Chiral modular functor with sewing and anomaly

Work over $\mathbb{C}$. Let $\mathsf{G}\mathsf{r}\mathsf{a}\mathsf{p}\mathsf{h}\mathsf{s}$ be the symmetric monoidal category whose objects are finite disjoint unions of corollas, a corolla being one vertex with a finite set of incident legs. A graph is specified by finite vertex and half-edge sets, a source map from half-edges to vertices, and an involution on half-edges: fixed half-edges are legs and paired half-edges are internal edges. A morphism $T\to T^{\mathrm{\prime }}$ is an isomorphism class of such a graph with its cut half-edges identified with the legs of the input corollas $T$ and its connected components, after contraction of internal edges, identified with the output corollas $T^{\mathrm{\prime }}$. Composition is graph substitution; disjoint union gives the monoidal product and the empty union its unit. Regard this category as a bicategory with only identity 2-morphisms. Here $\mathsf{C}\mathsf{a}\mathsf{t}$ is the bicategory of essentially small categories, functors and natural transformations, with Cartesian product.

The surface modular operad is a weak symmetric monoidal functor $\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}:\mathsf{G}\mathsf{r}\mathsf{a}\mathsf{p}\mathsf{h}\mathsf{s}\to \mathsf{C}\mathsf{a}\mathsf{t}$. For one corolla $T$, $\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}(T)$ is the groupoid of connected compact oriented smooth surfaces with boundary circles bijectively labeled by the legs of $T$ and supplied with orientation-preserving parametrizations by $S^1$; its morphisms are isotopy classes of orientation- and parametrization-preserving diffeomorphisms. For $T={⨆}_v{T}_v$, set $\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}(T)={\prod }_v\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}(T_v)$, with the terminal category at the empty union. The functor of a graph glues the paired boundary circles with opposite induced orientations and groups the glued surfaces by its connected components. The associativity, symmetry and unit identifications are the canonical identifications of these gluings, with the parametrized cylinder as operadic identity. Thus connected surfaces are objects of corolla values, not a category purported to be closed under disjoint union.

Let $\mathsf{F}\mathsf{o}\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{t}\mathsf{s}\subset \mathsf{G}\mathsf{r}\mathsf{a}\mathsf{p}\mathsf{h}\mathsf{s}$ have the same objects and only those graph morphisms whose connected components are trees: connected graphs with no cycles, equivalently with $e=v-1$ for their numbers of internal edges and vertices. Disjoint union and graph substitution preserve this condition. A cyclic operad here is a weak symmetric monoidal functor $\mathsf{F}\mathsf{o}\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{t}\mathsf{s}\to \mathsf{C}\mathsf{a}\mathsf{t}$ with an operadic identity as above. Define ${\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}}_0(T)$ to be the full subgroupoid of $\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}(T)$ in which every surface component has genus zero, using the product convention on disjoint corollas. Its operation functors are restricted to $\mathsf{F}\mathsf{o}\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{t}\mathsf{s}$. This gives a cyclic suboperad of $\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}{\mathrm{\mid }}_{\mathsf{F}\mathsf{o}\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{t}\mathsf{s}}$, with the same identity cylinder. In fact a connected sewing graph $\mathrm{\Gamma }$ produces genus ${\sum }_v{g}_v+e-v+1$ from vertex surfaces of genera $g_v$. Thus forests preserve genus zero; a self-sewing edge has $e=v=1$ and raises genus by one. No genus-zero suboperad on all of $\mathsf{G}\mathsf{r}\mathsf{a}\mathsf{p}\mathsf{h}\mathsf{s}$ is asserted.

Here finite category means a $\mathbb{C}$-linear abelian category $\mathcal{C}$ with finite-dimensional morphism spaces, finitely many simple objects up to isomorphism, enough projectives, and finite length for every object. Its Deligne tensor product $\mathcal{C}⊠\mathcal{C}$ is the finite category universal for right-exact $\mathbb{C}$-bilinear functors out of $\mathcal{C}\times \mathcal{C}$. Write ${\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^{\mathrm{f}\mathrm{d}}$ for the finite category of finite-dimensional complex vector spaces.

Boundary reversal is specified by a contravariant equivalence $\mathsf{D}:{\mathcal{C}}^{\mathrm{o}\mathrm{p}}\to \mathcal{C}$ and a natural biduality isomorphism $\eta :{\mathrm{i}\mathrm{d}}_{\mathcal{C}}\Rightarrow \mathsf{D}{\mathsf{D}}^{\mathrm{o}\mathrm{p}}$ satisfying $\mathsf{D}({\eta }_X)\circ {\eta }_{\mathsf{D}X}={\mathrm{i}\mathrm{d}}_{\mathsf{D}X}$. Reversing a boundary labeled by $X$ relabels it by $\mathsf{D}X$; a morphism $f:X\to Y$ is sent to $\mathsf{D}(f):\mathsf{D}Y\to \mathsf{D}X$, and two reversals are identified with the original label by ${\eta }_X$.

The evaluation pairing is a right-exact $\mathbb{C}$-linear functor $\kappa :\mathcal{C}⊠\mathcal{C}\to {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^{\mathrm{f}\mathrm{d}}$. Its coevaluation is a right-exact $\mathbb{C}$-linear functor $\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{v}:{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^{\mathrm{f}\mathrm{d}}\to \mathcal{C}⊠\mathcal{C}$; write $\mathrm{\Delta }=\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{v}(\mathbb{C})\in \mathcal{C}⊠\mathcal{C}$. The pairing is nondegenerate when the following two natural isomorphisms of functors $\mathcal{C}\to \mathcal{C}$ hold, using the canonical unit and associativity isomorphisms for $⊠$:

$({\mathrm{i}\mathrm{d}}_{\mathcal{C}}⊠\kappa )\circ (\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{v}⊠{\mathrm{i}\mathrm{d}}_{\mathcal{C}})\cong {\mathrm{i}\mathrm{d}}_{\mathcal{C}},(\kappa ⊠{\mathrm{i}\mathrm{d}}_{\mathcal{C}})\circ ({\mathrm{i}\mathrm{d}}_{\mathcal{C}}⊠\mathrm{c}\mathrm{o}\mathrm{e}\mathrm{v})\cong {\mathrm{i}\mathrm{d}}_{\mathcal{C}}\mathrm{.}$

The boundary-reversal equivalence is required to be the duality determined by this pairing: include a natural isomorphism ${\chi }_{X,Y}:\kappa (X,Y)\stackrel{\cong }{\to }{\mathrm{Hom}}_{\mathcal{C}}(X,\mathsf{D}Y{)}^{\ast },$ where the star is finite-dimensional complex duality. Require the pairing symmetry to agree, through $\chi$, with dualizing the natural map $f\mapsto \mathsf{D}(f)\circ {\eta }_X$ from ${\mathrm{Hom}}_{\mathcal{C}}(Y,\mathsf{D}X)$ to ${\mathrm{Hom}}_{\mathcal{C}}(X,\mathsf{D}Y)$. In particular, $\mathsf{D}$ is not independent label-permutation data unrelated to $\kappa$. The specified coevaluation and triangle isomorphisms are for this same evaluation pairing.

Symmetry is a specified natural isomorphism $\kappa \circ \tau \cong \kappa$, where $\tau$ exchanges the two Deligne factors, together with the induced symmetry $\tau (\mathrm{\Delta })\cong \mathrm{\Delta }$ compatible with both triangle isomorphisms. In the semisimple case, if $X_i$ ranges over representatives of the simple isomorphism classes and $X_i^{\vee }:=\mathsf{D}{X}_i$, then $\mathrm{\Delta }\cong {⨁}_i{X}_i^{\vee }⊠X_i$. Thus the first slot of each internal pair is exactly the boundary-reversed label.

A chiral modular functor consists of an extension of these $\mathsf{G}\mathsf{r}\mathsf{a}\mathsf{p}\mathsf{h}\mathsf{s}$-indexed Cat-valued modular operads $q:Q\to \mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}$ whose homotopy fiber over each $T\in \mathsf{G}\mathsf{r}\mathsf{a}\mathsf{p}\mathsf{h}\mathsf{s}$ and $\mathrm{\Sigma }\in \mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}(T)$ is the groupoid $Q_{\mathrm{\Sigma }}$ of pairs $(\tilde{\mathrm{\Sigma }},\theta )$ with $\tilde{\mathrm{\Sigma }}\in Q(T)$ and an isomorphism $\theta :q(\tilde{\mathrm{\Sigma }})\stackrel{\cong }{\to }\mathrm{\Sigma }$ in $\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}(T)$, with morphisms the arrows in $Q(T)$ commuting with these isomorphisms; require each $Q_{\mathrm{\Sigma }}$ to be nonempty and connected. For $T\in \mathsf{F}\mathsf{o}\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{t}\mathsf{s}$, let $Q_0(T)$ be the full subgroupoid of $Q(T)$ on objects whose images under $q_T$ lie in ${\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}}_0(T)$. Forest operations restrict to these subgroupoids, giving a cyclic operad $Q_0$ and a map $q_0:Q_0\to {\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}}_0$. The extension includes a section $s_0:{\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}}_0\to Q_0$ of cyclic operads: a weak symmetric monoidal natural transformation on $\mathsf{F}\mathsf{o}\mathsf{r}\mathsf{e}\mathsf{s}\mathsf{t}\mathsf{s}$ preserving the operadic identity, with a specified coherent isomorphism $q_0\circ s_0\cong {\mathrm{i}\mathrm{d}}_{{\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}}_0}$. It also includes the insertion-of-vacua condition below and a modular $Q$-algebra valued in $\mathrm{R}\mathrm{e}\mathrm{x}\mathrm{f}$ on all of $\mathsf{G}\mathsf{r}\mathsf{a}\mathsf{p}\mathsf{h}\mathsf{s}$. Here $\mathrm{R}\mathrm{e}\mathrm{x}\mathrm{f}$ is the symmetric monoidal bicategory whose objects are finite $\mathbb{C}$-linear abelian categories, whose 1-morphisms are right-exact $\mathbb{C}$-linear functors, whose 2-morphisms are natural transformations, and whose monoidal product is $⊠$. Choose a distinguished vacuum object $E\in \mathcal{C}$; no monoidal structure on $\mathcal{C}$ is assumed. A modular $Q$-algebra is a morphism $F:Q\to {\mathrm{End}}_{\mathcal{C}}^{\kappa }$ of modular operads, where ${\mathrm{End}}_{\mathcal{C}}^{\kappa }(T)=\mathrm{R}\mathrm{e}\mathrm{x}\mathrm{f}({\mathcal{C}}^{⊠\mathrm{Legs}(T)},{\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^{\mathrm{f}\mathrm{d}})$ for a corolla, extended by Cartesian products to disjoint unions. Its graph functors insert the coevaluation object at cut internal pairs, and its operadic identity is the pairing functor; the natural transformation $F$ preserves these identities and compositions up to the specified coherent isomorphisms. Concretely, the algebra assigns to a surface $\mathrm{\Sigma }$ with $n$ boundary circles a right-exact functor $F_{\mathrm{\Sigma }}:{\mathcal{C}}^{⊠n}\to {\mathrm{V}\mathrm{e}\mathrm{c}\mathrm{t}}_{\mathbb{C}}^{\mathrm{f}\mathrm{d}}$ and natural isomorphisms for parametrization-preserving mapping classes, disjoint union, capping by the vacuum label $E$, and sewing. Capping/insertion identifies the value with the extra boundary labeled by $E$ with the value before adding that boundary. If ${\mathrm{\Sigma }}_{\mathrm{g}\mathrm{l}}$ is obtained by gluing two oppositely oriented boundary circles of $\mathrm{\Sigma }$, sewing is the natural isomorphism $F_{{\mathrm{\Sigma }}_{\mathrm{g}\mathrm{l}}}(\dots )\cong F_{\mathrm{\Sigma }}(\dots ,\mathrm{\Delta },\dots )\mathrm{.}$ For semisimple $\mathcal{C}$ this reads $F_{{\mathrm{\Sigma }}_{\mathrm{g}\mathrm{l}}}(\dots )\cong {⨁}_i{F}_{\mathrm{\Sigma }}(\dots ,X_i^{\vee },X_i,\dots )$. Insertion of the distinguished vacuum object $E$ at a new marked boundary identifies that value with the value before adding the boundary. All these isomorphisms are required to be natural and coherent with iterated sewing, the identity cylinder, disjoint union, and the two pairing-coevaluation triangle isomorphisms.

The extension's insertion-of-vacua condition is an equivalence on its homotopy fibers, separate from the algebra's vacuum capping isomorphism. For a finite disjoint union $D^{\bigsqcup J}$ of parametrized oriented disks and an orientation-preserving embedding $\varphi :D^{\bigsqcup J}\hookrightarrow \mathrm{\Sigma }$, let ${\mathrm{\Sigma }}_{\varphi }$ be the complementary surface, with the new boundary circles retained and labeled for gluing. Let $e_D=s_0(D^{\bigsqcup J})\in Q_{D^{\bigsqcup J}}$ be the selected lift of the disks and let ${\gamma }_{\varphi }:Q_{D^{\bigsqcup J}}\times Q_{{\mathrm{\Sigma }}_{\varphi }}\to Q_{\mathrm{\Sigma }}$ be the graph-operation functor that glues the corresponding boundary circles. The induced functor ${\lambda }_{\varphi }:Q_{{\mathrm{\Sigma }}_{\varphi }}\to Q_{\mathrm{\Sigma }},x\mapsto {\gamma }_{\varphi }(e_D,x),$ is required to be an equivalence for every $\mathrm{\Sigma }$ and $\varphi$.

For the anomaly convention, let $2g-2+n>0$. Over a $\mathbb{C}$-scheme $S$, a stable $n$-pointed genus-$g$ family is a proper, flat, finitely presented morphism $\bar{\pi }:\bar{C}\to S$ of relative dimension one whose geometric fibers are connected nodal curves with arithmetic genus $g$, together with $n$ ordered, pairwise disjoint sections ${\bar{\sigma }}_i:S\to \bar{C}$ through the relative smooth locus, such that ${\omega }_{\bar{\pi }}({\sum }_i{\bar{\sigma }}_i(S))$ is relatively ample. Here arithmetic genus is $1-\chi ({\bar{C}}_s,{\mathcal{O}}_{{\bar{C}}_s})$. Let ${\bar{\mathcal{M}}}_{g,n,\mathbb{C}}$ be the moduli stack of these families, and let $\bar{\pi }:{\bar{\mathcal{C}}}_{g,n,\mathbb{C}}\to {\bar{\mathcal{M}}}_{g,n,\mathbb{C}}$ be its universal stable curve. The nodal boundary $\bar{\mathrm{\Delta }}$ is the divisor of singular fibers. The source stack ${\mathsf{M}}_{g,n}^f$ is the frame torsor over ${\bar{\mathcal{M}}}_{g,n,\mathbb{C}}$ whose points are stable curves together with nonzero tangent vectors at all markings; write ${\tilde{\mathcal{M}}}_{g,n}$ for it in this item. Concretely, it is the fiber product over ${\bar{\mathcal{M}}}_{g,n,\mathbb{C}}$ of the complements of the zero sections in the relative tangent lines ${\bar{\sigma }}_i^{\ast }{T}_{{\bar{\mathcal{C}}}_{g,n,\mathbb{C}}\mathrm{/}{\bar{\mathcal{M}}}_{g,n,\mathbb{C}}}$. Let $\pi :{\mathcal{C}}_{g,n}\to {\tilde{\mathcal{M}}}_{g,n}$ be the pullback of the universal curve, let ${\omega }_{\pi }$ be its relative dualizing line bundle, and let $\mathrm{\Lambda }=\det ({\pi }_{\ast }{\omega }_{\pi })$ be the Hodge line. The nodal boundary $\tilde{\mathrm{\Delta }}$ is the pullback of $\bar{\mathrm{\Delta }}$. The logarithmic Atiyah algebra $A_{\mathrm{\Lambda }}(-\tilde{\mathrm{\Delta }})$ is the sheaf of first-order differential operators on $\mathrm{\Lambda }$ whose scalar symbols are vector fields tangent to $\tilde{\mathrm{\Delta }}$; its anchor is the symbol map to $T_{{\tilde{\mathcal{M}}}_{g,n}}(-\log \tilde{\mathrm{\Delta }})$. By $\alpha A_{\mathrm{\Lambda }}(-\tilde{\mathrm{\Delta }})$ we mean the pushout of this Lie-algebroid extension along multiplication by $\alpha$ on its central copy of ${\mathcal{O}}_{{\tilde{\mathcal{M}}}_{g,n}}$. Use the analytifications of this stack, Hodge line and Lie algebroid for the holomorphic data below. To specify a chiral modular functor with anomaly parameter $\alpha \in \mathbb{C}$, require the preceding $(Q,F)$ to carry the following additional structure. For every stable $(g,n)$ and labels $X_1,\dots ,X_n\in \mathcal{C}$, supply a finite-rank holomorphic bundle ${\mathcal{B}}_{g,n}(X_{\bullet })$ on the analytic tangent-frame stack just defined, functorial in the labels, with an action $\mathrm{\nabla }:\alpha A_{\mathrm{\Lambda }}(-\tilde{\mathrm{\Delta }})⟶{\mathrm{Diff}}_{\mathbb{C}}^1({\mathcal{B}}_{g,n}(X_{\bullet }))\mathrm{.}$ Here the target consists of first-order operators with scalar symbol; require $\mathrm{\nabla }$ to be $\mathcal{O}$-linear, bracket preserving, to have symbol equal to the anchor, and to send the central section $f\in \mathcal{O}$ to $f{\mathrm{i}\mathrm{d}}_{\mathcal{B}}$. Thus for a local section $a$ with anchor $v$, ${\mathrm{\nabla }}_a(fs)=f{\mathrm{\nabla }}_a(s)+v(f)s$. This is the specified integrable logarithmic twisted connection, with precisely the $\alpha$-scaled Hodge extension as its domain.

The bundles must realize the given modular algebra, rather than an unrelated family: identify their smooth-curve fibers with the labeled values of $F$, and require their projective parallel transport to coincide with the mapping-class action of $F$ modulo invertible scalars. More explicitly, for a lift $\tilde{\mathrm{\Sigma }}$ put $K_{\tilde{\mathrm{\Sigma }}}=\ker ({\mathrm{Aut}}_{Q(T)}(\tilde{\mathrm{\Sigma }})\to {\mathrm{Aut}}_{\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}(T)}(\mathrm{\Sigma }))$. Require every element of $K_{\tilde{\mathrm{\Sigma }}}$ to act on each labeled value by an invertible scalar multiple of the identity. The chosen linear lifts of projective parallel transport must be the action on these values supplied by $Q$ and $F$, coherently under changing lifts. Projective transport means linear isomorphisms modulo invertible scalar multiples, also for zero fibers. Require the disjoint-union and sewing identifications to intertwine these twisted connections after the corresponding pullbacks, using normalization of nodal curves with the specified local plumbing for sewing. The vacuum-capping pullback has the precise domain specified next. These identifications and their compatibility are additional data; their existence is not deduced from connected extension fibers. This formulation allows extension presentations with extra kernel acting trivially.

For capping, assume that the target pair $(g,n)$ is stable and let $f:{\bar{\mathcal{M}}}_{g,n+1,\mathbb{C}}\to {\bar{\mathcal{M}}}_{g,n,\mathbb{C}}$ forget the last marking and stabilize. Write $r:C\to C^{\mathrm{s}\mathrm{t}}$ for this stabilization on families. Let $U_{g,n+1}$ be the open substack on which $r$ is an isomorphism near every retained marked section. Equivalently, remove the boundary divisors $D_{i,n+1}$, $1\le i\le n$, whose curves have a rational tail with exactly the retained marking $i$, the forgotten marking $n+1$, and its attaching node as special points. Set $U_{g,n+1}^f=U_{g,n+1}{\times }_{{\bar{\mathcal{M}}}_{g,n+1,\mathbb{C}}}{\tilde{\mathcal{M}}}_{g,n+1}$. With an empty set of retained markings this condition is vacuous. On this domain there is a framed capping morphism ${\xi }^f:U_{g,n+1}^f⟶{\tilde{\mathcal{M}}}_{g,n},(C;p_{\bullet };v_{\bullet })⟼(C^{\mathrm{s}\mathrm{t}};r(p_1),\dots ,r(p_n);d{r}_{p_1}{v}_1,\dots ,d{r}_{p_n}{v}_n)\mathrm{.}$ Require a natural vacuum-capping isomorphism $({\xi }^f{)}^{\ast }{\mathcal{B}}_{g,n}(X_1,\dots ,X_n)\stackrel{\cong }{\to }{{\mathcal{B}}_{g,n+1}(X_1,\dots ,X_n,E)\mid }_{U_{g,n+1}^f},$ intertwining the pulled-back logarithmic twisted connections via the Hodge-line identification induced by stabilization. On smooth curves it must induce the vacuum-capping map of $F$; require compatibility with disjoint union, sewing and successive cappings wherever the constituent framed morphisms are defined. This includes insertion at any new disjoint smooth point of any stable nodal family and also contraction of rational bridges away from the retained markings. The bundles and sewing data remain defined on the full stable-curve frame stacks. If a retained marking lies on a contracted tail, $d{r}_{p_i}{v}_i=0$, so this formula does not define a morphism to the nonzero-frame torsor there. The global unframed morphism $f$ still exists: its pullback of bundles on ${\tilde{\mathcal{M}}}_{g,n}$ has the globally valid source ${\bar{\mathcal{M}}}_{g,n+1,\mathbb{C}}{\times }_{{\bar{\mathcal{M}}}_{g,n,\mathbb{C}}}{\tilde{\mathcal{M}}}_{g,n}$, which remembers frames on the stabilized target, rather than frames on contracted components. It must not be identified with ${\tilde{\mathcal{M}}}_{g,n+1}$. The topological capping axiom for $F$ applies to every surface as before; it does not supply a nonexistent global morphism between these two frame torsors.

For a VOA realization the central-charge convention is $\alpha =c\mathrm{/}2$, where the Virasoro central element acts as $c\mathrm{i}\mathrm{d}$, and the vacuum is $V$. Identification of the bundles with its coinvariant sheaves, finite rank and the compatible anomaly and sewing structures must be established for that VOA. For the anomaly-equipped data just specified, mapping-class actions are linear on $Q$ and projective on $\mathsf{S}\mathsf{u}\mathsf{r}\mathsf{f}$.

When an analytic representative is used, the pointed-worldsheet and coordinate data are those of the supplied analytic-worldsheet definition, and a local plumbing parameter obeys $zw=q$ as in the supplied plumbing definition. These analytic data do not define the global algebraic stable-curve stack. The algebraic conformal-block input is the relative coinvariant quotient/sheaf defined by the outgoing chiral action, which by itself supplies no finite-rank, local-freeness, or base-change assertion. This definition specifies the structure and axioms; it does not assert that any given VOA supplies them.

The following schematic records only the topology of the two sewings. The marked circles joined by the internal edge are parametrized with opposite induced orientations; the edge is not a proof of the sewing isomorphism.

Coinvariants and conformal blocks on a pointed curve

Pointed analytic worldsheet and family

Plumbing family for punctured Riemann surfaces

Stable n-pointed algebraic curves and the moduli functor

Proper Deligne–Mumford stack of stable pointed curves

Virasoro algebra in the $L_m$ convention

The source-compatible structure definition is local. Existence of finite-dimensional coinvariant fibers, the projective connection, and coherent sewing for a particular VOA belongs to a separate theorem and is not implied here.