Geometric finite-group orbifold compactification data

Let $C$ be a specified unitary bosonic full-local CFT data package in the state-space/OPE presentation of the earlier full-local CFT definition. Its relevant data include a Hilbert space $\mathcal{H}$, a common dense invariant domain $\mathcal{D}$, left and right Virasoro actions, weak operator-valued local fields, radially convergent OPEs, vacuum data, locality, and genus-zero crossed-channel equality. Let $\mathcal{F}$ be its field-label space and $\sigma :\mathcal{F}\to \mathcal{D}$ its state-field bijection. Let $G$ be a finite group and $U:G\to U(\mathcal{H})$ a unitary representation preserving $\mathcal{D}$, the vacuum, both Virasoro actions, field adjoints, and every OPE coefficient. Require $U_{g}Y(a;z,\bar{z})U_g^{-1}=Y(U_{g}a;z,\bar{z})$, where $U_{g}a:={\sigma }^{-1}(U_g\sigma (a))$; thus the induced field-label maps obey the group law and preserve the specified local operator products. Require a specified compact smooth internal target $X$ with a smooth Riemannian metric $g_X$ and a group action $a:G\times X\to X$, writing $a_g(x)=a(g,x)$, such that each $a_g$ is a smooth diffeomorphism, $a_e={\mathrm{i}\mathrm{d}}_X$, $a_{gh}=a_g\circ a_h$, and $a_g^{\ast }{g}_X=g_X$. Supply a target realization of $C$ and a lift of this target action to its worldsheet fields inducing exactly $U_g$: on a target-coordinate configuration $f:\mathrm{\Sigma }\to X$ the action is $f\mapsto a_g\circ f$, where $\mathrm{\Sigma }$ is the worldsheet surface; the lift also specifies the action on any additional worldsheet fields. These realization and compatibility data are required, not inferred from the metric quotient.

A geometric finite-group orbifold compactification datum consists of $C$, $G$, $(X,g_X,a)$ and its specified worldsheet lift, the action $U$, a $g$-twisted Hilbert space ${\mathcal{H}}_g$ and dense invariant domain ${\mathcal{D}}_g$ for every $g\in G$, sector-field maps with their radial domains and branches, and unitary centralizer transports on every ${\mathcal{H}}_g$. All sector data satisfy the finite-group completion conditions, including its finite coordinate transports and compatible projected fields. In particular, identify ${\mathcal{H}}_e=\mathcal{H}$, ${\mathcal{D}}_e=\mathcal{D}$, and the untwisted fields, vacuum, adjoints and Virasoro actions with the parent data. The centralizer is $C_G(g):=\{h\in G:hg=gh\}$. After the supplied phase lift, its trace-insertion maps $V_h^g$ form an honest unitary representation on ${\mathcal{H}}_g$, preserve ${\mathcal{D}}_g$, and obey $V_h^e=U_h$. A projective initial lift is permitted only with a specified rephasing that yields these honest maps. Optional discrete torsion is a normalized group $2$-cocycle $\alpha :G\times G\to U(1)$, meaning $\alpha (e,g)=\alpha (g,e)=1$ and $\alpha (g,h)\alpha (gh,k)=\alpha (h,k)\alpha (g,hk)$; with no torsion set $\alpha =1$. For commuting $g,h$ put ${ϵ}_{\alpha }(g,h):=\alpha (g,h)\mathrm{/}\alpha (h,g)$ and define the projection action and projector by $W_h^g:={ϵ}_{\alpha }(g,h)V_h^g,P_g^{\alpha }:=\frac{1}{\mathrm{\mid }{C}_G(g)\mathrm{\mid }}{\sum }_{h\in C_G(g)}{W}_h^g\mathrm{.}$ The projected sector is ${\mathcal{H}}_g^{C_G(g),\alpha }:=\{v\in {\mathcal{H}}_g:W_h^{g}v=v\text{ for all }h\in C_G(g)\}$, and the candidate projected state space is ${⨁}_{[g]}{\mathcal{H}}_g^{C_G(g),\alpha }$, with projected dense domains and compatible torsion-dependent OPEs as supplied by the completion. Thus invariants refer to $W^g$, not to a projective initial lift or, when torsion is nontrivial, to the insertion representation $V^g$.

For a projected state of left/right conformal weights $(h,\bar{h})$, level matching means $h-\bar{h}\in \mathbb{Z}$. The phase-lift requirement is that the centralizer and twisted-OPE phase lifts exist and, after rephasing, satisfy the group law, associativity, and crossed-channel equations. For this interface, call these identities together with the specified torus modular-covariance identities the orbifold anomaly-consistency conditions; the phrase is only a name for these explicit compatibility conditions, not a claim to classify all anomalies of a complete string background. For each commuting pair $g,h$, the torus trace $Z_{g,h}(\tau ,\bar{\tau })={\mathrm{Tr}}_{{\mathcal{H}}_g}(V_h^g{Q}_g)$ uses the supplied torus evolution $Q_g=q^{L_0^g-c_L\mathrm{/}24}{\bar{q}}^{{\bar{L}}_0^g-c_R\mathrm{/}24}$, with $q^a=e^{2\pi i\tau a}$ and ${\bar{q}}^a=e^{-2\pi i\bar{\tau }a}$ defined by the supplied joint spectral data. It must be specified for $\mathrm{Im}\tau >0$, with its trace-class domain or regularization, and must obey the declared modular covariance under $(g,h)\mapsto (g^a{h}^c,g^b{h}^d)$ for $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in SL(2,\mathbb{Z})$. Require the torus amplitude to be $\mathrm{\mid }G{\mathrm{\mid }}^{-1}{\sum }_{gh=hg}{ϵ}_{\alpha }(g,h)Z_{g,h}$ and to agree with the trace on the projected sectors, using the completion's finite-sum-linear, conjugation-invariant regularization if needed. The twisted-sector OPE data include nonempty radial domains, coefficient maps, and branches, with convergent matrix coefficients and equality of iterated OPE channels. This item specifies genus-zero local data and genus-one traces only; it does not supply higher-genus amplitudes. Here higher-genus sewing data mean amplitudes on compact marked Riemann surfaces of genus at least two, together with factorization identities when punctures with local coordinates $z,w$ vanishing at their centers are glued by $zw=q$ for a complex parameter $q$ with $0<\mathrm{\mid }q\mathrm{\mid }<1$; the identity requires agreement with the convergent sum over a complete orthonormal basis of intermediate states and consistency between inequivalent gluing decompositions. A claim of a complete string background requires these data and review. Fixed loci $X^g=\{x\in X:g\cdot x=x\}$ are geometric data; when a fixed locus contributes localized states, those states occur in the corresponding twisted-sector description. Twisted sectors are required even for a fixed-point-free action, so the coarse orbit space alone does not encode the orbifold state space. A modular-invariant genus-one sum is not by itself a proof of twisted OPE locality or the higher-genus factorization identities just defined. The quotient notation $X\mathrm{/}G$ does not assert that the coarse quotient is smooth or that the worldsheet completion exists.

Anomaly-free finite-group orbifold CFT

The current CFT supplier is ready for this conditional interface. It does not prove that an arbitrary finite symmetry admits the required compatible phase lifts, modular-covariant traces, and twisted sectors with OPE data. Ginsparg's §§8.3–8.4 give the genus-one commuting-holonomy and twisted-sector picture; Belavin supplies one source-specific heterotic example. Neither source is used to infer a universal construction theorem. The R1 proposal's older note that the generic CFT supplier is blocked is stale relative to the inspected current carrier and readiness record; the actual conditional scope above controls.