to the measure.

For a fixed \lambda, approximate the indicator of its atom by polynomials p_n in L^2(|\mu|). Then |p_n|^2 converges to that indicator in L^1(|\mu|), because

Thus the nonnegative integrals \int |p_n|^2\,d\mu tend to c_\lambda.□

Proof. By universality, \Phi=L\mathcal Y_X for a module map L:X=M\boxtimes N\longrightarrow K. Both modules are now unitary. Such a map is bounded: decompose into the finitely many unitary simple summands; Schur’s lemma and uniqueness of an invariant form on each simple identify it with a finite family of linear maps between finite-dimensional multiplicity spaces. Theorem 6.1, including its rotated-diagonal identity, gives

The source N is strongly integrable by Theorem 5.3. This is precisely the transport domination hypothesis of Lemma 4.5, which gives the stated mode bounds and bounded annulus-cutoff smears. Proposition 5.1 then gives the asserted affiliation and strong intertwining property for homogeneous observables. For a finite sum of homogeneous observables, sum the weak commutation identities on the common smooth core before closing. The same bounded regularization and its adjoint preserve that core, so the graph argument of that proposition also applies to the closure of the sum. This proves the assertion for every v\in V.□

Proof. Put U=D^{\langle\sigma\rangle}. Cyclic-orbifold regularity [3, Theorem 5.24 and Corollary 5.25] makes U strongly rational. Let \mathcal C be its category of grading-restricted generalized modules, with its HLZ vertex tensor category structure, and write c_{X,Y} for its braiding. We use L(n)=L_n for Virasoro modes in the coordinate formulas below. Ordinary D-modules and the BDM modules T_\sigma(N) restrict to objects of \mathcal C:

This requires boundedness below and finite-dimensional weight spaces, not nonnegative module weights.

Regard D as a commutative algebra object in \mathcal C, and write \star for the relative product on \operatorname{Rep}_{\mathcal C}(D). This product is the cokernel imposing equality of the two D-actions. Right exactness allows the ambient associator and its inverse to descend through these quotients, giving an associator for \star [9, Theorems 2.51–2.53]. If \mu_Q denotes the action on an object Q, the categorical g-twisted condition is

For the canonical quotient q:X\boxtimes_U Y\to Q=X\star Y of objects in the g- and h-sectors, the target lies in the gh-sector: precompose this identity with 1_D\boxtimes q, use the hexagon and balancing, and then apply the two input identities. Right exactness makes that precomposition surjective [28, Proposition 2.4 and Corollary 2.5]. The field attached to \mu_Q has the vacuum, truncation, derivative, and grading properties; the displayed monodromy condition gives the equivariance for its prescribed twist. Analytic associativity reconstructs its twisted Jacobi identity [28, Propositions 4.11 and 4.13, Remark 4.12 and Theorem 4.14]. Thus these are actual twisted modules. Only the known 1- and \sigma-sectors are needed, not an exhaustive classification of algebra modules.

We spell out the relevant intertwiner dictionary. The two categorical action compatibilities are precisely analytic associativity and skew-associativity for the corresponding U-intertwiner \mathcal I [9, Theorem 3.44 and Proposition 3.50]. Suppose its charge X is ordinary and its source and target are \sigma-twisted. For \sigma a=(-1)^j a with j\in\{0,1\}, the matrix coefficients of

are branches of one analytic function g(z_1,z_2). For fixed nonzero z_2, multiplication by z_1^{j/2} cancels monodromy about 0. The iterate has integer powers about z_1=z_2, since X is ordinary. These two local loops generate the punctured-plane fundamental group, so z_1^{j/2}g(z_1,z_2) is single-valued. Source truncation gives a pole of finite order at 0, and ordinary-charge truncation gives one at z_1=z_2. At infinity, the finite weight support of the restricted-dual vector and the lower bound on intermediate target weights give polynomial growth. Thus the only possible poles are at 0,z_2,\infty. To pass from this fixed-z_2 rationality to a two-variable formal identity, take homogeneous inputs of weights h_x,h_y, a state a of weight d, and a restricted-dual vector of weight H. On compatible branches, the L(0)-bracket gives

The function u\mapsto u^{j/2}g(u,1) is rational, with poles only at 0,1,\infty; the prefactor is a rational power. Its three expansions therefore give the twisted Jacobi identity. The converse follows from formal duality [11, Proposition 2.17]. Semisimple L(0) and the L(0)-bracket remove logarithmic terms. Here the factor is z_1^{j/2}, as in Proposition 2.17; this corrects the interchange of the two exponents in the displayed cancellation factor in the proof of [11, Theorem 3.5]. Consequently \star represents exactly the needed twisted intertwining operators. On ordinary modules it is ordinary D-fusion [9, Theorem 3.54 and Corollary 3.55].

For an integer k\geq1, define the coefficients a_j (depending on k) by the formal coordinate identity

and set

These are the coordinate operators of [11, Section 5.1.1, equations (5.3)–(5.5)], with the chosen branches of the indicated powers.

Write I_V and I_D for the indicated spaces of ordinary and twisted intertwining operators, respectively. For ordinary V-modules M,N, the explicit mutually inverse maps of [11, Theorems 5.5, 5.9 and Corollary 5.10] give

Together with the BDM equivalence and the ordinary fusion universal property, this identifies the representing object:

This is the universal-property proof of [11, Theorem 6.3], applied only to grading-restricted targets.

For completeness, the finite-expansion step in that proof does not require integral module weights. If L(0)m=hm, write

The sum is finite by boundedness below, and

In the weak-associativity substitution this fractional monomial depends only on z_2 and multiplies both sides. The corresponding expansion of a state u\in V has integral powers. Taking a maximum over the finitely many coefficient pairs and absorbing their finitely many negative integral powers supplies the common exponent used in the proof of [11, Theorem 5.9].

To move the slot, pull back all three actions in an intertwiner by \sigma. This preserves twisted Jacobi and its universal property. For every ordinary \sigma-twisted object Q,

Thus E_Q:Q\longrightarrow Q^\sigma is an invertible module map. The charge pullback of M\otimes V is V\otimes M. Applying simultaneous pullback to (7.3) gives

This does not require M\otimes V itself to be flip-invariant.

Finally, put X=P\otimes V, Y=V\otimes P^\vee. Ordinary tensor-product factorization and the vacuum unit give X\star Y\cong P\otimes P^\vee. Its factorized intertwiner is

where u belongs to the first factor and v to the second. The relative associator and the two slot formulas now yield

The dictionary permits writing \boxtimes_D for these products. If V=\mathbb C, the flip is the identity and the statement is immediate.□

Proof of Theorem 7.1. For V=\mathbb C, the only simple module is the scalar module, so the assertion is immediate. Assume henceforth that V\ne\mathbb C, and fix an arbitrary simple V-module P. Its algebraic contragredient is denoted by P^\vee; neither module is assumed unitary.

The fixed-point algebra and a unitary twisted module. Set

Thus D=U\oplus J as a U-module. Tensor products preserve simplicity, regularity, CFT type and self-contragredience. In particular, D is a simple unitary strongly rational VOA. The flip preserves its unitary form and commutes with its PCT operator \Theta^D. The cyclic fixed-point theorem of Carnahan–Miyamoto [3, Corollary 5.25] applies to D: it is simple, nonnegatively graded and regular, with a nonsingular invariant bilinear form. It follows that U is regular. CFT type and the unitary structure are inherited, and the inherited form and PCT also give self-contragredience. Hence U is simple unitary strongly rational, and Theorem 6.1 applies to U.

Let T=T_\sigma(V) be the transposition-twisted D-module obtained from the vacuum V by the Barron–Dong–Mason construction [2, Theorem 3.9]. This construction preserves the unitary form [16, Lemma 10.12], so T|_U is unitary. Its conformal energy is

where c is the central charge of V. In particular, it has finite-dimensional energy spaces and a lower bound. Put

The tensor-product module theorem makes R simple over D; it too has finite-dimensional energy spaces and a lower bound.

The restrictions T|_U and R|_U have the same conformal vector and energy spaces as before restriction. They are grading-restricted U-modules of finite length. Indeed, regularity decomposes them into simple ordinary U-modules. There are finitely many simple types, and infinitely many copies of one type would make the energy space at its lowest weight infinite-dimensional. All fusion products over U used below therefore belong to its finite semisimple module category.

A positive form on the restriction of R. Here \boxtimes_D denotes the product representing twisted intertwining operators among grading-restricted ordinary or \sigma-twisted D-modules, as constructed in Lemma 7.2. That lemma gives

The ordinary labels P and P^\vee are still not assumed unitary. By rigidity, P\boxtimes_V P^\vee has a nonzero evaluation onto V. Applying T_\sigma to this morphism and using (7.4) yields a nonzero twisted intertwining operator I: the universal intertwiner’s coefficients generate its target, so composition with a nonzero target morphism cannot vanish. Its type is

We need this operator to detect every simple summand of R|_U. For two weak \sigma-twisted D-modules, the space of generalized intertwining operators between them is a weak D-module [11, Theorem 4.6]; write \mathcal H(T,T) for this space with source and target T. Moreover, [11, Proposition 4.8] identifies the charge map

as a homomorphism of weak D-modules. Its kernel is therefore a D-submodule. Since R is simple and I is nonzero, f_I is injective.

Restricting the observable states in the twisted Jacobi identity to U gives the ordinary Jacobi identity, with the same derivative identity and lower truncation. Thus I is an ordinary U-intertwining operator. If S\subset R|_U is any simple submodule, injectivity of f_I implies that I|_{S\times T} is nonzero. Rigidity over U then gives a nonzero, hence injective, morphism

The conjugate-module description of the contragredient gives a unitary form on T^\vee. Theorem 6.1, applied to U, makes T\boxtimes_U T^\vee unitary. Restricting its positive form unitarizes S. Taking the finite orthogonal sum over a simple decomposition of R|_U gives a positive U-invariant form h_1 on R.

The form h_1 gives the correct adjoints for fields in U. It does not yet give the correct adjoints for fields in J, although those fields already act algebraically on R. We will construct a second positive form from their action and then add the two forms. Dong and Lin constructed unitary order-two simple-current VOA extensions under compatible antiunitary-involution hypotheses [12, Theorem 3.3]. Carpi, Gaudio, Giorgetti and Hillier proved unitarity and complete unitarity of simple CFT-type extensions of a completely unitary base [4, Theorem 4.7]. Here the extension algebra D is already unitary, while the arbitrary module R initially has a positive form only after restriction to U. The argument below transfers that form to the existing D-action.

A second form from the sign action. Since V\ne\mathbb C, the flip is faithful and J\ne0: if v is independent of \Omega, then v\otimes\Omega-\Omega\otimes v is a nonzero vector of J. The orbifold constituent theorem [27, Corollary 4.8] identifies the tensor subcategory generated by the constituents of D with \operatorname{Rep}(\mathbb Z/2\mathbb Z). Here D is an ordinary VOA, so the abelian associativity cocycle in that theorem is trivial. In particular, J is a simple current, that is, a simple invertible object, and

The multiplication morphism

coming from Y^D|_{J\times J} is nonzero. For example, the invariant bilinear form of D restricts nondegenerately to J, so some product of two states in J has a nonzero vacuum coefficient. Thus m is an isomorphism.

Let

be the morphism corresponding to the action Y^R|_{J\times R}. Ordinary D-module associativity, expressed using the fusing isomorphism over U, says that

Here \alpha_{J,J,R}:(J\boxtimes_U J)\boxtimes_U R \longrightarrow J\boxtimes_U(J\boxtimes_U R) is the associator and \ell_R:U\boxtimes_U R\longrightarrow R is the left unit isomorphism; undecorated fusion signs in this identity are over U. The right side is an isomorphism, so L is epic. Tensoring by the invertible object J preserves composition length. The two sides of L have the same finite length, and hence L is an isomorphism.

Give J the form inherited from D. Apply Theorem 6.1 to J and (R,h_1), and transfer its positive invariant fusion form along L. This gives a second positive U-invariant form h_2 on R, for which Y^R|_{J\times R} is the transported universal intertwiner. We will show that h_1+h_2 is invariant for all of D.

Adjunction exchanges the two forms. The creation intertwiner of the U-module J is Y^D|_{J\times U} by skew symmetry. With the inherited forms, its adjoint annihilation is

This follows from unitarity of D and fixes the scalar normalization of the annihilation. The linear identification \bar J\cong J in (7.6) sends \bar j to \Theta^D j.

For w>0, define the antilinear charge transform on D by

The integer grading of D makes this expression unambiguous. For j,j'\in J and finite-energy r_1,r_2\in R, the exact transport identity of Theorem 6.1, followed by ordinary D-module associativity, gives

Initially, 0<z,w<1 and w^{-1}>z>w^{-1}-z>0. Pairings of completed vectors on the first line are the absolutely convergent sums over equal energies supplied by the transport identity. On the last line, h_1(\,\cdot\,,r_2) is a restricted-dual functional: h_1 is U-invariant and U has the same conformal vector as D. Thus convergent module associativity applies without already assuming D-invariance of h_1.

The vectors j_{(n)}r, with j\in J, r\in R and n\in\mathbb Z, span R. Their span is a U-submodule by the intertwiner commutator formula, and is the image of the surjective morphism L. Comparing Laurent coefficients in z in (7.7) therefore yields

For finite-energy x,y, the two sides are finite Laurent polynomials in w. Indeed, the forms pair equal energies, the states of D have integer weights, and e^{wL_1}j is a finite sum. Consequently, (7.8) extends from its nonempty initial interval to every w>0.

The charge transform satisfies

For completeness, \Theta^D commutes with the Virasoro operators and with the real integer-weight signs. Conjugating e^{wL_1} by (-w^2)^{L_0} gives e^{-w^{-1}L_1}, so

Apply (7.8) at radius w^{-1} with charge t_wj, exchange the two vector slots, and take complex conjugates. Using (7.9) gives the reverse relation

Adding (7.8) and (7.10) proves the invariant adjunction for h=h_1+h_2 on states of J. It already holds on U. Since D=U\oplus J, the form h is D-invariant; it is positive definite as a sum of positive definite forms. Only ordinary associativity and the normalized transport identity were used in this construction.

Finally, choose a nonzero homogeneous vector q\in P^\vee and put

This is positive definite. The fields of V\otimes\Omega act on P\otimes\mathbb Cq as the fields of V on P, and the PCT charge transform on v\otimes\Omega is the transform on v tensored with \Omega. The D-invariant adjunction therefore restricts to the V-invariant adjunction for h_P. This unitarizes the arbitrary simple module P. Finite direct sums of these forms prove the final assertion.□

Proof. We first verify the analytic inputs to Gui’s comparison and identify the underlying functor. The additional localizability conditions will be checked before invoking the tensor comparison.

Generating fields and charges. We verify the hypotheses of Gui’s comparison theorem [17, Section 3.4 and Corollary 3.23]. Regularity follows from rationality and C_2-cofiniteness. Theorem 7.1 supplies positive invariant forms on all simple modules; finite orthogonal sums then supply strong unitarity on the grading-restricted category. Theorem 3.1 gives energy bounds on every unitary module, and Corollary 3.4 gives strong locality of V.

Take a generating family E of homogeneous quasi-primary vectors in V as in the energy argument. Since strong locality holds for all vacuum fields, it holds in particular for E. Choose one representative of every simple module, equipped with its positive form, as the family of generating charges. This family certainly tensor-generates the algebraic category. In each charge module choose a nonzero lowest-weight vector a. It is quasi-primary because L_1 lowers conformal energy. Corollary 6.3 and Lemma 4.5 give polynomial energy bounds for every intertwining operator between unitary modules with this charge. Proposition 5.1 gives the strong intertwining property for every homogeneous observable, hence for E. The same fixed vector a works for every irreducible intertwining operator with that charge; it has not been chosen separately for different sources or targets.

These are exactly Condition A of [17, Section 3.4]. The chosen charge vectors are quasi-primary, so Condition B holds as well. Thus [17, Theorem 3.15] identifies the canonical positive fusion structures with a unitary modular tensor category. This agrees with the normalization of Theorem 6.1; positivity of an unspecified sesquilinear form is not being substituted for canonical positivity.

The underlying fully faithful functor. The functor on objects is the representation characterized by (5.4). On a module morphism it is its bounded extension to Hilbert completions. Such extensions are bounded because the modules are finite orthogonal sums of simple modules, and a map between simple unitary modules is either zero or a scalar multiple of a unitary isomorphism. The Hilbert adjoint is again a module map. The exact smear identity shows that these extensions intertwine the local algebra actions. They are therefore the morphism maps of the CWX functor, rather than of an independently chosen equivalence. Full faithfulness, including preservation of adjoints, is [17, Theorem 3.17]; see also [14, Theorem 4.24]. The uniqueness in Theorem 5.3 identifies the functor there with ours. This full-faithfulness result uses the observable-smear argument of [18, Theorem 4.3]: localized stress-energy smears recover L_0 and then the algebraic module maps. It precedes the construction of a categorical extension and does not use tensor comparison.

Localizability of finite sums and products. We use the corrected localizability requirements in [15, item 3]. Here is their verification for the operators entering Gui’s categorical partial extension. Write D_j=1+L_0 on the Hilbert completion of a unitary module. The charged energy bounds and the L_0-commutator formula give, for each smeared field T:\mathcal H_j^\infty\to\mathcal H_k^\infty and each nonnegative integer s, an integer r_s and a constant C_s such that

Indeed, commuting L_0 past a smear differentiates its smooth test and adds its fixed conformal-weight term; iteration reduces the estimate to the polynomial mode bounds. The same argument applies to the formal adjoint. This is the Sobolev estimate used in [17, Section 3.1 and Proposition 3.2].

Every bounded net morphism S between objects of the CWX image is, by the full faithfulness just established, the bounded extension of a VOA morphism. All intermediate objects here are completions of VOA fusion products, on which the local actions are already defined. Thus S and S^* commute with the respective L_0 operators and preserve all Sobolev spaces. Gui’s left and right operators are products of a charged smear with finitely many observable smears [17, Equations (3.18)–(3.19)]. Their finite compositions, and those compositions followed by a bounded net morphism, therefore have the form covered by the preceding estimates. Consequently each finite sum, with common source and target,

is smooth, has a densely defined formal adjoint, and satisfies \|B\xi\|\leq C\|D_j^r\xi\| for some integer r. In particular B is closable. This includes the finite sums of left or right fields in the corrected Definition 2.29 and the finite sums with a bounded morphism required by corrected Definition 2.41.

To verify localizability, let \mathscr D_0\subset\mathcal H_j^\infty be a dense linear subspace that is quasi-rotation invariant: for some \delta>0 there is a dense \mathscr D_\delta\subset\mathscr D_0 such that e^{itL_0}\mathscr D_\delta\subset\mathscr D_0 for |t|<\delta. Then \mathscr D_0 is a core for D_j^r [5, Lemma 7.2]. To see the regularization explicitly, choose \varphi\in C_c^\infty(-\delta,\delta) and set

The rapid decay of the Fourier transform of \varphi makes D_j^rR_\varphi bounded. Hence R_\varphi turns Hilbert-norm approximations from \mathscr D_\delta into Sobolev-norm approximations. For a vector in \mathscr D_\delta, its convolution is itself a Sobolev limit of Riemann sums in \mathscr D_0. Taking smooth approximate identities for \varphi now approximates every vector of \operatorname{Dom}D_j^r in that graph norm. The displayed bound for B turns this approximation into graph approximation for B on \mathcal H_j^\infty. Hence \overline{B|_{\mathscr D_0}}=\overline B, as required. Thus the strengthened finite-sum localizability hypotheses hold.

Connes fusion, braiding, and finite index. Full faithfulness alone would only allow us to transport the VOA tensor structure onto its image. The additional comparison with the already defined net tensor structure is [17, Corollary 3.23]. It supplies unitary natural maps

compatible with the unit and associativity constraints. Here \boxtimes_{A_V} denotes Connes fusion, while \boxtimes denotes VOA fusion. Denote the VOA and net braidings by c^V and c^{\mathrm{DHR}}, using the same circle orientation. The maps satisfy

Thus the comparison is braided and unitary for the specified fusion and braiding on the net side.

Finally, the source category is rigid. Applying this unitary tensor functor to its evaluation and coevaluation maps preserves the conjugate equations. Their images exhibit a dual for every object in the image. The equivalence between dualizability and finite index for the net sectors then gives the last assertion.□

Proof. Write A=A_V. Reconstruction will give a weak module on the finite-energy vectors. We first establish the separability and discrete rotation spectrum needed for that construction; regularity will then produce a simple module whose Hilbert closure is the whole sector. The vacuum Hilbert space is separable, since it is the completion of the countable sum of the finite-dimensional spaces V_n. The Hilbert space \mathcal K is also separable. To see this, choose a countable basis of intervals and, in the unit ball of each corresponding vacuum algebra, a countable \sigma-strong-* dense subset. Such subsets exist because these von Neumann algebras have separable preduals. Normality and additivity imply that the images of these countably many subsets generate all the represented local algebras. The norm-closed unital *-algebra they generate is separable and acts irreducibly. Every nonzero vector is therefore cyclic for this algebra, which proves the claim.

The canonical inner Möbius implementation in a locally normal representation has positive energy [33, Theorem 3.8]; its Virasoro action and the associated annulus action are the ones used in [20, Sections 2, 4 and 9]. Let H\geq0 be the self-adjoint generator of the lifted rotation group. Covariance implies that e^{2\pi iH} commutes with every represented local algebra. Irreducibility makes it scalar. Thus, for some h\in[0,1), the spectral theorem gives

In particular, H has pure point spectrum. Its eigenspaces \mathcal K_\lambda=\ker(H-\lambda) need not yet be finite dimensional. Set

This is a dense subspace of \mathcal K.

The VOA reconstructed from A_V is V, with its given conformal vector and inner product [20, Theorem 8.9(ii)]. Here strong locality supplies AQFT-locality in the terminology of that theorem, and the required finite-dimensional vacuum energy spaces are precisely the spaces V_n. We now apply Proposition 9.3, Definition 9.4, Theorem 9.5 and Corollary 9.6 of [20], together with Remark 9.8 of that paper. The remark extends all these conclusions to discrete rotation spectra with possibly infinite-dimensional eigenspaces. They provide a weak unitary V-module structure Y^M on M such that

1. (i)

   the conformal operator L_0^M equals H|_M;

2. (ii)

   for every homogeneous a\in V, smooth test function f, and interval I containing its support,

   \pi_I\!\left(\overline{Y(a,f)}\right) =\overline{Y^M(a,f)|_M}. \tag{9.1}

On the left, a normal representation acts on an affiliated closed operator through its spectral and polar data. In particular, (9.1) asserts equality of closed operators, and M is a core for the operator on either side. It does not assert only equality of formal matrix coefficients. This is the part of the reconstruction theorem needed below.

Strong rationality implies regularity: every weak V-module is an algebraic direct sum of ordinary simple modules [1, Theorem 4.5]. Choose a nonzero ordinary simple submodule S\subset M. Its inherited Hermitian form is positive and invariant. Because the two conformal operators agree, the homogeneous spaces

are finite dimensional. Let Q be the orthogonal projection of \mathcal K onto the Hilbert closure of S. Orthogonality of the energy spaces gives

We will show that Q commutes with the represented net. The domain argument is necessary here: algebraic invariance of S alone would not give reduction of the closed smeared fields.

Both each mode and its algebraic adjoint preserve S. Indeed, the unitary adjoint formula expresses the latter as a finite linear combination of modes of PCT-transformed states and their L_1-descendants. Consequently S^\perp\cap M is invariant under every mode, and

The projection Q also commutes with every energy projection. For x\in M, the component of Y^M(a,f)x at a fixed energy is a finite sum of mode terms, since x has only finitely many energy components. Comparing these components proves

Both sides are Hilbert-space vectors by the reconstruction theorem, so componentwise equality suffices. The argument extends by finite sums to inhomogeneous a.

Put T=\overline{Y^M(a,f)|_M}. If x\in\operatorname{Dom}T, choose x_j\in M with x_j\to x and Tx_j\to Tx. Then

Closedness implies Qx\in\operatorname{Dom}T and TQx=QTx. The same holds with 1-Q in place of Q. Hence the orthogonal decomposition defined by Q reduces T, and Q commutes with its bounded polar data. By (9.1), these are the data of the represented vacuum smear. Such data generate \pi_I(A(I)), because vacuum smears generate A(I) and \pi_I is normal. Thus Q commutes with every represented local algebra.

Irreducibility and S\ne0 now force Q=1. Each S_\lambda is closed in \mathcal K_\lambda, so this identity implies S_\lambda=\mathcal K_\lambda for every \lambda. Therefore M=S. In particular the energy multiplicities are finite and M is an ordinary simple unitary module. Finally, (9.1) identifies its strongly integrated representation with \pi. The identification uses the actual closed smeared fields, as required by the CWX construction.□

Proof. For the nontrivial VOA, Corollary 3.4 supplies the net and the hypotheses of Proposition 9.1. That proposition shows that every irreducible locally normal sector belongs to the CWX image. There are only finitely many such sectors, since V has only finitely many simple modules. They have finite statistical dimension: Theorem 8.1 sends their rigid module duals and duality maps to net duals and duality maps. No finite-index assumption was used in Proposition 9.1 to obtain this conclusion.

The irreducible diffeomorphism-covariant circle net A_V has the split property by [29, Theorem 5.4]. The dichotomy [26, Theorem 4.9] applies to a split conformal net all of whose irreducible sectors have finite statistical dimension. It says that such a net is completely rational or has uncountably many inequivalent irreducible sectors. The finite list above excludes the second alternative. Thus A_V is strongly additive and has finite \mu-index, as well as being irreducible and split.

For a locally normal representation on a separable Hilbert space, complete rationality gives a direct sum of irreducible sectors of finite statistical dimension [25, Corollary 39 and the proof of Corollary 40; Appendix B]. The modular-PCT hypothesis is automatic for conformal nets, as explained in Section 3, before Corollary 10, of [25].

The same conclusion holds without separability. Indeed, apply the countable-generator construction from Proposition 9.1 to the present representation \pi. The resulting separable unital C*-algebra D satisfies

For any vector \xi, the separable cyclic space \overline{D\xi} reduces D, hence also D'' and every represented interval algebra. A maximal orthogonal family of these cyclic reducing spaces exhausts the representation Hilbert space: any nonzero orthogonal complement would supply another such space. Compression to a reducing space preserves local normality. Apply the separable decomposition [25, Corollary 39 and the proof of Corollary 40; Appendix B] to each restriction and combine the resulting orthogonal sums.

In a dualizable representation only finitely many irreducible summands can occur. Any finite sub-sum of n nonzero summands has statistical dimension at least n and at most the dimension of the representation. This excludes infinite multiplicity even if the full decomposition has an uncountable index set. The CWX functor preserves finite direct sums, and Proposition 9.1 supplies every irreducible summand. It is therefore essentially surjective onto \operatorname{Rep}^{f}(A_V). Full faithfulness, the unitary tensor structure and compatibility with the DHR braiding were established in Theorem 8.1, so the functor is the asserted equivalence.

For V=\mathbb C\Omega, the scalar net has \mu-index one and its additive dualizable representation category is the category of finite-dimensional Hilbert multiplicity spaces. The same is true of finite-length unitary V-modules, with their ordinary tensor product and flip braiding. This gives the stated conclusion in the remaining case as well.□

Proof. If V=\mathbb C\Omega, the common zero conformal vector and CFT type force U=\mathbb C\Omega. A finite-index extension of the scalar net has finite-dimensional interval algebras. Since the interval factors of a nontrivial irreducible conformal net are type III [5, Section 3.3, property (7)], this extension is also scalar. The functors and tensorators then reduce to those of finite-dimensional Hilbert spaces. We may therefore assume V\ne\mathbb C\Omega.

Theorems 6.1 and 7.1 together give complete unitarity, and Corollary 6.3 gives energy bounds and strong intertwining for every irreducible unitary intertwining operator. These are Gui’s Condition I [14, Definition 4.15]. Theorem 9.2 adds complete rationality and essential surjectivity of the CWX functor, so V satisfies Condition I+ of [14, Definition 6.17]. We use Gui’s Wassermann tensorator on the CWX functor; its underlying functor is the one identified in Section 8.

Gui’s Theorems 3.13 and 5.3 classify the stated normalized net and unitary VOA extensions by normalized haploid commutative C^*-Frobenius algebras [14]. A finite-index net extension has a dualizable Q-system object, hence one in \Rep^f(A_V); every such object is a finite sum of VOA completions by Theorem 9.2, so is separable as required by Gui’s convention. The braided unitary base equivalence therefore transports all these algebra objects and their isomorphisms. For the algebraic VOA formulation, regularity from Section 2 and complete unitarity allow [4, Theorem 4.7], which gives the asserted unique unitary structure for simple CFT-type extensions.

Finally, Gui’s Corollary 6.5 and Theorem 6.18(2) pass Condition I+ to U. His Theorem 6.11 and Corollary 6.16 then give A_U=B_\Theta and the displayed equality of braided unitary tensor functors, with the full finite-index sector category as target [14].□