May 24, 2026
We investigate the renormalization group (RG) domain walls interpolating between the \(\mathbb{Z}_N\) parafermion theory (the critical \(N\)-state Potts model) and the Virasoro minimal model \(\mathcal{M}_{N+1}\). These flows are genuinely non-perturbative and an explicit construction of Gaiotto type RG domain wall remains elusive. We bypass this limitation by employing a bottom-up approach centered on the emergence of “phantom currents". By tracking the preserved non-invertible symmetries (\(\mathfrak{so}(3)_N\)) along the flow, we extract the exact spectrum of these currents localized on the defect. We demonstrate that the presence of a spin-1 phantom current allows the interface to be marginally deformed, dynamically generating a continuous defect conformal manifold. Furthermore, we show that an extra spin-2 operator, crucially as a \(W^{(3)}\)-algebra descendant of the spin-1 phantom current, rigidly constrains the UV-IR stress tensor mixing via the cluster decomposition principle. This algebraic framework enables the exact computation of the parameter-dependent transmission rate across the conformal manifold, which we observe strictly vanishes in the large-\(N\) limit as a consequence of macroscopic target space collapse.
USTC-ICTS/PCFT-26-31
Conformal interfaces and boundaries are fundamental objects in two-dimensional conformal field theories (CFTs). By gluing two potentially distinct CFTs together while preserving a diagonal Virasoro algebra, conformal interfaces govern the transmission and reflection of energy and charges, providing deep insights into entanglement, dualities, and defect dynamics. Among them, Renormalization Group (RG) domain walls hold a particularly special status. Generated by a relevant deformation restricted to half of the spacetime, an RG domain wall encapsulates the exact operator mixing between the ultraviolet (UV) and infrared (IR) fixed points. Consequently, the highly non-trivial and intricate dynamics occurring in the strongly coupled regime along the RG flow are exactly encoded within these defects. However, such RG domain walls rarely admit a purely algebraic description. Generically, these RG domain walls are intrinsically irrational, even when the UV and IR endpoints are both rational CFTs (RCFTs). In the seminal paper [1], a celebrated exact realization of such interfaces was given by Gaiotto, who constructed the RG domain walls between consecutive Virasoro minimal models1 \(\mathcal{M}_{N+2}^{UV}\rightarrow \mathcal{M}_{N+1}^{IR}\). The key insight in constructing this RG domain wall relies on the folding trick, embedding the product theory into an auxiliary theory \(\mathcal{T}_{\mathcal{B}}\) with an extended chiral algebra \(\mathcal{B}\), \[\begin{align} \mathcal{M}_{N+2}^{UV}\otimes\overline{\mathcal{M}_{N+1}^{IR}}\longrightarrow \mathcal{T}_{\mathcal{B}} \,. \end{align}\] Within this extended theory, the RG domain wall can be elegantly realized as a rational Cardy brane equipped with an additional \(\mathbb{Z}_2\) twist. This \(\mathbb{Z}_2\) automorphism plays a vital role, as it intricately mixes the UV and IR stress tensors, thereby providing a completely algebraic description of the otherwise irrational RG interface.
The fundamental reason why the UV and IR theories can be embedded into a larger chiral algebra stems from the preservation of generalized (non-invertible) symmetries along the RG flow (for comprehensive reviews on generalized and non-invertible symmetries and their applications, see e.g., [2]–[26]). Specifically, the relevant deformation triggering the RG flow preserves a large class of non-invertible symmetries, realized as topological defect lines (TDLs) or Verlinde lines. These preserved topological lines are completely transparent to the RG domain wall, allowing them to freely pass through the interface. Crucially, these TDLs can terminate on specific defect operators. Upon applying the folding trick, the presence of these topological lines allows for a generalized orbifolding procedure, or say equivalently gauging the non-invertible symmetries. From a modern perspective, this gauging procedure corresponds to condensing these Verlinde lines into the vacuum [27]–[32]. This topological condensation enlarges the vacuum state, thereby rigorously defining the extended chiral algebra \(\mathcal{B}\) of the auxiliary theory \(\mathcal{T}_{\mathcal{B}}\). During this process, the defect operators residing at the endpoints of the condensed topological lines are absorbed into the new vacuum, becoming genuine local operators in the extended theory. These newly emerged local operators are precisely what have recently been termed phantom currents [33], [34].
The presence of such phantom currents on a conformal interface (or RG domain wall) is not merely a kinematic curiosity. Rather, these currents encode profound non-perturbative dynamical properties of the defect. As recently demonstrated by Copetti et al. [33], the existence of a spin-1 phantom current provides an exactly marginal operator localized on the domain wall. Deforming the interface by this operator preserves its conformal invariance, thereby generating a continuous moduli space of defects, aptly named a defect conformal manifold. A fascinating physical consequence of this marginal deformation is that macroscopic observables, such as the energy reflection and transmission rates across the domain wall, vary continuously as functions of the moduli parameter. On the other hand, the presence of a spin-2 phantom current, e.g. those emerging in Gaiotto’s RG domain walls, plays a distinctly different but equally crucial role. In the unfolded picture, this spin-2 operator can participate in a highly non-trivial mixing between the stress tensor of the left (UV) theory \(\mathcal{T}_{UV}\), and that of the right (IR) theory \(\mathcal{T}_{IR}\), at the interface. As recently explored in [34], this mixing implies that the transmission rate is rigidly constrained by the OPE data of the phantom current and stress tensors. In particular, it is observed that, in the unfolded picture, the two stress tensors \(T_{UV}\) and \(\overline{T}_{IR}\) in the UV and IR theories need to strictly satisfy the cluster decomposition principle, \[\begin{align} T_{UV}(z)\cdot{\overline{T}}_{IR}(w)\sim0\,. \label{eq:cluster95decomp} \end{align}\tag{1}\] Here the equation hold in operator sense, so that it provides stringent constraints on the mixing coefficients among the spin-2 operators at the interface. Remarkably, the bottom-up approach allows one to determine macroscopic interface properties purely from local OPE data, completely bypassing the need for the Gaiotto’s non-trivial construction of the RG defect as certain twisted Cardy state in the folded picture.
In this paper, we provide a concrete and non-trivial realization of this mechanism by investigating the RG flow from the \(\mathbb{Z}_N\) parafermion \(\mathcal{P}_N\) (or the critical \(N\)-state Potts model) to the Virasoro minimal model \(\mathcal{M}_{N+1}\). Originally such a RG flow was discovered by Fateev and Zamolodchikov via the massless thermodynamic Bethe ansatz (TBA)[35]–[38]. This flow is triggered by the least relevant operator, namely the parafermion operator and its charge conjugation, with conformal weight \(h=1-\frac{1}{N}\), \[\begin{align} \mathcal{T}_{\mathcal{P}_N} \;\xrightarrow[]{\text{least rel. def.}} \;\mathcal{T}_{\mathcal{P}_N} + \lambda_1 \int d^2x\, \mathcal{O}_{\frac{N-1}{N}}(x) + \lambda_2 \int d^2x\, \mathcal{O}^{\ast}_{\frac{N-1}{N}}(x) \quad\xrightarrow[]{\text{RG flow}} \quad \mathcal{T}_{\mathcal{M}_{N+1}}\,, \label{eq:def} \end{align}\tag{2}\] where “\(\ast\)" is the \(\mathbb{Z}_N\)-charge conjugation operation. Unlike the well-known flows between consecutive minimal models, which become perturbative in the large-\(N\) limit, this RG flow is genuinely non-perturbative, as the central charges of the UV and IR theories approach \(c=2\) and \(c=1\) respectively as \(N\rightarrow\infty\). Consequently, whether this RG domain wall admits a fully algebraic description, such as a Gaiotto-like twisted Cardy brane, remains a mystery. Nevertheless, the flow preserves a rich spectrum of phantom currents with integer spins \[\begin{align} s(N) = \left\{ k^2 \;\middle|\; k \in \mathbb{Z}, \; 0 \le k \le \left\lfloor \frac{N}{2} \right\rfloor \right\} \end{align}\] allowing us to probe the interface dynamics purely from local OPE data. The presence of a spin-1 phantom current, \(J=\phi_{UV}\,\bar\phi_{IR}\), suggests that the RG interface can be marginally deformed to generate a defect conformal manifold. Specifically, the 1d local defect operator induced by \(J\) organizes the stress tensors and the spin-2 Virasoro primary \[\begin{align} W^-\propto h_{IR}\partial \phi_{UV}\bar\phi_{IR}-h_{UV}\phi_{UV}\partial\bar\phi_{IR} \end{align}\] into a \(\mathfrak{u}(1)\)-multiplet, forcing \(W^-\) to mix with the stress tensors \(T_{UV}\) and \(T_{IR}\) of the UV and IR theories at the interface. However, as shown in [34], if \(W^-\) is the sole spin-2 operator participating in the mixing, the cluster decomposition equation 1 strictly demands \(c_{UV}= c_{IR}\). This seemingly forbids the existence of a conformal manifold across our RG flow, where \(c_{UV}\neq c_{IR}\) is inherently the case.
However, beside the operator \(W^{-}\), there is an extra spin-2 operator due to the extended \(W^{(3)}\)-chiral algebra in the \(\mathbb{Z}_N\) parafermion UV theory. More specifically, in the folded picture, the system harbors a spin-2 operator, \[\begin{align} X=W^{(3)}_{-1}\phi_{UV}\,\bar\phi_{IR}\,, \end{align}\] which is the \(W\)-algebra descendant of the spin-1 phantom current. Although there is no fundamental spin-2 phantom current in the preserved spectrum (the spins jump from 1 to 4), we can show that \(X\) acts as a genuine, factorized Virasoro primary. It actively participates in the non-trivial mixing with the stress tensors at the interface alongside \(W^-\). This leads to a compelling two-step physical picture: Firstly, introducing the parafermion fields deformation on a half-plane generates a rigid RG interface \(\mathfrak D_N\) between the \(\mathbb{Z}_N\)-parafermion \(\mathcal{P}_N\) and \(\mathcal{M}_{N+1}\) model. Secondly, integrating the 1d local defect operator induced by the spin-1 phantom current along \(\mathfrak D_N\) marginally deforms the interface, yielding a defect conformal manifold \(\mathfrak D_N(\theta)\). The inclusion of the \(W\)-algebra descendant \(X\) ensures that the cluster decomposition equation 1 admits a continuous one-parameter family of non-trivial solutions for the mixing coefficients. This provides a highly non-trivial, explicit realization of the mechanism proposed in [33] in a strictly \(c_{UV}\neq c_{IR}\) case, demonstrating how the chiral algebra can dynamically sustain conformal manifolds on RG domain walls.
The remainder of this paper is organized as follows. In Section 2, we first briefly review the coset construction of the \(\mathbb{Z}_N\)-parafermion \(\mathcal{P}_N\) and the RG flow to minimal model \(\mathcal{M}_{N+1}\). We then prove that the preserved fusion category along the RG flow is \(\mathfrak{so}(3)_N\), i.e. the integer spin representation of \(\mathfrak{su}(2)_N\), from which the spectrum of the preserved phantom currents can be read off. In Section 3, we construct the RG domain wall and its conformal manifold between \(3\)-state Potts and \(\mathcal{M}_{4}\) model. The explicit construction of such defect is made possible by the fact that the \(3\)-state Potts can be obtained via an \(\mathbb{Z}_2\)-orbifolding of the minimal model \(\mathcal{M}_{5}\). Therefore one is able to establish the domain wall by stacking a topological interface onto the original Gaiotto wall between \(\mathcal{M}_{5}\) and \(\mathcal{M}_{4}\). In the rest of the section, we proceed to the general case \(N\). Although an explicit construction of the RG domain wall for generic \(N\) remains elusive, we show that the cluster decomposition equation will always admit a continuous one-parameter family of solutions, and thereby indicate the existence of the defect moduli. Finally, we compute the exact parameter-dependent transmission rate \(\mathcal{T}_N(\theta)\) across this RG domain wall from \(\mathcal{P}_N\) to \(\mathcal{M}_{N+1}\), \[\begin{align} \mathcal{T}_N(\theta)=\frac{2c_{UV\text{-}IR}}{c_{UV}+c_{IR}}=\frac{8(N+1)(1+\cos\theta)}{3(N+2)^2}\,, \end{align}\] as our main result, where \(\theta\) is the moduli parameter of the defect conformal manifold along the RG domain wall \(\mathfrak D_{N}(\theta)\).
In this section we collect the notation and modular data for the two coset RCFTs used below, see also in [39], [40]. A self-contained discussion on coset constructions, field identification, and selection rules is also given in App. 4.
The \(\mathbb{Z}_N\) parafermion theory is realized as the coset \[\begin{align} \mathcal{P}_N = \frac{\mathfrak{su}(2)_N}{\mathfrak{u}(1)_N}, \qquad c(\mathcal{P}_N) = \frac{2(N-1)}{N+2}. \label{eq:parafermion-coset} \end{align}\tag{3}\] We label the primary fields by pairs \[\begin{align} [t,s], \qquad t=1,\ldots,N+1, \qquad s\in \mathbb{Z}_{2N}. \end{align}\] The allowed labels obey the selection rule \[\begin{align} t+s\in 2\mathbb{Z}+1, \end{align}\] and are subject to the field identification \[\begin{align} [t,s]\sim [N+2-t,s+N], \qquad s \;\mathrm{mod}\;2N. \label{eq:field-identification} \end{align}\tag{4}\] Thus the primary fields of \(\mathcal{P}_N\) are equivalence classes of such labels. The conformal weight is \[\begin{align} h^{\mathcal{P}_N}_{[t,s]} = \frac{t^2-1}{4(N+2)} - \frac{s^2}{4N} \quad \mathrm{mod}\;1 . \end{align}\]
the modular \(S\)-matrix of the parafermion theory is \[\begin{align} S^{\mathcal{P}_N}_{[t,s],[t',s']} = \sqrt{\frac{4}{N(N+2)}}\, \sin\!\left(\frac{\pi tt'}{N+2}\right) \exp\!\left(-\frac{i\pi ss'}{N}\right). \label{eq:parafermion-S} \end{align}\tag{5}\]
The unitary Virasoro minimal model \(\mathcal{M}_{k+2}\) is realized as \[\begin{align} \mathcal{M}_{k+2} = \frac{\mathfrak{su}(2)_k\times \mathfrak{su}(2)_1}{\mathfrak{su}(2)_{k+1}}, \qquad c(\mathcal{M}_{k+2}) = 1-\frac{6}{(k+2)(k+3)}. \end{align}\]
Its primary fields are labeled by Kac labels with \(d\) labels the representation of \(\mathfrak{su}(2)_1\) \[\begin{align} [r,d,s], \qquad 1\le r\le k+1, \qquad d=1,2, \qquad 1\le s\le k+2, \end{align}\] with the field identification \[\begin{align} [r,d,s]\sim [k+2-r,3-d,k+3-s]. \end{align}\]
The conformal weight is \[\begin{align} h^{\mathcal{M}_{k+2}}_{[r,d,s]} = \frac{\big((k+3)r-(k+2)s\big)^2-1}{4(k+2)(k+3)} . \end{align}\]
In these conventions, the modular \(S\)-matrix is \[\begin{align} S^{\mathcal{M}_{k+2}}_{[r,d,s],[r',d',s']} = (-1)^{1+s r'+r s'} \sqrt{\frac{8}{(k+2)(k+3)}}\, \sin\!\left(\frac{\pi rr'}{k+2}\right) \sin\!\left(\frac{\pi ss'}{k+3}\right). \end{align}\]
Non-invertible symmetries in two-dimensional CFTs are naturally realized by TDLs[9], [13]. Since a TDL can be freely deformed on the CFT as long as it does not cross local operator insertions, it provides a sharp diagnostic of which generalized symmetries survive under a relevant deformation. Let \(\mathcal{L}\) be a topological line and \(|0\rangle\) the unique vacuum. Its expectation value is defined by \[\begin{align} \langle \mathcal{L}\rangle \equiv \langle 0|\mathcal{L}|0\rangle . \end{align}\] For a bulk primary operator \(\phi\), we say that \(\phi\) is neutral with respect to \(\mathcal{L}\), or equivalently that \(\mathcal{L}\) is transparent to \(\phi\), if \[\begin{align} \widehat{\mathcal{L}}|\phi\rangle = \langle \mathcal{L}\rangle |\phi\rangle . \label{eq:tdl-neutrality} \end{align}\tag{6}\] This condition states that the eigenvalue of \(\phi\) under the action of the defect coincides with the eigenvalue of the vacuum. Therefore, inserting \(\phi\) into the action does not obstruct the topological deformation of \(\mathcal{L}\). A TDL satisfying 6 with respect to the perturbing operator is preserved along the RG flow.
For diagonal RCFTs, the TDLs are the usual Verlinde lines. They are labelled by bulk primaries and act diagonally on the Hilbert space as \[\begin{align} \mathcal{L}_i |\phi_j\rangle = \frac{S_{ij}}{S_{0j}}|\phi_j\rangle , \label{eq:verlinde-line-action} \end{align}\tag{7}\] where \(S_{ij}\) is the modular \(S\)-matrix and \(0\) denotes the vacuum primary. Equivalently, in the three-dimensional topological field theory description, bulk primaries are represented by Wilson lines piercing the two-dimensional surface, while TDLs are Wilson lines lying on the surface. The eigenvalue in 7 is then the normalized Hopf-link invariant obtained when the defect line encircles the Wilson line associated with \(\phi_j\). For non-diagonal theories, the one-to-one correspondence between bulk primaries and TDLs need not hold. Nevertheless, the fusion category of topological lines still encodes the generalized symmetry data of the CFT and therefore imposes non-trivial constraints on possible RG flows.
We now apply this criterion to the parafermionic theory \(\mathcal{P}_N\). The most relevant parafermionic deformation is generated by the primary field of conformal weight \[\begin{align} h=1-\frac{1}{N}. \end{align}\] It may be represented by the fields \(\phi_{[1,2]}\) and \(\phi_{[1,2N-2]}\), or by an appropriate real linear combination of them. A topological line \(\mathcal{L}_{[t,s]}\) is preserved by this deformation precisely when its eigenvalue on the perturbing field agrees with its eigenvalue on the vacuum. Using 7 and 5 , this condition reduces to the simple phase constraint \[\begin{align} \exp\!\left(-\frac{2\pi i s}{N}\right)=1 . \end{align}\] Hence \[\begin{align} s=0 \quad \text{or}\quad s=N \qquad \mathrm{mod}\;2N . \label{eq:preserved-para} \end{align}\tag{8}\] Because the two choices are related by the field identification 4 , we may choose representatives with \(s=0\). The parity condition then requires \(t\) to be odd. Therefore the preserved topological lines are represented by \[\begin{align} \mathcal{L}_{[t,0]}, \qquad t\in 2\mathbb{Z}+1 . \label{eq:preserved-lines} \end{align}\tag{9}\] The other possibility is related by the field identification \[\begin{align} \mathcal{L}_{[t_k,N]}\sim \mathcal{L}_{[N+2-t_k,0]}. \label{eq:TDL-id} \end{align}\tag{10}\] We next determine the fusion rules among these preserved lines. Let \[\begin{align} \ell=t-1 . \end{align}\] Since the preserved representatives have \(t\in 2\mathbb{Z}+1\), they correspond precisely to the even-\(\ell\), or integer-spin, labels of \(\mathfrak{su}(2)_N\). Therefore \[\begin{align} \mathcal{L}_{[\ell_i+1,0]}\times\mathcal{L}_{[\ell_j+1,0]} = \bigoplus_{\substack{ \ell= |\ell_i-\ell_j|\\ \mathrm{step}\;2 }}^{ \min(\ell_i+\ell_j,\;2N-\ell_i-\ell_j) } \mathcal{L}_{[\ell+1,0]}, \qquad \ell_i,\ell_j,\ell\in 2\mathbb{Z} . \label{eq:SO3N-fusion} \end{align}\tag{11}\] Equivalently, \[\begin{align} N_{\ell_i,\ell_j}^{\;\;\ell_k} = \sum_{\substack{ \ell= |\ell_i-\ell_j|\\ \mathrm{step}\;2 }}^{ \min(\ell_i+\ell_j,\;2N-\ell_i-\ell_j) } \delta_{\ell_k,\ell}, \qquad \ell_i,\ell_j,\ell_k\in 2\mathbb{Z} . \end{align}\] Here we summarize the result, while the detailed calculation is given in App. 5. These are precisely the fusion rules of the integer-spin subcategory of \(\mathfrak{su}(2)_N\), usually denoted by \[\begin{align} \mathcal{C}_{\mathrm{pres}} \simeq \mathfrak{so}(3)_N . \end{align}\] In physical terms, the relevant perturbation does not preserve the full parafermionic defect category, but it does preserve the non-invertible subcategory generated by the lines 9 . This preserved category must be matched, or embedded, in the generalized symmetry category of the IR fixed point.
Let us illustrate the preservation of the deformation symmetry \(\mathcal{P}_3\). Following the notation used above, the relevant deformation is generated by the two fields of conformal weight \(h=2/3\), \[\begin{align} \mathcal{T}_{\mathcal{P}_3} \longrightarrow \mathcal{T}_{\mathcal{P}_3} +\lambda_1 \int d^2x\, \mathcal{O}_{\frac{2}{3}}(x) + \lambda_2 \int d^2x\, \mathcal{O}^\ast_{\frac{2}{3}}(x). \end{align}\] By the general selection rule, the topological lines preserved by this deformation are the lines labeled by \(\mathcal{L}_{[t,0]}\) with odd \(t\). For \(N=3\) this leaves \[\begin{align} \{\mathcal{L}_{[1,0]},\mathcal{L}_{[3,0]}\}\simeq \text{LY}. \end{align}\] Thus, in this example, the general deformation-preserved \(\mathfrak{so}(3)_N\) symmetry reduces at \(N=3\) to a preserved LY non-invertible symmetry.
This preserved LY symmetry strongly constrains the possible infrared behavior. A theory with an unbroken LY symmetry cannot end in a trivially gapped phase with a single featureless ground state. Hence the LY-preserving deformation has two possible types of infrared behavior: either it becomes gapped in a non-trivial way, or it remains gapless with a single conformal vacuum. The massless TBA analysis [38] selects a special integrable ray in the coupling space of this perturbation. In the standard conjugate normalization of the two perturbing operators, the gapless branch is characterized by equal magnitudes of the two couplings and by the relative phase \[\begin{align} |\lambda_1|=|\lambda_2|,\qquad \frac{\lambda_1}{\lambda_2}=e^{2\pi i/N}, \end{align}\] In the present case \(N=3\). Therefore the massless condition becomes \[\begin{align} \frac{\lambda_1}{\lambda_2}=e^{2\pi i/3}, \end{align}\] Thus the RG trajectory considered here is not a generic two-parameter LY-preserving deformation, but the \(N=3\) massless integrable ray. Since the preserved LY symmetry forbids a trivially gapped phase with a single featureless ground state, this identifies the deformation as a gapless LY-preserving flow rather than the massive parafermionic branch.
The possible conformal endpoints are then further restricted by \(c\)-theorem [41]. Since \[\begin{align} c(\mathcal{P}_3)=\frac{4}{5}, \end{align}\] any non-trivial unitary minimal-model endpoint must have smaller central charge. Among the relevant candidates below \(c=4/5\), this leaves \(\mathcal{M}_3\) and \(\mathcal{M}_4\). Defect category of the critical Ising model is of \(\text{TY}_2\) type and does not contain the required LY line. Therefore it cannot match the non-invertible symmetry preserved along the \(\mathcal{P}_3\) deformation. By contrast, the tricritical Ising model \(\mathcal{M}_4\) contains a LY sector. Hence the preserved symmetry can be matched in the RG flow, \[\begin{align} \mathcal{P}_3 \longrightarrow \mathcal{M}_4\,. \end{align}\]
By the folding trick, an RG domain wall between the UV theory \(\mathcal{T}_{UV}\) and the IR theory \(\mathcal{T}_{IR}\) can be equivalently regarded as a conformal boundary condition in the folded product theory. In the case we discuss, it is \[\begin{align} \mathcal{T}_\mathcal{A}=\frac{\mathfrak{su}(2)_N}{\mathfrak{u}(1)_N}\otimes \frac{\mathfrak{su}(2)_{N-1}\times\mathfrak{su}(2)_1}{\mathfrak{su}(2)_N} \;\longrightarrow\; \mathcal{T}_{\mathcal{B}}=\frac{\mathfrak{su}(2)_{N-1}\times\mathfrak{su}(2)_1}{\mathfrak{u}(1)_N}\,. \end{align}\] The boundary condition becomes elementary only after extending the chiral algebra of \(\mathcal{T}_{\mathcal{A}}\) to a larger rational algebra. We denote the corresponding extended theory by \(\mathcal{T}_{\mathcal{B}}\).
Let \(a,b,\ldots\) label irreducible sectors of the original folded theory \(\mathcal{T}_{\mathcal{A}}\), and let \(\mu,\nu,\ldots\) label irreducible sectors of the extended theory \(\mathcal{T}_{\mathcal{B}}\). Upon restricting a representation of \(\mathcal{T}_{\mathcal{B}}\) to the smaller chiral algebra of \(\mathcal{T}_{\mathcal{A}}\), its character decomposes as \[\begin{align} \chi^{\mathcal{B}}_{\mu}(\tau) = \sum_{a\in \mathcal{A}} n_{\mu}^{a}\, \chi^{\mathcal{A}}_{a}(\tau) \, , \label{eq:branching95character} \end{align}\tag{12}\] where \(n_{\mu}^{a}\in \mathbb{Z}_{\geq 0}\) is the branching multiplicity. Modular covariance requires the branching matrix to intertwine the modular \(S\)-matrices of the two theories: \[\begin{align} \sum_{a} n_{\mu}^{a}\, S^{\mathcal{A}}_{ab} = \sum_{\nu} S^{\mathcal{B}}_{\mu\nu}\, n_{\nu}^{b} \, . \label{eq:branching95S95intertwining} \end{align}\tag{13}\] This relation will be used repeatedly below to convert objects in the extended boundary theory into those in the folded product theory.
For the RG flow from the \(\mathbb{Z}_N\) parafermion theory \(\mathcal{P}_N\) to the minimal model \(\mathcal{M}_{N+1}\), the folded theory is of the form \[\begin{align} \mathcal{T}_{\mathcal{A}} = \mathcal{P}_{N}\otimes\overline{\mathcal{M}_{N+1}} \, . \end{align}\] The relevant extension can be described by the branching of the extended primaries into the product theory. Schematically, and suppressing the standard selection rules and field identifications, we write \[\begin{align} \phi^{\mathcal{B}}_{[t,d,s]} = \sum_{r} \left( {\phi}^{\mathcal{P}_N}_{[r,s]} \otimes \bar\phi^{\mathcal{M}_{N+1}}_{[t,d,r]} \right)\,. \label{eq:PN95branching} \end{align}\tag{14}\] The index \(r\) runs over the admissible \(\mathfrak{su}(2)_N\) labels. Equation 14 is the concrete realization of 12 in the present class of flows.
We now recall how the corresponding interface is written in the folded Ishibashi basis. For every pair \((\mu,a)\) with nonzero multiplicity \(n_{\mu}^{a}\), we introduce Ishibashi states \[\begin{align} |\!| \mathcal{A};\mu,a;i \rangle\!\rangle , \qquad i=1,\ldots,n_{\mu}^{a} \, , \end{align}\] where the extra label \(i\) resolves possible multiplicities in the branching. A Cardy boundary condition \(x\) of the extended theory \(\mathcal{T}_{\mathcal{B}}\) defines an interface in the unfolded picture: \[\begin{align} \mathcal{I}_x = \sum_{\mu,a} \sum_{i=1}^{n_{\mu}^{a}} \frac{S^{\mu,a}_{x}}{S^{\mathcal{A}}_{1a}} \, |\!| \mathcal{A};\mu,a;i |\!| . \end{align}\] A particularly important interface is \[\begin{align} \mathcal{I}_1 = \sum_{\mu,a} \sum_{i=1}^{n_{\mu}^{a}} \sqrt{ \frac{S^{\mathcal{B}}_{1\mu}}{S^{\mathcal{A}}_{1a}} } \, |\!| \mathcal{A};\mu,a;i |\!| . \label{eq:identity95interface} \end{align}\tag{15}\] The interface \(\mathcal{I}_1\) plays two complementary roles, which together motivate the bottom-up strategy of this paper. First, \(\mathcal{I}_1\) specifies which operators of the folded theory \(\mathcal{T}_\mathcal{A}\) condense into the new vacuum module of the extended theory \(\mathcal{T}_\mathcal{B}\). The Ishibashi sum in eq. 15 supported on pairs \((\mu,a)\) with non-vanishing branching multiplicity \(n^a_\mu\), and restricting to the vacuum sector \(\mu=1\) leaves precisely the multiplicities \(\{n_1^a\}\). These multiplicities catalogue the irreducible \(\mathcal{A}\)-sectors that contribute to the extended vacuum of \(\mathcal{B}\); Secondly, fusing \(\mathcal{I}_1\) with its orientation-reversed counterpart \(\widetilde{\mathcal{I}}_1\) collapses the Ishibashi sum into a sum of topological defect lines of \(\mathcal{T}_\mathcal{A}\), \[\begin{align} \widetilde{\mathcal{I}}_1\circ\mathcal{I}_1 = \sum_{a} n_{1}^{a}\, \mathcal{L}_{a} \, . \label{eq:interface95fusion95TDL} \end{align}\tag{16}\] Here \(\mathcal{L}_b\) denotes the topological defect line of \(\mathcal{T}_{\mathcal{A}}\) labeled by the sector \(b\). The last equality follows directly from the modular intertwining relation 13 . More importantly, the collection \[\begin{align} \mathcal{C}_{\rm pres}=\left\{\mathcal{L}_a\,|\,n_1^a>0\right\} \end{align}\] is exactly those TDLs penetrating the RG domain wall and coexist in both \(\mathcal{T}_{UV}\) and \(\mathcal{T}_{IR}\) in the unfolded picture. They forms a closed fusion subcategory of the symmetry category of \(\mathcal{T}_{\mathcal{A}}\), equipped with the structure of a commutative Frobenius algebra. The chiral extension \(\mathcal{A}\rightarrow \mathcal{B}\) is realized as the topological condensation (equivalently, the generalized gauging) of this subcategory inside \(\mathcal{T}_\mathcal{A}\): \[\begin{align} \mathcal{T}_\mathcal{B}=\mathcal{T}_\mathcal{A}/\mathcal{C}_{\rm pres}\,. \end{align}\] Condensing the lines \(\mathcal{L}_a\) enlarges the vacuum, and the defect operators previously living at the endpoints of these lines are absorbed into the new vacuum as genuine local fields of \(\mathcal{T}_\mathcal{B}\). These newly local fields are exactly the phantom currents, i.e. from eq. 14 that \[\begin{align} \phi^{\mathcal{B}}_{[1,1,0]} = \sum_{k=1}^{\lfloor N/2\rfloor} \left( {\phi}^{\mathcal{P}_N}_{[2k+1,0]} \otimes \bar\phi^{\mathcal{M}_{N+1}}_{[1,1,2k+1]} \right)\,. \label{eq:Bvac} \end{align}\tag{17}\] where one can honestly check the conformal weights of \({\phi}^{\mathcal{P}_N}_{[2k+1,0]}\) and \(\phi^{\mathcal{M}_{N+1}}_{[1,1,2k+1]}\) given by \[\begin{align} h_{[2k+1,0]}^{\mathcal{P}_N}=\frac{2k}{N+2}\,,\quad {\rm and}\quad h^{\mathcal{M}_{N+1}}_{[1,1,2k+1]}=k^2-\frac{2k}{N+2}\,, \end{align}\] indicating that the vacuum module \(\phi^{\mathcal{B}}_{[1,1,0]}\) of \(\mathcal{T}_\mathcal{B}\) contains a rich spectrum of phantom currents with integer spins \[\begin{align} s(N) = \left\{ k^2 \;\middle|\; k \in \mathbb{Z}, \; 0 \le k \le \left\lfloor \frac{N}{2} \right\rfloor \right\}\,. \end{align}\] The two viewpoints are dual descriptions of the same data: the operators entering the new vacuum of \(\mathcal{T}_\mathcal{B}\) (first viewpoint) are precisely the local fields created by line condensation in \(\mathcal{T}_\mathcal{A}\) (second viewpoint), and the controlling multiplicities \(\{n_1^a\}\) are common to both. In what follows we use whichever description is more convenient, the operator/branching description for spectral statements, and the line/condensation description for symmetry-theoretic ones.
The interface \(\mathcal{I}_1\) introduced above is, however, only kinematic data: it specifies the operator content compatible with the chiral extension \(\mathcal{A}\rightarrow\mathcal{B}\), but does not by itself realize the RG domain wall. A genuine RG wall, when it admits a fully algebraic description as a rational Cardy brane of \(\mathcal{T}_\mathcal{B}\), requires the additional input of an automorphism \(\sigma\) of the chiral algebra \(\mathcal{B}\). The \(\sigma\)-twist mixes the UV and IR stress tensors non-trivially at the wall and supplies the dynamical content that \(\mathcal{I}_1\) alone cannot encode. Concretely, \(\sigma\) defines a \(\sigma\)-twisted Cardy state in \(\mathcal{T}_\mathcal{B}\); Fusing this twisted brane with the interface \(\mathcal{I}_1\) yields the Gaiotto RG brane in the folded picture \(\mathcal{T}_\mathcal{A}\). For the consecutive minimal model flows \(\mathcal{M}_{N+2} \rightarrow \mathcal{M}_{N+1}\), the relevant \(\sigma\) is the \(\mathbb{Z}_2\) automorphism that swaps the two \(\mathfrak{su}(2)_1\) numerator factors of Gaiotto’s auxiliary theory \(\mathcal{T}_\mathcal{B}^{\,\prime}\), and the resulting wall reproduces the expected UV–IR operator mixing.
For the parafermion flow \(\mathcal{P}_N \rightarrow \mathcal{M}_{N+1}\) studied here, the corresponding automorphism is not known yet for general \(N\): the natural auxiliary theory \(\mathcal{T}_\mathcal{B}=\mathfrak{su}(2)_{N-1} \times \mathfrak{su}(2)_1/\mathfrak{u}(1)_N\) does not carry the manifest swap symmetry that powers Gaiotto’s construction. The lone accessible case is \(N = 3\), where the well-known isomorphism \(\mathcal{P}_3 \simeq \mathcal{M}_5 / \mathbb{Z}_2\) reduces the problem to a topological manipulation: fusing Gaiotto’s \(\mathcal{M}_5 \rightarrow \mathcal{M}_4\) wall with the \(\mathbb{Z}_2\)-orbifold topological interface relating \(\mathcal{M}_5\) and \(\mathcal{P}_3\) produces the desired RG domain wall. We carry out this construction explicitly in the first part of next section. For general \(N\) no such reduction is available, and a fully algebraic realization of the RG wall remains an open problem. Therefore, in the second part of next section, we adopt a complementary, bottom-up strategy: we exploit the local OPE data of the phantom currents to extract its macroscopic observables directly. As we shall show, the cluster decomposition condition 1 , together with the spin-1 and spin-2 phantom currents identified above, is sufficient to determine the transmission rate of the wall and the dimension of its defect conformal manifold — without committing to an explicit construction of the wall itself.
The \(\mathbb{Z}_3\)-parafermion model \(\mathcal{P}_3\) is well understood to be obtained from a \(\mathbb{Z}_2\)-orbifolding of the minimal model \(\mathcal{M}_{5}\). To construct the RG domain wall between \(\mathcal{P}_3\) and \(\mathcal{M}_{4}\), let’s first review the construction of the wall between \(\mathcal{M}_{5}\) and \(\mathcal{M}_{4}\). In [1], Gaiotto realized that the folded theory \(\mathcal{M}_{5}\otimes \overline{\mathcal{M}_{4}}\) can be embedded in the auxiliary theory \(\mathcal{T}^{\,\prime}_{\mathcal{B}}\), \[\begin{align} \mathcal{M}_{5}\otimes \overline{\mathcal{M}_{4}}=\frac{\mathfrak{su}(2)_3\times\mathfrak{su}(2)_1}{\mathfrak{su}(2)_4}\times \frac{\mathfrak{su}(2)_2\times\mathfrak{su}(2)_1}{\mathfrak{su}(2)_3} \;\longrightarrow\; \mathcal{T}^{\,\prime}_{\mathcal{B}}=\frac{\mathfrak{su}(2)_2\times\mathfrak{su}(2)_1\times\mathfrak{su}(2)_1}{\mathfrak{su}(2)_4}\,, \end{align}\] In \(\mathcal{T}^{\,\prime}_{\mathcal{B}}\), one can find the vacuum module character spelled as \[\begin{align} \chi_{\rm vac}^{\mathcal{B}'}=q^{-\frac{c_{UV}+c_{IR}}{24}}\left(1+3q^2\cdots\right)\,, \end{align}\] where the coefficient of \(q^2\) indicates that, beside the stress tensors \(T_{UV}\) and \(T_{IR}\), there is an additional spin-2 phantom current, \[\begin{align} X=\phi^{\mathcal{M}_5}_{7/5}\,\bar\phi^{\mathcal{M}_4}_{3/5}\,, \end{align}\] condensed in the vacuum. The theory \(\mathcal{T}_{\mathcal{B}}\) further admits a \(\mathbb{Z}_2\) automorphism by swapping the two \(\mathfrak{su}(2)_1\) factors on the numerator. In the folded theory, the RG domain wall turns out to be a RG boundary state. One then is able to convert the RG brane to a rational twisted Cardy brane defined by the automorphism mentioned above. Practically, the evaluation of any one-point correlator \(\phi^{\mathcal{M}_5}\bar \phi^{\mathcal{M} 4}\in \mathcal{T}^{\,\prime}_{\mathcal{B}}\) in presence of the RG brane can be further facilitated in the basis of a fermionized theory \(\widetilde{\mathcal{T}}^{\,\prime}_{\mathcal{B}}\). As shown in [1], in \(\mathcal{T}^{\,\prime}_{\mathcal{B}}\) there exists a spin-\(\frac{1}{2}\) simple current \(J_\psi\) for the purpose of fermionization, i.e. \[\begin{align} \mathcal{T}^{\,\prime}_{\mathcal{B}}\;\xrightarrow[{\rm resp.\;to}\;\;J_\psi]{\rm fermionization}\;\widetilde{\mathcal{T}}^{\,\prime}_{\mathcal{B}}=\mathcal{SM}_{4,\,6}\times\mathcal{T}_{\psi}\,, \end{align}\] where \(\mathcal{SM}_{4,\,6}\) is the second \(\mathcal{N}=1\) superconformal minimal model, and \(\mathcal{T}_{\psi}\) is the theory of free Majorana fermion. In the fermionic basis, one can express the stress tenors of the UV/IR theory and \(X\) as \[\begin{align} &T_{UV}=\frac{2}{5}T_{\mathcal{SM}}-\frac{\sqrt 6}{5}G\,\psi+\frac{4}{5}T_{\psi}\notag\\ &\overline{T}_{IR}=\frac{3}{5}T_{\mathcal{SM}}+\frac{\sqrt 6}{5}G\,\psi+\frac{1}{5}T_{\psi}\notag\\ &X=\frac{2}{\sqrt 7}T_{\mathcal{SM}}-\frac{\sqrt 6}{2\sqrt 7}G\,\psi-\frac{4}{\sqrt 7}T_{\psi}\,, \label{eq:N61195basis} \end{align}\tag{18}\] where \(T_{\mathcal{SM}}\), \(G\) are the stress tensor and spin-\(\frac{3}{2}\) supercurrent in \(\mathcal{S}\mathcal{M}_{4,\,6}\), \(T_\psi\) and \(\psi\) are the stress tenor and free fermion in \(\mathcal{T}_\psi\), and we also have normalized the operator \(X\) so that the OPE \(X(z)\cdot X(w)\sim\frac{1}{(z-w)^4}+\cdots\). The \(\mathbb{Z}_2\) automorphism simply maps \(\psi\rightarrow-\psi\).
Now we turn to the RG domain wall between \(\mathcal{P}_3\) and \(\mathcal{M}_{4}\). Using the folding trick, we can also embed the folded theory into a larger auxiliary one \[\begin{align} \mathcal{P}_3\otimes \overline{\mathcal{M}_{4}}=\frac{\mathfrak{su}(2)_3}{\mathfrak{u}(1)_3}\times \frac{\mathfrak{su}(2)_2\times\mathfrak{su}(2)_1}{\mathfrak{su}(2)_3} \;\longrightarrow\; \mathcal{T}_{\mathcal{B}}=\frac{\mathfrak{su}(2)_2\times\mathfrak{su}(2)_1}{\mathfrak{u}(1)_3}\,. \end{align}\] Recall that \(\mathcal{P}_3\) can be obtained by gauging a spin-3 primary \(W^{(3)}\) in \(\mathcal{M}_{5}\). In the embedding of \(\mathcal{M}_{5}\otimes \mathcal{M}_{4}\rightarrow \mathcal{T}_{\mathcal{B}}^{\,\prime}\), we have a spin-1 simple current \(J\) in the chiral algebra \(\mathcal{B}^{\,\prime}\) that can be decomposed in terms of the primaries in \(\mathcal{M}_{5}\otimes \overline{\mathcal{M}_{4}}\) as \[\begin{align} J=\left(\frac{2}{5}\right)_{\!\!\!\mathcal{M}_{5}}\!\!\!\!\otimes\left(\frac{3}{5}\right)_{\!\!\!\mathcal{M}_{4}} \oplus\;\; \left(3\right)_{\!\mathcal{M}_{5}}\!\otimes\left(0\right)_{\!\mathcal{M}_{4}} \end{align}\] The folded theory \(\mathcal{T}_B=\mathcal{P}_3\otimes \overline{\mathcal{M}_{4}}\) is then obtained by orbifolding the \(\mathbb{Z}_2^J\) symmetry generated by \(J\) , \[\begin{align} \mathcal{T}_{\mathcal{B}}=\mathcal{T}_{\mathcal{B}}^{\,\prime}/\mathbb{Z}_2^J\,. \end{align}\] In \(\mathcal{T}_{\mathcal{B}}\), as we have shown in Sec. 2, \[\begin{align} J=\phi^{\mathcal{P}_3}_{2/5}\,\bar\phi^{\mathcal{M}_4}_{3/5}\equiv \phi^{UV}_{2/5}\,\bar\phi^{IR}_{3/5}\,, \end{align}\] is the spin-1 phantom current. Meanwhile, for the spin-2 phantom current \(X\) in \(\mathcal{T}^{\,\prime}_{\mathcal{B}}\), notice that the operator \(\phi^{\mathcal{M}_5}_{7/5}\) won’t be primary in \(\mathcal{P}_3\), but a \(W^{(3)}\)-descendant of \(\phi^{\mathcal{P}_3}_{2/5}\). So the operator \(X\) now is Virasoro primary, but a descendant respect to chiral algebra \(\mathcal{B}\), i.e. \[\begin{align} X=\phi^{\mathcal{M}_5}_{7/5}\,\bar\phi^{\mathcal{M}_4}_{3/5}=\left(W^{(3)}_{-1}\phi^{\mathcal{P}_3}_{2/5}\right)\bar\phi^{\mathcal{M}_4}_{3/5}\equiv \left(W^{(3)}_{-1}\phi^{UV}_{2/5}\right)\bar\phi^{IR}_{3/5}\,. \end{align}\] Beside the spin-2 operator \(X\), as well as the two stress tensors, there are also two more spin-2 operators: the descendant of \(J\), \(\partial J\), and a new Virasoro (quasi-)primary \(W^{-}\) [33], \[\begin{align} W^{-}\equiv i\frac{1}{\sqrt 3}\phi^{UV}_{2/5}\,\partial\bar\phi^{IR}_{3/5}-i\frac{\sqrt 3}{2}\partial\phi^{UV}_{2/5}\,\bar\phi^{IR}_{3/5}\,\,, \label{eq:W95N613} \end{align}\tag{19}\] where we have normalized the OPE of \(W^{-}(z)\cdot W^{-}(w) \sim \frac{1}{(z-w)^4}+\cdots\). One can compute the vacuum character of \(\mathcal{T}_{\mathcal{B}}\), \[\begin{align} \chi_{\rm vac}^{\mathcal{B}}=q^{-\frac{\frac{4}{5}+\frac{7}{10}}{24}}\left(1+q+5q^2\cdots\right)\,, \end{align}\] and indeed finds that the coefficients in front of \(q\) and \(q^2\) matching with our previous operator counting.
To establish the RG domain wall, we have to further specify certain automorphism that helps define a twisted boundary condition for the RG brane in the folded picture. In the case of \(\mathcal{T}_{\mathcal{B}}=\frac{\mathfrak{su}(2)_2\times\mathfrak{su}(2)_1}{\mathfrak{u}(1)_3}\), it is not clear how to identify such an automorphism. However, since the orbifolding \(\mathcal{T}_{\mathcal{B}}=\mathcal{T}_{\mathcal{B}}^{\,\prime}/\mathbb{Z}_2^J\) keeps the two stress tensors operator \(T_{UV}\) and \(T_{IR}\) intact, an explicit twist connecting these two stress tensors would be transparent in the basis of a fermionic theory as before. Indeed, in \(\mathcal{T}'_{\mathcal{B}}\), one can find that the charges of \(J\) and \(J_\psi\) respect to each other are both unity, \[\begin{align} Q_{J}(J_\psi)=Q_{J_\psi}(J)=1 \end{align}\] so the currents \(J\) and \(J_\psi\) will be maintained in \(\mathcal{SM}_{4,\,6}\times\mathcal{T}_{\psi}\) and \(\mathcal{T}_{\mathcal{B}}\) respectively. We summarize the above procedure in the following commutative diagram: \[\begin{align} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/tinvsjfa.png}\tag{20}\end{figure} \notag \tag{21} \end{align}\]
It’s interesting to notice that \(\mathcal{SM}^D_{4,\,6}\), the \(D\)-type variant of \(\mathcal{SM}_{4,\,6}\), coincides with the first \(\mathcal{N}=2\) minimal model. In the vacuum module of a \(\mathcal{N}=2\) superconformal algebra, we have a spin-1 \(R\)-current \(J_R\), two spin-\(\frac{3}{2}\) supercurrents \(G_\pm\), and a spin-2 stress tensor \(T_{\mathcal{SM}}\). Combining with the vacuum module of \(\mathcal{T}_\psi\), overall we precisely have a spin-1 current \(J_R\), 5 spin-2 operator \(T_{\mathcal{SM}}\), \(T_\psi\), \(G_\pm\psi\), and \(\partial J\). For later convenience, we introduce the basis consistent with \(\mathcal{N}=1\) superconformal algebra, \[\begin{align} G=\frac{1}{\sqrt 2}(G_++G_-)\,,\quad {\rm and}\quad G_D=\frac{1}{\sqrt 2}(G_+-G_-)\,. \end{align}\] Clearly, the supercurrent \(G\) is neutral under the \(R\)-current \(J_R\). The fields \(T_{\mathcal{SM}}\) and \(G\) furnish a \(\mathcal{N}=1\) superconformal sub-algebra. The detailed full \(\mathcal{N}=2\) superconformal algebra OPEs are summarized in App. 6. As we’ve mentioned, the vacuum module, \[\begin{align} \mathcal{V}_{\rm vac}^{\mathcal{B}'}\subset \mathcal{V}^{\mathcal{N}=1}_{\rm vac}\otimes \mathcal{V}^{\psi}_{\rm vac}\subset \mathcal{V}^{\mathcal{N}=2}_{\rm vac}\otimes \mathcal{V}^{\psi}_{\rm vac}\,, \quad{\rm and}\quad \mathcal{V}_{\rm vac}^{\mathcal{B}'}\subset \mathcal{V}_{\rm vac}^{\mathcal{B}}\subset \mathcal{V}^{\mathcal{N}=2}_{\rm vac}\otimes \mathcal{V}^{\psi}_{\rm vac}\,, \end{align}\] remains the same according to the orbifolding/fermionization commutative diagram 21 . Therefore, for the three operators \(T_{UV}\), \(T_{IR}\) and \(X\), we still have the decomposition 18 . On the other hand, for the two extra operators, the spin-1 phantom current \(J\) and spin-2 \(W\) in \(\mathcal{V}^{\mathcal{B}}_{\rm vac}\) can be also identified with the \(\mathcal{N}=2\) superconformal algebra, and the vacuum module of free Majorana fermion \(\mathcal{T}_\psi\). We summarize them all as below: \[\begin{align} &J=\phi^{UV}_{2/5}\,\bar\phi^{IR}_{3/5}=\sqrt 3 J_R\,,\notag\\[1ex] &T_{UV}=\frac{2}{5}T_{\mathcal{SM}}-\frac{\sqrt 6}{5}G\,\psi+\frac{4}{5}T_{\psi}\notag\\[1ex] &\overline{T}_{IR}=\frac{3}{5}T_{\mathcal{SM}}+\frac{\sqrt 6}{5}G\,\psi+\frac{1}{5}T_{\psi}\notag\\[1ex] &X=\left(W^{(3)}_{-1}\phi^{UV}_{2/5}\right)\bar\phi^{IR}_{3/5}=\frac{2}{\sqrt 7}T_{\mathcal{SM}}-\frac{\sqrt 6}{2\sqrt 7}G\,\psi-\frac{4}{\sqrt 7}T_{\psi}\notag\\[1ex] &W^{-}=i\frac{1}{\sqrt 3}\phi^{UV}_{2/5}\,\partial\bar\phi^{IR}_{3/5}-i\frac{\sqrt 3}{2}\partial\phi^{UV}_{2/5}\,\bar\phi^{IR}_{3/5} =i\sqrt\frac{3}{2}G_D\psi \label{eq:N61295basis} \end{align}\tag{22}\]
In this fermionic basis, we can once again recast a operator \(\phi^{\mathcal{P}_3}\bar\phi^{\mathcal{M}_4}\in\mathcal{T}_{\mathcal{B}}\) in \(\mathcal{SM}^D_{4,\,6}\times\mathcal{T}_{\psi}\). Further applying the automorphism \(\psi\rightarrow-\psi\), it is easy to compute the one-point correlator of \(\phi^{\mathcal{P}_3}\bar\phi^{\mathcal{M}_4}\) in presence of the RG brane, and thus the UV-IR operator mixing as in [1]. However we will not pursue along this direction. For a study of generic RG domain wall between \(\mathcal{P}_N\) and \(\mathcal{M}_{N+1}\) we hope to report in the near future.
Now we turn to study the marginal deformation of the RG domain wall between \(\mathcal{P}_3\) and \(\mathcal{M}_{4}\), denoted by \(\mathfrak D_N\). Since there exists a spin-1 phantom current \(J\), as discussed in [33], the superconformal interface can be deformed by an 1d local operator on the defect , \[\begin{align} U_\theta=e^{i\theta Q_J}+h.c.\,,\quad{\rm with}\quad Q_J=\int_\gamma\frac{dz}{2\pi i}J(z) \end{align}\] The stress tensor \(T_{UV}\) and \(\overline{T}_{IR}\) will mix with the operator \(W\). A hallmark feature of such a defect having conformal manifold is that its reflection/transmission rates have to depend on a moduli parameter. So in what follows, we will compute the transmission rate by employing the cluster decomposition equation 1.
First, it is worth noticing that \(\mathfrak D_N\) can be obtained by fusing the Gaiotto wall of \(\mathcal{M}_5\) to \(\mathcal{M}_4\) to a topological interface induced from the \(\mathbb{Z}_2^J\)-orbifold. The transmission rate of \(\mathfrak D_N\) should be the same to that of the Gaiotto wall, as a topological interface is always transparent. Therefore, only \(T_{UV}\), \(\overline{T}_{IR}\) and the operator \(X\) can mix on the wall \(\mathfrak D_N\). As noted in [34], the reflections of \(T_{UV}\), \(\overline{T}_{IR}\) and \(X\) after hitting \(\mathfrak D_N\) are \[\begin{align} \begin{pmatrix} \overline{T}_{UV}\\[1ex] T_{IR}\\[1ex] \bar X \end{pmatrix} = \begin{pmatrix} \frac{1}{5}\,, &\frac{32}{35}\,, &-\frac{4}{5\sqrt 7}\\[1ex] \frac{4}{5}\,, &\frac{3}{35}\,, &\frac{4}{5\sqrt 7}\\[1ex] -\frac{2}{\sqrt 7}\,, &\frac{16}{7\sqrt 7}\,, & \frac{5}{7} \end{pmatrix} \cdot \begin{pmatrix} T_{UV}\\[1ex] \overline{T}_{IR}\\[1ex] X \end{pmatrix} \end{align}\]
Now when we employ the deformation by \(U_\theta\) on \(\mathfrak D_N\), one has to put new ansatz on the reflections of \(T_{UV}\) and \(\overline{T}_{IR}\) by \[\begin{align} & \overline{T}_{UV}=\alpha\,T_{UV}+\beta\, \overline{T}_{IR}+\gamma\,W+\delta\, X\notag\\[1ex] & T_{IR}=(1-\alpha)\,T_{UV}+(1-\beta)\, \overline{T}_{IR}-\gamma\,W-\delta\, X\,, \label{eq:ansatz} \end{align}\tag{23}\] where we have imposed the energy conservation equation \[\begin{align} \overline{T}_{UV}+T_{IR}=T_{UV}+\overline{T}_{IR}\,. \end{align}\] To further determine these mixing parameter \(\{\alpha,\,\beta,\,\gamma,\,\delta\}\), we need to know the OPE’s among the operators \(T_{UV}\), \(\overline{T}_{IR}\), \(W^{-}\) and \(X\). In the fermionic basis 22 , they are easily computed and the results are collected in App. 6. We then proceed to compute the OPE of \(\overline{T}_{UV}(z)\cdot\overline{T}_{UV}(w)\), \(T_{IR}(z)\cdot T_{IR}(w)\), and \(\overline{T}_{UV}(z)\cdot T_{IR}(w)\). The cluster decomposition requires the last one is zero \[\begin{align} \overline{T}_{UV}(z)\cdot T_{IR}(w)\sim 0\,, \label{eq:cluster95decomp952} \end{align}\tag{24}\] while, for the first two, we need them to satisfy \[\begin{align} &\overline{T}_{UV}(z)\cdot\overline{T}_{UV}(w)\sim\frac{\frac{2}{5}}{(z-w)^4}+\frac{\overline{T}_{UV}(w)}{(z-w)^2}+\frac{\partial\overline{T}_{UV}(w)}{z-w}\notag\\[1ex] &T_{IR}(z)\cdot T_{IR}(w)\sim\frac{\frac{7}{20}}{(z-w)^4}+\frac{T_{IR}(w)}{(z-w)^2}+\frac{\partial T_{IR}(w)}{z-w}\,. \label{eq:TT} \end{align}\tag{25}\] Eqs. 25 are guaranteed once the most singular term are satisfied, as the ansatz 23 are quasi-primary by definition. It leads to \[\begin{align} &\frac{2}{5}\alpha^2+\frac{7}{20}\beta^2+\gamma^2+\delta^2=\frac{2}{5}\notag\\[1ex] &\frac{2}{5}(1-\alpha)^2+\frac{7}{20}(1-\beta)^2+\gamma^2+\delta^2=\frac{7}{20}\,. \label{eq:TT95coef} \end{align}\tag{26}\] On the other hand, for eq. 24 , we collect the coefficients in front of all primaries, \[\begin{align} &\frac{1}{(z-w)^4}:\qquad \frac{2}{5}\alpha(1-\alpha)+\frac{7}{20}\beta(1-\beta)-\gamma^2-\delta^2=0\notag\\[1ex] &\frac{J(w)}{(z-w)^3}:\qquad -\gamma\,\delta+\delta\,\gamma=0\notag\\[1ex] &\frac{T_{UV}(w)}{(z-w)^2}:\qquad 2\alpha(1-\alpha)-\frac{5}{2}\gamma^2-\frac{7}{2}\delta^2=0\notag\\[1ex] &\frac{T_{IR}(w)}{(z-w)^2}:\qquad 2\beta(1-\beta)-\frac{20}{7}\gamma^2-\frac{12}{7}\delta^2=0\notag\\[1ex] &\frac{X(w)}{(z-w)^2}:\qquad \frac{7}{5}(1-2\alpha)\delta+\frac{3}{5}(1-2\beta)\delta-\frac{1}{\sqrt{7}}\gamma^2+\frac{3}{\sqrt 7}\delta^2=0\notag\\[1ex] &\frac{W^{-}(w)}{(z-w)^2}:\qquad (1-2\alpha)\gamma+(1-2\beta)\gamma-\frac{1}{\sqrt{7}}\gamma\,\delta-\frac{1}{\sqrt 7}\delta\,\gamma=0 \label{eq:CD95coef} \end{align}\tag{27}\] Solving eqs. 26 and 27 , one can find a family of solutions parametrized by the coefficient \(\gamma\), \[\begin{align} \alpha=\frac{1}{5}\left(3\pm\sqrt{4-25\gamma^2}\,\right)\,,\quad \beta=\frac{8}{35}\left(2\mp\sqrt{4-25\gamma^2}\,\right)\,,\quad \delta=\frac{1}{5\sqrt 7}\left(-2\pm\sqrt{4-25\gamma^2\,}\right)\,. \end{align}\] Since the RG flow is between unitary CFTs, we would expect the coefficients \(\alpha\) and \(\beta\) real. Therefore, it’s reasonable to re-parametrize \(\gamma\) as \[\begin{align} \gamma=\frac{2}{5}\sin\theta\,. \end{align}\] In terms of \(\theta\), we have \[\begin{align} \alpha=\frac{1}{5}(3-2\cos\theta)\,,\quad \beta=\frac{16}{35}(1+\cos\theta)\,,\quad \gamma=\frac{2}{5}\sin\theta\,,\quad \delta=-\frac{2}{5\sqrt 7}(1+\cos\theta)\,. \end{align}\] For \(\theta = 0\), we simply recover the Gaiotto’s RG domain wall as computed in [34]. To understand the physical properties of the defect conformal manifold, notice that the transmission rate \(\mathcal{T}\) across the interface is determined by the cross two-point function \(c_{UV\text{-}IR}\) between the stress tensors[42]–[44], \[\begin{align} {T}_{IR}(z)\cdot{T}_{UV}(w)\sim\frac{c_{UV\text{-}IR}}{2(z-w)^2}\,. \end{align}\] From our ansatz 23 , taking the expectation value with \({T}_{IR}\) and \(T_{UV}\) respectively, we obtain \[\begin{align} c_{UV\text{-}IR} = (1-\alpha)\,c_{UV} = \beta\,c_{IR}\,. \end{align}\] Therefore, the transmission rates in both directions are strictly proportional to \(\beta\). One can find that, with \(\theta\) increasing, \(\beta\) decreases and the defect becomes less transparent. At \(\theta = \pi\), we have \[\begin{align} \alpha = 1 \,,\quad \beta = 0 \,,\quad \gamma = 0 \,,\quad \delta = 0 \,, \end{align}\] meaning the transmission rate strictly vanishes. The conformal defect becomes completely reflective, reducing to a trivially factorized interface.
Now we proceed to the generic case of the RG flow from the \(\mathcal{P}_N\) to \(\mathcal{M}_{N+1}\) model. As we’ve shown in Sec. 2, the deformation 2 will preserve various non-invertible symmetries. Among them, the line \(\mathcal{L}_{[3,0]}\) in \(\mathcal{P}_N\) penetrates the RG wall and becomes the line \(\mathcal{L}_{[1,3]}\) in \(\mathcal{M}_{N+1}\). At the end of the two lines, there are attached defect operators \(\phi^{UV}_{[3,0]}\) and \(\phi^{IR}_{[1,3]}\) respectively. When one embeds the tensor product of the two theories into the auxiliary \(\mathcal{T}_{\mathcal{B}}\), \[\begin{align} \mathcal{P}_N\times\overline{\mathcal{M}_{N+1}}\longrightarrow\mathcal{T}_\mathcal{B}=\frac{\mathfrak {su}(2)_{N-1}\times\mathfrak{su}(2)_1}{\mathfrak{u}(1)_N}\,, \end{align}\] the tensor product of the two defect operators \[\begin{align} J=\phi^{UV}_{[3,0]}\,\bar\phi^{IR}_{[1,3]}\,, \end{align}\] as well as its descendants, condenses to the vacuum module of \(\mathcal{T}_\mathcal{B}\) and thus becomes a local operator. For generic \(N\), the vacuum module of \(\mathcal{T}_\mathcal{B}\) is \[\begin{align} \chi_{\rm vac}^{\mathcal{B}}=\sum_{k=0}^{\left\lfloor N/2\right\rfloor}\chi^{UV}_{[2k+1,0]}\,\chi^{IR}_{[1,2k+1]}=q^{-\frac{c_{UV}+c_{IR}}{24}}\left(1+q+5q^2\cdots\right)\,, \end{align}\] As before, the operator \(J\) corresponds to the linear \(q\)-term. We collect all four Virasoro (quasi-)primaries of spin-2: beside \(T_{UV}\) and \(\overline{T}_{IR}\), there is also a \(W^{-}\) operator defined as in eq. 19 , \[\begin{align} &W^{-}\equiv \sqrt\frac{1}{{-2h_{UV}h_{IR}}}\left(h_{UV}\,\phi^{UV}_{[3,0]}\,\partial\bar\phi^{IR}_{[1,3]}-h_{IR}\,\partial\phi^{UV}_{[3,0]}\,\bar\phi^{IR}_{[1,3]}\right)\,,\notag\\[2ex] {\rm with}\quad &h_{UV}=\frac{2}{N+2}\,,\quad {\rm and}\quad h_{IR}=\frac{N}{N+2}\,. \end{align}\] In addition, similar to the \(\mathcal{P}_3\) case, one can find a \(W^{(3)}\)-descendant, but Virasoro primary in the family of \(\phi^{UV}_{[3,0]}\). Notice that fields \(\phi^{UV}_{[3,0]}\) and \(W^{(3)}\) are \(\mathbb{Z}_N\)-charge conjugation even and odd respectively. It implies that the \(W^{(3)}\)-charge of \(\phi^{UV}_{[3,0]}\) is simply zero, \[\begin{align} W^{(3)}_0\,|[3,0]\rangle_{UV}=0\,. \end{align}\] Therefore the field \[\begin{align} \phi^{UV}_{w}\equiv W^{(3)}_{-1}\phi^{UV}_{[3,0]}\,, \end{align}\] must be Virasoro primary as \[\begin{align} L^{UV}_1|w\rangle_{UV}\propto W_{0}^{(3)}\,|[3,0]\rangle_{UV}=0\,. \end{align}\] So in \(\mathcal{T}_\mathcal{B}\), we have the fourth Virasoro (quasi-)primary \[\begin{align} X\equiv \phi^{UV}_w\,\bar\phi^{IR}_{[1,3]}\,. \end{align}\]
Now, we need the OPE for the operators \(T_{UV}\), \(\overline{T}_{IR}\), \(X\) and \(W^{-}\), to compute the solve the undetermined coefficients in the ansatz, \[\begin{align} & \overline{T}_{UV}=\alpha\,T_{UV}+\beta\, \overline{T}_{IR}+\gamma\,W+\delta\, X\notag\\[1ex] & T_{IR}=(1-\alpha)\,T_{UV}+(1-\beta)\, \overline{T}_{IR}-\gamma\,W-\delta\, X\,, \label{eq:ansatz2} \end{align}\tag{28}\]
The OPE of ansatz 28 for odd power of \(z-w\) trivially satisfies eq. 1 . First we claim that the OPE of \(X(z)\cdot X(w)\) and \(W^{-}(z)\cdot W^{-}(w)\) will not give non-trivial quasi-primaries at \(\mathcal{O}\left((z-w)^{-3}\right)\) and \(\mathcal{O}\left((z-w)^{-1}\right)\). Consider, for example, \[\begin{align} X(z)\cdot X(w)\sim \frac{1}{(z-w)^4}+\frac{O_3(w)}{(z-w)^3}+\frac{O_2(w)}{(z-w)^2}+\frac{O_1(w)}{z-w}\,. \end{align}\] The symmetry constraint \(X(z)\cdot X(w)=X(w)\cdot X(z)\) simply implies that \[\begin{align} O_3(w)=0\,,\quad {\rm and}\quad \mathcal{O}_1(w)=\frac{1}{2}\partial O_2(w)\,. \end{align}\] Therefore the diagonal terms at odd power of \(z-w\) provide no constraints. On the other hand the cross terms in eq. 1 , e.g. \[\begin{align} -\gamma\,\delta \;X(z)\cdot W^{-}(w)-\gamma\,\delta \;W^{-}(z)\cdot X(w)\bigg\vert_{(z-w)^{-1}}=\cdots-\gamma\,\delta\left(\frac{O_{prim}(w)}{z-w}+\frac{O_{prim}(z)}{w-z}\right)+\cdots\,,\notag \end{align}\] trivially satisfy the cluster decomposition principle, where \(O_{prim}\) are some certain spin-3 quasi-primaries if there exist.
At last, let’s consider the cross terms between stress tensors and \(X\) or \(W^{-}\) at \(\mathcal{O}\left((z-w)^{-1}\right)\). Suppose that there is a new quasi-primary \(O_{prim}\) of spin-3 between \(T_{UV}\) and \(X\), \[\begin{align} T_{UV}(z)\cdot X(w)\sim \cdots+\frac{O_{prim}}{(z-w)^3}+\cdots\,, \end{align}\] one must have \[\begin{align} \overline{T}_{IR}(z)\cdot X(w)\sim \cdots -\frac{O_{prim}}{(z-w)^3}+\cdots\,, \end{align}\] because operator \(X\) is a primary respect to the total stress tensor \(T=T_{UV}+\overline{T}_{IR}\) in the folded theory. Therefore the coefficient of \(O_{prim}\) in \(\overline{T}_{UV}(z)\cdot T_{IR}(w)\) is \[\begin{align} \left(-\alpha\,\delta +\beta\,\delta\,\right)\,\frac{1}{z-w}+\left((1-\alpha)\,\delta-(1-\beta)\,\delta\,\right)\,\frac{1}{w-z}=0 \end{align}\] Similar argument is applied to operator \(W^{-}\) as well. Overall, we will have no any non-trivial constraints at \(\mathcal{O}\left((z-w)^{-1}\right)\).
Therefore we only consider the OPE up to \(\mathcal{O}\left((z-w)^{-2}\right)\). The OPE among \(T_{UV}\) and \(\overline{T}_{IR}\) are standard \[\begin{align} &T_{UV}(z)\cdot T_{UV}(w)\sim \frac{c_{UV}}{(z-w)^4}+\frac{2T_{UV}(w)}{(z-w)^2}+\frac{\partial T_{UV}(w)}{z-w}\,,\quad {\rm with\;\;}c_{UV}=2\frac{N-1}{N+2}\notag\\[2ex] &T_{IR}(z)\cdot T_{IR}(w)\sim \frac{c_{IR}}{(z-w)^4}+\frac{2T_{IR}(w)}{(z-w)^2}+\frac{\partial T_{IR}(w)}{z-w}\,,\quad {\rm with\;\;}c_{IR}=1-\frac{6}{(N+1)(N+2)}\notag\\[2ex] &T_{UV}(z)\cdot \overline{T}_{IR}(w)\sim 0\,. \end{align}\] Because the operators factorize, the OPEs involving \(X\) or \(W^{-}\) with \(T_{UV}\) or \(T_{IR}\) can be straightforwardly obtained from the respective constituent theories \(\mathcal{P}_N\) or \(\mathcal{M}_{N+1}\), \[\begin{align} &T_{UV}(z)\cdot W^{-}(w) \sim \frac{\sqrt{\frac{1}{-2h_{UV}h_{IR}}}J(w)}{(z-w)^3}+\frac{\frac{1}{2}\sqrt{\frac{1}{-2h_{UV}h_{IR}}}\partial J(w)+W^{-}(w)}{(z-w)^2}\notag\\[2ex] &T_{IR}(z)\cdot W^{-}(w) \sim \frac{-\sqrt{\frac{1}{-2h_{UV}h_{IR}}}J(w)}{(z-w)^3}+\frac{-\frac{1}{2}\sqrt{\frac{1}{-2h_{UV}h_{IR}}}\partial J(w)+W^{-}(w)}{(z-w)^2}\notag\\[2ex] &T_{UV}(z)\cdot X(w)\sim \frac{(h_{UV}+1)X(w)}{(z-w)^2}\notag\\[2ex] &T_{IR}(z)\cdot X(w)\sim \frac{h_{IR}\,X(w)}{(z-w)^2} \end{align}\] In addition, for the OPE among operators \(X\) and \(W^{-}\), notice that \(\phi^{UV}_{[3,0]}\) and \(\bar\phi^{IR}_{[1,3]}\) do not talk to each other, and their own fusion channels are the same as those of TDLs \(\mathcal{L}_{[3,0]}^{UV}\) and \(\mathcal{L}^{IR}_{[1,3]}\). Therefore we have \[\begin{align} \left[\phi^{UV}_{[3,0]}\right]\cdot \left[\phi^{UV}_{[3,0]}\right]=\sum_{k=0}^{\left\lfloor N/2\right\rfloor}\left[\phi^{UV}_{[2k+1,0]}\right]\,,\qquad \left[\bar\phi^{IR}_{[1,3]}\right]\cdot \left[\bar \phi^{IR}_{[1,3]}\right]=\sum_{k=0}^{\left\lfloor N/2\right\rfloor}\left[\bar \phi^{IR}_{[1,2k+1]}\right]\,. \end{align}\] Recall that the operators \(\phi^{UV}_{[2k+1,0]}\) and \(\bar\phi^{IR}_{[1,2k+1]}\) have conformal weights \[\begin{align} h^{UV}_{[2k+1,0]}=\frac{2k}{N+2}\,,\qquad h^{IR}_{[1,2k+1]}=k^2-\frac{2k}{N+2}\,, \end{align}\] and all the operators \(\phi^{UV}_{[2k+1,0]}\,\bar\phi^{IR}_{[1,2k+1]}\) reside in the vacuum module. Therefore, for the fusion channels of \(W^{-}\) and \(X\), we only need to collect terms admitting integer pole structures in the OPE, and discard those with fractional branch cuts. Based on this observation, we have the following OPE’s, \[\begin{align} &X(z)\cdot X(w)\sim \frac{1}{(z-w)^4}+\frac{\frac{2(h_{UV}+1)}{c_{UV}}T_{UV}(w)+\frac{2h_{IR}}{c_{IR}}\overline{T}_{IR}(w)+C_{XXW^{-}}W^{-}(z)+C_{XXX}X(w)}{(z-w)^2}\notag\\[1ex] &X(z)\cdot W^{-}(w) \sim \frac{C_{JXW^{-}} J(w)}{(z-w)^3}+\frac{\frac{C_{JXW^{-}}}{2}\partial J(w)+C_{XW^{-}W^{-}}W^{-}(z)+C_{XXW^{-}}X(z)}{(z-w)^2}\notag\\[1ex] &W^{-}(z)\cdot W^{-}(w) \sim \frac{1}{(z-w)^4}\notag\\[1ex] &\qquad\qquad\qquad\quad\;+\frac{\frac{2}{c_{UV}}T_{UV}(w)+\frac{2}{c_{IR}}\overline{T}_{IR}(w)+C_{XW^{-}W^{-}}X(w)+C_{W^{-}W^{-}W^{-}}W^{-}(w)}{(z-w)^2}\,, \end{align}\] where the OPE coefficients \[\begin{align} &C_{XXW^{-}}=C_{W^{-}W^{-}W^{-}}=0\,,\quad C_{JXW^{-}}=i\sqrt{\frac{2N(N-2)}{(N-1)(N+4)}}\,,\notag\\[1ex] &C_{XXX}=\frac{3\sqrt2(N-4)}{\sqrt{(N-1)(N-2)(N+4)}}\,,\quad {\rm and}\quad C_{XW^{-}W^{-}}=\sqrt{\frac{2(N-2)}{(N-1)(N+4)}}\,. \end{align}\] The computation details on these OPE coefficients are summarized in the App. 7.
With these preparation, we can solve the constraints from OPE to find the mixing coefficients \(\{\alpha,\,\beta,\,\gamma,\,\delta\}\). As before, OPE between stress tensors themselves give \[\begin{align} &\frac{c_{UV}}{2}\alpha^2+\frac{c_{IR}}{2}\beta^2+\gamma^2+\delta^2=\frac{c_{UV}}{2}\notag\\[1ex] &\frac{c_{UV}}{2}(1-\alpha)^2+\frac{c_{IR}}{2}(1-\beta)^2+\gamma^2+\delta^2=\frac{c_{IR}}{2}\,, \end{align}\] and cluster decomposition 1 gives: \[\begin{align} &\frac{1}{(z-w)^4}:\qquad \frac{c_{UV}}{2}\,\alpha(1-\alpha)+\frac{c_{IR}}{2}\,\beta(1-\beta)-\gamma^2-\delta^2=0\notag\\[1ex] &\frac{J(w)}{(z-w)^3}:\qquad -\gamma\,\delta+\delta\,\gamma=0\notag\\[1ex] &\frac{T_{UV}(w)}{(z-w)^2}:\qquad 2\alpha(1-\alpha)-\frac{2}{c_{UV}}\gamma^2-\frac{2(h_{UV}+1)}{c_{UV}}\delta^2=0\notag\\[1ex] &\frac{T_{IR}(w)}{(z-w)^2}:\qquad 2\beta(1-\beta)-\frac{2}{c_{IR}}\gamma^2-\frac{2h_{IR}}{c_{IR}}\delta^2=0\notag\\[1ex] &\frac{X(w)}{(z-w)^2}:\qquad (h_{UV}+1)(1-2\alpha)\delta+h_{IR}(1-2\beta)\delta-C_{XW^{-}W^{-}}\,\gamma^2-C_{XXX}\delta^2=0\notag\\[1ex] &\frac{W^{-}(w)}{(z-w)^2}:\qquad (1-2\alpha)\gamma+(1-2\beta)\gamma-2C_{XW^{-}W^{-}}\,\gamma\,\delta=0 \end{align}\] By parameterizing \[\begin{align} \gamma=\frac{\sqrt{2(N-1)}}{N+2}\sin\theta\,, \end{align}\] we find \[\begin{align} \alpha=\frac{N-2\cos\theta}{N+2}\,,\quad \beta=\frac{4(N+1)(1+\cos\theta)}{(N+4)(N+2)}\,,\quad \delta=-\frac{\sqrt{2(N-2)(N-1)}}{(N+2)\sqrt{N+4}}(1+\cos\theta)\,. \end{align}\] Therefore, overall, we compute the transmission rate of the RG domain wall \(\mathfrak D_N(\theta)\) with defect conformal moduli parameter \(\theta\), \[\begin{align} \mathcal{T}_N(\theta)=\frac{2c_{UV\text{-}IR}}{c_{UV}+c_{IR}}=\frac{2\beta\,c_{IR}}{c_{UV}+c_{IR}}=\frac{8(N+1)(1+\cos\theta)}{3(N+2)^2} \end{align}\]
Similar to the case of \(N=3\), for generic \(N\), the RG domain wall also admits a moduli parameter \(\theta\) to interpolating the RG wall to an opaquely factorized conformal defect. Furthermore, it is interesting to notice that, for \(N\rightarrow \infty\), \(\beta\rightarrow 0\) indicates that the transmission rate \(\mathcal{T}_N\) goes to zero and the RG domain wall becomes purely factorized. The vanishing of the transmission rate in the large-\(N\) limit highlights the genuinely non-perturbative nature of the RG flow from the \(\mathbb{Z}_N\) parafermion theory to the minimal model \(\mathcal{M}_{N+1}\). Unlike the flow between consecutive minimal models—where the central charge difference \(\Delta c \sim \mathcal{O}(1/N^3)\) is parametrically small and the fixed points are close in theory space, yielding a highly transparent domain wall—the parafermion flow is characterized by a macroscopic loss of degrees of freedom. As \(N \to \infty\), the UV central charge approaches \(c_{UV} = 2\), corresponding geometrically to the \(O(3)\) non-linear sigma model, while the IR central charge approaches \(c_{IR} = 1\), corresponding to a free compact boson. This \(\Delta c \to 1\) drop reflects a severe dynamical collapse of the target space, wherein an entire spatial dimension acquires an infinite mass and is integrated out. Algebraically, this macroscopic decoupling manifests within the extended chiral algebra \(\mathcal{T}_{\mathcal{B}}\) of the folded theory: the critical structure constants governing the fusion of the phantom currents (such as \(C_{XXX}\) and \(C_{XW^-W^-}\)) scale as \(\mathcal{O}(1/\sqrt{N})\). Consequently, in the infinite \(N\) limit, the non-invertible symmetry structures that mediate the stress tensor mixing completely vanish. Lacking these non-zero three-point couplings, local chiral energy fluctuations cannot be coherently transmitted across the infinite distance in theory space, analytically forcing the RG domain wall to become completely reflective.
We would like to thank Yuya Kusuki and Yunqin Zheng for useful discussions. J.C. is supported by the Fujian Provincial Natural Science Foundation of China (No.2025J01004), and the National Natural Science Foundation of China (Grants No.12247103). T.C. is supported by the Xiamen University “Undergraduate Innovation Training Program" – Innovation Training (No.2025X1372).
Coset CFT provides one of the most systematic constructions of RCFT. They realize new chiral algebras by taking the commutant of a subalgebra inside a larger affine current algebra. Equivalently, starting from a Wess-Zumino-Witten theory with chiral algebra \(\mathfrak g_k\), one gauges a chiral subalgebra \(\mathfrak h_{k'}\subset \mathfrak g_k\). The resulting theory is denoted schematically by \[\begin{align} \frac{\mathfrak g_k}{\mathfrak h_{k'}} , \end{align}\] and its central charge is given by the difference \[\begin{align} c_{\text{coset}} = c(\mathfrak g_k)-c(\mathfrak h_{k'}) . \end{align}\] For \(\mathfrak{su}(2)_k\), we shall use \[\begin{align} c\!\left(\mathfrak{su}(2)_k\right) = \frac{3k}{k+2}. \end{align}\] For a coset theory, the numerator and denominator factors are represented by Chern-Simons theories with opposite orientations: \[\begin{align} \text{2D:}\quad \frac{\mathfrak{su}(2)_{k_1}\times\cdots\times \mathfrak{su}(2)_{k_n}}{\mathfrak{su}(2)_{k'_1}\times\cdots\times \mathfrak{su}(2)_{k'_m}} \qquad \longleftrightarrow \qquad \text{3D:}\quad \prod_{i=1}^{n} \mathfrak{su}(2)_{k_i} \times \prod_{j=1}^{m} \mathfrak{su}(2)_{-k'_j}. \end{align}\] The negative levels in the denominator encode the fact that these degrees of freedom are gauged. Equivalently, their modular data appear complex conjugated in the coset modular matrices. Before imposing the coset constraints, a Wilson line is labeled by a collection of integrable representations, \[\begin{align} \mathcal{W}_R = \mathcal{W}_{[\lambda_1,\ldots,\lambda_n;\,\mu_1,\ldots,\mu_m]}, \end{align}\] where \[\begin{align} \lambda_i=1,\ldots,k_i+1, \qquad \mu_j=1,\ldots,k'_j+1 . \end{align}\] Here we label \(\mathfrak{su}(2)_k\) integrable representations by \[\begin{align} a=1,\ldots,k+1, \qquad j=\frac{a-1}{2}. \end{align}\] For an abelian factor such as \(\mathfrak{u}(1)_k\), we use charge labels \[\begin{align} m\in \mathbb{Z}_{2k}. \end{align}\]
The physical spectrum is obtained only after imposing the constraints generated by a distinguished topological line, which we denote by \(\mathcal{W}_L\). This line is the identification line. In the three-dimensional picture, it is an invertible Wilson line generating a discrete one-form symmetry. Its action has two effects.
First, fusing \(\mathcal{W}_L\) with a Wilson line \(\mathcal{W}_R\) maps the latter to another Wilson line with shifted representation labels. Since the identification line is topological, the two configurations are physically equivalent. This produces the field identification relation \[\begin{align} R\sim L\times R . \end{align}\] Second, when \(\mathcal{W}_L\) is braided around \(\mathcal{W}_R\), the configuration acquires a monodromy phase \[\begin{align} \mathcal{M}_{L,R} = \frac{S_{L,R}S_{0,0}}{S_{L,0}S_{0,R}} . \end{align}\] Only fields with trivial monodromy, \[\begin{align} \mathcal{M}_{L,R}=1, \end{align}\] are mutually local with respect to the condensed line and therefore define admissible local operators in the coset theory. Thus, the familiar coset selection rules are reinterpreted as mutual-locality constraints with respect to the identification line.
A canonical example is the \(\mathbb{Z}_N\) parafermion theory 3 , Its three-dimensional realization is \[\begin{align} \mathfrak{su}(2)_N \times \mathfrak{u}(1)_{-N}. \end{align}\] We label Wilson lines by pairs \(\mathcal{W}_{[t,s]}\), where \[\begin{align} t=1,\ldots,N+1, \qquad s\in \mathbb{Z}_{2N}. \end{align}\] Because the \(\mathfrak{u}(1)\) factor appears with negative level in the coset, its contribution to the parafermion modular matrix is complex conjugated.
The identification line is \[\begin{align} \mathcal{W}_L = \mathcal{W}_{[N+1,N]} . \end{align}\] Fusion with this line gives the field identification 4 The monodromy of \(\mathcal{W}_L\) around a general line \(\mathcal{W}_{[t,s]}\) is \[\begin{align} \mathcal{M}_{L,[t,s]} = (-1)^{t+s-1}. \end{align}\] Therefore, the mutual-locality condition is \[\begin{align} \mathcal{M}_{L,[t,s]}=1 \qquad \Longleftrightarrow \qquad t+s\in 2\mathbb{Z}+1 . \end{align}\] 5 is calculated by the modular matrices of the two factors \[\begin{align} S^{\mathfrak{su}(2)_N}_{t,t'} = \sqrt{\frac{2}{N+2}}\, \sin\!\left(\frac{\pi tt'}{N+2}\right), \qquad S^{\mathfrak{u}(1)_N}_{s,s'} = \frac{1}{\sqrt{2N}} \exp\!\left(\frac{i\pi ss'}{N}\right), \label{eq:WZW-S-matrix} \end{align}\tag{29}\] Therefore, \[\begin{align} S^{\mathcal{P}_N}_{[t,s],[t',s']} = 2\, S^{\mathfrak{su}(2)_N}_{t,t'}\, \left(S^{\mathfrak{u}(1)_N}_{s,s'}\right)^* = \sqrt{\frac{4}{N(N+2)}}\, \sin\!\left(\frac{\pi tt'}{N+2}\right) \exp\!\left(-\frac{i\pi ss'}{N}\right). \end{align}\] The factor of \(2\) accounts for the length of the generic simple-current orbit.
It also provides a useful way to construct auxiliary rational theories with extended chiral algebras. The auxiliary theory appearing in the folded description of the RG interface between consecutive minimal models provides an important example. For the product theory \(\mathcal{M}_5\otimes \mathcal{M}_4\), it corresponds to the extended theory \[\begin{align} \mathcal{T}_{\mathcal{B}} = \frac{\mathfrak{su}(2)_2\times \mathfrak{su}(2)_1\times \mathfrak{su}(2)_1}{\mathfrak{su}(2)_4}. \end{align}\] At the level of central charges, this gives \[\begin{align} c(\mathcal{T}_{\mathcal{B}}) = c(\mathfrak{su}(2)_2)+2c(\mathfrak{su}(2)_1)-c(\mathfrak{su}(2)_4) = \frac{3}{2}, \end{align}\] The Wilson lines of \(\mathcal{T}_{\mathcal{B}}\) are labeled by four integers with identification \[\begin{align} [t,r,\widetilde{r},s]\sim [4-t,\;3-r,\;3-\widetilde{r},\;6-s]. \end{align}\] where \[\begin{align} t=1,2,3, \qquad r,\widetilde{r}=1,2, \qquad s=1,\ldots,5 . \end{align}\] Here \(t\) labels the \(\mathfrak{su}(2)_2\) factor, \(r\) and \(\widetilde{r}\) label the two \(\mathfrak{su}(2)_1\) factors, and \(s\) labels the denominator \(\mathfrak{su}(2)_4\) factor.
The mutual-locality condition with respect to the identification line is \[\begin{align} t+r+\widetilde{r}+s\in 2\mathbb{Z} . \end{align}\] Thus, the primary fields of \(\mathcal{T}_{\mathcal{B}}\) are equivalence classes of labels \([t,r,\widetilde{r},s]\) satisfying this parity constraint.
The conformal weight is computed from the coset formula \[\begin{align} h_{\mathcal{B}} = h^{\mathfrak{su}(2)_2}_{t} + h^{\mathfrak{su}(2)_1}_{r} + h^{\mathfrak{su}(2)_1}_{\widetilde{r}} - h^{\mathfrak{su}(2)_4}_{s} \quad \mathrm{mod}\;1, \end{align}\] with \[\begin{align} h^{\mathfrak{su}(2)_k}_{a} = \frac{a^2-1}{4(k+2)} . \end{align}\] Similarly, the modular matrix is obtained from the product of WZW modular matrices, with the denominator contribution complex conjugated. Since the \(\mathfrak{su}(2)\) modular matrices are real, this gives \[\begin{align} S^{\mathcal{B}}_{[t,r,\widetilde{r},s],[t',r',\widetilde{r}',s']} &= 2\, S^{\mathfrak{su}(2)_2}_{t,t'}\, S^{\mathfrak{su}(2)_1}_{r,r'}\, S^{\mathfrak{su}(2)_1}_{\widetilde{r},\widetilde{r}'}\, S^{\mathfrak{su}(2)_4}_{s,s'}\notag\\[1ex] &= \sqrt{\frac{8}{27}}\, \sin\!\left(\frac{\pi tt'}{4}\right) \sin\!\left(\frac{\pi rr'}{3}\right) \sin\!\left(\frac{\pi \widetilde{r}\widetilde{r}'}{3}\right) \sin\!\left(\frac{\pi ss'}{6}\right). \end{align}\] This construction makes explicit how the extended theory \(\mathcal{T}_{\mathcal{B}}\) inherits its rational data from the Chern–Simons description of the coset. In particular, its spectrum, conformal weights, field identifications, and modular transformations are all controlled by the same topological mechanism: the condensation of the identification line and the corresponding mutual-locality constraint.
In this appendix we spell out the fusion computation for the topological lines preserved by the parafermionic perturbation. The preserved lines found in the main text can be represented as \[\begin{align} \mathcal{L}_{[t,0]}, \qquad t\in 2\mathbb{Z}+1 . \end{align}\] Equivalently, writing \[\begin{align} \ell=t-1 , \end{align}\] the preserved labels are precisely the even-\(\ell\) labels of \(\mathfrak{su}(2)_N\).
We now compute their fusion rules. Let \[\begin{align} i=\mathcal{L}_{[t_i,0]}, \qquad j=\mathcal{L}_{[t_j,0]}, \qquad t_i,t_j\in 2\mathbb{Z}+1, \end{align}\] while \(k=\mathcal{L}_{[t_k,s_k]}\) is kept arbitrary. The Verlinde formula gives \[\begin{align} N_{ij}^{\;\;k} = \sum_{h\in\mathcal{P}_N} \frac{ S^{\mathcal{P}_N}_{ih} S^{\mathcal{P}_N}_{jh} \bigl(S^{\mathcal{P}_N}_{kh}\bigr)^{*} }{ S^{\mathcal{P}_N}_{[1,0]h} } . \label{eq:app-verlinde} \end{align}\tag{30}\] Using 5 , we obtain \[\begin{align} N_{ij}^{\;\;k} = \sum_{\mathcal{L}_{[t_h,s_h]}\in\mathcal{P}_N} \frac{4}{N(N+2)} \frac{ \sin\!\left(\frac{\pi t_i t_h}{N+2}\right) \sin\!\left(\frac{\pi t_j t_h}{N+2}\right) \sin\!\left(\frac{\pi t_k t_h}{N+2}\right) }{ \sin\!\left(\frac{\pi t_h}{N+2}\right) } \exp\!\left(\frac{\pi i s_k s_h}{N}\right). \label{eq:app-verlinde-expanded} \end{align}\tag{31}\]
To evaluate the sum explicitly, we lift the sum over equivalence classes to a sum over the covering labels \(\mathcal{L}_{[t_h,s_h]}\). The parity projection contributes \(\frac{1}{2}(1-(-1)^{t_h+s_h})\), and the quotient by the field identification contributes another factor of \(\frac{1}{2}\). Hence \[\begin{align} N_{ij}^{\;\;k} = \frac{1}{N(N+2)} \sum_{t_h=1}^{N+1} \sum_{s_h=0}^{2N-1} \bigl(1-(-1)^{t_h+s_h}\bigr) F_{ijk}(t_h) \exp\!\left(\frac{\pi i s_k s_h}{N}\right), \label{eq:app-lifted-sum} \end{align}\tag{32}\] where \[\begin{align} F_{ijk}(t_h) = \frac{ \sin\!\left(\frac{\pi t_i t_h}{N+2}\right) \sin\!\left(\frac{\pi t_j t_h}{N+2}\right) \sin\!\left(\frac{\pi t_k t_h}{N+2}\right) }{ \sin\!\left(\frac{\pi t_h}{N+2}\right) } . \end{align}\]
The \(s_h\)-sum is a finite Fourier sum. We have \[\begin{align} \sum_{s_h=0}^{2N-1} \exp\!\left(\frac{\pi i s_k s_h}{N}\right) = 2N\,\delta^{(2N)}_{s_k,0}, \end{align}\] and \[\begin{align} \sum_{s_h=0}^{2N-1} (-1)^{s_h} \exp\!\left(\frac{\pi i s_k s_h}{N}\right) = 2N\,\delta^{(2N)}_{s_k,N}, \end{align}\] where \(\delta^{(2N)}\) denotes the Kronecker delta modulo \(2N\). Therefore \[\begin{align} \sum_{s_h=0}^{2N-1} \bigl(1-(-1)^{t_h+s_h}\bigr) \exp\!\left(\frac{\pi i s_k s_h}{N}\right) =2N\left( \delta^{(2N)}_{s_k,0}-(-1)^{t_h}\delta^{(2N)}_{s_k,N} \right). \end{align}\] Substituting this into 32 gives \[\begin{align} N_{ij}^{\;\;k}=\delta^{(2N)}_{s_k,0}\, \mathcal{N}_{\ell_i,\ell_j}^{\;\;\ell_k} +\delta^{(2N)}_{s_k,N}\,\mathcal{N}_{\ell_i,\ell_j}^{\;\;N-\ell_k}, \label{eq:app-selection-rule} \end{align}\tag{33}\] where \[\begin{align} \ell_a=t_a-1, \qquad a=i,j,k, \end{align}\] and \[\begin{align} \mathcal{N}_{\ell_i,\ell_j}^{\;\;\ell_k} = \frac{2}{N+2} \sum_{t_h=1}^{N+1} \frac{ \sin\!\left(\frac{\pi(\ell_i+1)t_h}{N+2}\right) \sin\!\left(\frac{\pi(\ell_j+1)t_h}{N+2}\right) \sin\!\left(\frac{\pi(\ell_k+1)t_h}{N+2}\right) }{ \sin\!\left(\frac{\pi t_h}{N+2}\right) } . \label{eq:app-su2-verlinde-sum} \end{align}\tag{34}\] In deriving the second term in 33 , we used \[\begin{align} (-1)^{t_h+1} \sin\!\left(\frac{\pi(\ell_k+1)t_h}{N+2}\right) = \sin\!\left(\frac{\pi(N-\ell_k+1)t_h}{N+2}\right). \end{align}\]
The quantity \(\mathcal{N}_{\ell_i,\ell_j}^{\;\;\ell_k}\) is precisely the \(\mathfrak{su}(2)_N\) Verlinde coefficient. Equivalently, \[\begin{align} \mathcal{N}_{\ell_i,\ell_j}^{\;\;\ell_k} = \sum_{\substack{ \ell= |\ell_i-\ell_j|\\ \mathrm{step}\;2 }}^{ \min(\ell_i+\ell_j,\;2N-\ell_i-\ell_j) } \delta_{\ell_k,\ell}. \label{eq:app-su2-fusion} \end{align}\tag{35}\] Thus the \(s_h\)-sum imposes the selection rule 8 . The second possibility is not a new preserved object, because of 10 . In terms of \(\ell_k=t_k-1\), this identification sends \[\begin{align} \ell_k\longmapsto N-\ell_k . \end{align}\] Therefore, after choosing representatives with \(s=0\), the fusion rules of the preserved lines are \[\begin{align} \mathcal{L}_{[\ell_i+1,0]}\times\mathcal{L}_{[\ell_j+1,0]} =\bigoplus_{\substack{ \ell= |\ell_i-\ell_j|\\ \mathrm{step}\;2 }}^{ \min(\ell_i+\ell_j,\;2N-\ell_i-\ell_j) } \mathcal{L}_{[\ell+1,0]}, \qquad \ell_i,\ell_j,\ell\in 2\mathbb{Z} . \label{eq:app-preserved-fusion} \end{align}\tag{36}\] Equivalently, the fusion coefficients in the preserved sector are \[\begin{align} N_{\ell_i,\ell_j}^{\;\;\ell_k} = \sum_{\substack{ \ell= |\ell_i-\ell_j|\\ \mathrm{step}\;2 }}^{ \min(\ell_i+\ell_j,\;2N-\ell_i-\ell_j) } \delta_{\ell_k,\ell}, \qquad \ell_i,\ell_j,\ell_k\in 2\mathbb{Z} . \label{eq:app-SO3N-fusion} \end{align}\tag{37}\] These are exactly the fusion rules of the integer-spin subcategory of \(\mathfrak{su}(2)_N\). Hence the preserved defect category is \[\begin{align} \mathcal{C}_{\mathrm{pres}} \simeq \mathfrak{so}(3)_N . \end{align}\]
In this appendix, we collect the details of the \(\mathcal{N}=2\) superconformal algebra, and the OPE computation of cluster decomposition equation 1 in the basis of \(\mathcal{SM}^D_{4,\,6}\times\mathcal{T}_{\psi}\).
Recall that we introduce a \(\mathcal{N}=2\) basis consistent with \(\mathcal{N}=1\) superconformal algebra, \[\begin{align} G=\frac{1}{\sqrt 2}(G_++G_-)\,,\quad {\rm and}\quad G_D=\frac{1}{\sqrt 2}(G_+-G_-)\,. \end{align}\] In the \((G,\, G_D)\)-basis, we rewrite the standard \(\mathcal{N}=2\) superconformal algebra OPE as \[\begin{align} &T_{\mathcal{SM}}(z)\cdot T_{\mathcal{SM}}(w) \sim \frac{1/2}{(z-w)^4} + \frac{2T_{\mathcal{SM}}(w)}{(z-w)^2} + \frac{\partial T_{\mathcal{SM}}(w)}{z-w} \notag\\[1ex] &T_{\mathcal{SM}}(z)\cdot J_R(w) \sim \frac{J_R(w)}{(z-w)^2} + \frac{\partial J_R(w)}{z-w} \notag\\[1ex] &J_R(z)\cdot J_R(w) \sim \frac{1/3}{(z-w)^2} \notag\\[1ex] &T_{\mathcal{SM}}(z)\cdot G(w) \sim \frac{\frac{3}{2}G(w)}{(z-w)^2} + \frac{\partial G(w)}{z-w}\,,\quad T_{\mathcal{SM}}(z)\cdot G_D(w) \sim \frac{\frac{3}{2}G_D(w)}{(z-w)^2} + \frac{\partial G_D(w)}{z-w}\notag\\[1ex] &J_R(z)\cdot G(w) \sim 0\,,\quad J_R(z)\cdot G_D(w) \sim \frac{\sqrt{2} G_D(w)}{z-w}\notag\\[1ex] &G(z)\cdot G(w) \sim \frac{2/3}{(z-w)^3} + \frac{2T_{\mathcal{SM}}(w) }{z-w}\,,\quad G_D(z)\cdot G_D(w) \sim -\frac{2/3}{(z-w)^3} - \frac{2T_{\mathcal{SM}}(w) }{z-w}\,,\quad\notag\\[1ex] &G(z)\cdot G_D(w) \sim - \frac{2J_R(w)}{(z-w)^2} - \frac{ \partial J_R(w)}{z-w}\,, \label{eq:N61295OPE} \end{align}\tag{38}\] where we have put the central charge \(c=1\) of \(\mathcal{SM}_{4,\,6}^D\).
In the fermionic basis, we can spell all these spin-2 quasi-primaries as in eq.@eq:eq:N61295basis . We compute their OPEs from eq. 38 , and summarize below: \[\begin{align} &T_{UV}(z)\cdot T_{UV}(w)\sim \frac{\frac{2}{5}}{(z-w)^4}+\frac{2T_{UV}(w)}{(z-w)^2}+\frac{\partial T_{UV}(w)}{z-w}\notag\\[1ex] &T_{UV}(z)\cdot \overline{T}_{IR}(w)\sim 0\notag\\[1ex] &T_{UV}(z)\cdot X(w) \sim \frac{\frac{7}{5}X(w)}{(z-w)^2}\notag\\[1ex] &T_{UV}(z)\cdot W^{-}(w) \sim \frac{\frac{2\sqrt 3\,i}{5}J(w)}{(z-w)^3}+\frac{\frac{\sqrt 3\,i}{5}\partial J(w)+W^{-}(w)}{(z-w)^2}\notag\\[1ex] &\overline{T}_{IR}(z)\cdot \overline{T}_{IR}(w)\sim \frac{\frac{7}{20}}{(z-w)^4}+\frac{2\overline{T}_{IR}(w)}{(z-w)^2}+\frac{\partial \overline{T}_{IR}(w)}{z-w}\notag\\[1ex] &\overline{T}_{IR}(z)\cdot X(w) \sim \frac{\frac{3}{5}X(w)}{(z-w)^2}\notag\\[1ex] &\overline{T}_{IR}(z)\cdot W^{-}(w)\sim \frac{-\frac{2\sqrt 3\,i}{5}J(w)}{(z-w)^3}+\frac{-\frac{\sqrt 3\,i}{5}\partial J(w)+W^{-}(w)}{(z-w)^2}\notag\\[1ex] &X(z)\cdot X(w)\sim \frac{1}{(z-w)^4}+\frac{\frac{7}{2}T_{UV}(w)+\frac{12}{7}\overline{T}_{IR}(w)-\frac{3}{\sqrt 7}X(w)}{(z-w)^2}\notag\\[1ex] &X(z)\cdot W^{-}(w) \sim \frac{\sqrt{\frac{3}{7}}\,i J(w)}{(z-w)^3}+\frac{\sqrt{\frac{3}{7}}\frac{i}{2}\partial J(w)+\frac{1}{\sqrt{7}}W^{-}(z)}{(z-w)^2}\notag\\[1ex] &W^{-}(z)\cdot W^{-}(w) \sim \frac{1}{(z-w)^4}+\frac{\frac{5}{2}T_{UV}(w)+\frac{20}{7}\overline{T}_{IR}(w)+\frac{1}{\sqrt 7}X(w)}{(z-w)^2}\,. \end{align}\] Here we only collect the OPE terms up to \(\mathcal{O}((z-w)^{-2})\) because the \((z-w)^{-1}\) order contains spin-3 operators that are irrelevant to our discussion.
In this appendix, we provide the detailed computation of the OPE coefficients involving the phantom currents and their descendants. The relevant operators in the UV and IR theories are respectively \[\begin{align} \phi^{UV}\equiv\phi^{\mathcal{P}_N}_{[3,0]}, \qquad \phi^{IR}\equiv\phi^{\mathcal{M}_{N+1}}_{[1,3]} . \end{align}\] In the folded picture, the composite operators of interest are \[\begin{align} X=\phi_{w}^{UV} \bar\phi^{IR}, \qquad \phi_{w}^{UV}\equiv W^{(3)}_{-1}\phi^{UV} , \end{align}\] and \[\begin{align} W^-=\frac{ h^{UV}\phi^{UV}\partial\bar\phi^{IR} -h^{IR}\bar\phi^{IR}\partial\phi^{UV} }{\sqrt{-2h^{UV}h^{IR}}}, \quad J=\phi^{UV}\bar\phi^{IR} . \end{align}\] We shall first determine the relevant three-point coefficients in the \(\mathcal{P}_N\) sector, and then combine them with the corresponding minimal-model contribution.
The fundamental strategy is to lift the computation from the parafermion theory to the parent \(\mathfrak{su}(2)_N\) WZW model. Under the coset decomposition, a primary field \(\Phi_{j,m}\) in the \(\mathfrak{su}(2)_N\) theory factorizes into a parafermion primary \(\phi_{[t,s]}^{\mathcal{P}_N}\) and a free compact boson vertex operator from the \(\mathfrak{u}(1)_N\) sector. Consequently, the OPE coefficients in \(\mathcal{P}_N\) can be systematically extracted by computing the corresponding correlators in the \(\mathfrak{su}(2)_N\) WZW model, which are exactly known in terms of Wigner \(3j\)-symbols and Knizhnik-Zamolodchikov (KZ) structure constants[45], and subsequently stripping off the trivial \(\mathfrak{u}(1)_N\) free-boson contributions. Specifically, let’s first focus on the operator \[\begin{align} \phi^{UV} \equiv \phi^{\mathcal{P}_N}_{[3,0]}\,. \end{align}\] The affine \(\mathfrak{su}(2)_N\) representation decomposes as \[\begin{align} \mathcal{H}^{\mathfrak{su}(2)_N}_{t-1}=\bigoplus_{\substack{s=0,\dots,2N-1\\ t+s\;{\rm odd}}} \mathcal{H}^{\mathcal{P}_N}_{[t,s]} \otimes \mathcal{H}^{\mathfrak u(1)_N}_{s}. \end{align}\] Under this decomposition, the WZW quantum numbers are \[\begin{align} j=\frac{t-1}{2},\qquad m=\frac{s}{2}. \end{align}\] In particular, \[\begin{align} \phi^{UV} \quad\longleftrightarrow\quad \Phi_{j=1,m=0} \end{align}\] in the \(\mathfrak{su}(2)_N\) theory. Moreover, the conformal weights satisfy \[\begin{align} h^{UV}+h^{IR}=\frac{2}{N+2}+\frac{N}{N+2}=1 . \label{eq:spin-1} \end{align}\tag{39}\] We now construct the level-one descendant corresponding to \(\phi_{w}^{UV}=W^{(3)}_{-1}\phi^{UV}\). A natural basis for the relevant level-one subspace is \[\begin{align} v_+ = J^+_{-1}\Phi_{1,-1},\qquad v_0 = J^0_{-1}\Phi_{1,0},\qquad v_- = J^-_{-1}\Phi_{1,1} . \end{align}\] In the following, we suppress this common spin label and use the shorthand \[\begin{align} \Phi_m \equiv \Phi_{1,m}. \end{align}\] Thus all fields \(\Phi_m\) appearing below are understood to belong to the spin-one representation of \(\mathfrak{su}(2)_N\). We therefore write \[\begin{align} |\tilde{\phi}_{w}^{UV}\rangle=a v_+ + b v_0 + c v_- . \end{align}\] The state \(|\tilde{\phi}_{w}^{UV}\rangle\) is fixed by two conditions.
First, it should be annihilated by the positive \(\mathfrak u(1)\) mode, \[\begin{align} J^0_1|\tilde{\phi}_{w}^{UV}\rangle=0 . \end{align}\] Using \[\begin{gather} [J^+_m,J^-_n]=2J^0_{m+n}+Nm\delta_{m+n,0},\qquad [J^0_m,J^\pm_n]=\pm J^\pm_{m+n},\notag\\[0.5ex] [J^0_m,J^0_n]=\frac{N}{2}m\delta_{m+n,0}, \label{eq:A1-alg} \end{gather}\tag{40}\] this condition gives \[\begin{align} \sqrt 2\,a+\frac{N}{2}b-\sqrt 2\,c=0 . \end{align}\] Second, \(|\tilde{\phi}_{w}^{UV}\rangle\) should be quasi-primary in the \(\mathcal{P}_N\) theory, \[\begin{align} L^{\mathcal{P}_N}_1|\tilde{\phi}_{w}^{UV}\rangle=0 . \end{align}\] This gives \[\begin{align} a+c=0 . \end{align}\] Solving these two constraints, we may choose the representative \[\begin{align} |\tilde{\phi}_{w}^{UV}\rangle=-J^+_{-1}\Phi_{-1}+J^-_{-1}\Phi_{1}+\frac{4\sqrt 2}{N}J^0_{-1}\Phi_{0} . \end{align}\] After normalization, the state is \[\begin{align} |\phi_{w}^{UV}\rangle=\sqrt{\frac{N}{2(N-2)(N+4)}}\left(-J^+_{-1}\Phi_{-1}+J^-_{-1}\Phi_{1}+\frac{4\sqrt 2}{N}J^0_{-1}\Phi_{0}\right). \label{eq:D-state} \end{align}\tag{41}\] We next evaluate the required three-point coefficients. The chiral three-point function in the \(\mathfrak{su}(2)_N\) WZW theory takes the form [35], [45] \[\begin{align} \big\langle \Phi_{j_1,m_1}(z_1) \Phi_{j_2,m_2}(z_2) \Phi_{j_3,m_3}(z_3) \big\rangle &= C \begin{pmatrix} j_1 & m_1\\ j_2 & m_2\\ j_3 & m_3 \end{pmatrix} z_{12}^{D_{j_3}-D_{j_1}-D_{j_2}}z_{13}^{D_{j_2}-D_{j_3}-D_{j_1}}z_{23}^{D_{j_1}-D_{j_2}-D_{j_3}} , \end{align}\] where \[\begin{align} C \begin{pmatrix} j_1 & m_1\\ j_2 & m_2\\ j_3 & m_3 \end{pmatrix} = K(j_1,j_2,j_3) \begin{pmatrix} j_1 & j_2 & j_3\\ m_1 & m_2 & m_3 \end{pmatrix}. \end{align}\] Here the second factor is the Wigner \(3j\)-symbol, while \(K(j_1,j_2,j_3)\) contains the remaining \(N\)-dependent structure constant. Since only the case \(j_1=j_2=j_3=1\) is needed below, we denote \[\begin{align} K_N \equiv K(1,1,1). \end{align}\] The affine currents are expanded as \[\begin{align} J^a(z)=\sum_{n\in\mathbb{Z}}J^a_n z^{-n-1}, \end{align}\] Equivalently, if the current is expanded in a local coordinate around an insertion point \(z_i\), the modes are defined by \[\begin{align} J^a_n\mathcal{O}_i(z_i)=\oint_{z_i}\frac{dz}{2\pi i}(z-z_i)^nJ^a(z)\mathcal{O}_i(z_i). \label{eq:local-insertion} \end{align}\tag{42}\] We fix the three insertion points by global conformal symmetry as \[\begin{align} z_1=\infty,\qquad z_2=1,\qquad z_3=0, \end{align}\] and use the shorthand \[\begin{align} \big\langle \mathcal{O}_i(\infty)\mathcal{O}_j(1)\mathcal{O}_k(0) \big\rangle:=\lim_{z_1\to\infty}z_1^{2h_1}\big\langle\mathcal{O}_i(z_1)\mathcal{O}_j(1)\mathcal{O}_k(0)\big\rangle . \end{align}\] In the applications below, the fields are affine primaries \(\Phi_m\) and their level-one descendants \(v_i\). Thus the positive modes truncate as \[\begin{align} J^a_{n\geq2}v_i=0,\quad J^a_{n\geq1}\Phi_m=0. \label{eq:state-condition} \end{align}\tag{43}\] Consequently, in the contour manipulations below only the \(J^a_0\) and \(J^a_1\) terms can contribute.
Since \(J^a(z)\) is a meromorphic current, a contour encircling one insertion can be deformed to contours encircling the other insertions. We first consider \[\begin{align} \big\langle\mathcal{O}_i(\infty)\mathcal{O}_j(1)(J^a_{-1}\mathcal{O}_k)(0)\big\rangle. \end{align}\] By the local definition 42 , \[\begin{align} J^a_{-1}\mathcal{O}_k(0)=\oint_0\frac{dz}{2\pi i}\frac{1}{z}J^a(z)\mathcal{O}_k(0). \end{align}\] Hence \[\begin{align} \big\langle\mathcal{O}_i(\infty)\mathcal{O}_j(1)(J^a_{-1}\mathcal{O}_k)(0)\big\rangle=\oint_0\frac{dz}{2\pi i}\frac{1}{z}\big\langle\mathcal{O}_i(\infty)\mathcal{O}_j(1)J^a(z)\mathcal{O}_k(0)\big\rangle. \end{align}\] We now deform the contour around \(z=0\) to contours around \(z=1\) and \(z=\infty\). Near \(z=1\), \[\begin{align} \frac{1}{z}=\frac{1}{1+(z-1)}=1-(z-1)+(z-1)^2-\cdots . \end{align}\] Therefore, \[\begin{align} \oint_{1}\frac{dz}{2\pi i}\, \frac{1}{z}J^a(z) = J^a_0-J^a_1+J^a_2-\cdots . \end{align}\] After contour deformation, this contribution comes with an overall minus sign. Using 43 , it reduces to \[\begin{align} -J^a_0+J^a_1 . \end{align}\] For the contribution from infinity, we introduce the local coordinate \[\begin{align} z=\frac{1}{w}. \end{align}\] Since \(J^a\) is a weight-one current, \[\begin{align} J^a(z)\,dz=J^a(w)\,dw . \end{align}\] Thus \[\begin{align} \oint_{\infty}\frac{dz}{2\pi i}\, \frac{1}{z}J^a(z) =-\oint_{w=0}\frac{dw}{2\pi i}\,w J^a(w) =-J^a_1 . \end{align}\] The minus sign from this change of coordinate is canceled by the minus sign from the contour deformation. We obtain \[\begin{align} \big\langle\mathcal{O}_i(\infty)\mathcal{O}_j(1)(J^a_{-1}\mathcal{O}_k)(0)\big\rangle &=\big\langle(J^a_1\mathcal{O}_i)(\infty)\mathcal{O}_j(1)\mathcal{O}_k(0)\big\rangle -\big\langle\mathcal{O}_i(\infty)(J^a_0\mathcal{O}_j)(1)\mathcal{O}_k(0)\big\rangle\notag\\[1ex] &\quad+\big\langle\mathcal{O}_i(\infty)(J^a_1\mathcal{O}_j)(1)\mathcal{O}_k(0)\big\rangle . \label{eq:Ward-identities951} \end{align}\tag{44}\] Next, we consider \[\begin{align} \big\langle\mathcal{O}_i(\infty)(J^a_{-1}\mathcal{O}_j)(1)\mathcal{O}_k(0)\big\rangle=\oint_1\frac{dz}{2\pi i}\frac{1}{z-1}\big\langle\mathcal{O}_i(\infty)J^a(z)\mathcal{O}_j(1)\mathcal{O}_k(0)\big\rangle. \end{align}\] Deforming this contour to \(z=0\) and \(z=\infty\), we first expand the kernel near \(z=0\): \[\begin{align} \frac{1}{z-1}=-\frac{1}{1-z}=-1-z-z^2-\cdots . \end{align}\] Hence \[\begin{align} \oint_{0}\frac{dz}{2\pi i}\, \frac{1}{z-1}J^a(z)=-J^a_0-J^a_1-\cdots . \end{align}\] The contour deformation gives another minus sign, and therefore this part contributes \[\begin{align} J^a_0+J^a_1 . \end{align}\] For the contour at infinity, \[\begin{align} \oint_\infty\frac{dz}{2\pi i}\frac{1}{z-1}J^a(z) &=-\oint_{w=0}\frac{dw}{2\pi i}\frac{w}{1-w}J^a(w)\notag\\[1ex] &=-J^a_1-J^a_2-\cdots, \end{align}\] Again using 43 , only the \(J^a_1\) term survives after the contour deformation. Thus \[\begin{align} \big\langle\mathcal{O}_i(\infty)(J^a_{-1}\mathcal{O}_j)(1)\mathcal{O}_k(0)\big\rangle &=\big\langle(J^a_1\mathcal{O}_i)(\infty)\mathcal{O}_j(1)\mathcal{O}_k(0)\big\rangle +\big\langle\mathcal{O}_i(\infty)\mathcal{O}_j(1)(J^a_0\mathcal{O}_k)(0)\big\rangle\notag\\[1ex] &\quad+\big\langle\mathcal{O}_i(\infty)\mathcal{O}_j(1)(J^a_1\mathcal{O}_k)(0)\big\rangle . \label{eq:Ward-identities952} \end{align}\tag{45}\] Finally, we consider the function at infinity. In the local coordinate \(w=1/z\), the mode \(J^a_{-1}\) at infinity is represented by the kernel \(w^{-1}=z\). Therefore the relevant contour can be written as a contour at infinity with kernel \(zJ^a(z)\). Deforming it to the finite insertions gives \[\begin{align} \oint_{\infty}\frac{dz}{2\pi i}zJ^a(z)=\oint_{0}\frac{dz}{2\pi i}zJ^a(z)=J^a_1 \end{align}\] and, near \(z=1\), \[\begin{align} \oint_{\infty}\frac{dz}{2\pi i}zJ^a(z)=\oint_{1}\frac{dz}{2\pi i}(1+z-1)J^a(z)=J^a_0+J^a_1, \end{align}\] Consequently, \[\begin{align} \big\langle(J^a_{-1}\mathcal{O}_i)(\infty)\mathcal{O}_j(1)\mathcal{O}_k(0)\big\rangle &=\big\langle\mathcal{O}_i(\infty)\mathcal{O}_j(1)(J^a_1\mathcal{O}_k)(0)\big\rangle +\big\langle\mathcal{O}_i(\infty)(J^a_0\mathcal{O}_j)(1)\mathcal{O}_k(0)\big\rangle\notag\\[1ex] &\quad+\big\langle\mathcal{O}_i(\infty)(J^a_1\mathcal{O}_j)(1)\mathcal{O}_k(0)\big\rangle . \label{eq:Ward-identities953} \end{align}\tag{46}\] We first compute \(C_{\phi^{UV}\phi^{UV}\phi^{UV}_w}\). Since \[\begin{align} J^0_0\Phi^{UV}=J^0_0\Phi_0=0, \end{align}\] the \(J^0_{-1}\Phi_0\) term in 41 does not contribute. Using the Ward identities 44 , we find \[\begin{align} \big\langle\Phi^{UV}(\infty)\Phi^{UV}(1)(J^+_{-1}\Phi_{-1})(0)\big\rangle &=-\big\langle\Phi_0(\infty)(J^+_0\Phi_0)(1)\Phi_{-1}(0)\big\rangle\notag\\[1ex] &=-\sqrt 2\,K_N \begin{pmatrix} 1 & 1 & 1\\ 0 & 1 & -1 \end{pmatrix}\notag\\[1ex] &=-\frac{1}{\sqrt 3}K_N . \label{eq:three-point-1} \end{align}\tag{47}\] Similarly, \[\begin{align} \big\langle\Phi^{UV}(\infty)\Phi^{UV}(1)(J^-_{-1}\Phi_{1})(0)\big\rangle=\frac{1}{\sqrt 3}K_N . \label{eq:three-point-2} \end{align}\tag{48}\] Therefore, \[\begin{align} C_{\phi^{UV}\phi^{UV}\phi^{UV}_w}=\sqrt{\frac{N}{2(N-2)(N+4)}}\frac{2}{\sqrt 3}K_N . \label{eq:CLLw} \end{align}\tag{49}\] The coefficient \(C_{\phi^{UV}_w\phi^{UV}_w\phi^{UV}_w}\) can be computed in the same manner. Using the definition of \(\phi^{UV}_w\) in terms of \(v_\pm,\,v_0\), one finds \[\begin{align} C_{\phi^{UV}_w\phi^{UV}_w\phi^{UV}_w}=\left(\frac{N}{2(N-2)(N+4)}\right)^{\frac{3}{2}}(-C_{v_+v_+v_+}+C_{v_+v_+v_-}+\cdots) \end{align}\] where the ellipsis denotes the remaining terms obtained by expanding the three factors of \(\phi_{w}^{UV}\). By repeatedly applying the affine algebra 40 together with the Ward identities 44 , 45 , and 46 , each term can be reduced to a finite linear combination of the basic three-point functions of the form 47 and 48 .
As an illustration, let us consider the term \(C_{v_+v_+v_-}\). Throughout the following computation we fix the insertion points to be \(z_1=\infty\), \(z_2=1\), and \(z_3=0\), and suppress them from the notation. We obtain \[\begin{align} C_{v_+v_+v_-}&=\big\langle J^+_{-1}\Phi_{-1},v_+,v_-\big\rangle\notag\\[1ex] &=\big\langle\Phi_{-1},J^+_0v_+,v_-\big\rangle+\big\langle\Phi_{-1},J^+_1v_+,v_-\big\rangle+\big\langle\Phi_{-1},v_+,J^+_1v_-\big\rangle\notag\\[1ex] &=\sqrt{2}\big\langle\Phi_{-1},J^+_{-1}\Phi_0,J^-_{-1}\Phi_1\big\rangle+(N+2)\big\langle\Phi_{-1},J^+_{-1}\Phi_{-1},\Phi_1\big\rangle\notag\\[1ex] &=\sqrt{2}\bigl( -\big\langle\Phi_{-1},J^-_0J^+_{-1}\Phi_0,\Phi_1\big\rangle+\big\langle\Phi_{-1},J^-_1J^+_{-1}\Phi_0,\Phi_1\big\rangle \bigr)+(N+2)\big\langle\Phi_{-1},\Phi_{-1},J^+_0\Phi_1\big\rangle\notag\\[1ex] &=\sqrt{2}\bigl( 2\big\langle\Phi_{-1},J^0_{-1}\Phi_0,\Phi_1\big\rangle-\sqrt{2}\big\langle\Phi_{-1},J^+_{-1}\Phi_{-1},\Phi_1\big\rangle+N\big\langle\Phi_{-1},\Phi_0,\Phi_1\big\rangle \bigr)\notag\\[1ex] &=\sqrt{2}(N+2)\big\langle\Phi_{-1},\Phi_0,\Phi_1\big\rangle\notag\\[1ex] &=\frac{N+2}{\sqrt{3}}K_N \end{align}\] Carrying out the same reduction for all the remaining terms in the expansion, we finally obtain \[\begin{align} C_{\phi^{UV}_w\phi^{UV}_w\phi^{UV}_w}=\sqrt{\frac{N}{2(N-2)(N+4)}}\frac{\sqrt 6\,(N-4)}{2(N-2)}K_N . \label{eq:Cwww} \end{align}\tag{50}\] It remains to include the minimal-model factor. The field \(\phi^{IR}\) also corresponds to level-one \(\mathfrak{su}(2)_N\) sector. With the same normalization convention for \(K_N\), its three-point coefficient is \[\begin{align} C_{\phi^{IR}\phi^{IR}\phi^{IR}}=\frac{2\sqrt 3\,(N-2)}{\sqrt{N(N-1)}\,K_N}. \label{eq:CRRR} \end{align}\tag{51}\] Combining 50 and 51 , and using \(X=\phi_{w}^{UV}\phi^{IR}\), we obtain \[\begin{align} C_{XXX} =C_{\phi^{UV}_w\phi^{UV}_w\phi^{UV}_w}C_{\phi^{IR}\phi^{IR}\phi^{IR}}=\frac{3\sqrt2(N-4)}{\sqrt{(N-1)(N-2)(N+4)}}. \label{eq:CXXX} \end{align}\tag{52}\] The coefficient \(C_{W^-W^-X}\) factorizes in the same way into the left and right sectors. From the definition of \(W^-\), the relevant relative coefficient is \[\frac{h^{UV}h^{IR}}{2h^{UV}h^{IR}}=\frac{1}{2}.\] Therefore, \[\begin{align} C_{W^-W^-X} =\frac{1}{2}C_{\phi^{UV}\phi^{UV}\phi^{UV}_w}C_{\phi^{IR}\phi^{IR}\phi^{IR}}\notag=\sqrt{\frac{2(N-2)}{(N-1)(N+4)}} . \label{eq:CWWX} \end{align}\tag{53}\] Finally, the OPE between \(X\) and \(W^-\) generates \(J\). To determine the corresponding coefficient, we consider \(C_{XW^-J}\). The relevant factor is \[-i\sqrt{\frac{h^{IR}}{2h^{UV}}}=-i\sqrt{\left(\frac{N}{2+N}\right)\left(\frac{2+N}{4}\right)}=-i\sqrt{\frac{N}{4}} ,\] and hence \[\begin{align} C_{XW^-J} =-i\sqrt{\frac{N}{4}}C_{\phi^{UV}_w\phi^{UV}\phi^{UV}}C_{\phi^{IR}\phi^{IR}\phi^{IR}}=i\sqrt{\frac{2N(N-2)}{(N-1)(N+4)}} . \label{eq:CXWJ} \end{align}\tag{54}\]
As we only consider unitary models throughout the paper, the notation \(\mathcal{M}_{N+1}\equiv\mathcal{M}_{N+2,\,N+1}\) is used for brevity↩︎