Anomaly-free finite-group orbifold CFT

Let $T$ be a unitary bosonic full-local CFT data package in the state-space/OPE presentation defined by the earlier supplier Full local two-dimensional CFT data. For the relevant clauses, this means a Hilbert state space $\mathcal{H}$, a common dense invariant domain $\mathcal{D}$, left and right Virasoro actions, weak operator-valued distribution fields on $\mathcal{D}$, and radially convergent OPEs whose vacuum matrix coefficients obey locality and genus-zero crossed-channel equality. That supplier does not include higher-genus sewing. Let $G$ be a finite group and let $U:G\to U(\mathcal{H})$ be a unitary representation preserving $\mathcal{D}$, the vacuum, both Virasoro actions, field adjoints, and every OPE coefficient. Write $\mathcal{F}$ for the field-label space and $\sigma :\mathcal{F}\to \mathcal{D}$ for the supplied linear state-field bijection. On field labels define $U_{g}a:={\sigma }^{-1}(U_g\sigma (a))$ for $a\in \mathcal{F}$; this is well-defined because every $U_g$ preserves $\mathcal{D}$, and the induced label maps obey the same group law. Explicitly, $U_{g}Y(a;z,\bar{z})U_g^{-1}=Y(U_{g}a;z,\bar{z})$ on $\mathcal{D}$, with $U_g{U}_h=U_{gh}$ and $U_g{L}_{n}v=L_n{U}_{g}v$, $U_g{\bar{L}}_{n}v={\bar{L}}_n{U}_{g}v$ for every $g,h\in G$, $n\in \mathbb{Z}$, and $v\in \mathcal{D}$; here $L_n,{\bar{L}}_n$ are the two Virasoro mode actions supplied by the full-local data.

A finite-group orbifold completion is a conditional data package consisting of the following additional spaces, maps, and conditions. It is not a construction theorem for an arbitrary action.

1. Write $e$ for the identity of $G$. Identify ${\mathcal{H}}_e=\mathcal{H}$ and ${\mathcal{D}}_e=\mathcal{D}$, with precisely the parent vacuum, adjoints, Virasoro actions and finite coordinate transports. Require $Y_e=Y$ and, in clause 3, $V_h^e=U_h$ for every $h\in G$. For each $g\in G$, choose a Hilbert sector ${\mathcal{H}}_g$ and a dense linear domain ${\mathcal{D}}_g\subseteq {\mathcal{H}}_g$, invariant under the stated Virasoro modes and all stated sector-field maps. A field $Y_g(a;z,\bar{z})$ of the untwisted label $a$ acts on ${\mathcal{D}}_g$. Its matrix coefficients are weak distributions as defined in the full-local supplier. On each declared branch, analytic continuation once around the spatial $g$-twist satisfies ${\mathrm{AC}}_{z\circlearrowleft 0}{Y}_g(a;z,\bar{z})=Y_g(U_{g}a;z,\bar{z})$; this equation is the $g$-monodromy convention.

2. Set ${\mathcal{F}}_e=\mathcal{F}$ and ${\sigma }_e=\sigma$. For each $g$, choose field-label data ${\mathcal{F}}_g$ and a state-field bijection ${\sigma }_g:{\mathcal{F}}_g\to {\mathcal{D}}_g$. For every ordered pair $g,h\in G$ and $a\in {\mathcal{F}}_g$, specify weak operator-valued distributions $Y_{g,h}(a;z,\bar{z}):{\mathcal{D}}_h\to {\mathcal{D}}_{gh}$. Require $Y_{e,h}(a)=Y_h(a)$ for every untwisted label $a$, hence $Y_{e,e}=Y$, as equalities of weak distributions. For each ordered field pair specify a nonempty radial domain, for example $0<\mathrm{\mid }z-w\mathrm{\mid }<R_{a,b}$ with $R_{a,b}>0$, its branches, the common matrix-coefficient domain, and the OPE coefficient maps. Each OPE expansion must converge absolutely and locally uniformly on compact subsets of that domain. For the final rephased data of clause 4 and every triple of sectors, all iterated OPE trees must have equal matrix coefficients on their common domains after the declared analytic continuation, with the field order and branch factors fixed. These are genus-zero local and crossing conditions; higher-genus sewing is not asserted here.

3. For $h\in G$, specify unitary transport maps $V_h^g:{\mathcal{H}}_g\to {\mathcal{H}}_{hg{h}^{-1}}$ that preserve the stated domains, intertwine both Virasoro actions, and preserve the sector OPEs after the declared phase lift. Require $V_e^g=I$ and $V_k^{hg{h}^{-1}}{V}_h^g=V_{kh}^g$ for every $g,h,k\in G$. In particular, $V_h^g$ preserves ${\mathcal{H}}_g$ when $h\in C_G(g):=\{k\in G:kg=gk\}$. If an initial centralizer lift is projective, record its multiplier ${\mu }_g:C_G(g{)}^2\to U(1)$ and the rephasing ${\lambda }_g:C_G(g)\to U(1)$; require ${\mu }_g(h,k){\mu }_g(hk,\mathrm{\ell })={\mu }_g(k,\mathrm{\ell }){\mu }_g(h,k\mathrm{\ell })$ and ${\lambda }_g(h){\lambda }_g(k){\mu }_g(h,k)={\lambda }_g(hk)$. The rephased maps then form an honest representation. These $V$ maps are the insertion maps before the optional discrete-torsion modification in clause 6; the projected sector is defined there.

4. Record the scalar associator $\omega (g,h,k)\in U(1)$, if the chosen un-rephased twisted OPE products have one. Define its two channels coefficientwise, not by applying a state-field inverse to an unsmeared distribution. For $a\in {\mathcal{F}}_g$ and $b\in {\mathcal{D}}_h$, supply an index set $I_{a,b}$, labels $u_j\in {\mathcal{F}}_{gh}$, and scalar coefficient functions $C_j(z,\bar{z})$ on the declared branches such that, for every $\zeta \in {\mathcal{D}}_{gh}$, the analytic representative of the weak matrix coefficient is $⟨\zeta ,Y_{g,h}(a;z,\bar{z})b⟩={\sum }_{j\in I_{a,b}}{C}_j(z,\bar{z})⟨\zeta ,{\sigma }_{gh}(u_j)⟩\mathrm{.}$ For $c\in {\mathcal{D}}_k$ and $\xi \in {\mathcal{D}}_{ghk}$ define ${\mathcal{A}}_L:={\sum }_{j\in I_{a,b}}{C}_j(z_1-z_2,{\bar{z}}_1-{\bar{z}}_2)⟨\xi ,Y_{gh,k}(u_j;z_2,{\bar{z}}_2)c⟩\mathrm{.}$ Require this scalar sum to converge absolutely and locally uniformly on each common radial domain. Define ${\mathcal{A}}_R$ as the analytic representative there of the weak product distribution obtained by composing the smeared maps $Y_{g,hk}(a;f_1)Y_{h,k}({\sigma }_h^{-1}b;f_2)c$. Supply this product distribution and require both channels, integrated against compactly supported test densities in their domains, to agree with their corresponding smeared OPE expansions. If a direct-integral expansion is used, specify its measure and the corresponding absolutely convergent scalar integrals instead. Thus no unsmeared vector in ${\mathcal{D}}_{gh}$ is presumed. Fix the associator orientation on these scalar channels by ${\mathcal{A}}_L=\omega (g,h,k){\mathcal{A}}_R$, and require $\omega (h,k,\mathrm{\ell })\omega (g,hk,\mathrm{\ell })\omega (g,h,k)=\omega (gh,k,\mathrm{\ell })\omega (g,h,k\mathrm{\ell })\mathrm{.}$ The phase lift includes a normalized $2$-cochain $\beta :G^2\to U(1)$ with $\omega (g,h,k)=\frac{\beta (gh,k)\beta (g,h)}{\beta (g,hk)\beta (h,k)};$ rephasing the $(g,h)$ OPE map by $\beta (g,h{)}^{-1}$ makes the associativity equations strict. In this definition, anomaly-free means precisely that the stated centralizer and OPE phase lifts exist and that the rephased sector data satisfy the group law, associativity, and crossing equations above. No assertion is made that an arbitrary action admits such a lift.

5. For each sector supply self-adjoint closures of $L_0^g,{\bar{L}}_0^g$ with a joint spectral resolution and the parent central charges, and specify the torus evolution $Q_g(\tau ,\bar{\tau })=q^{L_0^g-c_L\mathrm{/}24}{\bar{q}}^{{\bar{L}}_0^g-c_R\mathrm{/}24}$, where $q^a=e^{2\pi i\tau a}$ and ${\bar{q}}^a=e^{-2\pi i\bar{\tau }a}$, with the operators defined by the supplied spectral data. For each commuting pair $g,h$, specify $Z_{g,h}(\tau ,\bar{\tau })={\mathrm{Tr}}_{{\mathcal{H}}_g}(V_h^g{Q}_g)$ on $\mathrm{Im}\tau >0$. Either require $Q_g$ to be trace-class there, or give a regularization linear in finite sums of insertions and invariant under the stated unitary sector transports. In both cases require simultaneous conjugation invariance of these traces. For $\gamma =\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in SL(2,\mathbb{Z})$ define $\gamma \cdot (g,h):=(g^a{h}^c,g^b{h}^d)$. After the chosen anomaly phase lift, require the exact covariance $Z_{g,h}(\gamma \tau ,\overline{\gamma \tau })=Z_{\gamma \cdot (g,h)}(\tau ,\bar{\tau })$ for every commuting pair. Thus $S$ sends $(g,h)$ to $(h,g^{-1})$ and $T$ sends it to $(g,gh)$.

6. Optional discrete torsion is a normalized group $2$-cocycle $\alpha :G^2\to U(1)$, meaning $\alpha (e,g)=\alpha (g,e)=1$ and $\alpha (g,h)\alpha (gh,k)=\alpha (h,k)\alpha (g,hk)$. For commuting $g,h$ put ${ϵ}_{\alpha }(g,h):=\alpha (g,h)\mathrm{/}\alpha (h,g)$; require this phase to be invariant under the displayed modular action and simultaneous conjugation. With no discrete torsion set $\alpha =1$. The projection action, distinguished from the trace insertion $V_h^g$, is $W_h^g:={ϵ}_{\alpha }(g,h)V_h^g(h\in C_G(g)),P_g^{\alpha }:=\frac{1}{\mathrm{\mid }{C}_G(g)\mathrm{\mid }}{\sum }_{h\in C_G(g)}{W}_h^g\mathrm{.}$ The cocycle identity gives ${ϵ}_{\alpha }(g,hk)={ϵ}_{\alpha }(g,h){ϵ}_{\alpha }(g,k)$ for $h,k\in C_G(g)$, so $W^g$ is an honest unitary representation and $P_g^{\alpha }$ is its orthogonal invariant projector. Explicitly the projected sector is ${\mathcal{H}}_g^{C_G(g),\alpha }:=\{v\in {\mathcal{H}}_g:W_h^{g}v=v\text{ for every }h\in C_G(g)\}=\{v\in {\mathcal{H}}_g:V_h^{g}v={ϵ}_{\alpha }(g,h{)}^{-1}v\text{ for every }h\in C_G(g)\}\mathrm{.}$ Supply the compatible torsion-dependent sector OPEs, adjoints and vacuum data and require their induced fields on the projected sum, with dense domain ${⨁}_{[g]}{P}_g^{\alpha }{\mathcal{D}}_g$, to satisfy the full-local supplier's unitary bosonic genus-zero conditions. This compatibility is additional completion data, not a consequence of modular covariance. The projected state space and torus amplitude are respectively ${\mathcal{H}}_{T\mathrm{/}G}:={⨁}_{[g]}{\mathcal{H}}_g^{C_G(g),\alpha },Z_{T\mathrm{/}G}(\tau ,\bar{\tau }):=\frac{1}{\mathrm{\mid }G\mathrm{\mid }}{\sum }_{gh=hg}{ϵ}_{\alpha }(g,h)Z_{g,h}(\tau ,\bar{\tau })\mathrm{.}$ Here $[g]$ ranges over one representative per conjugacy class. In the trace-class case conjugation invariance and $\mathrm{\mid }[g]\mathrm{\mid }=\mathrm{\mid }G\mathrm{\mid }\mathrm{/}\mathrm{\mid }{C}_G(g)\mathrm{\mid }$ identify this sum with ${\sum }_{[g]}{\mathrm{Tr}}_{{\mathcal{H}}_g}(P_g^{\alpha }{Q}_g)$, the trace on the projected state space. In a regularized case require the analogous identity with the supplied projected-sector regularization. Since ${ϵ}_{\alpha }(e,h)=1$, the projected untwisted sector is exactly the invariant subspace of the supplied parent $T$; for $G=\{e\}$ the genus-zero data reduce to $T$.

The finite coordinate interface is also part of the completion data. Require the group action on field labels to commute with the bulk field-label coordinate transports, and the action on states to intertwine their state-space transports and chart-vacuum data. Supply corresponding coordinate transports on every twisted sector and sector-field test-density space, respecting restrictions, sector OPEs, centralizer projections and the stated phase lifts. Require their induced transports on the projected sum to satisfy the full-local supplier's finite cocycles, test-density Jacobian law, centered-state limits and vacuum equations. These are hypotheses on a completion, not a claim that an arbitrary finite action integrates its Virasoro representations or automatically supplies the missing analytic data.

When these data and equalities hold, $T\mathrm{/}G$ denotes the resulting genus-zero full-local orbifold data together with its specified torus traces. A geometric quotient of a target space is not part of the definition and need not exist. Modular invariance of the weighted torus sum follows from the exact pair permutation and phase equality; this genus-one fact alone supplies neither the twisted OPE data nor higher-genus sewing.

Full local two-dimensional CFT data

Ginsparg, §§8.3–8.5

Sharpe, §4.2.1, equations (26)–(27)

Dong–Li–Mason, Introduction and Theorems 1 and 3

The Ginsparg construction is used for the finite-group projection and commuting-holonomy trace conventions. Dong–Li–Mason prove related existence and modular-covariance results for rational vertex operator algebras under finiteness hypotheses; those chiral results do not prove the general full-local completion required by this definition. Neither source makes every finite symmetry anomaly-free. The anomaly-free completion and its twisted OPE/sewing data remain explicit hypotheses.