arXiv subjects

Guillaume Baverez

Publications and source records attributed to Guillaume Baverez.

12 recordsLinked to original sources

arXiv2026

Non-Abelian multiplicative chaos on the circle

In this note, we introduce a generalisation of multiplicative chaos measures which is both non-Gaussian and non-Abelian. The renormalisation procedure takes as inputs an irreducible unitary representation of a compact connected Lie group, together with a Kac-Moody unitarising measure at some level $κ$, and outputs a random measure on the circle with values in the space of positive definite Hermitian endomorphisms of the representation space. So far, our construction is valid for the range of $κ$-values corresponding to the $L^2$-phase. The proof follows the usual route in the theory of multiplicative chaos, relying on an exact formula for the one-point function and a bound on the two-point function at colliding points. These expressions are derived using the rich algebraic structure of the theory: namely, we establish a one-dimensional version of the Knizhnik-Zamolodchikov equations.

math.PR↗

arXiv2026

The CFT of SLE loop measures and the Kontsevich--Suhov conjecture

This paper initiates the study of the conformal field theory of the SLE$_κ$ loop measure $ν$ for $κ\in(0,4]$, the range where the loop is almost surely simple. First, we construct two commuting representations $(\mathbf{L}_n,\bar{\mathbf{L}}_n)_{n\in\mathbb{Z}}$ of the Virasoro algebra with central charge $c_\mathrm{M}=1-6(\frac{2}{\sqrtκ}-\frac{\sqrtκ}{2})^2\leq1$ as (unbounded) first order differential operators on $L^2(ν)$. Second, we introduce highest-weight representations and characterise their structure: in particular, we prove the existence of vanishing singular vectors at arbitrary levels on the Kac table. Third, we prove an integration by parts formula for the SLE loop measure, and use it to define the Shapovalov form of the representation, a non degenerate (but \emph{not} positive definite) Hermitian form $\mathcal{Q}$ on $L^2(ν)$ with a remarkably simple geometric expression. The fact that $\mathcal{Q}$ differs from the $L^2(ν)$-inner product is a manifestation of non-unitarity. Finally, we write down a spectral resolution of $\mathcal{Q}$ using the joint diagonalisation of $\mathbf{L}_0$ and $\bar{\mathbf{L}}_0$. As an application of these results, we provide the first proof of the uniqueness of restriction measures, as conjectured by Kontsevich and Suhov. Our results lay the groundwork for an in-depth study of the CFT of SLE: in forthcoming works, we will define correlation functions on Riemann surfaces, and prove conformal Ward identities, BPZ equations, and conformal bootstrap formulas.

math.PR↗

arXiv2026

Semigroup of annuli in Liouville CFT

In conformal field theory, the semigroup of annuli with boundary parametrisation plays a special role, in that it generates the whole algebra of local conformal symmetries, the so-called Virasoro algebra. The subgroup of elements $\mathbb A_f=\mathbb D\setminus f(\mathbb D^\circ)$ for contracting biholomorphisms $f:\mathbb D\to f(\mathbb D)\subset \mathbb D^\circ$ with $f(0)=0$ is called the holomorphic semigroup of annuli. In this article, we construct a differentiable representation of the holomorphic semigroup on the space of bounded operators on the Hilbert space $\mathcal H$ of Liouville Conformal Field Theory. We show that it generates under differentiation the positive Virasoro elements $\mathbf L_n,\tilde{\mathbf L}_n$ for $n\geq 0$. We also construct a projective representation of the semigroup of annuli in the space of bounded operators on $\mathcal H$ in terms of Segal amplitudes and show that all Virasoro elements $\mathbf L_n,\tilde{\mathbf L}_n$ for $n\in\mathbb Z$ are generated by differentiation of these annuli amplitudes. Finally, we use this to show that the Segal amplitudes for Liouville theory are differentiable with respect to their boundary parametrisations, and the differential is computed in terms of Virasoro generators. This paper will serve, in a forthcoming work, as a fundamental tool in the construction of conformal blocks as globally defined holomorphic sections of a holomorphic line bundle on Teichmüller space and satisfying the Ward identities.

math.PR↗

arXiv2026

Higher equations of motion at level 2 in Liouville CFT

We prove conjectures of Zamolodchikov and Belavin-Belavin in Liouville conformal field theory (CFT), which are generalisations of the celebrated Belavin-Polyakov-Zamolodchikov equations known as the higher equations of motion. Algebraically, these equations give examples of non-zero singular states in Virasoro modules, which is a relatively rare phenomenon in the physical study of CFT. In probability theory, these equations and their variants have been instrumental in the rigorous derivation of the structure constants of Liouville CFT in the unit disc. The proof builds on a previous work of ours studying the analytic continuation of the Poisson operator of Liouville theory. The main novelty is that this operator admits poles on the Kac table, and the higher equations of motions are obtained via a residue computation.

math.PR↗

arXiv2026

Unitarising measures for Kac-Moody algebras

Given a compact connected Lie group $G$ with dual Coxeter number $\check h$ and a level $κ<-2\check h$, we introduce a probability measure $ν_κ$ on the space of holomorphic $\mathfrak g_{\mathbb C}$-valued $(1,0)$-forms in $\mathbb D$, in relation to the Kähler geometry of the loop group of $G$ and the action of a pair of Kac--Moody algebras at respective levels $κ$ and $-2\check h-κ>0$. We prove that $ν_κ$ is characterised by a covariance property making rigorous sense of the formal path integral ``$\mathrm dν_κ(γ)=e^{-\checkκ\mathscr{S}(γ)}Dγ$", where $Dγ$ is the non-existent Haar measure on the loop group and $\mathscr S$ is a Kähler potential for the right-invariant Kac--Moody metric. Infinitesimally, the covariance formula prescribes the Shapovalov forms of the Kac--Moody representations.

math.RT↗

arXiv2025

Irreducible Virasoro representations in Liouville conformal field theory

This paper studies the analytic continuation of Liouville eigenstates and shows that they assemble into irreducible highest-weight representations of the Virasoro algebra, for all values of the conformal weights. This builds on previous results from the first author and Guillarmou, Kupiainen, Rhodes & Vargas, where such representations were constructed except for the conformal weight on the Kac table. In order to extend these results to the degenerate weights, we find explicit analytic expressions for the Virasoro descendants and uncover the probabilistic meaning of the Kac table. In the algebraic approach to conformal field theory, the irreducibility is a crucial property that must be satisfied by the representations in the spectrum, and is usually taken as an axiom. Computationally, it leads to the celebrated null-vector (or BPZ) equations for correlation functions and conformal blocks, which are the cornerstone of the integrability of the theory.

math.PR↗

arXiv2025

Conformal welding and the matter--Liouville--ghost factorisation

We study the action of local conformal transformations on several measures related to the Gaussian free field and Schramm--Loewner evolutions. The main novelty of our work is a Cameron--Martin-type formula for the welding homeomorphism of the SLE loop measure ($κ\in(0,4]$); its proof relies on a rigorous interpretation (and computation) of the "Jacobian" of the conformal welding map, which we relate to the "$bc$-ghost system" from bosonic string theory. We also give an intrinsic definition of the trace of the GFF on SLE, and prove a characterisation of the free boundary GFF in $\mathbb{D}$. As an application, we introduce a new and intrinsic approach to the conformal welding of quantum surfaces.

math.PR↗

arXiv2025

Noise-like analytic properties of imaginary chaos

In this note we continue the study of imaginary multiplicative chaos $μ_β:= \exp(i βΓ)$, where $Γ$ is a two-dimensional continuum Gaussian free field. We concentrate here on the fine-scale analytic properties of $|μ_β(Q(x,r))|$ as $r \to 0$, where $Q(x,r)$ is a square of side-length $2r$ centred at $x$. More precisely, we prove monofractality of this process, a law of the iterated logarithm as $r \to 0$ and analyse its exceptional points, which have a close connection to fast points of Brownian motion. Some of the technical ideas developed to address these questions also help us pin down the exact Besov regularity of imaginary chaos, a question left open in [JSW20]. All the mentioned properties illustrate the noise-like behaviour of the imaginary chaos. We conclude by proving that the processes $x \mapsto |μ_β(Q(x,r))|^2$, when normalised additively and multiplicatively, converge as $r \to 0$ in law, but not in probability, to white noise; this suggests that all the information of the multiplicative chaos is contained in the angular parts of $μ_β(Q(x,r))$.

math.PR↗

arXiv2024

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

In this work, we construct a representation of the Virasoro algebra in the canonical Hilbert space associated to Liouville conformal field theory. The study of the Virasoro operators is performed through the introduction of a new family of Markovian dynamics associated to holomorphic vector fields defined in the disk. As an output, we show that the Hamiltonian of Liouville conformal field theory can be diagonalized through the action of the Virasoro algebra. This enables to show that the scattering matrix of the theory is diagonal and that the family of the so-called primary fields (which are eigenvectors of the Hamiltonian) admits an analytic extension to the whole complex plane, as conjectured in the physics literature.

math.PR↗

arXiv2020

On the $\log$-regularity of SLE$_4$

We prove that the welding homeomomorphism of SLE$_4$ is almost surely $\log$-regular. In a previous version of this work, we had erroneously deduced its removability from this property. Nevertheless, the $\log$-regularity does provide some information and could lead to future developments.

math.PR↗

arXiv2019

Modular bootstrap agrees with path integral in the large moduli limit

Based on the rigorous path integral formulation of Liouville Conformal Field Theory initiated by David-Kupiainen-Rhodes-Vargas on the Riemann sphere and David-Rhodes-Vargas on the torus of modulus $τ$, we give the exact asymptotic behaviour of the 1-point toric correlation function as $\mathrm{Im}\:τ\to\infty$. In agreement with formulae predicted within the bootstrap formalism of theoretical physics, our results feature an $(\mathrm{Im}\:τ)^{-3/2}$ decay rate and we identify the derivative of DOZZ formula in the limit.

math.PR↗

arXiv2018

Fusion asymptotics for Liouville correlation functions

David-Kupiainen-Rhodes-Vargas introduced a probabilistic framework based on the Gaussian Free Field and Gaussian Multiplicative Chaos in order to make sense rigorously of the path integral approach to Liouville Conformal Field Theory (LCFT). We use this setting to compute fusion estimates for the four-point correlation function on the Riemann sphere, and find that it is consistent with predictions from the framework of theoretical physics known as the conformal bootstrap. This result fits naturally into the famous KPZ conjecture which relates the four-point function to the expected density of points around the root of a large random planar map weighted by some statistical mechanics model. From a purely probabilistic point of view, we give non-trivial results on negative moments of GMC. We give exact formulae based on the DOZZ formula in the Liouville case and asymptotic behaviours in the other cases, with a probabilistic representation of the limit. Finally, we show how to extend our results to boundary LCFT, treating the cases of the fusion of two boundary or bulk insertions as well as the absorption of a bulk insertion on the boundary.

math.PR↗