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arXiv:2109.02672

Symmetry enhancement in RCFT

Ken Kikuchi

Published Sep 6, 2021• 13 citations
arXivHTMLPDF

Abstract

We propose when and why symmetry enhancements happen in massless renormalization group (RG) flows to two-dimensional rational conformal field theories (RCFTs). We test our proposal against known RG flows from unitary minimal models. We also suggest which sign of the relevant coupling triggers the massless RG flow. The other sign triggers massive RG flows to topological quantum field theories (TQFTs). We comment on their ground state degeneracies.

DOI
10.48550/arxiv.2109.02672

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We propose when and why symmetry enhancements happen in massless renormalization group (RG) flows to two-dimensional rational conformal field theories (RCFTs). We test our proposal against known RG flows from unitary minimal models. We also suggest which sign of the relevant coupling triggers the massless RG flow. The other sign triggers massive RG flows to topological quantum field theories (TQFTs). We comment on their ground state degeneracies.
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A bstract We consider topological defect lines (TDLs) in two-dimensional conformal field theories. Generalizing and encompassing both global symmetries and Verlinde lines, TDLs together with their attached defect operators provide models of fusion categories without braiding. We study the crossing relations of TDLs, discuss their relation to the ’t Hooft anomaly, and use them to constrain renormalization group flows to either conformal critical points or topological quantum field theories (TQFTs). We show that if certain non-invertible TDLs are preserved along a RG flow, then the vacuum cannot be a non-degenerate gapped state. For various massive flows, we determine the infrared TQFTs completely from the consideration of TDLs together with modular invariance.
In this Letter we continue the investigation of RG flows between Virasoro minimal models of two-dimensional conformal field theories that are protected by noninvertible symmetries. RG flows leaving unbroken a subcategory of noninvertible symmetries are associated with anomaly matching conditions that we employ systematically to map the space of flows between minimal models beyond the Z_{2}-symmetric proposed recently in the literature. We introduce a family of nonlinear integral equations that appear to encode the exact finite-size, ground-state energies of these flows, including nonintegrable cases, such as the recently proposed M(kq+I,q)→M(kq-I,q). Our family of NLIEs encompasses and generalizes the integrable flows known in the literature: ϕ_{(1,3)}, ϕ_{(1,5)}, ϕ_{(1,2)} and ϕ_{(2,1)}. This work uncovers a new interplay between exact solvability and noninvertible symmetries. Furthermore, our nonperturbative description provides a nontrivial test for all the flows conjectured by anomaly matching conditions, but so far not observed by other means.
We derive universal constraints on (1+1)d(1+1)d(1+1)d rational conformal field theories (CFTs) that can arise as transitions between topological theories protected by a global symmetry. The deformation away from criticality to the trivially gapped phase is driven by a symmetry preserving relevant deformation and under renormalization group flow defines a conformal boundary condition of the CFT. When a CFT can make a transition between distinct trivially gapped phases the spectrum of the CFT quantized on an interval with the associated boundary conditions has degeneracies at each energy level. Using techniques from boundary CFT and modular invariance, we derive universal inequalities on all such degeneracies, including those of the ground state. This establishes a symmetry enriched ccc-theorem, effectively a lower bound on the central charge which is strictly positive, for this class of CFTs and symmetry protected flows. We illustrate our results for the case of flows protected by SU(M)/ZMSU(M)/\mathbb{Z}_{M}SU(M)/ZM​ symmetry. In this case, all SPT transitions can arise from the WZW model SU(M)1,SU(M)_{1},SU(M)1​, and we develop a dictionary between conformal boundary conditions and relevant operators.
We study the classification of the gapped phases or massive renormalization group (RG) flows dual to the massless RG flows under changing the sign of the coupling constants. Whereas our formalism is based on combining Higgs- or Nambu-Goldstone-type arguments with Cardy's smeared boundary conformal field theories (SBCFTs), several puzzling structures arise. More specifically, the established Higgs or Nambu-Goldstone type arguments on the duality imply that the natural basis for the gapped states should be constructed from a set of smeared Ishibashi states, which are unphysical in boundary critical phenomena. Hence, the module of the gapped phases can be outside of that of boundary critical phenomena, whereas one can still calculate characterizing quantities by applying SBCFTs to the models. For example, we demonstrate that the massive RG flow dual to the massless RG flow from the tricritical Ising model to the Ising model, one of the simplest massless RG flows, has this unusual structure. This can be regarded as a quantum field-theoretic analogue of order-disorder coexistence in lattice models. More generally, the resultant gapped phases usually spontaneously break non-group-like symmetry (or noninvertible symmetry). Our work provides systematic quantum field theoretic descriptions of such unusual phases with spontaneous symmetry breaking of non-group-like (or noninvertible) symmetries.