Explicit cylinder sewing inverse by logarithmic coefficient extraction

Let $\tilde{y}$ be F2's logarithmic lift of $q{\mathrm{\partial }}_q$, and put $A(q)={\sum }_i\mathrm{Res}{h}_i(q,t_i)Y_i(\omega ,t_i)d{t}_i$ on W. Let $\kappa (q)$ be the holomorphic Schwarzian scalar constructed in F3 for this lift and a meromorphic projective coefficient on the normalization. The operator $q{\mathrm{\partial }}_q-A(q)+\kappa (q)$ preserves the Ward ideal on the smooth locus by F2. It is a connection there after division by q. In one complex dimension it has no curvature two-form. For completeness, every cylinder sewing is parallel for its dual connection: the sphere Ward identity makes the completed vector corresponding to chi supported on diagonal weights, with equal nilpotent component actions. After multiplication by $p_1^{L_{+}(0)}{p}_2^{L_{-}(0)}$ it depends only on $q=p_1{p}_2$ and satisfies the annular residue balance, by moving a mode across the sphere pairing and conjugating by these powers. The two paired raw conformal residues of the lift therefore add with coefficients $a+b=1$ to its q derivative. The normalization conformal Ward identity of F3 then gives $q{\mathrm{\partial }}_{q}S(w)=S(-A(q)w)+\kappa (q)S(w)$, precisely dual parallelism. This is the same finite-coefficient calculation as F3, with the sphere pairing in place of its identity coevaluation. There is no assumption that an arbitrary sewing is parallel for the uncorrected Virasoro operator.

We first establish the finite-order extension actually needed below; F2 is not assumed to supply an outgoing-jet matching inverse. Fix a finite state cutoff and an incoming pole bound $k\ge k_0$, and set $E=V_{\le d,C}\otimes {\omega }_{C\mathrm{/}\mathrm{\Delta }}(k{D}_{\mathrm{i}\mathrm{n}})$ on a small disk about $q=0$. By F2 there are actual total holomorphic sections $s_1,\dots ,s_a$ whose central restrictions form a basis of $H^0(C_0,E_0)$. Put $C_h=(C_0,({\mathcal{O}}_C\mathrm{/}(q^{h+1})){\mathrm{\mid }}_{C_0})$ and $E_h=E\mathrm{/}{q}^{h+1}E$; these symbols denote a finite ringed-space thickening only for this local argument, not an assumed base-change theorem for nonreduced bases. Multiplication by $q$ is injective on $E$: in a node chart $q=xy$ is a nonzero divisor in $\mathbb{C}\{x,y\}$, and off the node it is a coordinate. Consequently the kernel of $E_h\to E_0$ is canonically $q{E}_{h-1}$. For any global section $t$ of $E_h$, express $t{\mathrm{\mid }}_{C_0}={\sum }_i{a}_{i,0}{s}_i{\mathrm{\mid }}_{C_0}$ and subtract this actual section. The difference divides uniquely by $q$ as a global section of $E_{h-1}$. Repeat finitely many times. This writes $t={\sum }_{j=0}^h{q}^j{\sum }_i{a}_{i,j}{s}_i$ modulo $q^{h+1}$, so the polynomial combination on the right is an actual holomorphic extension. Uniqueness of division makes the local quotients glue. The argument uses actual central section surjectivity and a finite induction; no convergence of a formal correction or Banach matching inverse is required.

The normalization interpolation used below also has a direct finite-dimensional proof. At either outgoing point allow a specified finite pole order and prescribe finitely many Laurent coefficients in a finite state-weight truncation. Let $Z$ be the sum of the two outgoing points. The kernel of this finite jet map is a finite-rank bundle whose filtration quotients are fixed line bundles further twisted by a fixed multiple of $-Z$ and by $k{D}_{\mathrm{i}\mathrm{n}}$. Since $D_{\mathrm{i}\mathrm{n}}$ meets every normalization component, increasing $k$ makes every quotient degree exceed the corresponding canonical degree. F5 gives zero $H^1$ for the kernel, first for its line quotients and then by the finite filtration. The jet exact sequence is therefore surjective on global sections. A linear right inverse exists because its target is finite dimensional. This proves simultaneous independent interpolation at both branches with finite incoming poles; it does not assume any coefficient belongs to D.

Given a block ${\psi }_{q_0}$ on any smooth fiber, extend it uniquely to a parallel block $\psi$ on the universal cover of the punctured q-disk, with its initial value specified at the chosen lift $\mathrm{\ell }={\mathrm{\ell }}_1+{\mathrm{\ell }}_2$ of $q_0$. Locally this is the finite matrix equation in a block-bundle frame: Picard iterated integrals converge on each relatively compact disk by their factorial bound. Finite chains of such disks give continuation along any compact path; uniqueness on overlap and subdivision of homotopies give the solution on the simply connected cover. F2's fixed finite generators give its evaluation column a finite Euler system $q{\mathrm{\partial }}_{q}f=B(q)f$ with B holomorphic on the whole disk: reduce the finitely many images under the operator in step 1.1 modulo the global Ward span. This construction is valid even if this generating column has redundant coordinates.

We prove the regular-singular expansion needed here. Take a basis of the rank-s block space at $q_0$ and let Y(q) be its d-by-s evaluation matrix on the finite generators. Parallel continuation around zero gives an invertible s-by-s monodromy matrix T, with $Y(t+2\pi i)=Y(t)T$ for $q=e^t$. Choose a matrix C with $e^{2\pi iC}=T$: on each Jordan block write $T=\lambda (I+N)$ and use a logarithm of the nonzero scalar $\lambda$ and the finite series for $\log (I+N)$. Then $H(q)=Y(q)e^{-tC}$ is single valued and holomorphic on the punctured disk. It has at most a pole at zero. Indeed boundedness of B gives along every radial segment $\mathrm{\parallel }Y(q)\mathrm{\parallel }\le C_1\mathrm{\mid }q{\mathrm{\mid }}^{-C_2}$ by the integral inequality and its elementary exponential iteration. A bounded interval of arguments adds a uniform finite factor. The same polynomial bound, times a finite logarithmic power, holds for $e^{-tC}$ by its Jordan form. Thus $\mathrm{\parallel }H(q)\mathrm{\parallel }\le C_3\mathrm{\mid }q{\mathrm{\mid }}^{-C_4}$; the Cauchy integral for its Laurent coefficients makes all sufficiently negative coefficients zero. Hence $Y=H(q)e^{tC}$ has a convergent finite-exponent, uniformly finite-log expansion. It converges on the full punctured disk, since H is holomorphic there and meromorphic at zero.

Every W vector is a holomorphic combination of the same finite generators modulo global Ward relations, by F2. Multiplying their expansions by those holomorphic coefficients therefore gives $\psi (w)={\sum }_{n\in E+\mathbb{N}}{\sum }_{l=0}^L{\psi }_{n,l}(w)q^n(\log q{)}^l,$ with one finite exponent-coset set and one L for all w. Combine equal exponents and choose distinct coset representatives. Coefficients are linear functionals on W and all indicated series converge absolutely locally uniformly. Such coefficients are unique: the deck shift on the finite log space has distinct eigenvalues $e^{2\pi ie}$ for distinct cosets, so finite polynomial spectral projectors separate the cosets. In one coset repeated finite differences in $\log q\mapsto \log q+2\pi i$ separate its logarithmic powers, and Taylor uniqueness separates the remaining integral powers. The same argument proves that the Ward identities for any total holomorphic nodal current $v(q)={\sum }_{k\ge 0}{q}^k{v}_k$ give ${\sum }_{k\ge 0}{\psi }_{m-k,l}(v_k\cdot w)=0.$ For fixed w the residue action of v(q) has values in a fixed finite W-span, by its finite state and pole cutoffs. Therefore its convergent Taylor series may be paired termwise and regrouped; the displayed coefficient sum is finite by the common lower exponent bound.

Each coefficient ${\psi }_{n,l}$ belongs to the actual D of F1. If $\psi =0$, this is immediate; otherwise the finite-exponent expansion from step 3.1 has a nonempty representative set. Let $\mathcal{E}$ be the finite set of exponent representatives furnished by steps 3.1–4.1, put $\beta ={\min }_{\alpha \in \mathcal{E}}\mathrm{Re}\alpha$, and let $b$ be a common real lower bound for the generalized weights of D, supplied by F1. Set $\underset{‾}{b}=\min \{\beta ,b\}$. For the fixed complex exponent n choose an integer $h\ge 0$ with $h+1>\mathrm{Re}n-\underset{‾}{b}$. This real-part inequality ensures that an $O(q^{h+1})$ error cannot contribute to the coefficient at exponent n: by step 4.1 every term it multiplies has exponent $\alpha +m$ with $\alpha \in \mathcal{E}$ and $m\in \mathbb{N}$, so every resulting exponent has real part at least $\beta +h+1\ge \underset{‾}{b}+h+1>\mathrm{Re}n$. The bound also incorporates the module lower bound b; it does not require identifying the target coefficient as a module weight. Let sigma be any normalization current in the outgoing lattice at levels $(h,h)$, with arbitrary finite incoming poles. In the node current frame its two branch coefficients vanish to order at least h+1: a state of weight d contributes $x^{d+h}dx=x^{h+1}(x^{d}dx\mathrm{/}x)$, and similarly on the other branch. Define the node section by the sum of these two one-variable series. On the exterior take sigma constant in q. Its matching error is $O(q^{h+1})$, since every crossed series monomial is $q^m{x}^{-m}$ or $q^m{y}^{-m}$ with $m\ge h+1$. These exterior and node sections therefore glue modulo $q^{h+1}$ to a section of $E_h$, after choosing a finite state cutoff and an incoming pole bound containing $\sigma$ and at least $k_0$. Step 1.2 extends it to an actual total current $v(q)=\sigma +O(q^{h+1})$ on the exterior, with a finite incoming pole bound. Step 4.1 then says ${\psi }_{n,l}(\sigma \cdot w)=0$. Since h is fixed for the coefficient, this holds for every sigma in that lattice. Thus ${\psi }_{n,l}\in (W\mathrm{/}{J}_{h,h}{)}^{\ast }\subset D$. This proves membership, not merely formal outgoing truncation in W*.

If $\mathrm{\Gamma }(\chi )=0$ at the chosen nonzero parameter, its cylinder sewing is a parallel section for step 1.1 and hence zero everywhere by finite ODE uniqueness. Sphere Ward for $L(0)$ makes chi supported on diagonal joint weights, with matching nilpotent actions. Its constant-log coefficient at $q^n$ is the vector ${\chi }_{[n,n]}\in D_{[n,n]}$ evaluated on W. Expansion uniqueness from step 4.1 and the injection $D\subset W^{\ast }$ make every such vector zero, hence chi zero. This proves injectivity independently of a dimension comparison.

We next prove the annular balance relation for these D-valued coefficients. For every state u and every power series f(x,y), ${\mathrm{Res}}_{x}f(x,q\mathrm{/}x)Y_{+}(x^{L(0)-1}u,x){\sum }_n{\psi }_{n,l}{q}^{n}dx={\mathrm{Res}}_{y}f(q\mathrm{/}y,y)Y_{-}(y^{L(0)-1}Ju,y){\sum }_n{\psi }_{n,l}{q}^{n}dy,$ where $J=e^{L(1)}(-1{)}^{L(0)}$. Here is the finite-jet justification. Work modulo each exponent coset times $q^{h+1}$; only finitely many coefficient vectors occur, all in D by step 5.1. Choose a branch jet order Q so large that every error at order Q annihilates those finitely many vectors under both outgoing residue actions, using their uniform state-weight pole cutoff in F1. On the normalization, lift the desired branch jets of the node current $f(x,y)x^{L(0)}udx\mathrm{/}x$ coefficientwise through q order h. Step 1.3 gives these lifts with finite incoming poles: for the finitely many q-coefficients under consideration prescribe the finite Laurent jets of the desired branch expressions through order Q in the normalized node frame. Their negative powers are bounded by the q-truncation, so this is a finite interpolation problem. Their branch errors have order Q. Add these two regular one-branch error series to the local node normal form, so the same-branch errors match exactly. Choose $Q>h$ as well. The remaining crossed errors are then $O(q^{h+1})$, so the tuple is an actual section modulo $q^{h+1}$. Step 1.2 extends it to a total nodal current without changing its first h exterior coefficients, after enlarging its finite incoming pole bound if necessary. Step 4.1 annihilates its incoming residue convolution. Apply the universal evaluation block of F1 to the same normalization coefficients: its incoming residues are minus the two outgoing actions. The chosen Q kills the branch errors on our finite coefficient vectors, leaving exactly the displayed difference, with the sign $dx\mathrm{/}x=-dy\mathrm{/}y$. This proves the identity modulo every h, hence coefficientwise, for arbitrary f; only finitely many coefficients of f enter each truncation.

Define degree-indexed modes $u[k]=\mathrm{Res}{z}^{k+\mathrm{wt}u-1}Y(u,z)dz$, and define $u^o[k]$ with the opposite field. In step 6.1 use $f=x^k$ if $k\ge 0$ and $f=y^{-k}$ if $k<0$, and make the substitution $y=z^{-1}$ on its right side. The opposite-field formula and residue inversion give $u_{+}[k]{\psi }_{n,l}=u_{-}^o[k]{\psi }_{n-k,l}\mathrm{.}$ This explicitly keeps the shifted mode convention: $u[k]$ lowers weight by k, while $u^o[k]$ raises it by k. For the conformal vector this includes $L_{+}(0){\psi }_{n,l}=L_{-}(0){\psi }_{n,l}$.

Write the lift near its fixture as $a(x,y)x{\mathrm{\partial }}_x+b(x,y)y{\mathrm{\partial }}_y$, with $a+b=1$. The Schwarzian Ward identity from F3 applied on the normalization to each coefficient field of the lift, and the parallel equation in step 1.1, give $q{\mathrm{\partial }}_q\psi ={\sum }_{m\ge 0}{q}^m(\nu (v_m{)}_{+}+\nu (v_m{)}_{-})\psi \mathrm{.}$ Indeed the incoming term from the parallel equation is $-\psi (A(q)w)+\kappa \psi (w)$; the raw conformal Ward scalar says this is the paired outgoing action. Step 6.1 with $u=\omega$ moves the minus residue to the plus branch, where its coefficient adds to a. Their sum is one, so the right side is $L_{+}(0)\psi$, and also $L_{-}(0)\psi$. Comparing coefficients gives $(L_{\pm }(0)-n){\psi }_{n,l}=(l+1){\psi }_{n,l+1}\mathrm{.}$ The finite chain in l proves ${\psi }_{n,l}\in D_{[n,n]}$, and its nilpotent conformal parts satisfy $N_{\pm }{\psi }_{n,l}=(l+1){\psi }_{n,l+1}$. This derives both weights and logarithmic coefficients, rather than presuming that an expansion exponent is a module weight.

Define $\chi \in (D^{\mathrm{\prime }}{)}^{\ast }$ by $\chi (d^{\mathrm{\prime }})={\sum }_n⟨{\psi }_{n,0},d^{\mathrm{\prime }}⟩$. This is a finite sum for every d', since step 8.1 puts its summands in distinct diagonal joint weight blocks and d' has finite support. Step 7.1, summed over n and transposed in the restricted pairing, says $\chi (u_{+}^o[k]d^{\mathrm{\prime }})=\chi (u_{-}[k]d^{\mathrm{\prime }})$. These are precisely the two-point sphere Ward identities in coordinates $1\mathrm{/}z,z$: Laurent monomials and the opposite transformation exhaust the global meromorphic currents there. Thus $\chi \in {\mathrm{C}\mathrm{B}}_N(D^{\mathrm{\prime }})$. Moreover $p_1^{L_{+}(0)}{p}_2^{L_{-}(0)}{\psi }_{n,0}=q^n{\sum }_{l=0}^L(\log q{)}^l{\psi }_{n,l},q=p_1{p}_2,$ by the nilpotent recurrence in step 8.1. Finite weight-basis contraction in the sewing tensor therefore gives $\mathrm{\Gamma }(\chi )=\psi {\mathrm{\mid }}_{q_0}$ exactly. Ordinary convergence in F3 justifies its analytic evaluation. This is the promised explicit right inverse.

Finally the representing property for D identifies ${\mathrm{C}\mathrm{B}}_F(W\otimes D_Q(V{)}^{\mathrm{\prime }})$ with $\mathrm{Hom}(D_Q(V{)}^{\mathrm{\prime }},D)$. Restricted transposition identifies this with $\mathrm{Hom}(D^{\mathrm{\prime }},D_Q(V))$. The representing property for $D_Q(V)$ followed by vacuum evaluation identifies the latter space with ${\mathrm{C}\mathrm{B}}_N(D^{\mathrm{\prime }})$, using F4's actual propagation isomorphism. On arguments these maps are simply composition with $\omega ={\gamma }_Q(\mathbf{1},-)$, so finite coefficient contraction shows that their composite with Gamma is exactly the asserted cylinder insertion sewing. It is therefore an isomorphism as well. No rationality, simple-object expansion or finite tensor-category theorem is needed.