On the correspondence between graded-local conformal nets and vertex operator superalgebras with applications

This thesis deals with two different mathematical axiomatisations of chiral Conformal Field Theory (CFT). On the one hand, there is the axiomatisation via Haag-Kastler axioms, in an operator algebraic setting, dealing with families of von Neumann algebras, also known as graded-local conformal net theory. On the other hand, there is the axiomatisation via vertex operator superalgebras (VOSAs). In [CKLW18], the authors establish a correspondence between these two settings, taking into account Bose theories only, that is, local conformal net theory and vertex operator algebra (VOA) theory respectively. In this thesis, we extend the correspondence given in [CKLW18] to the case of graded-local conformal nets and VOSAs, in order to include Fermi theories too. Furthermore, we present completely new results about theories with non-trivial grading. As an application of the correspondence between local conformal nets and VOAs, we classify all the subtheories of specific concrete models, known as even rank-one lattice chiral CFTs. This thesis has been supervised by Dr. Robin Hillier and it is based on two papers [CGH19], published as Editor's Pick on Journal of Mathematical Physics, and [CGH21], still in preparation, of the author jointly with S. Carpi and R. Hillier. References: [CGH19] S. Carpi, T. Gaudio and R. Hillier. Classication of unitary vertex subalgebras and conformal subnets for rank-one lattice chiral CFT models. J. Math. Phys. 60(9), 093505, pp. 20 (2019). [CGH21] S. Carpi, T. Gaudio and R. Hillier. From vertex operator superalgebras to graded-local conformal nets and back. In preparation. [CKLW18] S. Carpi, Y. Kawahigashi, R. Longo and M. Weiner. From vertex operator algebras to conformal nets and back. Mem. Amer. Math. Soc. 254(1213), pp. vi+85 (2018)