what follows. We therefore hope to provide a bird's-eye view of the different directions that research into conformal field theories (CFTs) with boundaries or defects is taking. Quantum field theory (QFT) lies at the heart of much of modern theoretical physics: it describes systems in particle physics, condensed matter physics, and even quantum gravity, via holographic duality. Combined with the machinery of the renormalization group (RG), one can in principle systematically study an enormous variety of phenomena at different length scales in QFT-and therefore in nature. CFTs occupy privileged places in the space of QFTs-they lie at the endpoints of RG flows. Therefore, they often characterize the ultraviolet (UV) and infrared (IR) limits of QFTs. Moreover, CFTs describe critical phenomena and the worldsheet theory in string theory. They also provide important testing grounds for integrability, duality, and other general phenomena in QFT. CFTs are powerful because they are highly symmetric. In particular, CFTs are scale-invariant by definition, and so all correlation lengths are infinite. CFTs are also invariant under translations, rotations, boosts, and inversions. These symmetries constrain correlation functions, sometimes completely, providing a powerful non-perturbative approach to many aspects of QFT. However, no real-world system has infinite size-boundary effects are always important. Moreover, all real-world systems involve impurities, differently-ordered regions separated by domain walls, and other types of defects that break translational and rotational symmetries to subgroups 2. In short, the application of CFT to the real world necessarily entails studying boundary CFTs (bCFTs) and defect CFTs (dCFTs). The reduced symmetry of bCFTs and dCFTs compared to CFTs loosens spectral constraints, making correlation functions richer and more intricate. For example, in a bCFT, scalar operators can have non-zero one-point functions, which are generally forbidden in a CFT. However, bCFTs and dCFTs often retain enough symmetry to provide calculable non-perturbative information about many systems. For example, bCFT provides a solution to the single-impurity Kondo problem [1] and a fully non-perturbative definition of D-branes and other spacetime defects in string theory [2]. Remarkably, bCFT and dCFT can also provide insight into RG flows beyond the critical endpoints. Indeed, consider a CFT with a relevant deformation that triggers an RG flow to another CFT. Now, imagine integrating that relevant deformation over half the spacetime. Then, in the IR, the result is an interface between the UV and IR CFTs, called an RG interface or RG domain wall [11]. Such a construction, though clearly sacrificing some of the spacetime symmetry, is potentially very powerful: the problem of classifying RG flows between CFTs maps onto the problem of classifying defects between CFTs-which should be simpler, because much of the machinery of CFT can be brought to bear. Similarly, if the relevant deformation produces a mass gap, then the ...

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 halfinteger 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 [1]. We then act 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.

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We consider a class of conformal defects in Virasoro minimal models that have been defined as fixed points of the renormalisation group and calculate the leading contribution to the reflection coefficient for these defects. This requires several structure constants of the operator algebra of the defect fields, for which we present a derivation in detail. We compare our results with our recent work on conformal defects in the tricritical Ising model.