CFT: Algebraic, Topological and probabilistic approaches in Conformal Field Theory

Within the probabilistic approach, a more appropriate way to define and treat a CFT lies within the system of axioms introduced by G. Segal, where one associates to Riemann surfaces with marked points and parametrized boundaries a set of operators on a Hilbert space representing the space of fields on the circle. It is required that these operators have conformal covariance and compose well under gluing of surfaces. This axiomatic way of defining a CFT turns out to be well designed for the probabilistic construction of CFTs, since the operators associated to surfaces with boundary can be understood as conditional expectations of a measure associated to the classical action of the considered CFT (when this action exists).

An important aspect of our program is a non-perturbative approach to the correlation functions of a full, local CFT. In the VOA picture, it starts from the observation that correlation functions should be specific elements in (tensor products of) vector spaces of conformal blocks. Such systems of conformal blocks can be constructed from representation theories of vertex algebras. These systems may be regarded as vector bundles with a (projectively) flat connection and are thus described, up to isomorphism, by their monodromies which are, in turn, representations of mapping class groups. These representations in fact combine into a modular functor which can be constructed using tools from quantum topology, for instance a (possibly non-semisimple) 3d TQFT for which the input is a modular tensor category of a given LCFT. 

In the probabilistic approach, one gives a sense to the path integral as an expectation of certain observables of a random field, in order to define the correlation functions. Then, the question is to uncover the chiral algebras (e.g., the Virasoro VOA or its extensions given by the W-algebras) in the probabilistic expressions and develop an operator product expansion to express, in the end, the correlation functions in terms of 3-point conformal blocks on the sphere. When it can be made rigorous, this approach gives a probabilistic representation of the conformal blocks and proves their convergence, which is sometimes very hard from a purely algebraic approach. As an example, for the Liouville theory, which is unitary, the probabilistic approach proved to be efficient. New recent results also show that non-unitary theories could also be constructed by probabilistic methods, and it looks very promising and important to see how this connects to algebraic methods or even if it can help discovering new features in the algebraic approach.