The compact-boson reflection orbifold and $c=1$ moduli branch

Let $\rho$ send the charge label $Q=(m,n)$ to $-Q$ and every left/right oscillator mode to its negative. By [F1], the charges and quadratic stress tensors are fixed up to sign reversal, so ${\rho }^2=1$ and both stress tensors are invariant. By [F2], $ϵ(-Q,-Q^{\mathrm{\prime }})=(-1{)}^{(-m)(-n^{\mathrm{\prime }})}=ϵ(Q,Q^{\mathrm{\prime }})$. The OPE in [F3] has unchanged exponents because each is a product of two sign-reversed charges; descendant oscillator signs are likewise preserved in every normal-ordered product. Since the compact-boson fields are generated by these charge fields and oscillators, $\rho$ is an order-two stress-preserving automorphism of the complete compact-boson OPE data.

By [F1], the target coordinate has period $2\pi r$. Reflection fixed points satisfy $x\equiv -x(\mathrm{m}\mathrm{o}\mathrm{d}2\pi r)$, hence $2x\in 2\pi r\mathbb{Z}$. Modulo $2\pi r$, exactly $x=0$ and $x=\pi r$ solve this equation; they are the two twisted ground-state labels in [F5].

The commuting-pair formula in [F4] gives $Z_{\mathrm{o}\mathrm{r}\mathrm{b}}(r;\tau )=\frac{1}{2}(Z_{1,1}+Z_{1,\rho }+Z_{\rho ,1}+Z_{\rho ,\rho })\mathrm{.}$ Here $Z_{1,1}=Z_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}(r;\tau )$. By [F2] and [F5], reflection exchanges every nonzero charge sector with its distinct negative-charge sector, so each such pair contributes zero to $Z_{1,\rho }$. Only the zero-charge oscillator space remains; [F5] makes reflection multiply each level-$n$ oscillator by $-1$, so its chiral trace is $t(q)=q^{-1\mathrm{/}24}{\prod }_{n\ge 1}(1+q^n{)}^{-1}$. The product identity from [F9] is $2\eta \mathrm{/}{\theta }_2=q^{-1\mathrm{/}12}{\prod }_{n\ge 1}(1+q^n{)}^{-2}=t(q{)}^2$. The independent right-moving trace is $\overline{t(q)}$, hence $Z_{1,\rho }=\mathrm{\mid }t(q){\mathrm{\mid }}^2=2\mathrm{\mid }\eta \mathrm{/}{\theta }_2\mathrm{\mid }$.

By [F5], each of the two twisted sectors has half-integer oscillators and ground weight $1\mathrm{/}16$ in each chirality. Its chiral vacuum exponent is $1\mathrm{/}16-1\mathrm{/}24=1\mathrm{/}48$. Summing the geometric oscillator towers gives $u_{-}(q)=q^{1\mathrm{/}48}{\prod }_{n\ge 1}(1-q^{n-1\mathrm{/}2}{)}^{-1}$ without insertion and $u_{+}(q)=q^{1\mathrm{/}48}{\prod }_{n\ge 1}(1+q^{n-1\mathrm{/}2}{)}^{-1}$ with the centralizer insertion from [F5]. The product identities $\eta \mathrm{/}{\theta }_4=q^{1\mathrm{/}24}{\prod }_{n\ge 1}(1-q^{n-1\mathrm{/}2}{)}^{-2}=u_{-}(q{)}^2$ and $\eta \mathrm{/}{\theta }_3=q^{1\mathrm{/}24}{\prod }_{n\ge 1}(1+q^{n-1\mathrm{/}2}{)}^{-2}=u_{+}(q{)}^2$ follow from [F9]. The right-moving chiral factors are their complex conjugates, so one fixed point contributes $\mathrm{\mid }{u}_{-}{\mathrm{\mid }}^2=\mathrm{\mid }\eta \mathrm{/}{\theta }_4\mathrm{\mid }$ without insertion and $\mathrm{\mid }{u}_{+}{\mathrm{\mid }}^2=\mathrm{\mid }\eta \mathrm{/}{\theta }_3\mathrm{\mid }$ with insertion. The two fixed points therefore yield $Z_{\rho ,1}=2\mathrm{\mid }\eta \mathrm{/}{\theta }_4\mathrm{\mid }$ and $Z_{\rho ,\rho }=2\mathrm{\mid }\eta \mathrm{/}{\theta }_3\mathrm{\mid }$. The insertion sign in the latter trace is $(-1{)}^N$ from [F5], so each occupation-number sum has denominator $1+q^{n-1\mathrm{/}2}$. Thus $Z_{\mathrm{o}\mathrm{r}\mathrm{b}}=\frac{1}{2}{Z}_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}+\mathrm{\mid }\eta \mathrm{/}{\theta }_2\mathrm{\mid }+\mathrm{\mid }\eta \mathrm{/}{\theta }_3\mathrm{\mid }+\mathrm{\mid }\eta \mathrm{/}{\theta }_4\mathrm{\mid }\mathrm{.}$ All products converge absolutely for $\mathrm{Im}\tau >0$.

The family hypothesis [F10] supplies at every radius the exact torus covariance of [F4]. With $G=\{1,\rho \}$ and ${\rho }^2=1$, all four pairs $(g,h)\in G^2$ commute. Under $S$, the covariance sends $(g,h)$ to $(h,g^{-1})=(h,g)$, a permutation of these four pairs; under $T$ it sends $(g,h)$ to $(g,gh)$, also a permutation. Therefore the finite sum $\frac{1}{2}{\sum }_{(g,h)\in G^2}{Z}_{g,h}$ in Step 1.3 is invariant under both generators and hence under $SL(2,\mathbb{Z})$. Applying the explicit equality in Step 1.4 identifies that invariant sum with the displayed orbifold partition function. This argument uses the exact covariance already assumed for the compatible completion, not the countable-choice-dependent theta or circle transformation results [F6, F7].

By [F1], the parent compact-boson state-field bijection sends the left and right level-one oscillator states to the parent current fields in the Statement. The action in [F5] negates each corresponding oscillator state; the equivariance clause in [F4] therefore negates each parent current field and fixes the product state. The latter lies in the projected untwisted domain and has the invariant field $J\bar{J}$ in the compatible orbifold completion. Its weights are $(1,1)$ because its state has one left and one right oscillator excitation. Since [F4] requires the group action to preserve the Virasoro fields and the sector data use the same $c_L=c_R=1$ stress tensors from [F1], projection and twisting retain these central charges. On a Euclidean worldsheet patch $\mathrm{\Sigma }\subset \mathbb{C}$ with coordinate $z=\sigma +it$, a smooth classical configuration used to specify the normalization is a map $X_{\mathrm{c}\mathrm{l}}:\mathrm{\Sigma }\to \mathbb{R}\mathrm{/}(2\pi r\mathbb{Z})$ with local real lifts ${\tilde{X}}_{\mathrm{c}\mathrm{l}}$. Set $S_g[{\tilde{X}}_{\mathrm{c}\mathrm{l}}]=(g\mathrm{/}(2\pi )){\int }_{\mathrm{\Sigma }}\mathrm{\partial }{\tilde{X}}_{\mathrm{c}\mathrm{l}}\bar{\mathrm{\partial }}{\tilde{X}}_{\mathrm{c}\mathrm{l}}d\sigma dt$, where $\mathrm{\partial }=({\mathrm{\partial }}_{\sigma }-i{\mathrm{\partial }}_t)\mathrm{/}2$ and $\bar{\mathrm{\partial }}=({\mathrm{\partial }}_{\sigma }+i{\mathrm{\partial }}_t)\mathrm{/}2$. Restrict to configurations with finite Dirichlet integral and to $g>0$, so this functional and its coupling derivative are finite. This classical functional fixes a coupling convention only; it does not assert that quantum fields are smooth or construct a path integral. Put $j_{\mathrm{c}\mathrm{l}}=i\mathrm{\partial }{\tilde{X}}_{\mathrm{c}\mathrm{l}}$ and ${\bar{j}}_{\mathrm{c}\mathrm{l}}=i\bar{\mathrm{\partial }}{\tilde{X}}_{\mathrm{c}\mathrm{l}}$. Then ${\mathrm{\partial }}_g{S}_g{\mathrm{\mid }}_{g=1}=(1\mathrm{/}(2\pi )){\int }_{\mathrm{\Sigma }}\mathrm{\partial }{\tilde{X}}_{\mathrm{c}\mathrm{l}}\bar{\mathrm{\partial }}{\tilde{X}}_{\mathrm{c}\mathrm{l}}d\sigma dt=-(1\mathrm{/}(2\pi )){\int }_{\mathrm{\Sigma }}{j}_{\mathrm{c}\mathrm{l}}{\bar{j}}_{\mathrm{c}\mathrm{l}}d\sigma dt$. By the Gaussian action-coordinate convention stipulated in the Statement, the corresponding infinitesimal CFT-data deformation is represented by the state field $J\bar{J}$; this is a model assumption, not an inference from a formal path integral. Rescaling $Y=\sqrt{g}X$ changes the period from $2\pi r$ to $2\pi \sqrt{g}r$. The resulting radius $r_g=\sqrt{g}r$ satisfies $d{r}_g\mathrm{/}dg{\mathrm{\mid }}_{g=1}=r\mathrm{/}2$, so this action-coordinate tangent is a nonzero scalar multiple of the radius tangent of the untwisted Gaussian family. By [F10], compatible completions with the same fixed labels, domains, and consistency equations are supplied at every positive radius. Thus the tangent lies in the projected orbifold family and is exactly marginal under the definition in the Statement. The all-radius persistence uses [F10] as an explicit model hypothesis.