Segal's axioms and bootstrap for Liouville Theory

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 A.A. Belavin, A.M Polyakov and A.B. Zamolodchikov. In Segal's formulation the basic objects of CFT, the correlation functions of conformal primary fields, are viewed as functions (in fact sections of a line bundle) 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. Namely, to a Riemann surface Σ with marked points and boundary given by a union of parametrized circles, we assign a Hilbert-Schmidt operator A Σ , called the amplitude of Σ, acting 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 Σ is a sphere with b ∈ {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 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. Such holomorphic factorization of correlation functions has been conjectured in physics since the 80's and we give here its first rigorous derivation for a nontrivial CFT.

The moduli of annuli in random conformal geometry

“…In our previous work [ARS21], based on the conformal bootstrap of boundary Liouville CFT, we gave a conjecture ([ARS21, Conjecture 1.4]) on the joint distribution of L γ φ (∂ 0 C τ ), L γ φ (∂ 1 C τ ) and A γ φ (C τ ) under the Liouville field measure LF τ . This conjecture was recently proved by Wu [Wu22] based on the rigorous bootstrap framework for Liouville CFT developed in [GKRV20,GKRV21]. The starting point of our proof of Theorems 1.3 and 1.4 is a corollary of Wu's theorem, which is interesting in its own right.…”

“…In mathematics, the so-called imaginary DOZZ formula conjectured in [IJS16] was proved in [AS21], and the annulus partition function is proved in this paper. It would be a breakthrough to carry out a functional analytic conformal bootstrap program for CLE similar to the Liouville case [GKRV20,GKRV21]. It would already be very interesting to give a purely CLEbased proof of results in our Section 1.4, as for the κ = 4 case in [ALS20].…”

The Virasoro structure and the scattering matrix for Liouville conformal field theory

“…In the context of the probabilistic construction of LCFT, the central charge c L is just a scalar and the conformal bootstrap has recently been established by the last 4 authors in the papers [KRV20, GKRV20,GKRV22]. To implement this programme, a Hilbert space H = L 2 (H −s (T), µ 0 ) was introduced where H −s (T) is the Sobolev space of order −s < 0 on the unit circle T = {z ∈ C | |z| = 1} and µ 0 is the distribution of the Gaussian Free Field on C restricted to T times the Lebesgue measure for the zero Fourier mode on T; the Hilbert space H is therefore a space of fields ϕ on the circle.…”

What is a random surface?

“…The second paper derives an analog of Plancharel's theorem (which states that the Fourier transform preserves 𝐿 2 norm) along with a certain "spectrum" relevant to this context [119] . The final paper completes the bootstrap program with an extension to 𝑛-point functions and higher genus surfaces [120]. We recommend that the reader take a look at the introduction to [119], which summarizes this viewpoint and situates it within the larger enterprise of quantum field theory.…”

“…The answer is that after such a conditioning or such a weighting, 𝜙 is no longer Gaussian, and the 𝑛-point correlation computation transforms from easy to doable but only barely. On the other hand, in order to understand some fundamental things like the law of conformal modulus of four points (or 𝑛 points) sampled independently from the measure on an LQG sphere, one has to address the harder question, and this is precisely what is done in [119,120,158]. Although these results are rather recent, they have already inspired a tremendous amount of activity, establishing exact solvability for many problems that could previously only be addressed more qualitatively.…”