arXiv subjects

Baojun Wu

Publications and source records attributed to Baojun Wu.

14 recordsLinked to original sources

arXiv2026

$\mathrm {U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$ Turaev-Viro invariants for cusped $3$-manifolds

We define a family of Turaev-Viro type invariants for hyperbolic $3$-manifolds with cusp ends, extending the invariants introduced in \cite{LMSWY} for hyperbolic $3$-manifolds with totally geodesic boundary. These invariants are constructed from what we call the ideal $\mathrm{U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$-$6j$ symbols, which are variants of the $6j$-symbols associated with the positive representations of the modular double of $\mathrm{U}_q\mathfrak{sl}(2;\mathbb R)$. We also prove that these invariants decay exponentially, with the exponential decay rate determined by the hyperbolic volume of the manifold.

math.GT↗

arXiv2026

Ward identities: a geometric point of view and applications

The work of Baverez, Guillarmou, Kupiainen, and Rhodes [BGKR24] is the starting point of this paper. It shows that analytic changes of boundary parametrizations act differentiably on Liouville amplitudes, with derivative given by Virasoro operators and a scalar anomaly term. We compute this scalar term. Its holomorphic part is a Schwarzian boundary integral, which gives a geometric explanation of the Virasoro central term. We then derive local Ward identities on disks, annuli, and pairs of pants. They give finite recursions for descendant matrix coefficients. From these recursions we recover the polynomial factorization of normalized pair-of-pants coefficients. In the annular zero-weight limit, we recover the Shapovalov form. For a pair of pants with two incoming boundaries, we recover the formal chiral vertex-operator coefficients. We also give a geometric proof of smoothness in the bulk insertion points and derive the genus-zero arbitrary level BPZ equations for degenerate bulk insertions.

math-ph↗

arXiv2026

Exact solution of three-point functions in critical loop models

We propose an exact formula for three-point functions on the sphere in critical loop models with primary fields $V_{(r,s)}$ characterized by $2r$ legs and a parameter $s$ that describes diagonal fields for $r=0$ and the momentum of legs for $r>0$. We demonstrate its validity in three ways: the conformal bootstrap method for 4-point functions, a transfer-matrix study of the lattice model, and a probabilistic method based on conformal loop ensemble and Liouville quantum gravity. This work provides a crucial missing piece for solving critical loop models and reveals a deep unity between three fundamental approaches to 2D statistical physics: transfer matrix, conformal field theory, and probability theory.

cond-mat.stat-mech↗

arXiv2025

Asymptotics of $b$-$6j$ symbols and anti-de Sitter tetrahedra

In this paper, we study the asymptotics of the $6j$-symbols for the principal series of the modular double of $\mathrm U_q\mathfrak{sl}(2;\mathbb R)$, and of their analytic extension -- what we call the $b$-$6j$ symbols, relating them in various cases to the volume of truncated hyperideal tetrahedra in the hyperbolic and the anti-de Sitter geometry. To the best of our knowledge, this is the first time that the anti-de Sitter geometry appears in the asymptotics of quantum invariants. In addition, based on the connection to conformal field theory, we reveal a correspondence between the edge lengths and the dihedral angles of truncated hyperideal anti-de Sitter tetrahedra and the Fenchel-Nielsen coordinates of hyperbolic four-holed spheres. We also provide a concrete instance of $3D/2D$ holography, in the spirit of the AdS/CFT correspondence.

math-ph↗

arXiv2025

Three-point connectivity constant for $q$-state Potts spin clusters

Recently, Ang--Cai--Sun--Wu (2024) determined the three-point connectivity constant for two-dimensional critical percolation, confirming a prediction of Delfino and Viti (2010). In this paper, we address the analogous problem for planar critical $q$-state Potts spin clusters. We introduce a continuum three-point connectivity constant and compute it explicitly. Under the scaling-limit conjecture for Potts spin clusters, this quantity coincides with the scaling limit of the properly normalized probability that three points lie in the same spin cluster. The resulting formula agrees with the imaginary DOZZ formula up to an explicit $q$-dependent constant with a geometric interpretation. This answers a question from Delfino--Picco--Santachiara--Viti (2013). The proof exploits the coupling between CLE and LQG, together with the BCLE descriptions of $q$-state Potts scaling limits due to Miller--Sheffield--Werner (2017) and K\"ohler-Schindler and Lehmk\"uhler (2025).

math.PR↗

arXiv2025

Turaev-Viro invariant from the modular double of $\mathrm {U}_{q}\mathfrak{sl}(2;\mathbb R)$

We define a family of Turaev-Viro type invariants of hyperbolic $3$-manifolds with totally geodesic boundary from the $6j$-symbols of the modular double of $\mathrm U_{q}\mathfrak{sl}(2;\mathbb R)$, and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the $1$-loop term the adjoint twisted Reidemeister torsion of the double of the manifolds.

math.GT↗

arXiv2025

Mixing rate exponent of planar Fortuin-Kasteleyn percolation

Duminil-Copin and Manolescu (2022) recently proved the scaling relations for planar Fortuin-Kasteleyn (FK) percolation. In particular, they showed that the one-arm exponent and the mixing rate exponent are sufficient to derive the other near-critical exponents. The scaling limit of critical FK percolation is conjectured to be a conformally invariant random collection of loops called the conformal loop ensemble (CLE). In this paper, we define the CLE analog of the mixing rate exponent. Assuming the convergence of FK percolation to CLE, we show that the mixing rate exponent for FK percolation agrees with that of CLE. We prove that the CLE$_\kappa$ mixing rate exponent equals $\frac{3 \kappa}{8}-1$, thereby answering Question 3 of Duminil-Copin and Manolescu (2022). The derivation of the CLE exponent is based on an exact formula for the Radon-Nikodym derivative between the marginal laws of the odd-level and even-level CLE loops, which is obtained from the coupling between Liouville quantum gravity and CLE.

math.PR↗

arXiv2024

SLE Loop Measure and Liouville Quantum Gravity

As recently shown by Holden and two of the authors, the conformal welding of two Liouville quantum gravity (LQG) disks produces a canonical variant of SLE curve whose law is called the SLE loop measure. In this paper, we demonstrate how LQG can be used to study the SLE loop measure. Firstly, we show that for $\kappa\in (8/3,8)$, the loop intensity measure of the conformal loop ensemble agrees with the SLE loop measure as defined by Zhan (2021). The former was initially considered by Kemppainen and Werner (2016) for $\kappa\in (8/3,4]$, and the latter was constructed for $\kappa\in (0,8)$. Secondly, we establish a duality for the SLE loop measure between $\kappa$ and $16/\kappa$. Thirdly, we obtain the exact formula for the moment of the electrical thickness for the shape (probability) measure of the SLE loop, which in the regime $\kappa\in (8/3,8)$ was conjectured by Kenyon and Wilson (2004). This relies on the exact formulae for the reflection coefficient and the one-point disk correlation function in Liouville conformal field theory. Finally, we compute several multiplicative constants associated with the SLE loop measure, which are not only of intrinsic interest but also used in our companion paper relating the conformal loop ensemble to the imaginary DOZZ formulae.

math.PR↗

arXiv2024

Conformal Bootstrap for surfaces with boundary in Liouville CFT. Part 1: Segal axioms

This paper is the first part of the proof of the conformal bootstrap for Liouville conformal field theory on surfaces with a boundary, devoted to Segal's axioms in this context. We introduce the notion of Segal's amplitudes on surfaces with corners and prove the gluing property for such amplitudes. The semi-group of half-annuli and its generator are studied and we develop the necessary material for proving its spectral decomposition using scattering theory in the companion paper \cite{GRW2}. The Segal gluing properties and the spectral decomposition allows us to prove the conformal bootstrap formula for correlation functions of Liouville conformal field theory with a boundary. This has several important applications to the study of conformal blocks (analyticity and convergence) in \cite{remypreprint}, in the construction of a unitary representation of mapping class group in the space of conformal blocks, and the study of random moduli \cite{ARSmoduliRPM} in Liouville quantum gravity.

math-ph↗

arXiv2023

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↗

arXiv2023

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↗

arXiv2023

Liouville conformal field theory on Riemann surface with boundaries

In this note, we give a unified rigorous construction for the Liouville conformal field theory on compact Riemann surface with boundaries for $γ\in (0,2]$ and prove a certain type of Markov property. We also prove some fusion-type estimates in the Boundary LCFT. This note will serve as a reference for a program leading to the conformal bootstrap for the Riemann surface with boundaries and several related projects.

math.PR↗

arXiv2022

Conformal Bootstrap on the Annulus in Liouville CFT

This paper is the first of a series of works on the conformal bootstrap in Liouville conformal field theory (CFT) with boundaries. We focus here on the case of the annulus with two boundary insertions, each of which lies on the different connected components of the boundary. In the course of proving the bootstrap formula, we established several properties on the corresponding annulus conformal blocks: 1) we show that they converge everywhere on the spectral line and they are continuous with respect to the spectrum and the primary weights. 2) we relate them to their torus counterparts by rigorously implementing Cardy's doubling trick for boundary CFT, 3) we solve a conjecture of Martinec on the annulus partition function, 4) we also extend the bootstrap formula to the one-point case. As an application of our bootstrap result, we give an exact formula for the bosonic LQG partition function of the annulus when $\gamma\in (0,2)$. Our paper serves as a key ingredient in the recent derivation of the random moduli for the Brownian annulus by Ang, Remy, and Sun (2022). We also solve several other conjectures relate to torus conformal blocks which arise from physics literature.

math.PR↗

arXiv2021

Integrability of Conformal Loop Ensemble: Imaginary DOZZ Formula and Beyond

The scaling limit of the probability that $n$ points are on the same cluster for 2D critical percolation is believed to be governed by a conformal field theory (CFT). Although this is not fully understood, Delfino and Viti (2010) made a remarkable prediction on the exact value of a properly normalized three-point probability. It is expressed in terms of the imaginary DOZZ formula of Schomerus, Zamolodchikov and Kostov-Petkova, which extends the structure constants of minimal model CFTs to continuous parameters. Later, similar conjectures were made for scaling limits of random cluster models and O$(n)$ loop models, representing certain three-point observables in terms of the imaginary DOZZ formula. Since the scaling limits of these models can be described by the conformal loop ensemble (CLE), such conjectures can be formulated as exact statements on CLE observables. In this paper, we prove Delfino and Viti's conjecture on percolation as well as a conjecture of Ikhlef, Jacobsen and Saleur (2015) on the nesting loop statistics of CLE. Our proof is based on the coupling between CLE and Liouville quantum gravity on the sphere, and is inspired by the fact that after reparametrization, the imaginary DOZZ formula is the reciprocal of the three-point function of Liouville CFT. Recently, Nivesvivat, Jacobsen and Ribault systematically studied a CFT with a large class of CLE observables as its correlation functions, including the ones from these two conjectures. We believe that our framework admits sufficient flexibility to exactly solve the three-point functions for CLE observables with natural geometric interpretations, including those from this CFT. As a demonstration, we solve the case corresponding to three points lying on the same loop, where the answer is a variant of the imaginary DOZZ formula.

math-ph↗