# The Fyodorov–Bouchaud formula and Liouville conformal field theory

```@article{Remy2020TheFF,
title={The Fyodorov–Bouchaud formula and Liouville conformal field theory},
author={Guillaume Remy},
journal={Duke Mathematical Journal},
year={2020}
}```
• G. Remy
• Published 18 October 2017
• Mathematics
• Duke Mathematical Journal
In a remarkable paper in 2008, Fyodorov and Bouchaud conjectured an exact formula for the density of the total mass of (sub-critical) Gaussian multiplicative chaos (GMC) associated to the Gaussian free field (GFF) on the unit circle. In this paper we will give a proof of this formula. In the mathematical literature this is the first occurrence of an explicit probability density for the total mass of a GMC measure. The key observation of our proof is that the negative moments of the total mass…

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### The distribution of Gaussian multiplicative chaos on the unit interval

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The Annals of Probability
• 2020
We consider the sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0,1] and prove an exact formula for the fractional moments of the total mass of this measure.

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### Probabilistic conformal blocks for Liouville CFT on the torus

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• 2020
Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced by Polyakov in the context of string theory. Conformal blocks are objects underlying the integrable

### Higher order BPZ equations for Liouville conformal field theory

Inspired by some intrinsic relations between Coulomb gas integrals and Gaussian multiplicative chaos, this article introduces a general mechanism to prove BPZ equations of order \$(r,1)\$ and \$(1,r)\$

### The moduli of annuli in random conformal geometry

• Mathematics, Physics
• 2022
. We obtain exact formulae for three basic quantities in random conformal geometry that depend on the modulus of an annulus. The ﬁrst is for the law of the modulus of the Brownian annulus describing

### Ward Identities in the \$\$\mathfrak {sl}_3\$\$ Toda Conformal Field Theory

• Mathematics
Communications in Mathematical Physics
• 2022
Abstract. Toda conformal field theories are natural generalizations of Liouville conformal field theory that enjoy an enhanced level of symmetry. In Toda conformal field theories this higher-spin

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## References

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• Mathematics
Annals of Mathematics
• 2020
Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called DOZZ formula, for the 3 point structure constants of Liouville

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• Mathematics
Communications in Mathematical Physics
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In 1983 Belavin, Polyakov, and Zamolodchikov (BPZ) formulated the concept of local conformal symmetry in two dimensional quantum field theories. Their ideas had a tremendous impact in physics and

### The distribution of Gaussian multiplicative chaos on the unit interval

• Mathematics
The Annals of Probability
• 2020
We consider the sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0,1] and prove an exact formula for the fractional moments of the total mass of this measure.

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• G. Remy
• Mathematics
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In this work we construct Liouville quantum gravity on an annulus in the complex plane. This construction is aimed at providing a rigorous mathematical framework to the work of theoretical physicists

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The purpose of these notes, based on a series of 4 lectures given by the author at IHES, is to explain the recent proof of the DOZZ formula for the three point correlation functions of Liouville

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• Mathematics
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In this paper, we construct Liouville Quantum Field Theory (LQFT) on the toroidal topology in the spirit of the 1981 seminal work by Polyakov [Phys. Lett. B 103, 207 (1981)]. Our approach follows the

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• Mathematics
• 2015
In this work, we continue the constructive probabilistic approach to the Liouville Quantum Field theory (LQFT) started in [8]. We give a rigorous construction of the stress energy tensor in LQFT and

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• Mathematics
The Annals of Probability
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In this short note, we derive a precise tail expansion for Gaussian multiplicative chaos (GMC) associated to the 2d GFF on the unit disk with zero average on the unit circle. More specifically, we

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• Mathematics
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AbstractConsider a bounded planar domain D, an instance h of the Gaussian free field on D, with Dirichlet energy (2π)−1∫D∇h(z)⋅∇h(z)dz, and a constant 0≤γ<2. The Liouville quantum gravity measure on

### Path integral for quantum Mabuchi K-energy

• Physics, Mathematics
Duke Mathematical Journal
• 2022
We construct a path integral based on the coupling of the Liouville action and the Mabuchi K-energy on a one-dimensional complex manifold. To the best of our knowledge this is the first rigorous