Locally C1-cofinite genus-zero open-closed OPE data

Fix a positive-graded chiral vertex operator algebra $V={⨁}_{n\ge 0}{V}_n$ with $V_0=\mathbb{C}\mathbf{1}$ and put $V_{>0}={⨁}_{n>0}{V}_n$. Here positive-graded has the exact conformal-vector, Virasoro, finite-dimensional weight-space, and CFT-type conventions of Conformal vertex algebra of CFT type. Use the ordinary non-super mode convention. The source category $V{\text{-mod}}_{\mathrm{f}\mathrm{.}\mathrm{g}\mathrm{.}}$ consists of $V$-modules $W$ with locally finite $L_W(0)$, finite-dimensional generalized eigenspaces $W_{[h]}$, and a finite set ${\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_m\in \mathbb{C}$ such that $W={⨁}_{j=1}^m{⨁}_{n\ge 0}{W}_{[{\mathrm{\Delta }}_j+n]}$. Also require $C_1(W)={\mathrm{span}}_{\mathbb{C}}\{u_{-1}w:u\in V_{>0},w\in W\}$ to have finite codimension and the source restricted-dual module $W^{\vee }={⨁}_h{W}_{[h]}^{\ast }$ to be finitely generated by a finite subset under the $V$-modes. The restricted-dual space and opposite-field action are those in Contragredient module of a generalized module. That item defines the candidate action but explicitly leaves its module axioms to a separate theorem; The restricted dual with opposite action is a contragredient module supplies the actual module used here. Finite-coset support, finite-dimensional generalized weight spaces, $C_1$-cofiniteness, and dual finite generation are separate clauses; the R2 $C_1$-quotient condition alone is not this source category.

For any such generalized modules $W_1,W_2,W_3$, write ${\mathcal{I}}_{\log }\left(\genfrac{}{}{0px}{}{W_3}{W_1{W}_2}\right)$ for the logarithmic intertwining-operator space of Logarithmic intertwining operator. Define ${\mathcal{I}}^0\left(\genfrac{}{}{0px}{}{W_3}{W_1{W}_2}\right):=\{\mathcal{Y}\in {\mathcal{I}}_{\log }\left(\genfrac{}{}{0px}{}{W_3}{W_1{W}_2}\right):(w_1{)}_{n;k}{w}_2=0\text{ for every }{w}_1\in W_1,w_2\in W_2,n\in \mathbb{C},k\ge 1\},$ where $\mathcal{Y}(w_1,x)w_2={\sum }_{n\in \mathbb{C}}{\sum }_{k\ge 0}(w_1{)}_{n;k}{w}_2{x}^{-n-1}(\log x{)}^k$ is the source expansion. Thus ${\mathcal{I}}^0$ is the no-log subspace on generalized modules. It is not an assertion that generalized modules are ordinary; when all three modules are ordinary it agrees with the ordinary intertwining-operator space defined in Intertwining-operator space and fusion multiplicity. All OPE coefficients below are required to lie in these no-log spaces, while analytic continuation of their blocks is allowed to acquire logarithmic terms.

The locally $C_1$-cofinite full bulk data over $V$ consist of a countable index set $I_c$ and pairs $(M_i,{\bar{M}}_i)$ of source-category modules, with $F_c={⨁}_{h,\bar{h}\in \mathbb{R}}{F}_{h,\bar{h}}^c$ satisfying integral-spin support ($F_{h,\bar{h}}^c=0$ unless $h-\bar{h}\in \mathbb{Z}$), common lower bounds on $h,\bar{h}$, and finite-dimensional cumulative spaces ${⨁}_{h+\bar{h}\le H}{F}_{h,\bar{h}}^c$. Require $F_c\cong {⨁}_{i\in I_c}{M}_i\otimes {\bar{M}}_i$ as a $V\otimes V$-module, the two chiral copies of $V$ to embed in the left and right kernels of the full translation modes, and for each $i,j\in I_c$ a finite set $I_c(i,j)\subset I_c$ containing every intermediate label in the full OPE of $M_i\otimes {\bar{M}}_i$ with $M_j\otimes {\bar{M}}_j$. For each pair $(i,j)$ the module-valued OPE coefficient lies in ${⨁}_{k\in I_c(i,j)}{\mathcal{I}}^0\left(\genfrac{}{}{0px}{}{M_k}{M_i{M}_j}\right)\otimes {\mathcal{I}}^0\left(\genfrac{}{}{0px}{}{{\bar{M}}_k}{{\bar{M}}_i{\bar{M}}_j}\right)\mathrm{.}$ The two factors are respectively for the left and right copies of $V$. Define the source graded dual by $F_c^{\vee }={⨁}_{h,\bar{h}}(F_{h,\bar{h}}^c{)}^{\ast }$.

The locally $C_1$-cofinite boundary data over $V$ consist of a countable index set $I_o$ and a decomposition $F_o={⨁}_{h\in \mathbb{R}}{F}_h^o\cong {⨁}_{k\in I_o}{N}_k$ into source-category modules, with finite-dimensional cumulative spaces ${⨁}_{h\le H}{F}_h^o$, and a finite set $I_o(k,\mathrm{\ell })\subset I_o$ supporting each boundary OPE of $N_k$ with $N_{\mathrm{\ell }}$. Require $L^o(0){\mathrm{\mid }}_{F_h^o}=hI$ for every $h\in \mathbb{R}$, where $L^o(0)$ is the conformal zero mode of the prescribed boundary $V$-action specified below. Thus the displayed boundary decomposition is its conformal eigenspace grading, not an independent regrading of the underlying module. For each pair $(k,\mathrm{\ell })$ the boundary OPE coefficient lies in ${⨁}_{m\in I_o(k,\mathrm{\ell })}{\mathcal{I}}^0\left(\genfrac{}{}{0px}{}{N_m}{N_k{N}_{\mathrm{\ell }}}\right)$. Set $F_o^{\vee }={⨁}_h(F_h^o{)}^{\ast }$. For each $i\in I_c$ the bulk-to-boundary operator restricted to $M_i\otimes {\bar{M}}_i$ lies in the finite direct sum ${⨁}_{j\in B(i)}{\mathcal{I}}^0\left(\genfrac{}{}{0px}{}{N_j}{M_i{\bar{M}}_i}\right)$ for a finite set $B(i)\subset I_o$, where this notation denotes the no-log generalized-module intertwiner space just defined.

The full and boundary algebra axioms are additional premises distinct from local $C_1$-cofinite decompositions. The precise full-vertex-algebra clauses are Moriwaki Definition 3.1 and its energy-momentum-tensor conditions preceding Proposition 3.5; the boundary clauses are equation (3.10) and Definition 3.15. Definitions 3.6 and 3.19 add the finite-support/local-$C_1$ clauses to these algebras; they do not replace the algebra axioms. Adopt the following explicit content.

For the bulk, give a nonzero vacuum ${\mathbf{1}}_c\in F_{0,0}^c$ and a linear formal field map $Y_c(a;z,\bar{z})={\sum }_{r,s\in \mathbb{R}}{a}_{(r,s)}{z}^{-r-1}{\bar{z}}^{-s-1}\mathrm{.}$ For each fixed pair of states, its exponent support is bounded below in each variable and finite below every bound on the sum of the two exponents; only integer differences of the two exponents occur. For homogeneous $a\in F_{h,\bar{h}}^c$ its modes obey the grading rule $a_{(r,s)}{F}_{k,\bar{k}}^c\subset F_{h+k-r-1,\bar{h}+\bar{k}-s-1}^c$. Require $Y_c({\mathbf{1}}_c;z,\bar{z})=I$, $Y_c(a;z,\bar{z}){\mathbf{1}}_c\in F_c[\text{\hspace{0.17em}⁣}[z,\bar{z}]\text{\hspace{0.17em}⁣}]$, and constant term $a$. The coefficient fields use the actual full mode products, not a product inferred from the unrelated R5 field-label space.

For every three homogeneous bulk states $a,b,c$ and every homogeneous restricted-dual functional $u$, require absolute convergence of the product $⟨u,Y_c(a;z_1,{\bar{z}}_1)Y_c(b;z_2,{\bar{z}}_2)c⟩$ when $\mathrm{\mid }{z}_1\mathrm{\mid }>\mathrm{\mid }{z}_2\mathrm{\mid }>0$, of the iterate $⟨u,Y_c(Y_c(a;z_0,{\bar{z}}_0)b;z_2,{\bar{z}}_2)c⟩$ when $\mathrm{\mid }{z}_2\mathrm{\mid }>\mathrm{\mid }{z}_0\mathrm{\mid }>0$, and of the reversed product when $\mathrm{\mid }{z}_2\mathrm{\mid }>\mathrm{\mid }{z}_1\mathrm{\mid }>0$. They must be the respective channel expansions of one single-valued real-analytic function ${\mu }_{u;a,b,c}$ on $\{(z_1,z_2)\in {\mathbb{C}}^2:z_1,z_2\mathrm{\ne }0,z_1\mathrm{\ne }{z}_2\}$, with $z_0=z_1-z_2$ in the iterate. This is the three-channel full-algebra axiom, independent of the later bulk-to-boundary bootstrap.

Define $D_{l}a$ and $D_{r}a$ as the coefficients of $z$ and $\bar{z}$ in $Y_c(a;z,\bar{z}){\mathbf{1}}_c$. Give conformal vectors ${\omega }_l\in F_{2,0}^c$ and ${\omega }_r\in F_{0,2}^c$ with $D_r{\omega }_l=D_l{\omega }_r=0$. Require $Y_c({\omega }_l;z,\bar{z})={\sum }_n{L}^l(n)z^{-n-2}$ and $Y_c({\omega }_r;z,\bar{z})={\sum }_n{L}^r(n){\bar{z}}^{-n-2}$ to give commuting Virasoro actions with the declared central charges, $L^l(-1)=D_l$, $L^r(-1)=D_r$, $L^l(0){\mathrm{\mid }}_{F_{h,\bar{h}}^c}=hI$, and $L^r(0){\mathrm{\mid }}_{F_{h,\bar{h}}^c}=\bar{h}I$. The lower bounds and finite cumulative weight spaces are the ones required above. These are the energy-momentum data of the full VOA; they are not inferred merely from the decomposition as a $V\otimes V$-module.

For the boundary, give a nonzero ${\mathbf{1}}_o\in F_0^o$ and a linear map $Y_o(a,z)={\sum }_{r\in \mathbb{R}}{a}_{(r)}^o{z}^{-r-1}$ with $Y_o(a,z)b\in F_o((z^{\mathbb{R}}))$: for each real bound $H$, only finitely many exponents at most $H$ occur. For the prescribed $V$-module action $a_{(n)}^o$ and $v\in F_o$, require $[L^o(-1),Y_o(v,z)]={\mathrm{\partial }}_z{Y}_o(v,z),[a_{(n)}^o,Y_o(v,z)]={\sum }_{k\ge 0}\left(\genfrac{}{}{0px}{}{n}{k}\right)Y_o(a_{(k)}^{o}v,z)z^{n-k},$ $Y_o(a_{(n)}^{o}v,z)={\sum }_{k\ge 0}\left(\genfrac{}{}{0px}{}{n}{k}\right)(a_{(n-k)}^o{Y}_o(v,z)(-z{)}^k-Y_o(v,z)a_{(k)}^o(-z{)}^{n-k})\mathrm{.}$ Here $a\in V$, $n\in \mathbb{Z}$, and coefficientwise lower truncation makes each action meaningful. Require $Y_o({\mathbf{1}}_o,z)=I$, $Y_o(v,z){\mathbf{1}}_o\in F_o[\text{\hspace{0.17em}⁣}[z]\text{\hspace{0.17em}⁣}]$ with constant term $v$, and the following ordered associativity axiom. For $a,b,c\in F_o$ and $u\in F_o^{\vee }$, the product and iterate series converge absolutely on their product and iterate nested domains, respectively, and restrict to one real-analytic function on $\{(x_1,x_2)\in {\mathbb{R}}^2:x_1>x_2>0\}$. Its product expression is $⟨u,Y_o(a,x_1)Y_o(b,x_2)c⟩$, and its iterate expression is $⟨u,Y_o(Y_o(a,x_1-x_2)b,x_2)c⟩$ on $x_2>x_1-x_2>0$. Complex powers use the declared branches. Boundary commutativity is not an extra axiom.

To type the chiral embeddings, specify injective linear maps ${\iota }_l,{\iota }_r:V\to F_c$ and ${\iota }_o:V\to F_o$ carrying the vacuum to ${\mathbf{1}}_c,{\mathbf{1}}_c,{\mathbf{1}}_o$ and the conformal vector to ${\omega }_l,{\omega }_r,{\omega }_o$, respectively. The left image lies in $\ker D_r$ and the right image in $\ker D_l$. Require the full-field identities $Y_c({\iota }_l(a);z,\bar{z})=Y_{F_c}^l(a,z)$ and $Y_c({\iota }_r(a);z,\bar{z})=Y_{F_c}^r(a,\bar{z})$, where the right sides are the specified $V$-module actions. Their mode products on the respective images satisfy ${\iota }_l(a_{(n)}b)=({\iota }_{l}a{)}_{(n,-1)}{\iota }_{l}b$ and ${\iota }_r(a_{(n)}b)=({\iota }_{r}a{)}_{(-1,n)}{\iota }_{r}b$. On the boundary require $Y_o({\iota }_o(a),z)=Y_{F_o}(a,z)$, the specified module field. Thus the mode actions and product-preservation interfaces used below are actually supplied. The two chiral actions commute, and $L^o(n)$ are the conformal-vector modes of the boundary $V$-action.

Write $Y_c,Y_o,Y_b$ for the full, boundary, and bulk-to-boundary OPE maps, respectively; $Y_b$ is specified explicitly below. Thus $a_{(n)}^l,a_{(n)}^r$ denote the two chiral mode actions on $F_c$, $a_{(n)}^o$ the boundary mode action on $F_o$, and $L^l(-1),L^r(-1),L^o(-1)$ their translation modes.

More explicitly, the bulk-to-boundary field is a linear map $Y_b(\cdot ,z):F_c\to F_o[\text{\hspace{0.17em}⁣}[z^{\mathbb{R}}]\text{\hspace{0.17em}⁣}]$, $v\mapsto {\sum }_{r\in \mathbb{R}}{B}_r(v)z^{-r-1}$, with the source's mode actions $a_{(n)}^l,a_{(n)}^r$ on $F_c$ and $a_{(n)}^o$ on $F_o$. It satisfies, for $a\in V$, $v\in F_c$, and $n\in \mathbb{Z}$, $Y_b(L^l(-1)v,z)={\mathrm{\partial }}_z{Y}_b(v,z),a_{(n)}^o{Y}_b(v,z)=Y_b(a_{(n)}^{r}v,z)+{\sum }_{k\ge 0}\left(\genfrac{}{}{0px}{}{n}{k}\right)Y_b(a_{(k)}^{l}v,z)z^{n-k},$ and $Y_b(a_{(n)}^{l}v,z)={\sum }_{k\ge 0}\left(\genfrac{}{}{0px}{}{n}{k}\right)(a_{(n-k)}^o{Y}_b(v,z)(-z{)}^k-Y_b(a_{(k)}^{r}v,z)(-z{)}^{n-k})\mathrm{.}$ The sums are interpreted coefficientwise with the module truncation convention in the source. These are the bulk-boundary covariance identities of Moriwaki, Definition 3.22 and equation (3.15); its Definition 3.28 imposes the finite support $B(i)$. These finite-support requirements are separate from $C_1$-cofiniteness of an individual module.

We fix the source's explicit tree domains. For a binary tree $A$ with $n\ge 2$ leaves labelled by distinct coordinates $z_1,\dots ,z_n$, let $V(A)$ and $E(A)$ be its internal vertices and internal edges, and let $t_A$ be the root. For each internal vertex $v$, let $R(v)$ be the label of the rightmost leaf below $v$, and let $L(v)$ be the label of the rightmost leaf in the left child subtree of $v$. Put $z_v=z_{L(v)}-z_{R(v)}$. If $e$ joins a child vertex $d(e)$ to its parent $u(e)$, put ${\zeta }_e=z_{d(e)}\mathrm{/}{z}_{u(e)}$. Let $r_A$ be the rightmost leaf label, set $z_A=z_{r_A}$ and $x_A=z_{t_A}$, and use the tree-coordinate map ${\mathrm{\Psi }}_A=(z_A,x_A,({\zeta }_e{)}_{e\in E(A)})$. A positive radius tuple $p=(p_e{)}_{e\in E(A)}$ is $A$-admissible when the full product region in these coordinates, with $z_A\in \mathbb{C}$, $x_A\in {\mathbb{C}}^{\times }$, and $0<\mathrm{\mid }{\zeta }_e\mathrm{\mid }<p_e$, maps into the distinct-point configuration space. Let $C_{\mathrm{c}\mathrm{u}\mathrm{t}}=\mathbb{C}\setminus {\mathbb{R}}_{\le 0}$ and $D_{p_e}^{\mathrm{c}\mathrm{u}\mathrm{t}}=\{\zeta \in C_{\mathrm{c}\mathrm{u}\mathrm{t}}:0<\mathrm{\mid }\zeta \mathrm{\mid }<p_e\}$. Define $U_A={\bigcup }_{p\mathrm{A}\text{-}\mathrm{a}\mathrm{d}\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{b}\mathrm{l}\mathrm{e}}{\mathrm{\Psi }}_A^{-1}\text{\hspace{0.17em}⁣}(\mathbb{C}\times C_{\mathrm{c}\mathrm{u}\mathrm{t}}\times {\prod }_{e\in E(A)}{D}_{p_e}^{\mathrm{c}\mathrm{u}\mathrm{t}})\subset X_n(\mathbb{C})\mathrm{.}$ For the zero-input vacuum tree $\mathrm{\varnothing }$, set $X_0(\mathbb{C})=\{()\}$ and $U_{\mathrm{\varnothing }}=X_0(\mathbb{C})$. For the one-input identity tree $\bullet$, set $X_1(\mathbb{C})=\mathbb{C}$ and $U_{\bullet }=\mathbb{C}$. These cases have no root-scale or internal-edge coordinates. For a two-coloured tree $E$ with $r$ bulk and $s$ boundary leaves, define its doubled chiral tree $\tilde{E}$ recursively using its typed vertices. An open leaf is retained. Open multiplication becomes ordinary binary multiplication of the doubled open subtrees. A closed subtree $A$ entering the bulk-to-boundary vertex $\tau$ becomes the binary tree $(A,\bar{A})$, where $\bar{A}$ is a copy of the entire closed subtree with barred leaf labels. Inserting a closed subtree at a closed leaf therefore inserts $A$ and $\bar{A}$ at the corresponding paired chiral leaves. The coordinate tuple below interleaves the labels $(i,\bar{i})$; their order in a doubled tree need not be adjacent. In particular $\tau ({\mu }_c(1,2))$ doubles to $((1,2),(\bar{1},\bar{2}))$, whereas ${\mu }_o(\tau (1),\tau (2))$ doubles to $((1,\bar{1}),(2,\bar{2}))$; these are distinct channel trees. Set $X_n(\mathbb{C})=\{(z_1,\dots ,z_n)\in {\mathbb{C}}^n:z_i\mathrm{\ne }{z}_j\text{ for }i\mathrm{\ne }j\},X_{r,s}(\mathbb{H})=\{(z_1,\dots ,z_r,x_1,\dots ,x_s):\mathrm{Im}{z}_i>0,x_j\in \mathbb{R},z_i\mathrm{\ne }{z}_k,x_j\mathrm{\ne }{x}_{\mathrm{\ell }},z_i\mathrm{\ne }{x}_j\},$ $\mathrm{\Phi }(z_1,\dots ,z_r,x_1,\dots ,x_s)=(z_1,{\bar{z}}_1,\dots ,z_r,{\bar{z}}_r,x_1,\dots ,x_s),{\tilde{U}}_E=U_{\tilde{E}},U_E={\mathrm{\Phi }}^{-1}({\tilde{U}}_E)\cap X_{r,s}(\mathbb{H})\mathrm{.}$ For a bulk tree $A_c$ with $r$ bulk leaves, use its ordinary nested domain $U_{A_c}\subset X_r(\mathbb{C})$ in each chiral coordinate copy and define ${\mathrm{\Phi }}_c(z_1,\dots ,z_r)=(z_1,\dots ,z_r;{\bar{z}}_1,\dots ,{\bar{z}}_r),U_{A_c}^c={\mathrm{\Phi }}_c^{-1}(U_{A_c}\times U_{A_c})\subset X_r(\mathbb{C})\mathrm{.}$ Thus the holomorphic and antiholomorphic bulk OPE series live on $U_{A_c}\times U_{A_c}$, and their restriction is real analytic on $U_{A_c}^c$. For zero or one bulk leaf the corresponding closed OPE is the declared vacuum or state-field map on $U_{\mathrm{\varnothing }}^c=\{()\}$ or $U_{\bullet }^c=\mathbb{C}$, respectively, with no internal projection sum. The same pullback convention gives $U_E=\{()\}$ for the zero-input open vacuum tree and $U_E=\mathbb{R}$ for the single open leaf; a single bulk leaf entering a bulk-to-boundary vertex still doubles to two chiral leaves and uses the two-leaf domain. Branches of all real powers use the principal logarithm on $C_{\mathrm{c}\mathrm{u}\mathrm{t}}$.

The genus-zero bulk-boundary bootstrap data are these five channel conditions:

1. The bulk vacuum maps to the boundary vacuum.
2. For $a_1,a_2\in F_c$, a restricted-dual functional $u\in F_o^{\vee }$, and $z_1,z_2\in \mathbb{H}$ with $\mathrm{Re}{z}_1>\mathrm{Re}{z}_2$, the two series $⟨u,e^{{\bar{z}}_2{L}^o(-1)}{Y}_o\text{\hspace{0.17em}⁣}(Y_b(a_1,z_1-{\bar{z}}_1),{\bar{z}}_1-{\bar{z}}_2)Y_b(a_2,z_2-{\bar{z}}_2)⟩$ and $⟨u,e^{{\bar{z}}_2{L}^o(-1)}{Y}_b\text{\hspace{0.17em}⁣}(Y_c(a_1,z_1-z_2,{\bar{z}}_1-{\bar{z}}_2)a_2,z_2-{\bar{z}}_2)⟩$ are absolutely convergent respectively when $\frac{\mathrm{\mid }{z}_1-{\bar{z}}_1\mathrm{\mid }+\mathrm{\mid }{z}_2-{\bar{z}}_2\mathrm{\mid }}{\mathrm{\mid }{\bar{z}}_1-{\bar{z}}_2\mathrm{\mid }}<1,\frac{\mathrm{\mid }{z}_1-z_2\mathrm{\mid }+\mathrm{\mid }{\bar{z}}_1-{\bar{z}}_2\mathrm{\mid }}{\mathrm{\mid }{z}_2-{\bar{z}}_2\mathrm{\mid }}<1,$ and are restrictions of one single-valued real-analytic function on $X_{2,0}(\mathbb{H})$.
3. For $a\in F_c$, $b\in F_o$, $u\in F_o^{\vee }$, $z\in \mathbb{H}$, and $x\in \mathbb{R}$, the series $⟨u,e^{x{L}^o(-1)}{Y}_o\text{\hspace{0.17em}⁣}(Y_b(a,z-\bar{z}),\bar{z}-x)b⟩,⟨u,e^{\bar{z}{L}^o(-1)}{Y}_o\text{\hspace{0.17em}⁣}(b,x-\bar{z})Y_b(a,z-\bar{z})⟩$ are absolutely convergent respectively on $\mathrm{Re}z>x$, $\mathrm{\mid }z-\bar{z}\mathrm{\mid }<\mathrm{\mid }\bar{z}-x\mathrm{\mid }$ and on $x>\mathrm{Re}z$, $\mathrm{\mid }z-\bar{z}\mathrm{\mid }<\mathrm{\mid }x-\bar{z}\mathrm{\mid }$, and are restrictions of one single-valued real-analytic function on $X_{1,1}(\mathbb{H})$.

The configuration space is $X_{r,s}(\mathbb{H})=\{(z_1,\dots ,z_r,x_1,\dots ,x_s):\mathrm{Im}{z}_i>0,x_j\in \mathbb{R},\text{ and all displayed points are distinct}\}$; when open leaves are ordered, use the chamber $x_1>\cdots >x_s$. The five equations counted by the source are the vacuum identity and the two ordered expansions in each of conditions 2 and 3. This definition records the exact analytic and finite-support package; it does not assert existence of a physical CFT or a path-integral measure.

C1-cofinite generalized module in the ungraded HLZ convention

Contragredient module of a generalized module

The restricted dual with opposite action is a contragredient module

Strongly graded generalized module in the HLZ setting

Intertwining-operator space and fusion multiplicity

Logarithmic intertwining operator

Full local two-dimensional CFT data

Bordered analytic worldsheet with marked collars

Conformal boundaries, boundary-changing fields, and defects

Binary-tree insertion coordinates are biholomorphic

Conformal vertex algebra of CFT type

Module for an ordinary vertex algebra

The quoted C1 quotient, dual finite generation, finite intermediate-label support, boundary finite support, and finite bulk-to-boundary support are distinct clauses. The R2 quotient definition supplies only the first. The no-log requirement is imposed inside the logarithmic intertwiner spaces for generalized modules; this keeps the tree-block and theorem types aligned without treating generalized modules as ordinary. In particular, the higher-genus sewing and the annulus Cardy condition are not included.

Moriwaki's Theorem 3.30 uses the analytic convergence theorem for iterated intertwining-operator products and the two-colored parenthesized permutation-braid action. Those results are not included in this definition; their local proof status is recorded in the consuming theorem. This item specifies the data and bootstrap interfaces, not the convergence theorem.