The presence of boundaries in quantum theory is ubiquitous and central to many parts of the theory like, e.g., in condensed matter or field theory. A precise mathematical description of a boundary and its effects to the theory is a fundamental question in mathematical physics. E.g., Boundaries have been studied intensively in the setting of Euclidean quantum field theory in several articles by Fuchs et al. Due to the lack of a Wick rotation in the presence of a boundary, the theory has to be developed on its own in the context of relativistic quantum field theory.
The article under review describes in detail and with full mathematical precision boundaries and its effects in the frame of two-dimensional conformal quantum field theories. In this context, the boundary is a timelike plane of co-dimension 1 in Minkowski space (typically, the \(x=0\) plane). In particular, the authors consider so-called phase boundaries which share the same stress-energy tensor on both sides, while the bulk field content may be different on each side. Using the language of algebraic quantum field theory they focus on how the presence of boundaries intertwines with the principle of Einstein’s causality. Mathematically, the authors base their analysis on modular C*-tensor categories based on the Doplicher-Haag-Roberts category of chiral superselection sectors. Moreover, the article contains an analysis of defects and a classification of boundary conditions in terms of central projections associated with non-local extensions of the local theory. The article contains a long and informative introduction which also relates the results presented with alternative approaches.