Review on the probabilistic construction and Conformal bootstrap in Liouville Theory

Abstract

In the paper, we review the recent construction of the Liouville conformal field theory (CFT) from probabilistic methods, and the formalization of the conformal bootstrap. This model has offered a fruitful playground to unify the probabilistic construction of the path integral, the geometric axiomatics of CFT by Segal and the representation theoretical content of the conformal bootstrap. We explain and extract the main steps and ideas behind the construction and resolution of this non-compact CFT.

Figures (15)

• Figure 1: Triangulation of the Riemann sphere sampled from the uniform probability measure.@T.Budd
• Figure 2: A surface cut along a circle $\mathcal{C}$
• Figure 3: In/out boundaries on $\mathbb{A}_T$
• Figure 4: Disks $\mathbb{D}$ and $\lambda \mathbb{D}$, with marked point $z=0$
• Figure 5: $\mathbb{A}_q$ glued to $q\mathbb{A}_{q'}$ is $\mathbb{A}_{qq'}$

Theorems & Definitions (21)

• Theorem 3.1
• Proposition 3.2
• Proposition 3.3
• Theorem 3.4
• Theorem 4.1
• Theorem 5.1
• Proposition 6.1
• Proposition 6.2
• Proposition 6.3
• Theorem 6.4