Quantum field theory is the branch of theoretical physics that studies subatomic particles and their interactions. Although experimentally verified, it lacks a rigorous mathematical foundation. One of the first axiomatizations for quantum field theory appeared in the work of Wightman in the 1950s, where observables are quantum fields. Later, in the 1960s, Araki, Haag and Kastler proposed the formalism of Algebraic Quantum Field Theory where one considers instead families of bounded operator algebras, known as Araki–Haag–Kastler nets.

Constructing physically relevant models in a chosen formalism is usually a difficult task. Thus, first one may attempt to construct models which are richer in symmetries, e.g., conformal field theories in low-dimensional space-time.

After providing a brief overview on the Wightman and the Algebraic Quantum Field Theory formalisms, we focus on the construction of quantum fields in the Wightman axiomatization for a class of 2-dimensional conformal field theory. In this setting, we construct Wightman fields as products of charged primary fields for the left and right chiral component under assumptions on the representation theory of the chiral components.

On the other hand, the Osterwalder–Schrader reconstruction results provide conditions to be verified by correlation functions in the Euclidean space to give rise to a quantum field theory in the Wightman formalism. In this setting, we show that a class of 2-dimensional conformal field theories described using full Vertex Operator Algebras satisfies the conditions of the Osterwalder–Schrader reconstruction, providing another way to construct quantum fields in the Wightman formalism.

This talk is based on arXiv:2301.12310, arXiv:2407.18222, and arXiv:2506.01008.