Hermitian quasi-primary generators determine the unitary PCT structure

For homogeneous $a\in V_d$, put $R(z)=\exp (zL(1))(-z^{(-2)}{)}^{L(0)}$ and define the anti-linear-in-a field

D(a,z)=[Y(R(z)a,z^(−1))]†.

The formal variable is fixed under dagger and scalar coefficients are conjugated; product order is reversed. More explicitly,

D(a,z)=Σ_n D(a)n z^(−n−1), D(a)n=(−1)^d Σ(j≥0) (1/j!) [(L(1)^j a)(2d−j−n−2)]†.

This j sum is finite, because L(1) lowers the nonnegative grading. Each summand before dagger has energy shift n+1−d; hence $D(a{)}_n$ has shift d−n−1. In particular $D(a{)}_{n}v=0$ for all sufficiently large n on each finite-grade-support v. Thus D(a,z) is an actual vectorwise lower-truncated field, even if the untransposed opposite field is upper truncated.

Here are the needed opposite identities, with their local derivation so that no named unitarity or reconstruction proposition is imported. The state Möbius commutators are

[L(0),Y(a,t)]=Y(L(0)a,t)+t∂_tY(a,t), [L(1),Y(a,t)]=Y(L(1)a,t)+2tY(L(0)a,t)+t²∂_tY(a,t).

They follow by taking the conformal-vector residues of Jacobi. Integrating them as formal power-series differential equations gives

$b^{L(0)}Y(a,t)b^{(-L(0))}=Y(b^{L(0)}a,bt)$, $\exp (sL(1))Y(a,t)\exp (-sL(1))=Y(\exp (s(1-st)L(1))(1-st{)}^{(-2L(0))}a,t\mathrm{/}(1-st))$.

Substitution yields the exact iterate transformation

$R(z)Y(a,x)b=Y(R(z+x)a,-x\mathrm{/}[z(z+x)])R(z)b$,

expanded in nonnegative powers of x. These identities are verified by differentiating both sides in s or the scaling parameter and comparing their value at zero; the coefficient recursions determine the series uniquely. The finite state exponential introduces no infinite-state-support issue.

For completeness the opposite Jacobi expansion follows coefficientwise. For homogeneous states a,b,u and a homogeneous restricted functional λ, the two products of Y(a,t1),Y(b,t2) paired with λ,u are expansions of one rational function with poles only at $t1=0,t2=0,t1=t2$. Indeed choose N such that $a_{(r)}b=0$ for $r\ge N$. Jacobi gives locality after multiplying by $(t1-t2{)}^N$. Truncation bounds the negative t2 exponent in the first ordering and the negative t1 exponent in the second. The homogeneous weight balance fixes their sum; therefore after multiplying by that power the common scalar series has only finitely many monomials and is Laurent polynomial. The iterate expansion follows by the same Jacobi residue after $t1=t2+t0$, first multiplying by a sufficient power of t1 to allow substitution on u. This is purely a formal Laurent-series argument.

Apply this rational-function statement to each of the finitely many states in R(z1)a,R(z2)b and substitute $t_i=z_i^{(-1)}$. Inversion reverses the product expansion order. Set $z1=z2+z0$ in the iterate expansion and use the preceding iterate transformation; since $t1-t2=-z0\mathrm{/}(z1z2)$, the inner state becomes Y(a,z0)b. The elementary formal delta identity, multiplied by each Laurent monomial times a power of (z1−z2), now gives opposite Jacobi for $Y^o(a,z)=Y(R(z)a,z^{(-1)})$, with both product orders reversed. Equivalently it is the ordinary module Jacobi identity with $Y^o(b,z2)Y^o(a,z1)$ in the first product and $Y^o(a,z1)Y^o(b,z2)$ in the second. All statements are separated by the restricted functionals, so they are coefficientwise identities on V. This proof uses only nonnegative grading, finite-dimensional grades, and the established vertex-algebra Jacobi, not $V0=C1$.

Taking the graded adjoint reverses those product orders back. Consequently D satisfies the ordinary Jacobi identity with iterate D(Y(a,z0)b,z2). Taking its defining nth-product residues gives

D(a_(n)b,z)=D(a,z)_(n)D(b,z) (every integer n).

The sums are finite on a fixed vector by lower truncation and locality, exactly as in the reconstruction lemma's mode formula. This identity is the full graded-product adjoint assertion; it is not obtained by incorrectly commuting a raw adjoint past a normal ordering.

The opposite Möbius identities follow directly from the three sl2 commutators and differentiation of R(z):

$[Y^o(a,z),L(1)]={\mathrm{\partial }}_z{Y}^o(a,z)$, $[Y^o(a,z),L(0)]=Y^o(L(0)a,z)+z{\mathrm{\partial }}_z{Y}^o(a,z)$, $[Y^o(a,z),L(-1)]=Y^o(L(1)a,z)+2z{Y}^o(L(0)a,z)+z^2{\mathrm{\partial }}_z{Y}^o(a,z)$.

For example R'(z)=R(z)L(−1)+z^(−2)L(−1)R(z) gives the first identity using translation covariance. The other two follow by substituting the finite adjoint sl2 identities for R. Taking daggers and using Möbius unitarity gives

[L(−1),D(a,z)]=∂_zD(a,z), [L(0),D(a,z)]=D(L(0)a,z)+z∂_zD(a,z), [L(1),D(a,z)]=D(L(1)a,z)+2zD(L(0)a,z)+z²∂_zD(a,z).

Define $\theta (a)=D(a{)}_{(-1)}1$. Since D(a) is a field and is translation covariant, D(a,z)1 solves ${\mathrm{\partial }}_{z}v=L(-1)v$ as a lower-bounded Laurent series. Its least possible negative coefficient must vanish by comparing the derivative's least power with the right side; induction excludes all negative coefficients. Thus

D(a,z)1=exp(zL(−1))theta(a).

The two families D(a),D(b) are mutually local: this follows either from their Jacobi just established, or by taking adjoints in the finitely many localities for the state components L(1)^j a,L(1)^k b and using (z1^(−1)−z2^(−1))^N=(z2−z1)^N/(z1z2)^N. For a quasi-primary generator a of weight d, the explicit mode formula and Hermitian ENERGY adjoints give

D(a,z)=(−1)^dY(a,z), theta(a)=(−1)^d a.

Therefore every D(b) is local with each original generating field Y(a), not just with the other D fields. The uniqueness argument in the formal reconstruction lemma applies to D(b)−Y(theta(b)): its vacuum series is zero; locality propagates its vanishing from 1 to each generating mode word by cancelling a power of z−w in V((z))((w)); the words span V. Hence

D(b,z)=Y(theta(b),z) for every b∈V.

In particular theta is defined intrinsically on states, not by choosing an expression for a state as a word. The nth-product identity in the opposite-field calculation above and state-field injectivity give

theta(a_(n)b)=(theta a)_(n)(theta b), theta(1)=1.

Theta is anti-linear. Its square fixes each generator, the vacuum, and all their mode words, hence theta²=1. Thus theta is an anti-linear vertex-algebra automorphism. Taking the constant vacuum coefficient in the displayed Möbius identities shows theta L(j)=L(j)theta for j=−1,0,1. In particular it preserves each grade and translation. None of these steps assumes that the original FJ involution intertwines all reconstructed products; only its Hermitian generator combinations were used.

Put $h_{\theta }(u,v)=⟨\theta v,\theta u⟩$. This is a positive Hermitian form, linear in its first argument. For a Hermitian generator a of weight d put $\epsilon =(-1{)}^d$. The automorphism identity and grade preservation give $\theta a_n=\epsilon a_n\theta$. Hence, using $a_n^{†}=a_{-n}$,

$h_{\theta }(a_{n}u,v)=⟨\theta v,\epsilon a_n\theta u⟩=⟨\epsilon a_{-n}\theta v,\theta u⟩=h_{\theta }(u,a_{-n}v)$.

Thus the same generator-mode adjoints hold for $h_{\theta }$ as for the original Hilbert form. Their vacuum functionals agree. For w of positive weight, $⟨\theta w,1⟩=⟨1,w⟩=0$ by orthogonal grades. For w of weight zero the explicit formula in the opposite-field calculation above reduces to $\theta (w)=(w_{(-1)}{)}^{†}1$, and therefore

$h_{\theta }(1,w)=⟨\theta w,1⟩=⟨1,w_{(-1)}1⟩=⟨1,w⟩$.

The equality extends by linearity in the appropriate slots to every w. Now write the first argument of either form as a finite generating mode word, move its leftmost mode to the second argument using the common adjoint relation, and induct on that word's length. The terminal expression has first argument 1, where the forms agree. Since these words span V, $h_{\theta }=h$ on all V. Equivalently,

〈theta u,theta v〉=〈v,u〉.

Theta is therefore antiunitary and involutive. This proof did not need V0=C1 or simplicity.

First $T=L(-1)$ kills V0: for $v\in V0$, $L(1)v=0$ by grading and $\mathrm{\mid }\mathrm{\mid }Tv\mathrm{\mid }{\mathrm{\mid }}^2=⟨v,L(1)Tv⟩=⟨v,2L(0)v+TL(1)v⟩=0$. Let ω be the given conformal vector and put $\delta =\theta (\omega )-\omega$. Since theta commutes with L(0),L(1), δ is quasi-primary of weight two. Both conformal vectors have zero mode T, because theta preserves all vertex products and commutes with T. Therefore ${\delta }_{(0)}=0$ as an operator.

Skew symmetry gives Y(δ,z)ω=exp(zT)Y(ω,−z)δ. Its z^(−1) coefficient is

δ_(0)ω=−Tδ+T(2δ)−(T²/2)L(1)δ+(T³/6)L(2)δ=Tδ.

The third term vanishes by quasi-primarity; L(2)δ lies in V0, so the fourth term vanishes by TV0=0; all subsequent terms have negative weight and vanish. Thus Tδ=0. Möbius unitarity then gives 0=||Tδ||²=〈δ,L(1)Tδ〉=4||δ||². Positivity implies δ=0. Consequently theta(ω)=ω, and theta commutes with every Virasoro mode as well. No full Virasoro adjoint relation was assumed to prove this.

The field identity D(theta a,z)=Y(a,z), theta²=1, and finite-block double adjunction give

Y(a,z)†=Y(exp(zL(1))(−z^(−2))^L(0) theta(a),z^(−1)).

For homogeneous $a\in V_d$ this is the explicit coefficient identity

(a_(n))†=(−1)^d Σ_(j≥0) (1/j!) (L(1)^j theta(a))_(2d−j−n−2).

It has finite j support, and is the adjoint formula for arbitrary descendants, not only quasi-primary generators. For a homogeneous product $c=a_{(r)}b$ of weight $e=wt(a)+wt(b)-r-1$, substitute $\theta (c)=(\theta a{)}_{(r)}(\theta b)$ in the same formula to obtain its full product-mode adjoint, with factor $(-1{)}^e$ and index 2e−j−n−2. If $c=0$ the formula is zero, independently of its chosen expression. Every raw composition has $(AB{)}^{†}=B^{†}{A}^{†}$ on finite graded blocks; the opposite transformation and state-product identity above explain how this reversal becomes the vertex product formula.

Define B(u,v)=〈u,theta v〉. It is bilinear; antiunitarity and theta²=1 imply B(u,v)=B(v,u), and it is nondegenerate. Since theta fixes ω it commutes with R(z). Therefore the full adjoint identity yields, coefficientwise,

$B(Y(a,z)b,c)=⟨Y(a,z)b,\theta c⟩=⟨b,Y^o(\theta a,z)\theta c⟩=⟨b,\theta Y^o(a,z)c⟩=B(b,Y(R(z)a,z^{(-1)})c)$.

This is the complete invariant bilinear identity required by the local unitary-VOA definition. The Hilbert form remains its given positive normalized form. In particular ω is fixed, L(n)†=L(−n) follows by applying the descendant formula to ω, and all reconstructed vertex fields have the required adjoint on the algebraic finite-energy domain. Hilbert closure and common-core assertions for smeared fields are the separate analytic energy-bound arguments; none are silently inferred from this formal identity.