Source-defined exceptional $N=1$ minimal-model target data

The source-defined exceptional target means the two-dimensional minimal-model target reported in [A1], with central charge convention

$$c_m=\frac{3}{2}(1-\frac{8}{m(m+2)}),m=12,$$

and the exceptional modular invariant labelled $(E_6,D_8)$. Its $N=1$ super-Virasoro algebra from [F1] is a subalgebra, not its full chiral algebra. The reported full chiral algebra is the extension $SW(3\mathrm{/}2,5\mathrm{/}2)$: besides the $N=1$ super-stress-tensor generators of spins $3\mathrm{/}2$ and $2$, it has an additional superprimary of weight $5\mathrm{/}2$, with components of spins $5\mathrm{/}2$ and $3$.

Here a superprimary of weight $h$ means a nonzero homogeneous vector $v$ in an NS representation of the supplied $N=1$ algebra such that $L_{0}v=hv$, $L_{n}v=0$ for every integer $n>0$, and $G_{r}v=0$ for every positive half-integer $r$. The source reports the additional $h=5\mathrm{/}2$ generator as a fermionic superprimary $U$; its superpartner $G_{-1\mathrm{/}2}U$ has weight $3$ by the $[L_0,G_{-1\mathrm{/}2}]$ relation and is reported as the bosonic spin-$3$ current.

The reported extended modules are indexed by the following table (source §3.2, Table 2). An extended module is a representation of the reported extended chiral algebra; the listed weights are the $L_0$ eigenvalues of its super-Virasoro primary vectors, each generating its tower of descendants. The orbit column uses the second Kac index $s$ with the equivalence $s\sim 14-s$ on $\{1,3,5,7,9,11,13\}$. These are reported representation data, not a construction or existence proof of the extended algebra.

The fixed orbit $\{7\}$ is resolved into two inequivalent modules in each sector. Their characters coincide, but their lowest states have opposite nonzero eigenvalues of the spin-$3$ current's zero mode $W_0$; the subscripts $\pm$ label that pair, not fermion parity. Thus each sector has five extended modules. For the NS labels $a\in A=\{\mathbf{1},{\widehat{\mathrm{\Phi }}}_{1,3},\widehat{X},{\mathrm{\Phi }}_{+},{\mathrm{\Phi }}_{-}\}$, write $B_a$ for the module and $h_a$ for its smallest weight in the table. A descendant is obtained by a finite word in negative current modes from a lowest state; its level $\mathrm{\ell }$ is its $L_0$ eigenvalue minus $h_a$. The NS character is the trace ${\widehat{\chi }}_a(\tau )={\mathrm{Tr}}_{B_a}\exp (2\pi i\tau (L_0-c_{12}\mathrm{/}24))$ in the source's convergent upper-half-plane domain. The reported NS partition function is $Z_{\mathrm{N}\mathrm{S}}={\sum }_{a\in A}\mathrm{\mid }{\widehat{\chi }}_a{\mathrm{\mid }}^2$ (equation (34)). This statement concerns the reported spectrum, not a reconstruction of the full local package.

The chiral-fermion-parity operator $\omega$ on $B_a$ has eigenvalue ${\omega }_a(-1{)}^{2\mathrm{\ell }}$ on level $\mathrm{\ell }\in \frac{1}{2}{\mathbb{Z}}_{\ge 0}$, where

$$({\omega }_{\mathbf{1}},{\omega }_{{\widehat{\mathrm{\Phi }}}_{1,3}},{\omega }_{\widehat{X}},{\omega }_{{\mathrm{\Phi }}_{+}},{\omega }_{{\mathrm{\Phi }}_{-}})=(+1,-1,-1,+1,+1)\mathrm{.}$$

The right operator $\tilde{\omega }$ uses the right level instead. Thus the signs of the lowest states are not constant signs on entire modules. In the source's three-point convention, $N_{abc}^{+}$ and $N_{abc}^{-}$ count even and odd chiral three-point structures, respectively, and $N_{abc}$ is their sum; a structure is even or odd according to the parity of its number of supercurrent top components. The reported grading is $N_{abc}^{\pm }=N_{abc}(1\pm {\omega }_a{\omega }_b{\omega }_c)\mathrm{/}2$ (equations (36), (44), (46)). These NS signs do not assign a single parity to a Ramond module with odd zero modes.

Distinguish the conjugation operator $C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}}$ from the central mode $C$ of [F1]. On the diagonal NS sectors $B_a\otimes {\tilde{B}}_a$, $C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}}$ is the source's simultaneous left/right lift of the current automorphism $(T,G,U,W)\mapsto (T,G,-U,-W)$. Explicitly, its lift intertwines every current mode $V_n$ with $(AV{)}_n$, where $A$ is that sign automorphism; it maps the $a$ sector to the $\mathcal{C}(a)$ sector, with $\mathcal{C}$ fixing $\mathbf{1},{\widehat{\mathrm{\Phi }}}_{1,3},\widehat{X}$ and exchanging ${\mathrm{\Phi }}_{+},{\mathrm{\Phi }}_{-}$. The lift is normalized to square to one, fix the lowest state of each fixed diagonal sector, and exchange the normalized lowest states of the resolved sectors without a phase. Its action on current descendants must satisfy that intertwining relation. On the four super-Virasoro towers in each fixed NS module, in the weight order of the table, the source gives relative lift signs $(+,-,+,-)$ for $\mathbf{1}$ and ${\widehat{\mathrm{\Phi }}}_{1,3}$, and $(+,-,-,+)$ for $\widehat{X}$; on a left/right tower pair $C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}}$ acts by the product of the tower signs. The source leaves the relative exchange sign $\eta \in \{+1,-1\}$ on the upper resolved tower of weight $51\mathrm{/}14$ undetermined (equations (51)–(52), §3.7); the lower resolved tower is normalized to exchange with sign $+1$. That unresolved datum is retained, rather than supplied by a label permutation. It is not a constant sign on every tower or sector. These are source-reported lift data (§3.6), whose existence is not deduced from the label permutation alone. The source's R-parities are $R=\omega C_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}}$ and $\tilde{R}=\tilde{\omega }{C}_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{j}}$ (equation (48)); the commuting factors give involutions. On the comparison superfields the chosen $R$ has superspace action

$$\mathrm{\Phi }(x,\theta ,\tilde{\theta })⟼-\mathrm{\Phi }(x,-\theta ,\tilde{\theta }),\mathrm{\Phi }=(X,Y)\mathrm{.}$$

Here $x$ is a local Euclidean coordinate on the two-dimensional worldsheet, and $\theta ,\tilde{\theta }$ are independent Grassmann-odd coordinates, meaning ${\theta }^2={\tilde{\theta }}^2=0$ and $\theta \tilde{\theta }=-\tilde{\theta }\theta$. These are source conventions for the target comparison. Modular and fusion data alone do not supply the full local OPE package in [F2].