Types of quantum field thories
Standard AQFT references on conformal nets include
and lots of others
A somewhat novel take on conformal nets is presented and studied in
where conformal nets are arranged into a tricategory with symmetric monoidal structure that is a delooping of the bicategory of von Neumann algebras and bimodules between these. Among other things, this work connects the AQFT notion of conformal nets with the FQFT notion of cobordism representations.
Chris Douglas, André Henriques, Topological modular forms and conformal nets , in Branislav Jurčo, Hisham Sati, Urs Schreiber (eds.), Mathematical Foundations of Quantum Field and Perturbative String Theory
and in terms of actual local nets in
Articles that discuss the representation theory of conformal nets include