Scope of the two-Weyl-fermion self-duality example

Arias-Tamargo et al. study the theory $\mathcal{T}$ of two free right-moving complex Weyl fermions in two dimensions. In their convention the single-component quantum fields ${\psi }_1,{\psi }_2$ on $(t,x)\in {\mathbb{R}}^{1,1}$ have Lagrangian density $\mathcal{L}=i{\sum }_{j=1}^2{\psi }_j^{†}({\mathrm{\partial }}_t+{\mathrm{\partial }}_x){\psi }_j$. Right-moving means $({\mathrm{\partial }}_t+{\mathrm{\partial }}_x){\psi }_j=0$ as an operator-valued distribution; the fields are not ordinary pointwise functions. Use natural units $\mathrm{\hslash }=c=1$, so each field has length dimension $-1\mathrm{/}2$. Their coprime pair $(p,q)=(1,2)$ gives the cyclic group $G=G(1,2)\cong \mathbb{Z}\mathrm{/}5\mathbb{Z}$, generated by ${\psi }_1\mapsto e^{4\pi i\mathrm{/}5}{\psi }_1,{\psi }_2\mapsto e^{-2\pi i\mathrm{/}5}{\psi }_2\mathrm{.}$ The source reports that this action is anomaly-free, meaning that it admits the consistent finite gauging $\mathcal{T}\mathrm{/}G$: sum over flat $G$ gauge backgrounds and project onto invariant states. Its twisted sectors on a spatial circle with dimensionless coordinate $\sigma <mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2\pi$ have boundary conditions ${\psi }_1(\sigma +2\pi )=(-1{)}^{1-u}{e}^{-4\pi i\alpha \mathrm{/}5}{\psi }_1(\sigma )$ and ${\psi }_2(\sigma +2\pi )=(-1{)}^{1-u}{e}^{2\pi i\alpha \mathrm{/}5}{\psi }_2(\sigma )$, with $\alpha \in \{0,\dots ,4\}$ and $u=0$ (Neveu–Schwarz) or $u=1$ (Ramond). On their charge eigenspaces labelled by $n_1,n_2\in \mathbb{Z}-u\mathrm{/}2$, the commuting charges $Q_1,Q_2$ have eigenvalues $n_1+2n_2$ and $2n_1-n_2+\alpha$. The possible extensions of the generator to these sectors are $\exp (2\pi i[Q_2+t(2Q_1-Q_2)]\mathrm{/}5)$, with $t\in \{1,3,5,7,9\}$. The source's modular-consistency condition (2.50) is $2pt\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}5)$, hence selects $t_{\ast }=3$. It requires the torus amplitudes, including spin structures and gauge holonomies, to transform with the same phases as those of $\mathcal{T}$, not bosonic modular invariance.

A self-duality isomorphism here is a bijection between the local and extended operators of $\mathcal{T}\mathrm{/}G$ and $\mathcal{T}$ preserving conformal weights and operator-product coefficients. Choose the source's reference $D$ in (2.67), with the phases fixed in (3.15), which exchanges the two $G$-neutral charge generators without extra phases. Gauging only on one side of a line, with gauge trivialization fixed on the interface (the source's Dirichlet boundary condition), and then applying $D$ gives the defect $N$. Write $L_g$ for the symmetry line implementing $g\in G$, $\mathbf{1}=L_0$, $\bar{N}$ for orientation reversal, and $\otimes$ for spatially ordered fusion; $\cong$ means an invertible junction identification and $\oplus$ a direct sum of line sectors. The reported fusion rules are $L_g\otimes L_h\cong L_{g+h},L_g\otimes N\cong N\cong N\otimes L_g,N\otimes \bar{N}\cong {⨁}_{g\in G}{L}_g\mathrm{.}$ For this phase-fixed reference choice the source also reports $N\cong \bar{N}$, hence $N\otimes N\cong {⨁}_{g\in G}{L}_g$. These are the Tambara–Yamagami fusion rules for $G$; specifying a category additionally requires coherent associativity data, which these equations alone do not determine.

Changing $D$ by composition with a symmetry of $\mathcal{T}$ changes $N$ by fusion with its invertible symmetry line. Define $\widehat{G}=\mathrm{Hom}(G,U(1))$, the character group whose Wilson lines in the gauged theory evaluate a character on gauge holonomy, and $G^{\mathrm{\prime }}=D(\widehat{G})$, its image among symmetry lines of $\mathcal{T}$. The reference choice gives $G^{\mathrm{\prime }}=G$; other choices in the source give $G^{\mathrm{\prime }}\mathrm{\ne }G$ and fail the same Tambara–Yamagami rules. These are source-reported results for this chiral fermion theory and these choices, not locally proved fusion rules or empirical observations, and imply no universal rule for compact-boson defects or arbitrary noninvertible lines.

Arias-Tamargo et al., §2 (2.1), §§2.2–2.5, 3.1–3.4; PDF v1 equation numbering