CONFSTAT · Conformally invariant and near critical models in statistical field theory

The goal of this proposal is to study the interplay between models of modern probability theory and quantum field theory (QFT). The proposal has two main lines of research:- Conformal field theory (CFT) and the Ising and Liouville models. In the physics literature, there is a far reaching conjecture that the scaling limits of critical lattice models can be described in two ways: either as a CFT or as a fixed point of a renormalization group transformation (RGT). Mathematically, this remains far from understood. In this proposal, these questions will be studied in two of the most canonical CFTs: the Ising model and the Liouville model. Basic properties of CFTs will be established for the critical Ising model and the Liouville model will be showed to be a fixed point of a (RGT). - Scaling limits of near critical models and bosonization. In the physics community, it is believed that also near critical models of statistical mechanics should have scaling limits that are described by QFTs that are perturbations of CFTs. Moreover, these QFTs are special in the sense that despite losing the conformal invariance of CFTs, they retain some of the exact solvability and enjoy other interesting properties. In this proposal, these themes are explored in the setting of the near-critical Ising model (a perturbation of the critical Ising CFT), the sine-Gordon (SG) model, and the massive Thirring model (MTM). An important theme being bosonization, namely that correlation functions of the near critical Ising model and the MTM are conjectured to be given in terms of SG correlation functions. In addition to this, fundamental properties of the SG model will be studied, such as operator product expansions, exponential decay of correlations, and construction of the model on Riemann surfaces.The problems solved in this program will lay the foundations of a mathematically rigorous understanding of QFT in statistical mechanics and some of the most important near critical models.