Defects in the Tri-critical Ising model

TL;DR

The work develops a folding-based, interface-enabled route to construct new, non-topological, non-factorising defects in the tri-critical Ising model by relating TCIM defects to boundary conditions in the folded SVIR_3^⊗2 theory and employing TCIM–SVIR interfaces. By classifying NS defects in SVIR_3 and exploiting embeddings into sVir_{10} (c=7/5), the authors identify both known and new defects, including two novel D^± constructions, and organize them within SW(3/2) and SW(10) extended algebras. The approach provides a consistent framework that yields TCIM defects through I D’ I† with integer overlapping coefficients, and it suggests potential matches or contrasts with Gang–Yamaguchi’s earlier proposals, while highlighting the role of Ramond sectors for a complete GSO-projected defect set. The results advance the understanding of TCIM defect spectra and demonstrate a concrete, calculable method to generate and classify defects via folded fermionic theories and interfaces, with implications for conformal boundary state techniques and extended algebras. Overall, the paper presents a principled, constructive strategy to uncover and characterize new TCIM defects, enriching the landscape of conformal defect theory.

Abstract

We consider two different conformal field theories with central charge c=7/10. One is the diagonal invariant minimal model in which all fields have integer spins; the other is the local fermionic theory with superconformal symmetry in which fields can have half-integer spin. We construct new conformal (but not topological or factorised) defects in the minimal model. We do this by first constructing defects in the fermionic model as boundary conditions in a fermionic theory of central charge c=7/5, using the folding trick as first proposed by Gang and Yamaguchi. We then acting on these with interface defects to find the new conformal defects. As part of the construction, we find the topological defects in the fermionic theory and the interfaces between the fermionic theory and the minimal model. We also consider the simpler case of defects in the theory of a single free fermion and interface defects between the Ising model and a single fermion as a prelude to calculations in the tri-critical Ising model.

Figures (5)

• Figure 1: The equivalence between a boundary condition $B$ in the folded fermionic model, a defect $D'$ in the fermionic model and a defect $D$ in the TCIM. $I$ is an interface defect.
• Figure 2: The trace over the space of fields on which the $D_{-+}$ defect can end is related by a modular transformation to the trace with the defect inserted
• Figure 3: The $\psi$ field in the free-fermion model is realised as a defect-creation operator in the Ising model. The associated space of three-defect junctions is one-dimensional
• Figure 4: The Dynkin diagrams of $D_6$ and $E_6$ showing the bi-colouration and the map $r$
• Figure 5: Different boundary conditions on SVIR$_3{}^{\otimes 2}$ result in different geometrical set-ups for $\mathrm{SVIR}_3$.