Segal's axioms and bootstrap for Liouville Theory

TL;DR

The paper constructs a rigorous probabilistic realization of Segal’s CFT axioms for Liouville CFT on arbitrary genus surfaces, expressing LCFT correlation functions as holomorphic conformal blocks integrated against the DOZZ structure constants over the LCFT spectrum. By developing Segal amplitudes for building blocks (pairs of pants, annuli, disks) and proving a gluing (composition) property, it provides a complete conformal bootstrap at the level of higher-genus surfaces, with blocks holomorphically depending on plumbing coordinates. A key achievement is the rigorous construction of normalized conformal blocks and their holomorphic factorisation, which, together with Ward identities and DOZZ data, yields the full correlation functions on all surfaces. The framework yields both a probabilistic and geometric picture of LCFT, enabling explicit spectral decompositions, Virasoro descendant structure, and a projective mapping class group action in the space of conformal blocks, with broad implications for random geometry and 2D quantum gravity.

Abstract

In 1987 Graeme Segal gave a functorial definition of Conformal Field Theory (CFT) that was designed to capture the mathematical essence of the Conformal Bootstrap formalism pioneered in physics by Belavin-Polyakov-Zamolodchikov. In Segal's formulation the basic objects of CFT, the correlation functions of conformal primary fields, are viewed as functions on the moduli space of Riemann surfaces with marked points which behave naturally under gluing of surfaces. In this paper we give a probabilistic realization of Segal's axioms in Liouville Conformal Field Theory (LCFT) which is a CFT that plays a fundamental role in the theory of random surfaces and two dimensional quantum gravity. Then we use Segal's axioms to express the correlation functions of LCFT in terms of the basic objects of LCFT: its {\it spectrum} and its {\it structure constants}, determined in earlier works by the authors. As a consequence, we obtain a formula for the correlation functions as multiple integrals over the spectrum of LCFT, the structure of these integrals being associated to a pant decomposition of the surface. The integrand is the modulus squared of a function called conformal block: its structure is encoded by the commutation relations of an algebra of operators called the Virasoro algebra and it depends holomorphically on the moduli of the surface with marked points. The integration measure involves a product of structure constants, which have an explicit expression, the so called DOZZ formula.

Figures (8)

• Figure 1: The plumbed surfaces $\Sigma_{\bf q}$ with four pairs of pants $\mathcal{P}_1,\dots,\mathcal{P}_4$ and six annuli $\mathbb{A}_{q_1},\dots,\mathbb{A}_{q_6}$ of modulus ${\bf q}=(q_1,\dots,q_6)$ between the pairs of pants.
• Figure 2: Diagram of the proof of Theorem \ref{['th:fullexpressioncorrel']}
• Figure 3: Multigraphs associated to marked Riemann surfaces
• Figure 4: The plumbing with parameter $q=e^{-t+i\theta}$, viewed as gluing an annulus ${\mathbb{A}_q}$ to $\partial \mathcal{D}_1$ and $\partial \mathcal{D}_2$ with a twist of angle $\theta$. The length for the flat metric $|dz|^2/|z|^2$ of the annulus is $t$.
• Figure 5: Surface of genus $2$ with no marked points decomposed into $2$ pairs of pants

Theorems & Definitions (103)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Proposition 2.2
• Definition 3.1
• Lemma 4.1
• proof
• Definition 4.2: Amplitudes
• Remark 4.3