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Operator Algebras and Conformal Field Theory

Final Report Summary - OACFT (Operator Algebras and Conformal Field Theory)

Stand alone description of the project and its outcomes
1. Structure of superconformal nets and classification of the discrete series [Carpi, Kawahigashi, Longo]. We study the general structure of Fermi conformal nets of von Neumann algebras on S1, in particular a Jones index can be associated with a representation. We classify the superconformal nets with central charge less than 3/2. The Operator Algebraic description of SUSY in CFT is new.
2. Construction of a spectral triple associated with representations in CFT [Carpi, Hiller, Kawahigashi, Longo]. We construct infinite dimensional spectral triples associated with representations of the super-Virasoro algebra. For the first time, Connes Noncommutative Geometry enters naturally in the Conformal Net framework. This work is highly interdisciplinary.
3. Study of noncommutative Geometrical structures in CFT [Carpi, Hillier, Kawahigashi, Longo, Xu]. In this work we have made a Noncommutative Geometrical analysis of the supersymmetric representations for the N=2 SUSY conformal net. In particular we have shown the important role played by a Spectral Flow. One important outcome is the description/clarification of the characters of the N=2 Super-Virasoro algebra. The Noncommutative Geometry that emerges furnish a new and unconventional description of certain physical representations.
4. Relations between Local Conformal Nets and Vertex Algebras [Carpi, Longo, Weiner]. We have sufficient criteria to pass from a Vertex Algebra to a Conformal Net, yet this work will need further investigation, partly due to the technical difficulties of the project, partly because the team has focused on recently emerged interesting tasks.
5. Analysis of the internal structure of nets and subnets.
5A. Jones index of subnet and braiding [Carpi, Kawahigashi, Longo]. Restrictions on possible index values.
5B. Structure of the set of KMS states [Carpi, Longo, Tanimoto, Weiner]. We show here the uniqueness of the locally normal temperature state for a completely rational net and classify the temperature states in important models.
6. Themalization effects in boundary CFT [Longo, Martinetti, Rehren]. Description of the modular group in Boundary CFT; the geometric behavior here depends on the distance to the boundary too.
7. Operator Algebras and Boundary Quantum Field Theory [Longo, Witten]. We have built up local, time translation covariant Boundary QFT nets starting with a local conformal net A and a related semigroup of unitaries. By an analog of the Beurling-Lax theorem, a class of such unitaries are attached to scattering functions. The methods and the ideas are completely new and unconventional and open a new ways to construct models. It is a remarkable collaboration by scientists with a very different background an the output is intrinsically quite interdisciplinary.
8. Operator Algebras and Boundary Conformal Field Theory [Carpi, Kawahigashi, Longo, Rehren].
8A. Themalization effects in boundary CFT [Longo, Martinetti, Rehren]. Description of the modular group in Boundary CFT; the geometric behavior here depends on the distance to the boundary too.
8B. Boundary QFT on different spacetimes [Longo, Rehren]. Description and construction of Boundary CFT and QFT nets on the interior of the Lorentz hyperboloid; the geometric behavior here depends on the distance to the boundary too.
8C. Adding a boundary condition [Carpi, Kawahigashi, Longo]. We have provided an Operator Algebraic procedure to construct all Boundary CFT nets that are locally isomorphic to a given conformal net on the two-dimensional Minkowski plane (completely rational case). These results prompt to a classification of Boundary CFT (rational case) and have been the subject of various scientific talks.