Probabilistic interpretation of quantum field theories
Martin Hairer
TL;DR
This work develops a Segal-inspired, rigorous framework for two-dimensional conformal field theories by starting from the 0D path integral and the Gaussian free field, and then building a 2D theory via cobordisms and boundary amplitudes. It introduces half-densities and $\,\zeta$-regularized determinants to formulate finite, functorial amplitudes that satisfy locality and coercivity, culminating in a monoidal functor $\mathcal{F}^V: \\mathcal{C}^{(2)} \to \\mathbf{Hil}$ that encodes boundary data and cobordism composition. The notes then specialize to Liouville theory, showing how conformal changes induce a simple shift of the field by $Q\varphi$ and how the central charge $c = 1 + 6Q^2$ arises from the conformal anomaly, with the Seiberg bounds $\sum_i \alpha_i > 2Q$ guaranteeing a finite, well-defined theory with insertions. Overall, the paper provides a coherent, rigorous pathway from the free field to Liouville CFT within Segal’s axiomatic framework, laying groundwork for the bootstrap/DOZZ program and offering tools for further mathematical analysis of 2D quantum fields.
Abstract
In this note we provide a gentle introduction to the concepts and intuition behind the recent breakthrough results on the mathematically rigorous construction of a non-trivial 2D conformal field theory, namely the so-called Liouville theory. This gives us the opportunity to review Segal's axioms for conformal field theories and to discuss in some detail how the free field fits into them.