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Antti Kupiainen

Publications and source records attributed to Antti Kupiainen.

At least 19 recordsLinked to original sources
arXiv2026

Energy field of critical Ising model and examples of singular fields in QFT

The goal of this paper is to prove singularity of three natural fields in QFT with respect to their natural base measure. The fields we consider are the following ones: (1) The near-critical limit of the $2d$ Ising model (in the $β$-direction) is locally singular w.r.t the critical scaling limit of $2d$ Ising. (N.B. In the $h$-direction it is not locally singular). (2) The $2d$ Hierarchical Sine-Gordon field is singular w.r.t the $2d$ hierarchical Gaussian Free Field for all $β\in[β_{L^2}, β_{BKT})$. (3) The Hierarchical $Φ^4_3$ field is singular w.r.t the $3d$ hierarchical GFF. Item (1) gives the first strong indication that the energy field of critical $2d$ Ising model does not exist as a random Schwarz distribution on the plane. Item (2) has been proved to be singular for the non-hierarchical $2d$ Sine-Gordon sufficiently far from the BKT point in [GM24] while item (3) is proved to be singular for the non-hierarchical $3d$ $Φ^4_3$ field in [BG21, OOT21, HKN24]. We believe our way to detect a singular behaviour at all scales is very much down to earth and may be applicable in all settings where one has a good enough control on the so-called effective potentials.

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

Conformal welding of independent Gaussian multiplicative chaos measures

We solve the classical conformal welding problem for a composition of two random homeomorphisms generated by independent Gaussian multiplicative chaos measures with small parameter values. In other words, given two such measures on the boundary of the unit disk we show that there exist conformal maps to complementary domains on the Riemann sphere such that the pushforward of the normalised measures agree on their common boundary.

math.PR↗
arXiv2025

Segal's axioms and bootstrap for Liouville Theory

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.

math.PR↗
arXiv2025

Probabilistic construction of the $\mathbb{H}^3$-Wess-Zumino-Witten conformal field theory and correspondence with Liouville theory

Wess-Zumino-Witten (WZW) models are among the most basic and most studied Conformal Field Theories (CFT). They have had a huge influence not only in physics but also in mathematics, in representation theory and geometry. However their rigorous probabilistic construction and analysis starting from the path integral is still missing and all their properties have been obtained algebraically from their postulated affine Lie algebra symmetry. Initially considered as taking values in a compact semisimple Lie Group G, the WZW model also has a "dual" formulation where the group $G$ is replaced by the homogenous space $G^{\mathbb{C}}/G$, where $G^{\mathbb{C}}$ is the complexification of $G$, and it has been argued that the former can be (re-)constructed from the latter. For $G={\rm SU}(2)$, the space ${\rm SL}(2,\mathbb{C})/{\rm SU}(2)$ can be identified with the three dimensional hyperbolic space $\mathbb{H}^3$ and, in physics, the corresponding CFT has been studied as the simplest example of the AdS/CFT correspondence. A surprising correspondence between the $\mathbb{H}^3$-WZW CFT and the Liouville CFT was found by Ribault and Teschner and later generalised by Hikida and Shomerus. This correspondence has been dubbed by Gaiotto-Teschner as the "quantum analytic Langlands correspondence" since the analytic Langlands correspondence of Etingof, Frenkel and Kazhdan seems to emerge in its formal semi classical limit. In this paper we give a rigorous construction of the path integral for the $\mathbb{H}^3$-WZW model on a closed Riemann surface $Σ$, twisted by an arbitrary smooth gauge field on $Σ$. Using the probabilistic path integral we prove a correspondence between the correlation functions of the primary fields of the $\mathbb{H}^3$ model and those of Liouville CFT extending the expressions proposed by Ribault-Teschner and by Hikida-Schomerus to this general setup.

math-ph↗
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↗
arXiv2024

Review on the probabilistic construction and Conformal bootstrap in Liouville Theory

In the paper, we review the recent construction of the Liouville conformal field theory (CFT) from probabilistic methods, and the formalization of the conformal bootstrap. This model has offered a fruitful playground to unify the probabilistic construction of the path integral, the geometric axiomatics of CFT by Segal and the representation theoretical content of the conformal bootstrap. We explain and extract the main steps and ideas behind the construction and resolution of this non-compact CFT.

math-ph↗
arXiv2023

Compactified Imaginary Liouville Theory

On a given Riemann surface, we construct a path integral based on the Liouville action functional with imaginary parameters. The construction relies on the compactified Gaussian Free Field (GFF), which we perturb with a curvature term and an exponential potential. In physics this path integral is conjectured to describe the scaling limit of critical loop models such as Potts and O(n) models. The potential term is defined by means of imaginary Gaussian Multiplicative Chaos theory. The curvature term involves integrated 1-forms, which are multivalued on the manifold, and requires a delicate regularisation in order to preserve diffeomorphism invariance. We prove that the probabilistic path integral satisfies the axioms of Conformal Field Theory (CFT) including Segal's gluing axioms. We construct the correlation functions for this CFT, involving electro-magnetic operators. This CFT has several exotic features: most importantly, it is non unitary and has the structure of a logarithmic CFT. This is the first mathematical construction of a logarithmic CFT and therefore the present paper provides a concrete mathematical setup for this concept.

math-ph↗
arXiv2020

Conformal bootstrap in Liouville Theory

The conformal bootstrap hypothesis is a powerful idea in theoretical physics which has led to spectacular predictions in the context of critical phenomena. It postulates an explicit expression for the correlation functions of a conformal field theory in terms of its 3-point correlation functions. In this paper we give the first mathematical proof of the conformal bootstrap hypothesis in the context of Liouville theory, a 2-dimensional conformal field theory studied since the eighties in theoretical physics and constructed recently by F. David and the three last authors using probability theory. The proof is based on a probabilistic construction of the Virasoro algebra highest weight modules through spectral analysis of an associated self adjoint operator akin to harmonic analysis on non compact Lie groups but in an infinite dimensional setup.

math.PR↗
arXiv2020

Stress-Energy in Liouville Conformal Field Theory

We construct the stress-energy tensor correlation functions in probabilistic Liouville Conformal Field Theory (LCFT) on the two-dimensional sphere by studying the variation of the LCFT correlation functions with respect to a smooth Riemannian metric. In particular, we derive conformal Ward identities for these correlation functions. This forms the basis for the construction of a representation of the Virasoro algebra on the canonical Hilbert space of the LCFT. In \cite{ward} the conformal Ward identities were derived for one and two stress-energy tensor insertions using a different definition of the stress-energy tensor and Gaussian integration by parts. By defining the stress-energy correlation functions as functional derivatives of the LCFT correlation functions and using the smoothness of the LCFT correlation functions proven in \cite{Oik} allows us to control an arbitrary number of stress-energy tensor insertions needed for representation theory.

math-ph↗
arXiv2019

Integrability of Liouville theory: proof of the DOZZ Formula

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 Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e. to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first rigorous integrability result on Gaussian Multiplicative Chaos measures.

math.PR↗
arXiv2018

The DOZZ Formula from the Path Integral

We present a rigorous proof of the Dorn, Otto, Zamolodchikov, Zamolodchikov formula (the DOZZ formula) for the 3 point structure constants of Liouville Conformal Field Theory (LCFT) starting from a rigorous probabilistic construction of the functional integral defining LCFT given earlier by the authors and David. A crucial ingredient in our argument is a probabilistic derivation of the reflection relation in LCFT based on a refined tail analysis of Gaussian multiplicative chaos measures.

hep-th↗
arXiv2017

Local Conformal Structure of Liouville Quantum Gravity

Liouville Conformal Field Theory (LCFT) is an essential building block of Polyakov's formulation of non critical string theory. Moreover, scaling limits of statistical mechanics models on planar maps are believed by physicists to be described by LCFT. A rigorous probabilistic formulation of LCFT based on a path integral formulation was recently given by the present authors and F. David in \cite{DKRV}. In the present work, we prove the validity of the conformal Ward identities and the Belavin-Polyakov-Zamolodchikov (BPZ) differential equations (of order $2$) for the correlation functions of LCFT. This initiates the program started in the seminal work of Belavin-Polyakov-Zamolodchikov \cite{BPZ} in a probabilistic setup for a non-trivial Conformal Field Theory. We also prove several celebrated results on LCFT, in particular an explicit formula for the 4 point correlation functions (with insertion of a second order degenerate field) leading to a rigorous proof of a non trivial functional relation on the 3 point structure constants derived earlier in the physics literature by Teschner \cite{Tesc}. The proofs are based on exact identities which rely on the underlying Gaussian structure of LCFT combined with estimates from the theory of critical Gaussian Multiplicative Chaos and a careful analysis of singular integrals (Beurling transforms and generalizations). As a by-product, we give bounds on the correlation functions when two points collide making rigorous certain predictions from physics on the so-called "operator product expansion" of LCFT.

math.PR↗
arXiv2016

Quantum Fields and Probability

I review some recent work where ideas and methods from Quantum Field Theory have proved useful in probability and vice versa. The topics discussed include the use of Renormalization Group theory in Stochastic Partial Differential Equations driven by space-time white noise and the use of the theory of Gaussian Multiplicative Chaos in the study of two dimensional Liouville Conformal Field theory.

math.PR↗
arXiv2016

Towards rigorous analysis of the Levitov-Mirlin-Evers recursion

This paper aims to develop a rigorous asymptotic analysis of an approximate renormalization group recursion for inverse participation ratios $P_q$ of critical powerlaw random band matrices. The recursion goes back to the work by Mirlin and Evers [37] and earlier works by Levitov [32, 33] and is aimed to describe the ensuing multifractality of the eigenvectors of such matrices. We point out both similarities and dissimilarities of LME recursion to those appearing in the theory of multiplicative cascades and branching random walks and show that the methods developed in those fields can be adapted to the present case. In particular the LME recursion is shown to exhibit a phase transition, which we expect is a freezing transition, where the role of temperature is played by the exponent $q$. However, the LME recursion has features that make its rigorous analysis considerably harder and we point out several open problems for further study

math-ph↗
arXiv2016

Fluctuation Relation for Qubit-Calorimetry

Motivated by proposed thermometry measurement on an open quantum system, we present a simple model of an externally driven qubit interacting with a finite sized, fermion environment acting as calorimeter. The derived dynamics is governed by a stochastic Schrödinger equation coupled to the temperature change of the calorimeter. We prove a fluctuation relation and deduce from it a notion of entropy production. Finally, we discuss the first and second law associated to the dynamics.

quant-ph↗
arXiv2016

Energy fluctuations of finite free-electron Fermi gas

We discuss the energy distribution of free-electron Fermi-gas, a problem with a textbook solution of Gaussian energy fluctuations in the limit of a large system. We find that for a small system, characterized solely by its heat capacity $C$, the distribution can be solved analytically, and it is both skewed and it vanishes at low energies, exhibiting a sharp drop to zero at the energy corresponding to the filled Fermi sea. The results are relevant from the experimental point of view, since the predicted non-Gaussian effects become pronounced when $C/k_B \lesssim 10^3$ ($k_B$ is the Boltzmann constant), a regime that can be easily achieved for instance in mesoscopic metallic conductors at sub-kelvin temperatures.

cond-mat.mes-hall↗