arXiv research

Derives stress-energy identities in Liouville theory on compact surfaces.

Research proves the semi-classical limit of Liouville conformal field theory, describing deterministic geometry from random fluctuations.

Motivated by the supersymmetric extension of Liouville theory in the recent physics literature, we couple the standard Liouville functional with a spinor field term. The resulting functional is conformally invariant. We study geometric and analytic aspects of the resulting Euler-Lagrange equations, culminating in a blo…

Using probabilistic methods, we first define Liouville quantum field theory on Riemann surfaces of genus $\mathbf{g}\geq 2$ and show that it is a conformal field theory. We use the partition function of Liouville quantum field theory to give a mathematical sense to Polyakov's partition function of noncritical bosonic s…

Local gradient estimates for eigenfunctions on conformal solitons improve Liouville theorems.

Researchers construct a probabilistic model for a WZW theory on hyperbolic space and link it to Liouville theory.

Our purpose is to pursue the rigorous construction of Liouville Quantum Field Theory on Riemann surfaces initiated by F. David, A. Kupiainen and the last two authors in the context of the Riemann sphere and inspired by the 1981 seminal work by Polyakov. In this paper, we investigate the case of simply connected domains…

Suggestions concerning the generalization of the geometric quantization to the case of nonlinear field theories are given. Results for the Liouville field theory are presented.

The operator realizing a Dehn twist in quantum Teichmuller theory is diagonalized and continuous spectrum is obtained. This result is in agreement with the expected spectrum of conformal weights in quantum Liouville theory at c>1. The completeness condition of the eigenvectors includes the integration measure which app…

In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville Conformal Field Theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full…

In the present paper we prove Liouville-type theorems: non-existence theorems for conformal mappings of complete Riemannian manifolds. In addition, we give an application of these results to the theory of conharmonic transformations. A part of these results was announced in our reports on the conferences "Differential …

In this note, we study Liouville type theorem for conformal Gaussian curvature equation (also called the mean field equation) $-Δu=K(x)e^u, in R^2$ where $K(x)$ is a smooth function on $R^2$. When $K(x)=K(x_1)$ is a sign-changing smooth function in the real line $R$, we have a non-existence result for the finite to…

Discrete version of Liouville's theorem for simplicial complexes.

Quantizes geodesic lengths in Teichmüller spaces using algebraic methods.

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.

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 construction of such an object and this is done by means of probabilistic tools. Both functionals play an important role resp…

Random Gaussian fields on 4D Riemannian manifolds with conformal invariance.

Holographic principle matches deformed Liouville theory action.

We establish a general Liouville type theorem for conformally invariant fully nonlinear equations.

Optimal Liouville theorem for half-Euclidean space equations.

Enhances conformal geometry in higher dimensions with infinite-dimensional algebra.

We present some further results on Liouville type theorems for some conformally invariant fully nonlinear equations.

This paper proves Liouville theorems for conformally invariant fully nonlinear equations.

We review the construction of homological evolutionary vector fields on infinite jet spaces and partial differential equations. We describe the applications of this concept in three tightly inter-related domains: the variational Poisson formalism (e.g., for equations of Korteweg-de Vries type), geometry of Liouville-ty…

We study the qualitative behavior of nonlinear Dirac equations arising in quantum field theory on complete Riemannian manifolds. In particular, we derive monotonicity formulas and Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term.

Study shows that deformed Liouville metrics on tori remain Liouville.

Study of gauge theory blowups and Painlevé VI identity.

The study confirms Liouville-type theorems for positive harmonic functions on manifolds with nonnegative Ricci curvature and strictly convex boundary.

The Yamabe problem concerns finding a conformal metric on a given closed Riemannian manifold so that it has constant scalar curvature. This paper concerns mainly a fully nonlinear version of the Yamabe problem and the corresponding Liouville type problem.

Krasnov (arXiv: hep-th/0005106) identified the renormalized volume of a Schottky 3-manifold with the action of the Liouville theory on the conformal infiinity. We try to compute the renormalized volume in terms of more transparent geometric quantities.

Length metrics can be closely approximated by conformally flat metrics.

We establish Liouville type theorems for degenerate conformally invariant equations.

Plane triangulations remain rigid under discrete conformal changes.

Solves a conjecture using a new formula on conformally Einstein manifolds.

It is the purpose of the present paper to outline an introduction in theory of embeddings in the manifold Osc^{2}M. First, we recall the notion of 2-osculator bundle. The second section is dedicated to the notion of submanifold in the total space of the 2-osculator bundle, the manifold Osc^{2}M. A moving frame is const…

It is the purpose of the present paper to outline an introduction in theory of embeddings in the manifold Osc^{2}M. First, we recall the notion of 2-osculator bundle. The second section is dedicated to the notion of submanifold in the total space of the 2-osculator bundle, the manifold Osc^{2}M. A moving frame is const…

Topological conformal field theories are defined using only basic results from the theory of quasiconformal mappings.

Machine learning explores symmetries in field theory and algebra.

The paper extends topological field theory to noncompact surfaces using symmetric powers.

The paper defines analogs of volume and action for curves in flag manifolds.

Study constructs scattering theory for massless Dirac field on Kerr spacetime.

We prove the equivalence of several natural notions of conformal maps between sub-Riemannian manifolds. Our main contribution is in the setting of those manifolds that support a suitable regularity theory for subelliptic $p$-Laplacian operators. For such manifolds we prove a Liouville-type theorem, i.e., 1-quasiconform…

Exact solution found for two-body financial dealer model using kinetic theory.

We formulate the unitary rational orbifold conformal field theories in the algebraic quantum field theory framework. Under general conditions, we show that the orbifold of a given unitary rational conformal field theories generates a unitary modular category. Many new unitary modular categories are obtained. We also sh…

Classifies geodesic flows on projective plane with potential field.

The paper connects currents and entropy in hyperbolic 3-manifolds.

We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when w…

The universal Liouville action equals the renormalized volume of a hyperbolic 3-manifold.