Braided categories of endomorphisms as invariants for local quantum field theories (Q2413038)

This paper is a contribution to the study of completely rational chiral conformal field theories in two dimensions within the setting of algebraic QFT. Some of the paper's ideas may be of interest for algebraic QFT more generally. Motivated by the difficulties of defining a QFT through construction of the vacuum representation of a local net of observables, the authors investigate weaker structures which encode relevant information about a two-dimensional chiral CFT. In particular, they demonstrate that the \textit{braided action} of the braided monoidal DHR category [\textit{S. Doplicher}, \textit{R. Haag} and \textit{J. E. Roberts}, ``Local observables and particle statistics I'', Commun. Math. Phys. 23, No. 3, 199--230 (1971; \url{doi:10.1007/BF01877742}); \textit{K. Fredenhagen} et al., Rev. Math. Phys. Spec. Issue, 113--157 (1992; Zbl 0774.46041)] on a local algebra \(\mathcal{A}(I_0)\) for a fixed interval \(I_0\) is useful in this regard. The concept of \textit{braided action} is introduced in this paper, and refers to keeping track of how the braiding isomorphisms of the DHR category are represented as morphisms of the monoidal category \(\mathrm{End}(\mathcal{A}(I_0))\). The braided action encodes in particular which pairs of DHR endomorphisms braid trivially. This leads to the definition of \textit{abstract point} in Section 6, of which the points of \(I_0\) are particular examples. These abstract points are studied in the later sections. Together with comparability relations between abstract points introduced in Section 9, they are employed in order to prove that the braided action is enough to uniquely determine every prime conformal net, under additional assumptions whose status seems to remain unclear (Proposition 10.1). Here, the notion of \textit{prime conformal net} is introduced in Section 8, characterizing those completely rational chiral CFTs which cannot be decomposed into simpler pieces. On the other hand, determining \textit{which} braided actions of unitary modular tensor categories on the hyperfinite \(III_1\) factor arise in this way is mentioned as a completely open problem. Besides the concept of prime conformal net, also the duality relations derived in Sections 4 and 5 between (local or global) subalgebras and full subcategories of the DHR category may be of independent interest for the study of chiral CFTs.