GKO coset CFT with a specified affine embedding

Fix compact WZW data as in Compact Wess–Zumino–Witten model and its left-right completion for a finite family of compact, connected, simply connected simple target groups $G_i$, with complex simple Lie algebras ${\mathfrak{g}}_i$, positive integral levels $k_i$, and basic forms $B_i$ whose long roots have squared length $2$. Let $\widehat{\mathfrak{g}}={⨁}_{i=1}^r{\widehat{{\mathfrak{g}}_i}}_{k_i}$ be the direct sum of affine current algebras. For each $i$, let $V^{k_i}({\mathfrak{g}}_i)$ be the universal affine vertex algebra supplied by Universal affine vertex algebras exist at every complex level, using the state-field convention of Vertex algebra in the state–field convention. The tensor product ${\mathcal{U}}_G={⨂}_{i=1}^r{V}^{k_i}({\mathfrak{g}}_i)$ has the componentwise vacuum and state-field operations.

Let $H$ be a specified compact connected subgroup of ${\prod }_i{G}_i$ whose complexified Lie algebra $\mathfrak{h}$ is simple, and let $\iota :\mathfrak{h}\to {⨁}_i{\mathfrak{g}}_i$ be the differential of its inclusion, with components ${\iota }_i$. Choose the invariant form $B_{\mathfrak{h}}$ normalized so long roots have squared length $2$. For each component supply its embedding index $I_i$ defined by $B_i({\iota }_{i}x,{\iota }_{i}y)=I_i{B}_{\mathfrak{h}}(x,y)$. The induced denominator level is $\mathrm{\ell }={\sum }_i{I}_i{k}_i$, the central coefficient in the diagonal current bracket $J_x^{\mathfrak{h}}(n)={\sum }_i{J}_{{\iota }_{i}x}^{(i)}(n)$.

Choose a numerator full-local package exactly as in Full local two-dimensional CFT data, with state space ${\mathcal{H}}_G$, common dense invariant domain ${\mathcal{D}}_G$, field-label space ${\mathcal{F}}_G$, state-field map ${\sigma }_G$, and local fields $Y_G$. Let ${\mathcal{F}}_G^{\mathrm{h}\mathrm{o}\mathrm{l}}$ be the subspace of labels whose local fields depend holomorphically on $z$ on their declared domains. The numerator chiral algebra is a specified complex vertex algebra ${\mathcal{A}}_G$ in the state-field convention, together with a vertex-algebra homomorphism ${\mathrm{\Phi }}_G:{\mathcal{U}}_G\to {\mathcal{A}}_G$. This includes the exact choice of ambient algebra and its current-generated subalgebra; if the choice is a quotient or an extension, specify its kernel or extension map. Specify an injective linear map ${\mathrm{ȷ}}_G:{\mathcal{A}}_G\to {\mathcal{F}}_G^{\mathrm{h}\mathrm{o}\mathrm{l}}$ preserving the vacuum, translation, every vertex product, and the corresponding local OPE.

For each numerator sector $V_{\lambda }^G$, the common domain ${\mathcal{D}}_G$, and the regular module ${\mathcal{A}}_G$, supply ordinary ${\mathcal{A}}_G$-module structures in the sense of Module for an ordinary vertex algebra, compatible with ${\mathrm{ȷ}}_G$ and the full-local field map: on ${\mathcal{D}}_G$ and each sector, the field in the package labelled by ${\mathrm{ȷ}}_G(a)$ is the same operator-valued distribution as the module field of $a$. Require that for each $x\in {\mathfrak{g}}_i$, the local field labelled by ${\mathrm{ȷ}}_G{\mathrm{\Phi }}_G(\mathbf{1}\otimes \cdots \otimes x_{-1}\mathbf{1}\otimes \cdots \otimes \mathbf{1})$ is exactly the left WZW current field identified in Compact Wess–Zumino–Witten model and its left-right completion, and its modes on ${\mathcal{D}}_G$, every $V_{\lambda }^G$, and the regular module ${\mathcal{A}}_G$ are exactly $J_n^{(i),x}$. The restrictions to the numerator current algebras are required to be restricted affine-current modules: for each fixed vector and current, all sufficiently large positive modes annihilate that vector. These homomorphisms, field identifications, and module actions are part of the model data, not consequences of naming a numerator chiral algebra.

Specify a vertex-algebra homomorphism ${\mathrm{\Phi }}_H:V^{\mathrm{\ell }}(\mathfrak{h})\to {\mathcal{A}}_G$ whose current generator is ${\mathrm{\Phi }}_H(x_{-1}\mathbf{1})={\sum }_{i=1}^r{\mathrm{\Phi }}_G(\mathbf{1}\otimes \cdots \otimes ({\iota }_{i}x{)}_{-1}\mathbf{1}\otimes \cdots \otimes \mathbf{1})\mathrm{.}$ The denominator module action is the restriction along ${\mathrm{\Phi }}_H$; its modes on ${\mathcal{D}}_G$, ${\mathcal{A}}_G$, and every selected numerator module must be exactly $J_x^{\mathfrak{h}}(n)$. These restrictions are restricted affine-current modules of level $\mathrm{\ell }$. Give the analogous right-moving chiral algebra, embedding, and module data for a full left-right coset.

Require $k_i+h_i^{\vee }\mathrm{\ne }0$ and $\mathrm{\ell }+h_{\mathfrak{h}}^{\vee }\mathrm{\ne }0$. Let ${\omega }_{\mathfrak{h}}=\frac{1}{2(\mathrm{\ell }+h_{\mathfrak{h}}^{\vee })}{\sum }_a(e_a{)}_{-1}(e_a{)}_{-1}\mathbf{1}\in V^{\mathrm{\ell }}(\mathfrak{h})$ for a $B_{\mathfrak{h}}$-orthonormal basis $(e_a)$. Supply the numerator Sugawara states ${\omega }_i\in {\mathcal{A}}_G$ as images of the factor Sugawara states, and set ${\omega }_H={\mathrm{\Phi }}_H({\omega }_{\mathfrak{h}})$ for the embedded denominator Sugawara state. Normalize their vertex fields as $Y_{{\mathcal{A}}_G}({\omega }_i,z)={\sum }_n{L}_n^{(i)}{z}^{-n-2}$ and $Y_{{\mathcal{A}}_G}({\omega }_H,z)={\sum }_n{L}_n^H{z}^{-n-2}$. Their actions on ${\mathcal{A}}_G$ and each selected numerator module must be the Sugawara modes of Sugawara modes and their normal ordering. Set ${\omega }_G={\sum }_i{\omega }_i$, $T_G(z)={\sum }_i{Y}_{{\mathcal{A}}_G}({\omega }_i,z)$, $T_H(z)=Y_{{\mathcal{A}}_G}({\omega }_H,z)$, and $L_n^G={\sum }_i{L}_n^{(i)}$. Define the coset stress state and modes by ${\omega }_{G\mathrm{/}H}={\omega }_G-{\omega }_H$, $Y_{{\mathcal{A}}_G}({\omega }_{G\mathrm{/}H},z)={\sum }_n{L}_n^{G\mathrm{/}H}{z}^{-n-2}$, and $L_n^{G\mathrm{/}H}=L_n^G-L_n^H$. The lemma on this page applies on the common restricted modules and on the regular module ${\mathcal{A}}_G$; it proves the Virasoro relation and commutation with every embedded denominator-current mode under the specified noncritical hypotheses. By the field-mode and OPE compatibility just imposed, those vanishing mode commutators are exactly the absence of singular terms in $Y_{{\mathcal{A}}_G}({\omega }_{G\mathrm{/}H},z)J_x^{\mathfrak{h}}(w)$. Thus ${\omega }_{G\mathrm{/}H}$ lies in the explicitly defined commutant ${\mathrm{Com}}_{{\mathcal{A}}_G}({\mathrm{\Phi }}_H(V^{\mathrm{\ell }}(\mathfrak{h})))=\{a\in {\mathcal{A}}_G:J_x^{\mathfrak{h}}(n)a=0\text{ for every }x\in \mathfrak{h}\text{ and }n\ge 0\}\mathrm{.}$ Equivalently, its local field has regular OPE with every embedded denominator current. Its central charge is that of Sugawara embeddings give a Virasoro coset stress tensor.

For each selected numerator module $V_{\lambda }^G$, specify its denominator-module decomposition, for example $V_{\lambda }^G\text{\hspace{0.17em}⁣}{\downarrow }_{V^{\mathrm{\ell }}(\mathfrak{h})}\cong {\widehat{⨁}}_{\nu \in S_{\lambda }}{B}_{\lambda ,\nu }\widehat{\otimes }{V}_{\nu }^H$. Here $V_{\nu }^H$ is a specified denominator module and $B_{\lambda ,\nu }$ is the corresponding multiplicity space; if restriction is not semisimple, give the actual replacement decomposition and topology. The sets $S_{\lambda }$ are the selection rules and must be stated. Supply any field-identification equivalence relation on the pairs $(\lambda ,\nu )$, its fixed-point resolutions into resolved sectors, and the chosen left-right pairing; none is inferred from the notation $G\mathrm{/}H$.

Write ${\mathcal{C}}_L={\mathrm{Com}}_{{\mathcal{A}}_G}({\mathrm{\Phi }}_H(V^{\mathrm{\ell }}(\mathfrak{h})))$ and let ${\mathcal{C}}_R$ be the commutant in the specified right-moving numerator algebra. Supply their inherited vertex-algebra structures and the following compatible branching actions. If $d_{\lambda ,\nu }:B_{\lambda ,\nu }\widehat{\otimes }{V}_{\nu }^H\to V_{\lambda }^G$ denotes a summand inclusion in the specified decomposition, give $B_{\lambda ,\nu }$ a ${\mathcal{C}}_L$-module field $Y_{\lambda ,\nu }^B$ such that, on the declared common algebraic domains, for every $a\in {\mathcal{C}}_L$, $Y_{V_{\lambda }^G}(a,z)d_{\lambda ,\nu }(b\otimes v)=d_{\lambda ,\nu }(Y_{\lambda ,\nu }^B(a,z)b\otimes v)\mathrm{.}$ Thus the commutant acts on the multiplicity factor and as the identity on the denominator factor, with no action mixing denominator labels. Require the analogous identity for ${\mathcal{C}}_R$ and the right branching spaces. The module structures obey the vacuum, lower-truncation and module identities of Module for an ordinary vertex algebra. If the chosen restriction uses a nonsemisimple replacement, specify its actual multiplicity objects and action-intertwining maps instead of the displayed summand inclusions; do not infer such a decomposition. Every field identification and fixed-point resolution used below must intertwine these actions, so each actual resolved branching sector has a specified left or right commutant-module action.

Choose the full-local package for ${\mathcal{H}}_{G\mathrm{/}H}$ with notation ${\mathcal{D}}_{G\mathrm{/}H},{\mathcal{F}}_{G\mathrm{/}H},{\sigma }_{G\mathrm{/}H},Y_{G\mathrm{/}H},L_n^{\mathrm{l}\mathrm{o}\mathrm{c}},{\bar{L}}_n^{\mathrm{l}\mathrm{o}\mathrm{c}}$. Specify injective linear field-label maps ${\mathrm{ȷ}}_L:{\mathcal{C}}_L⟶{\mathcal{F}}_{G\mathrm{/}H}^{\mathrm{h}\mathrm{o}\mathrm{l}},{\mathrm{ȷ}}_R:{\mathcal{C}}_R⟶{\mathcal{F}}_{G\mathrm{/}H}^{\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{i}},$ where the two target spaces consist of labels with fields depending only on $z$ and only on $\bar{z}$, respectively. Both maps preserve the vacuum label, translation, all vertex products and the corresponding local OPE, with ${\sigma }_{G\mathrm{/}H}{\mathrm{ȷ}}_L(\mathbf{1})={\sigma }_{G\mathrm{/}H}{\mathrm{ȷ}}_R(\mathbf{1})={\mathrm{\Omega }}_{G\mathrm{/}H}$. Require compatible commuting ${\mathcal{C}}_L$- and ${\mathcal{C}}_R$-module actions on ${\mathcal{D}}_{G\mathrm{/}H}$. On a selected resolved paired sector ${\mathcal{B}}_p^L\otimes {\mathcal{B}}_q^R$, with its separately specified multiplicity factor, these actions are $Y_{{\mathcal{B}}_p^L}(a,z)\otimes I$ and $I\otimes Y_{{\mathcal{B}}_q^R}(\bar{a},\bar{z})$, acting trivially on the multiplicity factor. They extend to the declared completed state space on its common domain. For all $a\in {\mathcal{C}}_L$, $\bar{a}\in {\mathcal{C}}_R$, require $Y_{G\mathrm{/}H}({\mathrm{ȷ}}_L(a);z,\bar{z})=Y_{{\mathcal{D}}_{G\mathrm{/}H}}^L(a,z),Y_{G\mathrm{/}H}({\mathrm{ȷ}}_R(\bar{a});z,\bar{z})=Y_{{\mathcal{D}}_{G\mathrm{/}H}}^R(\bar{a},\bar{z})\mathrm{.}$ These are equalities of the weak local-field distributions and their radial mode expansions on the package's declared domains, not merely equalities of characters.

Finally require that these maps identify the coset stress states with the stress fields of this very full-local package: $Y_{G\mathrm{/}H}({\mathrm{ȷ}}_L({\omega }_{G\mathrm{/}H});z,\bar{z})={\sum }_{n\in \mathbb{Z}}{L}_n^{\mathrm{l}\mathrm{o}\mathrm{c}}{z}^{-n-2},Y_{G\mathrm{/}H}({\mathrm{ȷ}}_R({\bar{\omega }}_{G\mathrm{/}H});z,\bar{z})={\sum }_{n\in \mathbb{Z}}{\bar{L}}_n^{\mathrm{l}\mathrm{o}\mathrm{c}}{\bar{z}}^{-n-2}\mathrm{.}$ On each branching space these modes are exactly those induced by $L_n^G-L_n^H$ (and the right-moving difference), and on ${\mathcal{D}}_{G\mathrm{/}H}$ they are the corresponding resolved-sector actions, with identity on all other factors. In particular the package central charges are the respective Sugawara central-charge differences. The field maps, compatible branching actions and stress-field equalities are requirements on the chosen completion; the chiral coset lemma supplies the Virasoro relations, not the existence of such a full-local completion.

The full coset state space ${\mathcal{H}}_{G\mathrm{/}H}$ is a specified direct sum or direct integral of the resolved left and right branching multiplicity spaces with stated multiplicities, a unique vacuum pair of multiplicity one, the topology, and a common domain. It must satisfy the full local state-space/OPE conditions, including local fields and crossed-channel equality. A full-CFT record also specifies its left-right torus pairing and either a convergent torus trace or an explicitly declared regularization. If modular invariance is claimed, the associated torus amplitude must be invariant under $\tau \mapsto (a\tau +b)\mathrm{/}(c\tau +d)$ and its conjugate for every integer matrix with $ad-bc=1$. Unitarity and reflection positivity require their own inner-product, adjoint, and positivity data. Higher-genus sewing is additional data or a separate theorem.

Compact Wess–Zumino–Witten model and its left-right completion

Full local two-dimensional CFT data

Restricted affine-current module

Sugawara modes and their normal ordering

Dual Coxeter number for the normalized invariant form

Sugawara modes satisfy Virasoro at every noncritical complex level

Universal affine vertex algebras exist at every complex level

Vertex algebra in the state–field convention

Module for an ordinary vertex algebra

Restricted affine-current fields are mutually local

Sugawara embeddings give a Virasoro coset stress tensor

This is the algebraic affine coset interface. A gauged WZW action, gauge fixing, anomaly cancellation, and a measure on fields are not identified with these current-algebra data. A central-charge calculation alone supplies neither the branching modules nor a full local coset CFT.