Reflection-orbifold sectors of the compact boson

Use the following explicit convention for the half-integer bosonic Fock spaces. For each $j\in \{0,1\}$, let ${\mathcal{H}}_{\rho ,j}^{\mathrm{a}\mathrm{l}\mathrm{g}}=\mathbb{C}[t_s,{\bar{t}}_s:s\in {\mathbb{Z}}_{\ge 0}+\frac{1}{2}]\mid j⟩,$ where each polynomial uses finitely many variables and $\mid j⟩$ is the constant monomial. For $s>0$, set ${\alpha }_{-s}f=t_{s}f$, ${\alpha }_{s}f=s\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{t}_s$, ${\bar{\alpha }}_{-s}f={\bar{t}}_{s}f$, and ${\bar{\alpha }}_{s}f=s\mathrm{\partial }f\mathrm{/}\mathrm{\partial }{\bar{t}}_s$. In particular, every positive mode annihilates $\mid j⟩$. Give distinct monomials the orthogonal inner product $⟨{\prod }_{s>0}{t}_s^{u_s}{\bar{t}}_s^{{\bar{u}}_s}\mid j⟩,{\prod }_{s>0}{t}_s^{v_s}{\bar{t}}_s^{{\bar{v}}_s}\mid j⟩⟩={\mathbf{1}}_{u=v,\bar{u}=\bar{v}}{\prod }_{s>0}{s}^{u_s+{\bar{u}}_s}{u}_s!{\bar{u}}_s!\mathrm{.}$ The products are finite on monomials. This positive inner product makes ${\alpha }_s^{\ast }={\alpha }_{-s}$ and ${\bar{\alpha }}_s^{\ast }={\bar{\alpha }}_{-s}$ on the algebraic core; ${\mathcal{H}}_{\rho ,j}$ is its Hilbert completion.

For a Ginsparg radius $r>0$, the compact target is the circle $S_r^1=\mathbb{R}\mathrm{/}(2\pi r\mathbb{Z})$. Let $\rho$ act by $[X]\mapsto [-X]$, $V_{m,n}\mapsto V_{-m,-n}$, and $a_k,{\bar{a}}_k\mapsto -a_k,-{\bar{a}}_k$ for every integer mode $k$, including $k=0$. The untwisted invariant sector is the $+1$ eigenspace of $\rho$ in the Hilbert space ${\mathcal{H}}_r$ of Compact free boson as a full local CFT.

A spatially $\rho$-twisted lift of the target coordinate obeys $X(\sigma +2\pi )=-X(\sigma )+2\pi rj$, where $j\in \{0,1\}$ represents the two classes modulo $2$. These sectors are localized at the fixed classes $[0]$ and $[\pi r]$. For each $j$, let ${\mathcal{H}}_{\rho ,j}^{\mathrm{a}\mathrm{l}\mathrm{g}}$ be the bosonic Fock space generated from one ground vector $\mathrm{\mid }j⟩$ by left and right modes with $s\in \mathbb{Z}+\frac{1}{2}$, satisfying $[{\alpha }_s,{\alpha }_t]=s{\delta }_{s+t,0}I$ and $[{\bar{\alpha }}_s,{\bar{\alpha }}_t]=s{\delta }_{s+t,0}I$; left and right modes commute. The ground vectors have norm one, and the Fock inner product is fixed by ${\alpha }_s^{\ast }={\alpha }_{-s}$, ${\bar{\alpha }}_s^{\ast }={\bar{\alpha }}_{-s}$; let ${\mathcal{H}}_{\rho ,j}$ be the Hilbert completion. There are no momentum-winding zero modes. If $n_s,{\bar{n}}_s\in {\mathbb{Z}}_{\ge 0}$ for $s>0$ are the occupation numbers of ${\alpha }_{-s},{\bar{\alpha }}_{-s}$, set $E_L={\sum }_{s}s{n}_s$, $E_R={\sum }_{s}s{\bar{n}}_s$, $N_L={\sum }_s{n}_s$, and $N_R={\sum }_s{\bar{n}}_s$. The ground weights are $(1\mathrm{/}16,1\mathrm{/}16)$, and a descendant has weights $(h,\bar{h})=(1\mathrm{/}16+E_L,1\mathrm{/}16+E_R)$. The centralizer lift ${\rho }_{\mathrm{t}\mathrm{w}}$ fixes both $\mathrm{\mid }j⟩$ and acts on a descendant by $(-1{)}^{N_L+N_R}$. Thus it has order two; the projection keeps descendants with an even total number of oscillator creation operators, not a parity exponent of the half-integer energy level.

The reflection-orbifold Hilbert space is the direct sum of the $\rho$-invariant untwisted subspace and the $+1$ eigenspaces of ${\rho }_{\mathrm{t}\mathrm{w}}$ in ${\mathcal{H}}_{\rho ,0}\oplus {\mathcal{H}}_{\rho ,1}$, where ${\mathcal{H}}_{\rho ,j}$ is the Hilbert completion of ${\mathcal{H}}_{\rho ,j}^{\mathrm{a}\mathrm{l}\mathrm{g}}$. On each twisted Fock basis, let $L_0$ and ${\bar{L}}_0$ act diagonally with the weights $h,\bar{h}$ specified above. The two twisted ground states are created by fields ${\sigma }_0,{\sigma }_{\pi r}$ with the stated reflection monodromy. Put $J(z)=a(z)={\sum }_{k\in \mathbb{Z}}{a}_k{z}^{-k-1}$ and $\bar{J}(\bar{z})=\bar{a}(\bar{z})={\sum }_{k\in \mathbb{Z}}{\bar{a}}_k{\bar{z}}^{-k-1}$. For each insertion pair use the square-root branch $(z-w{)}^{1\mathrm{/}2}=\exp (\frac{1}{2}\mathrm{Log}(z-w))$ on $z-w\in \mathbb{C}\setminus (-\mathrm{\infty },0]$, and its conjugate for $\bar{z}-\bar{w}$. Normalize the excited twist fields ${\tau }_j,{\bar{\tau }}_j$ so that $J(z){\sigma }_j(w,\bar{w})\sim {\tau }_j(w,\bar{w})\mathrm{/}(z-w{)}^{1\mathrm{/}2}$ and $\bar{J}(\bar{z}){\sigma }_j(w,\bar{w})\sim {\bar{\tau }}_j(w,\bar{w})\mathrm{/}(\bar{z}-\bar{w}{)}^{1\mathrm{/}2}$ on these branches.

For $g\in \{1,\rho \}$ let ${\mathcal{H}}_1={\mathcal{H}}_r$ and ${\mathcal{H}}_{\rho }={\mathcal{H}}_{\rho ,0}\oplus {\mathcal{H}}_{\rho ,1}$. Define $U_1^{(g)}=I$; define $U_{\rho }^{(1)}=\rho$ on ${\mathcal{H}}_1$ and $U_{\rho }^{(\rho )}={\rho }_{\mathrm{t}\mathrm{w}}$ on ${\mathcal{H}}_{\rho }$. The torus trace in spatial sector $g$ with temporal insertion $h$ is $Z_{g,h}(r;\tau )={\mathrm{Tr}}_{{\mathcal{H}}_g}\text{\hspace{0.17em}⁣}\left(U_h^{(g)}{q}^{L_0-1\mathrm{/}24}{\bar{q}}^{{\bar{L}}_0-1\mathrm{/}24}\right),q=e^{2\pi i\tau },\bar{q}=e^{-2\pi i\bar{\tau }},\mathrm{Im}\tau >0.$ The reflection-orbifold torus amplitude is the commuting-holonomy sum $Z_{\mathrm{o}\mathrm{r}\mathrm{b}}(r;\tau )=\frac{1}{2}{\sum }_{g,h\in \{1,\rho \}}{Z}_{g,h}(r;\tau )\mathrm{.}$ In particular, the Fock definitions give the absolutely convergent products $Z_{\rho ,1}=2q^{1\mathrm{/}48}{\bar{q}}^{1\mathrm{/}48}{\prod }_{k\ge 1}(1-q^{k-1\mathrm{/}2}{)}^{-1}(1-{\bar{q}}^{k-1\mathrm{/}2}{)}^{-1},Z_{\rho ,\rho }=2q^{1\mathrm{/}48}{\bar{q}}^{1\mathrm{/}48}{\prod }_{k\ge 1}(1+q^{k-1\mathrm{/}2}{)}^{-1}(1+{\bar{q}}^{k-1\mathrm{/}2}{)}^{-1}\mathrm{.}$ On the untwisted space, $Z_{1,1}=Z_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}$ and $Z_{1,\rho }=q^{-1\mathrm{/}24}{\bar{q}}^{-1\mathrm{/}24}{\prod }_{k\ge 1}(1+q^k{)}^{-1}(1+{\bar{q}}^k{)}^{-1},$ because nonzero charge sectors are exchanged in pairs by $\rho$ and only the zero-charge sector contributes to this trace. The untwisted circle amplitude is $Z_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}(r;\tau )={\mathrm{Tr}}_{{\mathcal{H}}_r}(q^{L_0-1\mathrm{/}24}{\bar{q}}^{{\bar{L}}_0-1\mathrm{/}24})$. For $\mathrm{Im}\tau >0$, define $\eta (\tau )=q^{1\mathrm{/}24}{\prod }_{k\ge 1}(1-q^k),{\theta }_3(\tau )={\sum }_{j\in \mathbb{Z}}{q}^{j^2\mathrm{/}2},{\theta }_4(\tau )={\sum }_{j\in \mathbb{Z}}(-1{)}^j{q}^{j^2\mathrm{/}2},{\theta }_2(\tau )={\sum }_{j\in \mathbb{Z}}{q}^{(j+1\mathrm{/}2{)}^2\mathrm{/}2}\mathrm{.}$ Use the convergent trace sums for $\mathrm{Im}\tau >0$; no value at $r=1$ is assumed as part of this definition. This definition does not assert higher-genus sewing or a path-integral construction.

Compact free boson as a full local CFT

Ginsparg, Sections 8.3–8.4

The twisted-sector ground-state multiplicity is two because the circle reflection has the two fixed classes $[0]$ and $[\pi r]$. Ginsparg derives the radius-one trace comparison in Section 8.4, equation (8.24); neither that evaluation nor the identity with the independent-Majorana spin sum is assumed here.