A mathematical formulation of two-dimensional conformal field theory

The mathematical construction of non-trivial quantum field theory in four dimensions, known as the "Yang-Mills existence and mass gap problem", is a very important issue in mathematical sciences. There are many examples of rigorous quantum field theories in two dimensions, although the four dimensions have not yet been solved. In particular, two-dimensional conformal field theory, which is a quantum field theory with conformal symmetry, has good properties and can be formulated mathematically using algebraic structures formed by "products of a field and a field" (operator product expansion).

In this talk, this algebraic formulation (full vertex algebra) will be explained. Various construction methods and concrete examples (construction using codes, construction from quantum groups, and construction by deformation) will then be discussed.

All the talk here is mathematical, but I will try to speak in a way that is motivated by physics as much as possible throughout the talk. I hope to receive various comments from the viewpoints of other fields.