Abstract
In 1987, 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 and 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 realisation of Segal’s axioms in Liouville Conformal Field Theory (LCFT), a CFT that plays a fundamental role in the theory of random surfaces and two-dimensional quantum gravity. Namely, to a Riemann surface $\Sigma$ with marked points and boundary given by a union of parameterised circles, we associate a Hilbert-Schmidt operator $\mathcal{A}_\Sigma$, called the amplitude of $\Sigma$, which acts on a tensor product of Hilbert spaces assigned to the boundary circles. We show that this correspondence is functorial: gluing of surfaces along boundary circles maps to a composition of the corresponding operators. Correlation functions of LCFT, constructed probabilistically in earlier works by the authors and F. David, can then be expressed as compositions of the amplitudes of simple building blocks where $\Sigma$ is a sphere with $b\in {1,2,3}$ disks removed (hence $b$ boundary circles) and $3-b$ marked points. These amplitudes in turn are shown to be determined by basic objects of LCFT: its spectrum and its 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 related to a pant decomposition of the surface. The integrand is the square modulus 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. Such a holomorphic factorisation of correlation functions has been conjectured in physics since the 80’s and we give here its first rigorous derivation for a non-trivial CFT.
