2-dimensional conformal field theories: an axiomatic approach | Colóquio do DFMA

Resumo: Quantum field theory is a branch of theoretical physics that regards subatomic particles and their interactions. Despite its remarkable experimental success, the theory still lacks a fully rigorous mathematical foundation. One of the earliest attempts to axiomatize quantum field theory was made by Wightman in the 1950s, taking quantum fields as the fundamental objects. In the 1960s, Araki, Haag and Kastler introduced a different approach, known as Algebraic Quantum Field Theory. In this framework, rather than working directly with quantum fields, one considers families of algebras of bounded operators, known as Araki-Haag-Kastler nets.
Constructing physically relevant models within a given formalism is often a difficult task. One may therefore begin by considering models with additional symmetries, such as conformal field theories in low-dimensional spacetime.
After providing a brief overview on the Wightman and the Algebraic Quantum Field Theory formalisms, we focus on the rigorous construction of quantum fields in the Wightman axiomatization for a class of 2-dimensional conformal field theories starting from the left and right chiral components and under some assumptions on the representation theory of the chiral components.
On the other hand, the Osterwalder-Schrader reconstruction results provide conditions to be verified by n-point functions in 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 theories in the Wightman formalism.

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

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