Strongly graded contragredient duality for Möbius and conformal modules

We first establish opposite-field identities in this exact generality. For $v\in V_{(d)}^{(\alpha )}$, its opposite modes are $v_n^o=(-1{)}^d{\sum }_{r\ge 0}\frac{(L(1{)}^{r}v{)}_{2d-r-n-2}}{r!}\mathrm{.}$ The sum is finite by F1. Each summand has auxiliary-degree shift $\alpha$ and generalized-weight shift $n+1-d$. For every fixed $w$, module truncation shows $v_n^{o}w=0$ for all sufficiently negative $n$. The opposite vacuum is the identity. These statements use neither a global bound on $W$ nor a lower bound uniform in auxiliary degree.

The Möbius brackets on states imply the covariance identities $b^{L(0)}Y(u,t)b^{-L(0)}=Y(b^{L(0)}u,bt),e^{aL(1)}Y(u,t)e^{-aL(1)}=Y(e^{a(1-at)L(1)}(1-at{)}^{-2L(0)}u,\frac{t}{1-at})\mathrm{.}$ For the second identity, differentiation in $a$ on each side gives the same formal differential equation, using $[L(1),Y(u,t)]=Y(L(1)u,t)+2tY(L(0)u,t)+t^2{\mathrm{\partial }}_{t}Y(u,t)$; the constant terms agree. Recursion in $a$ proves equality coefficientwise. The first follows by the weight commutator. Combining them, with all powers expanded in nonnegative powers of $t$, gives $B(s)Y(u,t)v=Y(B(s+t)u,-\frac{t}{s(s+t)})B(s)v\mathrm{.}$ This calculation uses the state-field Möbius axioms themselves, so it does not import a regular-Jacobi result restricted to CFT-type algebras.

The three Lie brackets give $L(-1)B=-B(x^{2}L(-1)+2xL(0)+L(1)),L(0)B=B(L(0)+x^{-1}L(1)),L(1)B=-x^{-2}BL(1)\mathrm{.}$ Also $B^{\mathrm{\prime }}=B(-2x^{-1}L(0)-x^{-2}L(1))=BL(-1)+x^{-2}L(-1)B$. The chain rule with module translation proves ${\mathrm{\partial }}_x{Y}_W^o(v,x)=Y_W^o(L(-1)v,x)$. Substituting the three displayed identities in the module Möbius brackets proves $[Y_W^o(v,x),L_W(-j)]={\sum }_{k=0}^{j+1}\left(\genfrac{}{}{0px}{}{j+1}{k}\right)x^k{Y}_W^o(L(j-k)v,x),j=-1,0,1.$ Finally scaling $L(1)$ across $(-x^2{)}^{L(0)}$ gives $B(x^{-1})B(x)=1$. All state exponentials used here are finite by F1.

Here is the rational-expansion justification of opposite Jacobi. Fix homogeneous $u,v,w,f$ with $f\in W^{\mathrm{\prime }}$. Module Jacobi makes the scalar product orders and iterate the three expansions of a rational function in $t_1,t_2$ with poles only at $t_1{t}_2(t_1-t_2)=0$. Indeed choose $N$ such that $u_{k}v=0$ for $k\ge N$. The commutator residue of Jacobi gives equality of the two product expansions after multiplication by $(t_1-t_2{)}^N$. In the first expansion truncation on $w$ bounds the negative $t_2$-powers; in the second it bounds the negative $t_1$-powers. The exact generalized-weight shift and the one output weight selected by $f$ fix the sum of the two exponents. Consequently the common multiplied series has both exponents bounded below and is a Laurent polynomial. Multiplying Jacobi by an additional sufficiently large power of $t_1$, taking its $t_1$-residue and using truncation on $w$ permits $t_1=t_2+t_0$ and gives the iterate expansion of that same rational function. Finite sums handle arbitrary inputs. Auxiliary degrees merely make some coefficients zero; no sum over degrees is introduced.

Transposition of the mode shift in step 1.1 sends $f\in (W^{\mathrm{\prime }}{)}_{[h]}^{(\beta )}$ to degree $\alpha +\beta$ and weight $h+d-n-1$. The target is dual to $W_{[h+d-n-1]}^{(-\alpha -\beta )}$. As $n\to +\mathrm{\infty }$ this runs down ONE fixed integer string in ONE fixed auxiliary degree, so strong lower truncation makes it zero. Finite support of $f$ and $v$ gives one common bound. Thus the transpose preserves the restricted direct sum and is a lower-truncated field on it. This is the precise replacement for the unavailable global lower-bound argument.

Apply step 2.1 to the finite states $B(x_1)u,B(x_2)v$ and put $t_i=x_i^{-1}$. Inversion reverses the order of the radial formal expansions. In the iterate expansion $x_1=x_2+x_0$, step 1.2 identifies the transformed inner state and the difference $t_1-t_2=-x_0\mathrm{/}(x_1{x}_2)$. Thus the opposite products in reversed order and $Y_W^o(Y(u,x_0)v,x_2)$ are precisely the three expansions of one rational function with poles only on $x_1{x}_2(x_1-x_2)=0$. The formal delta identity for these expansions follows by linearity from the basic delta identity, multiplied by the respective expansions of $x_1^p{x}_2^q(x_1-x_2{)}^r$, $p,q,r\in \mathbb{Z}$: on delta support replace $x_1-x_2$ by $x_0$ and $x_1$ by $x_2+x_0$. Every rational function in question is a finite sum of such monomials. The separating restricted functionals therefore prove opposite Jacobi, with reversed operator products, coefficientwise on $W$.

Pair the candidate module axioms against $w\in W$. Opposite vacuum transposes to vacuum. Transposition reverses the two opposite products in step 3.1, producing ordinary module Jacobi; the expansions are well defined by step 2.2. Translation and the three Möbius brackets transpose from step 1.3. Since transposition reverses a commutator while $j\mapsto -j$ reverses the ${\mathfrak{s}\mathfrak{l}}_2$ bracket, $L^{\mathrm{\prime }}(j)=L(-j{)}^t$ is the required representation. In the conformal case $B(x)\omega =x^{-4}\omega$, so all $L^{\mathrm{\prime }}(n)=L(-n{)}^t$ are the conformal-field coefficients. The same two reversals preserve the Virasoro central term.

On each finite component the transpose of $L(0)-h$ is nilpotent, so its dual is the generalized-weight-$h$ space. Its dimension and lower truncation in each fixed string agree with those of $W^{(-\beta )}$. There is no assertion of a nilpotence exponent uniform in all components. This verifies every strong-grading requirement for $W^{\mathrm{\prime }}$. Finite-dimensional evaluation identifies $W_{[h]}^{(\beta )}$ with the corresponding double dual. Applying the defining action twice and using $B(x^{-1})B(x)=1$ proves that evaluation commutes with the fields; $L^{\mathrm{\prime }}(j)=L(-j{)}^t$ proves the same for Möbius operators.

For a grading-preserving module map $g:W\to U$, restriction of $g^t$ maps $U^{\mathrm{\prime }}$ to $W^{\mathrm{\prime }}$ because a component maps to that same component. Pairing the field actions and using $g{Y}_W^o=Y_U^{o}g$ proves that $g^{\mathrm{\prime }}$ intertwines all fields, and transposing $gL(j)=L(j)g$ proves the Möbius condition. Transposition reverses compositions and preserves identities. Hence it is the asserted contravariant functor, compatible with the evaluation isomorphisms.