Gauging or extending bulk and boundary conformal field theories: Application to bulk and domain wall problem in topological matter and their descriptions by (mock) modular covariant

TL;DR

The paper tackles the problem of relating RG flows in 1+1D CFTs to the bulk topological orders in 2+1D by employing gauging (group extensions) and domain walls within smeared BCFTs and extended BCFT/CFT frameworks.A core methodological advance is the construction of Z_N extended BCFTs, anomaly-free vs anomalous gauging, and a universal modular-partition-function formalism that captures massive and massless RG flows via folding and defect/interface techniques.Key contributions include a mass-condensation criterion tied to noninvertible symmetries, a new class of nonchiral edge CFTs (OECFT) for coupled topological orders, and a systematic way to analyze gapped/charged domain walls and their anyon transport phenomenology.The results unify bulk TO classifications with BCFT/CCFT data using simple-current extensions and modular invariants, offering a practical framework for both theoretical exploration and numerical/numerical-lattice studies of topological phases.

Abstract

We study gauging operations (or group extensions) in (smeared) boundary conformal field theories (BCFTs) and bulk conformal field theories, and their applications to various phenomena in topologically ordered systems. We apply the resultant theories to the correspondence between the renormalization group (RG) flow of CFTs and the classification of topological quantum field theories in the testable information of general classes of partition functions. One can obtain the bulk topological properties of $2+1$ dimensional topological ordered phase corresponding to the massive RG flow of $1+1$ dimensional systems, or smeared BCFT. We present an obstruction of mass condensation for smeared BCFT analogous to the Lieb-Shultz-Mattis theorem for noninvertible symmetry. Related to the bulk topological degeneracies in $2+1$ dimensions and quantum phases in $1+1$ dimensions, we construct a new series of BCFT. We also investigate the implications of the massless RG flow of $1+1$ dimensional CFT to $2+1$ dimensional topological order, which corresponds to the earlier proposal by L. Kong and H. Zheng in [Nucl. Phys. B 966 (2021), 115384], arXiv:1912.01760, closely related to the integer-spin simple current by Schellekens and Gato-Rivera. We study the properties of the product of two CFTs connected by the two kinds of massless flows. The (mock) modular covariants appearing in the analysis seem to contain new ones. By applying the folding trick to the coupled model, we provide a general method to solve the gapped and charged domain wall. One can obtain the general phenomenology of the transportation of anyons through the domain wall. Our work gives a unified direction for the future theoretical and numerical studies of the topological phase based on the established data of classifications of conformal field theories or modular invariants.

Figures (11)

• Figure 1: Summary of the relationship between RG flow in CFTs and the notions in TQFTs. The upper figure shows $D$-dimensional TQFT with two boundaries. A FQH system in cylinder geometry is a typical example. The massive RG flow of CFTs or the (extended) smeared BCFTsCardy:2017ufe in this work determine the structure of bulk topological order with a close connection to the entanglement entropyLi_2008Qi_2012Lencses:2018paaDas:2015ohaLou:2019heg. In TQFT, the massive RG flow in CFTs corresponds to the gluing operation of the two boundaries. On the other hand, the massless RG flow of CFTs implies the (Witt) equivalence relation between group extended TQFTs (or extended SFCsBischoff:2019jhodavydov2021braidedpicardgroupsgraded). Applying the interface or domain wall known as a gapped or symmetry-preserving domain wallKaidi:2021gbs in TQFT corresponds to the massless RG flow. By analyzing $BCFT_{D}$ with $D\ge 3$ as an ancillary CFT Nishioka:2022ook, a similar phenomenology will be true in general. We also note that the existing Witt equivalence for MTC (not SFC) seems too strong when applying to realistic settingsKawahigashi:2015lxaKawahigashi:2021hdsMoller:2024plbMoller:2024xtt (we stress that this does not mean their arguments are wrong. This implies the significance of extension for application to more realistic settings). One needs to introduce (or relax) the equivalence relation to the gauged bulk or chiral CFT.
• Figure 2: Folding trick applied to coupled models. By applying the folding trickWong:1994np, one can obtain the correspondence between a coupled model ($CFT_{D}\otimes CFT'_{D}$) and anyon transformation law of two theories($D_{RG}: CFT_{D}\rightarrow \overline{CFT'_{D}}$). Similar arguments can be seen in Kaidi:2021gbsZhao:2023wtg, for example. Because of the folding, the orientation or chirality of one of the two bulk CFTs is reversed. The coupled models in this work are the multicomponent or multilayered TOs. The mapping $D_{\text{RG}}$ corresponding to the symmetry-preserving domain wallKaidi:2021gbs or RG domain wallBrunner:2007urGaiotto:2012np classifies the TOs, possibly including FQHEs with nonchiral anyon. By shrinking one of the two systems, $CFT'_{D}$ in the righthand side of the figure, one can obtain the mismatch of bulk theory $CFT_{D}$ and boundary theory $CFT'_{D}$. This explains recent controversies on the thermal Hall conductance in experimental settingsMross_2018Wang2017TopologicalOF.
• Figure 3: Mass-condensation or ordering in BCFT and RG. In a CFT without boundary, the vacuum expectation value of a bulk primary field $\Phi_{\alpha}$ should vanish. Under the operator-state correspondence, this vanishing corresponds to the vanishing of the expectation value of the LSM twist operator $U_{\alpha}$ corresponding to the many-body elementary excitation. Under RG, the ground state expectation value of the LSM operator becomes finite and this is a mass-condensation in the QFT side. The same condensation can be mimicked by introducing boundary to the same CFT, and this represents (massive) QFT/BCFT correspondence summarized by CardyCardy:2017ufe.
• Figure 4: The connection between symmetry and defect under modular $S$ transformation. The bondary states $|\beta\rangle\rangle$ can also be considered as defect condensation $\mathcal{Q}_{\beta}|0\rangle\rangle$Graham:2003ncFukusumi:2020irh. Hence, in a large class of models, the combination of defects and symmetry governs the RG to the gapped phase.
• Figure 5: Existence of a disordered state and its implication to an anomaly-free model. If we assume an $Z_{N}$ symmetric anomaly-free theory flows to a nondegenerate disordered phase labelled by $a$, the anomaly freeness imposes the existence of zero modes $(a,p)$. These zero modes only appear when applying the $Z_{N}$ extension to the resultant gapped system and are absent in the original spin or bosonic system.