Diagonal rational annulus and Cardy-state conventions

Fix a conformal VOA $V$ of CFT type and central charge $c$ whose vacuum module is irreducible, and set $M_0=V$. Here CFT type means $V={⨁}_{n\ge 0}{V}_n$, with finite-dimensional $L(0)$-eigenspaces $V_n$ and $V_0=\mathbb{C}\mathbf{1}$; it is part of the conformal VOA convention used by the ordinary-module and contragredient suppliers, not a consequence of vacuum irreducibility. Choose a finite label set $I$ with vacuum label $0$ and, for each $i\in I\setminus \{0\}$, a chosen irreducible ordinary module $M_i$. The labels select the finite channels of this annulus datum; they need not exhaust all irreducibles of a module category. The word “rational” in this datum names the finite-channel conventions below, not an assertion that the entire ordinary-module category is semisimple or finite. Such stronger category hypotheses, when needed in a consuming theorem, must be supplied there. For each label choose an involution $i\mapsto i^{\ast }$ and a module isomorphism $J_i:M_i^{\vee }\to M_{i^{\ast }}$, where $M_i^{\vee }={⨁}_h(M_i{)}_h^{\ast }$ is the restricted contragredient. The candidate restricted-dual action is defined in Contragredient module of a generalized module; The restricted dual with opposite action is a contragredient module supplies its actual module axioms. Thus the dual label is an actual module isomorphic to the restricted contragredient, not merely a formal label or candidate action. Write $\overline{M_{i^{\ast }}}$ for a right-moving copy of $M_{i^{\ast }}$ on which the modes are denoted ${\bar{L}}_n$; the overline specifies the right-moving action and does not conjugate scalar coefficients.

Let ${\chi }_i(\tau ):={\chi }_{M_i}(\tau )$ be the ordinary character of Ordinary VOA character, with its parameter $\tau \in \mathbb{H}$ and nome $q(\tau )=e^{2\pi i\tau }$. Assume these character series converge locally uniformly on $\mathbb{H}$, so they define analytic functions there; convergence is an input to this definition, not supplied by the formal character definition. Let $S=(S_{ij}{)}_{i,j\in I}$ be a specified matrix in the convention ${\chi }_i(-1\mathrm{/}\tau )={\sum }_j{S}_{ij}{\chi }_j(\tau )$ wherever this transformation is assumed. The diagonal closed sector data are ${\mathcal{H}}_{\mathrm{c}\mathrm{l}}={⨁}_{i\in I}{M}_i\otimes \overline{M_{i^{\ast }}}\mathrm{.}$

For each $i$, choose bases $e_{i,h,\alpha }$ of the finite-dimensional weight spaces $(M_i{)}_h$ and let $e^{i,h,\alpha }\in (M_i{)}_h^{\ast }$ be the dual bases. Use the module isomorphism $J_i$ to define the formal Ishibashi vector in the product weight completion ${\prod }_h(M_i{)}_h\otimes \overline{(M_{i^{\ast }}{)}_h}$ of its sector by $\mathrm{\mid }i⟩⟩:={\sum }_{h,\alpha }{e}_{i,h,\alpha }\otimes \overline{J_i(e^{i,h,\alpha })}\mathrm{.}$ The vector is independent of the chosen weight bases. The actual contragredient action has $L_{M_i^{\vee }}(m)$ equal to the transpose of $L_{M_i}(-m)$, so the canonical coevaluation tensor is annihilated by $L_n\otimes 1-1\otimes L_{M_i^{\vee }}(-n)$; transporting the right factor by $J_i$ proves the Virasoro gluing equations $(L_n\otimes 1-1\otimes {\bar{L}}_{-n})\mathrm{\mid }i⟩⟩=0(n\in \mathbb{Z})\mathrm{.}$ Its formal bra is normalized by $⟨⟨i\mathrm{\mid }q({\tau }_c{)}^{(L_0+{\bar{L}}_0-c\mathrm{/}12)\mathrm{/}2}\mathrm{\mid }j⟩⟩={\delta }_{ij}{\chi }_i({\tau }_c)\mathrm{.}$ Here $⟨⟨i\mathrm{\mid }$ is the algebraic dual under the weight-block evaluation pairings, with distinct sectors orthogonal and the pairing on each displayed basis tensor normalized by $⟨⟨i\mathrm{\mid }{e}_{i,h,\alpha }\otimes \overline{J_i(e^{i,h,\beta })}⟩={\delta }_{\alpha \beta }$. The overlap then sums $\dim (M_i{)}_h{q}^{h-c\mathrm{/}24}$ over the weight blocks, which is ${\chi }_i({\tau }_c)$. The completion and pairing are formal weight-block operations; no Hilbert-space norm or infinite-dimensional trace is asserted.

A diagonal rational annulus datum additionally specifies $\delta >0$, a width-one cylinder $[0,1]\times (\mathbb{R}\mathrm{/}\delta \mathbb{Z})$, boundary labels $a,b\in I$, finite-dimensional multiplicity spaces $U_{ab}^j$ with $n_{ab}^j=\dim U_{ab}^j\in {\mathbb{Z}}_{\ge 0}$, and the open state space ${\mathcal{H}}_{ab}={⨁}_{j\in I}{U}_{ab}^j\otimes M_j$. Set ${\tau }_o=i\delta \mathrm{/}2,{\tau }_c=2i\mathrm{/}\delta =-1\mathrm{/}{\tau }_o,q_o=q({\tau }_o)=e^{-\pi \delta },q_c=q({\tau }_c)=e^{-4\pi \mathrm{/}\delta }\mathrm{.}$ The notation ${\chi }_i(q_o)$ and ${\chi }_i(q_c)$ is shorthand for ${\chi }_i({\tau }_o)$ and ${\chi }_i({\tau }_c)$, respectively. The open-channel trace and normalized closed-channel overlap are $Z_{ab}^{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}}(q_o)={\sum }_{j\in I}{n}_{ab}^j{\chi }_j({\tau }_o),Z_{ab}^{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}(q_c)={\sum }_{i\in I}⟨a\mathrm{\mid }i⟩⟩⟨⟨i\mathrm{\mid }b⟩⟩{\chi }_i({\tau }_c)\mathrm{.}$

For a Cardy-state construction, require $S$ to be symmetric and $S_{0i}\mathrm{\ne }0$ for every $i$, choose one square root $d_i$ with $d_i^2=S_{0i}$, and set $B_b^i=S_{bi}\mathrm{/}{d}_i,⟨a\mathrm{\mid }i⟩⟩=S_{a^{\ast }i}\mathrm{/}{d}_i,\mathrm{\mid }b⟩={\sum }_i{B}_b^i\mathrm{\mid }i⟩⟩,⟨a\mathrm{\mid }={\sum }_i⟨a\mathrm{\mid }i⟩⟩⟨⟨i\mathrm{\mid }\mathrm{.}$ The corresponding boundary states are required to satisfy $(L_n-{\bar{L}}_{-n})\mathrm{\mid }b⟩=0,⟨a\mathrm{\mid }(L_n-{\bar{L}}_{-n})=0(n\in \mathbb{Z}),$ which follows coefficientwise from the Ishibashi gluing equations. The Cardy annulus condition is the equality $Z_{ab}^{\mathrm{o}\mathrm{p}\mathrm{e}\mathrm{n}}(q_o)=Z_{ab}^{\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}}(q_c)$ under ${\tau }_c=-1\mathrm{/}{\tau }_o$. The definitions above specify its two channel expressions; the equality and nonnegative integral multiplicities are separate properties to prove or assume for a particular datum. No arbitrary Virasoro-gluing state is asserted to satisfy this condition, and no existence of a full local boundary CFT follows from it.

Conformal boundaries, boundary-changing fields, and defects

Ordinary VOA character

Contragredient module of a generalized module

The restricted dual with opposite action is a contragredient module

Intertwining-operator space and fusion multiplicity

The nome normalization follows Cardy's unit-width strip convention. The selected orientation makes the Cardy open multiplicity with labels $(a,b)$ equal to $\dim \mathcal{I}\left(\genfrac{}{}{0px}{}{M_a}{M_b{M}_j}\right)$ when the rational fusion-space identification is supplied. The character matrix $S$ is kept distinct from any categorical Hopf-link matrix. Symmetry, nonzero vacuum row, analytic convergence, and the existence of these data are inputs to the Cardy construction; only the separate consuming theorem can prove them from stronger rationality hypotheses.