Exactly solvable conformal field theories

TL;DR

This work surveys exact solvability of two-dimensional conformal field theories through the bootstrap, detailing how Virasoro symmetry and degenerate fields tightly constrain spectra and correlation functions. It formats solvable CFTs into diagonal (Liouville, generalized minimal models, A-series) and non-diagonal (D-series) classes, plus loop CFTs with extended spectra including O(n), PSU(n), and Potts models, and then develops analytic (BPZ equations, shift equations) and numerical bootstrap tools. The analytic bootstrap uses degenerate fields to derive differential equations and explicit structure constants via double Gamma functions; the numerical bootstrap implements crossing symmetry with truncations and interchiral blocks to handle infinite spectrums, illustrating strategies to solve loop-like theories. The article also links loop CFTs to statistical loop models through combinatorial maps, provides explicit 4-point constants in several cases, and discusses how limits among Liouville, generalized minimal models, and Runkel–Watts theories illuminate a broader landscape of exactly solvable CFTs. Overall, it outlines a cohesive program for solving a broad family of 2d CFTs using bootstrap, degeneracy, and combinatorial structures, while highlighting remaining open problems and directions for loop theories. The results have implications for statistical physics, quantum gravity in two dimensions, and the broader understanding of conformal bootstrap in non-unitary and non-diagonal settings.

Abstract

We review 2d CFT in the bootstrap approach, and sketch the known exactly solvable CFTs with no extended chiral symmetry: Liouville theory, (generalized) minimal models, limits thereof, and loop CFTs, including the $O(n)$, Potts and $PSU(n)$ CFTs. Exact solvability relies on local conformal symmetry, and on the existence of degenerate fields. We show how these assumptions constrain the spectrum and correlation functions. We discuss how crossing symmetry equations can be solved analytically and/or numerically, leading to analytic expressions for structure constants in terms of the double Gamma function. In the case of loop CFTs, we sketch the corresponding statistical models, and derive the relation between statistical and CFT variables. We review the resulting combinatorial description of correlation functions, and discuss what remains to be done for solving the CFTs.