Conformal field theory of dipolar SLE(4) with mixed boundary condition

通过量子曲面的共形焊接得到的SLE气泡测度

The SLE Bubble Measure via Conformal Welding of Quantum Surfaces

本文证明SLE气泡测度自然产生于两个Liouville量子引力圆盘的共形焊接，并给出分解公式，进而计算SLE气泡共形半径的矩，并将Liouville共形场论中的体-边界关联函数与量子边界长度和量子面积联系起来。

Conformal weldings of random surfaces: SLE and the quantum gravity zipper

We construct a conformal welding of two Liouville quantum gravity random surfaces and show that the interface between them is a random fractal curve called the Schramm-Loewner evolution (SLE), thereby resolving a variant of a conjecture of Peter Jones. We also demonstrate some surprising symmetries of this construction, which are consistent with the belief that (path decorated) random planar maps have (SLE-decorated) Liouville quantum gravity as a scaling limit. We present several precise conjectures and open questions.

Conformal restriction, highest-weight representations and SLE

We show how to relate Schramm-Loewner Evolutions (SLE) to highest-weight representations of infinite dimensional Lie Algebras using the conformal restriction properties studied by Lawler, Schramm and Werner in the paper arXiv:math.PR/0209343. This confirms the prediction from theoretical physics and conformal field theory that two-dimensional critical systems are related to such degenerate representations.

随机表面的共形焊接与SLE配分函数

SLE partition functions via conformal welding of random surfaces

本文通过共形焊接典型LQG表面，展示了SLE配分函数如何作为随机模量密度中的乘法因子出现，覆盖多重SLE、虚几何流线和边界格林函数，提供了构造SLE配分函数的替代途径。

Integrability of SLE via conformal welding of random surfaces

We demonstrate how to obtain integrable results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called $\mathrm{SLE}_κ(ρ_-;ρ_+)$. Our proof is built on two connections between SLE, LCFT, and mating-of-trees. Firstly, LCFT and mating-of-trees provide equivalent but complementary methods to describe natural random surfaces in LQG. Using a novel tool that we call the uniform embeddin…

On Convergence to SLE$_6$ I: Conformal Invariance for Certain Models of the Bond-Triangular Type

Following the approach outlined in [26], convergence to SLE$_6$ of the Exploration Processes for the correlated bond-triangular type models studied in [11] is established. This puts the said models in the same universality class as the standard site percolation model on the triangular lattice [27]. In the context of these models, the result is proven for all domains with boundary Minkowski dimension less than two. Moreover, the proof of convergence applies in the context of general critical 2D percolation models and for general domains, under the stipulation that Cardy's Formula can be establi…

量子圆盘与多重SLE的共形焊接：非简单情形

Conformal welding of quantum disks and multiple SLE: the non-simple case

本研究将量子圆盘的共形焊接推广至非简单情形，基于SLE与刘维尔共形场论的联系构造了多重SLE，并证明了其配分函数的存在性与性质。

Conformal fields, restriction properties, degenerate representations and SLE

In this note, we show how to relate the Schramm-Loewner Evolution processes (SLE) to highest-weight representations of the Virasoro Algebra. The conformal restriction properties of SLE that have been recently studied in the paper arXiv:math.PR/0209343 by G. Lawler, O. Schramm and the second author play an instrumental role. In this setup, various considerations from conformal field theory can be interpreted and reformulated via SLE. This enables to make a concrete link between the two-dimensional discrete critical systems from statistical physics and conformal field theory.

Conformal invariance in random cluster models. II. Full scaling limit as a branching SLE

In the second article of this series, we establish the convergence of the loop ensemble of interfaces in the random cluster Ising model to a conformal loop ensemble (CLE) --- thus completely describing the scaling limit of the model in terms of the random geometry of interfaces. The central tool of the present article is the convergence of an exploration tree of the discrete loop ensemble to a branching SLE$(16/3,-2/3)$. Such branching version of the Schramm's SLE not only enjoys the locality property, but also arises logically from the Ising model observables.

The SLE loop via conformal welding of quantum disks

We prove that the SLE$_κ$ loop measure arises naturally from the conformal welding of two $γ$-Liouville quantum gravity (LQG) disks for $γ^2 = κ\in (0,4)$. The proof relies on our companion work on conformal welding of LQG disks and uses as an essential tool the concept of uniform embedding of LQG surfaces. Combining our result with work of Gwynne and Miller, we get that random quadrangulations decorated by a self-avoiding polygon converge in the scaling limit to the LQG sphere decorated by the SLE$_{8/3}$ loop. Our result is also a key input to recent work of the first and third coauthors on …

Coupling of Gaussian free field with general slit SLE

We consider a coupling of the Gaussian free field with slit holomorphic stochastic flows, called ($δ,σ$)-SLE, which contains known SLE processes (chordal, radial, and dipolar) as particular cases. In physical terms, we study a free boundary conformal field theory with one scalar bosonic field, where Green's function is assumed to have some general regular harmonic part. We establish which of these models allow coupling with ($δ,σ$)-SLE, or equivalently, when the correlation functions induce local ($δ,σ$)-SLE martingales (martingale observables).

SLE martingales in coset conformal field theory

Scharmm-Loewner evolution (SLE) and conformal field theory (CFT) are popular and widely used instruments to study critical behavior of two-dimensional models, but they use different objects. While SLE has natural connection with lattice models and is suitable for strict proofs, it lacks computational and predictive power of conformal field theory. To provide a way for the concurrent use of SLE and CFT we consider CFT correlation functions which are martingales with respect to SLE. We establish connection between parameters of Schramm-Loewner evolution on coset space and algebraic data of coset…

SLE_k growth processes and conformal field theories

SLE_k stochastic processes describe growth of random curves which, in some cases, may be identified with boundaries of two dimensional critical percolating clusters. By generalizing SLE_k growths to formal Markov processes on the central extension of the 2d conformal group, we establish a connection between conformal field theories with central charges c_k=(3k-8)(6/k-1)/2 and zero modes -- observables which are conserved in mean -- of the SLE_k stochastic processes.

Conformal restriction: the chordal case

We characterize and describe all random subsets $K$ of a given simply connected planar domain (the upper half-plane $\H$, say) which satisfy the ``conformal restriction'' property, i.e., $K$ connects two fixed boundary points (0 and $\infty$, say) and the law of $K$ conditioned to remain in a simply connected open subset $D$ of $\H$ is identical to that of $Φ(K)$, where $Φ$ is a conformal map from $\H$ onto $D$ with $Φ(0)=0$ and $Φ(\infty)=\infty$. The construction of this family relies on the stochastic Loewner evolution (SLE) processes with parameter $κ\le 8/3$ and on their distortion under …

Conformal removability of non-simple Schramm-Loewner evolutions

We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $κ\in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_κ$ is a.s. connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_1,\ldots,U_n$ with $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. We show that the range of an SLE$_κ$ for $κ\in {\mathcal K}$ is a.s…

Conformal transformations and the SLE partition function martingale

We present an implementation in conformal field theory (CFT) of local finite conformal transformations fixing a point. We give explicit constructions when the fixed point is either the origin or the point at infinity. Both cases involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We use this to build coherent state representations and to derive a close analog of Wick's theorem for the Virasoro algebra. This allows to compute the conformal partition function in non trivial geometries obtained by removal of hulls from the upper half plane. This is then applied to stochastic…

Conformal Field Theories of Stochastic Loewner Evolutions

Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems f…

Conformal invariance of planar loop-erased random walks and uniform spanning trees

We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D…

Conformal invariance of boundary touching loops of FK Ising model

In this article we show the convergence of a loop ensemble of interfaces in the FK Ising model at criticality, as the lattice mesh tends to zero, to a unique conformally invariant scaling limit. The discrete loop ensemble is described by a canonical tree glued from the interfaces, which then is shown to converge to a tree of branching SLEs. The loop ensemble contains unboundedly many loops and hence our result describes the joint law of infinitely many loops in terms of SLE type processes, and the result gives the full scaling limit of the FK Ising model in the sense of random geometry of the …

多连通域中高斯自由场的共形场论

Conformal field theory of Gaussian free fields in a multiply connected domain

该研究基于高斯自由场构建多连通域下中心荷$c≤1$的共形场论框架，引入广义Eguchi-Ooguri方程推导Ward方程显式形式，借助BPZ方程实现对应SLE的共形场论表述并构造其鞅可观测量。

Conformal welding of quantum disks

Two-pointed quantum disks with a weight parameter $W > 0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their boundary lengths gives another quantum disk decorated with an independent chordal $\mathrm{SLE}_κ(ρ_-;ρ_+)$ curve. This is the finite-volume counterpart of the classical result of Sheffield (2010) and Duplantier-Miller-Sheffield (2014) on the welding of infinite-area two-pointed quantum surfaces called quantum wedges, which is fundamental to the mating-of-trees…

Non-simple SLE curves are not determined by their range

We show that when observing the range of a chordal SLE$_κ$ curve for $κ\in (4,8)$, it is not possible to recover the order in which the points have been visited. We also derive related results about conformal loop ensembles (CLE): (i) The loops in a CLE$_κ$ for $κ\in (4,8)$ are not determined by the CLE$_κ$ gasket. (ii) The continuum percolation interfaces defined in the fractal carpets of conformal loop ensembles CLE$_κ$ for $κ\in (8/3, 4)$ (we defined these percolation interfaces in previous work, and showed there that they are SLE$_{16/κ}$ curves) are not determined by the CLE$_κ$ carpet th…

SLE, CFT and zig-zag probabilities

The aim of these notes is threefold. First, we discuss geometrical aspects of conformal covariance in stochastic Schramm-Loewner evolutions (SLEs). This leads us to introduce new ``dipolar'' SLEs, besides the known chordal, radial or annular SLEs. Second, we review the main features of our approach connecting SLEs to conformal field theories (CFTs). It is based on using boundary CFTs to probe the SLE hulls. Finally, we study zig-zag probabilities and their relation with CFT correlation functions. We suggest a putative link between the braiding of SLE samples and that of CFT correlation functio…

无穷小共形限制与 Virasoro 代数的酉化测度

Infinitesimal conformal restriction and unitarizing measures for Virasoro algebra

本文利用 SLE$_κ$ 环路测度构造了中心荷 $c(κ) \le 1$ 的 Virasoro 代数自然表示，引入非退化双线性 Hermite 形式证明其不定酉性，并基于无穷小共形限制性质完成证明。

SLE曲线量子长度的初等方法

An elementary approach to quantum length of SLE

本文通过初等方法证明了SLE_κ曲线的左右侧√κ-量子长度相等，利用了高斯乘法混沌与共形Minkowski内容的关系。

Radial SLE martingale-observables

We implement a version of radial conformal field theory in a family of statistical fields generated by central charge modification of the Gaussian free field and show that the correlation functions of such fields under the insertion of one-leg operator form a collection of radial SLE martingale-observables. We apply the renormalization procedure to the multi-point vertex fields with the neutrality condition to expand this collection and study its basic properties.

From CLE(κ) to SLE(κ,ρ)'s

We show how to connect together the loops of a simple Conformal Loop Ensemble (CLE) in order to construct samples of chordal SLE(κ) processes and their SLE(κ,ρ) variants, and we discuss some consequences of this construction.

刘维尔共形场论与量子拉链

Liouville conformal field theory and the quantum zipper

该研究针对源自刘维尔共形场论（LCFT）的刘维尔量子引力曲面，用LCFT和SLE给出曲面与曲线的显式描述，证明了LCFT的边界BPZ方程，还证明了κ>8时全平面SLE_κ的可逆性，并将证明LCFT曲面空间在共形焊接下封闭。

Uniqueness of the welding problem for SLE and Liouville quantum gravity

We give a simple set of geometric conditions on curves $η$, $\tildeη$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $φ\colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $η$ with $φ(η) = \tildeη$ then $φ$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $η$ is a non-space-filling SLE$_κ$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $φ$ is a h…

SLE环路测度与刘维尔量子引力

SLE Loop Measure and Liouville Quantum Gravity

本文利用刘维尔量子引力（LQG）研究SLE环路测度，证明了共形环路系综与SLE环路测度的等价性，建立了κ与16/κ的对偶性，并导出了电厚度矩的精确公式。

A multifractal boundary spectrum for SLE$_κ(ρ)$

We study SLE$_κ(ρ)$ curves, with $κ$ and $ρ$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed "angle" and determine the almost sure Hausdorff dimension of these sets. This is done by studying the moments of the spatial derivatives of the conformal maps $g_t$, by employing the Girsanov theorem and using imaginary geometry techniques to derive a correlation estimate.

多重弦SLE($κ$)与量子Calogero–Moser系统

Multiple chordal SLE($κ$) and quantum Calogero-Moser system

本文研究带标记边界点的多重弦SLE($κ$)，利用共形场论Coulomb气体方法构造两类配分函数解，并证明其对应量子Calogero–Moser本征态。

Stochastic geometry of critical curves, Schramm-Loewner evolutions, and conformal field theory

Conformally-invariant curves that appear at critical points in two-dimensional statistical mechanics systems, and their fractal geometry have received a lot of attention in recent years. On the one hand, Schramm has invented a new rigorous as well as practical calculational approach to critical curves, based on a beautiful unification of conformal maps and stochastic processes, and by now known as Schramm-Loewner evolution (SLE). On the other hand, Duplantier has applied boundary quantum gravity methods to calculate exact multifractal exponents associated with critical curves. In the first par…

Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries

We provide multiple Schramm-Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess-Zumino-Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE trac…

共形反常的普适性

Universality of the conformal anomaly

该研究证明局部实一维模函子的普适性质，将其与普适Liouville作用关联，为共形场论和Schramm-Loewner演化中中心荷的出现提供数学解释，还证明布朗环测度作为SLE典范限制函数的唯一性。

Gaussian free field and conformal field theory

In these mostly expository lectures, we give an elementary introduction to conformal field theory in the context of probability theory and complex analysis. We consider statistical fields, and define Ward functionals in terms of their Lie derivatives. Based on this approach, we explain some equations of conformal field theory and outline their relation to SLE theory.

A simple construction of Werner measure from chordal SLE$_{8/3}$

We give a direct construction of the conformally invariant measure on self-avoiding loops in Riemann surfaces (Werner measure) from chordal $\text{SLE}_{8/3}$. We give a new proof of uniqueness of the measure and use Schramm's formula to construct a measure on boundary bubbles encircling an interior point. After establishing covariance properties for this bubble measure, we apply these properties to obtain a measure on loops by integrating measures on boundary bubbles. We calculate the distribution of the conformal radius of boundary bubbles encircling an interior point and deduce from it expl…

里ouville量子引力中的径向共形焊接

Radial conformal welding in Liouville quantum gravity

研究扩展了里ouville量子引力中两个三点表面共形焊接的理论，得到径向SLE曲线并通过虚几何描述。

Hypergeometric SLE: Conformal Markov Characterization and Applications

This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter family of random curves---Hypergeometric SLE---denoted by hSLE$_κ(ν)$ for $κ\in (0,4]$ and $ν<κ-6$. The proof relies crucially on Dubédat's commutation relation [Dub07] and a uniqueness result proved in [MS16b]. The classification indicates that hypergeometric SLE is the only possible scaling limit of the interfaces in critical lattice models (conjectured or…

Multiple backward Schramm--Loewner evolution and coupling with Gaussian free field

It is known that a backward Schramm--Loewner evolution (SLE) is coupled with a free boundary Gaussian free field (GFF) with boundary perturbation to give conformal welding of quantum surfaces. Motivated by a generalization of conformal welding for quantum surfaces with multiple marked boundary points, we propose a notion of multiple backward SLE. To this aim, we investigate the commutation relation between two backward Loewner chains, and consequently, we find that the driving process of each backward Loewner chain has to have a drift term given by logarithmic derivative of a partition functio…

共形缝合与物质--Liouville--鬼场因子化

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

本文研究局部共形变换在GFF和SLE相关测度上的作用，主要创新是为SLE loop measure的缝合同胚提出Cameron--Martin型公式，并引入量子曲面共形缝合的新方法。

Ising Interfaces and Free Boundary Conditions

We study the interfaces arising in the two-dimensional Ising model at critical temperature, without magnetic field. We show that in the presence of free boundary conditions between plus and minus spins, the scaling limit of these interfaces can be described by a variant of SLE, called dipolar SLE(3). This generalizes a celebrated result of Chelkak and Smirnov and proves a conjecture of Bauer, Bernard and Houdayer. We mention two possible applications of our result.

Almost sure multifractal spectrum of SLE

Suppose that $η$ is a Schramm-Loewner evolution (SLE$_κ$) in a smoothly bounded simply connected domain $D \subset {\mathbb C}$ and that $ϕ$ is a conformal map from ${\mathbb D}$ to a connected component of $D \setminus η([0,t])$ for some $t>0$. The multifractal spectrum of $η$ is the function $(-1,1) \to [0,\infty)$ which, for each $s \in (-1,1)$, gives the Hausdorff dimension of the set of points $x \in \partial {\mathbb D}$ such that $|ϕ'( (1-ε) x)| = ε^{-s+o(1)}$ as $ε\to 0$. We rigorously compute the a.s. multifractal spectrum of SLE, confirming a prediction due to Duplantier. As corollar…

A glimpse of the conformal structure of random planar maps

We present a way to study the conformal structure of random planar maps. The main idea is to explore the map along an SLE (Schramm--Loewner evolution) process of parameter $ κ= 6$ and to combine the locality property of the SLE_{6} together with the spatial Markov property of the underlying lattice in order to get a non-trivial geometric information. We follow this path in the case of the conformal structure of random triangulations with a boundary. Under a reasonable assumption called (*) that we have unfortunately not been able to verify, we prove that the limit of uniformized random planar …

Imaginary geometry III: reversibility of SLE_κ for κ\in (4,8)

Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix κ\in (4,8) and let η be an SLE_κprocess from x to y in D. We prove that the law of the time-reversal of ηis, up to reparameterization, an SLE_κprocess from y to x in D. More generally, we prove that SLE_κ(ρ_1;ρ_2) processes are reversible if and only if both ρ_i are at least κ/2-4, which is the critical threshold at or below which such curves are boundary filling. Our result supplies the missing ingredient needed to show that for all κ\in (4,8) the so-called conformal loop ensembles CLE_κ are canonicall…

关于Loewner能量、共形限制与Werner自回避环测度的一个注记

A note on Loewner energy, conformal restriction and Werner's measure on self-avoiding loops

本文建立了黎曼球面上Jordan曲线Loewner能量与SLE_{8/3}型Werner测度间的表达式，证明基于共形映射下能量变换公式，类比SLE限制性质。

Conformal welding problem, flow line problem, and multiple Schramm--Loewner evolution

A quantum surface (QS) is an equivalence class of pairs $(D,H)$ of simply connected domains $D\subsetneq\mathbb{C}$ and random distributions $H$ on $D$ induced by the conformal equivalence for random metric spaces. This distribution-valued random field is extended to a QS with $N+1$ marked boundary points (MBPs) with $N\in\mathbb{Z}_{\ge 0}$. We propose the conformal welding problem for it in the case of $N\in\mathbb{Z}_{\ge 1}$. If $N=1$, it is reduced to the problem introduced by Sheffield, who solved it by coupling the QS with the Schramm--Loewner evolution (SLE). When $N \ge 3$, there natu…

A Fast Algorithm for Simulating the Chordal Schramm-Loewner Evolution

The Schramm-Loewner evolution (SLE) can be simulated by dividing the time interval into N subintervals and approximating the random conformal map of the SLE by the composition of N random, but relatively simple, conformal maps. In the usual implementation the time required to compute a single point on the SLE curve is O(N). We give an algorithm for which the time to compute a single point is O(N^p) with p<1. Simulations with kappa=8/3 and kappa=6 both give a value of p of approximately 0.4.

Schramm Loewner Evolution and Liouville Quantum Gravity

We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.

Conformal Loop Ensembles: The Markovian characterization and the loop-soup construction

For random collections of self-avoiding loops in two-dimensional domains, we define a simple and natural conformal restriction property that is conjecturally satisfied by the scaling limits of interfaces in models from statistical physics. This property is basically the combination of conformal invariance and the locality of the interaction in the model. Unlike the Markov property that Schramm used to characterize SLE curves (which involves conditioning on partially generated interfaces up to arbitrary stopping times), this property only involves conditioning on entire loops and thus appears a…

SLE and the free field: Partition functions and couplings

Schramm-Loewner Evolutions ($\SLE$) are random curves in planar simply connected domains; the massless (Euclidean) free field in such a domain is a random distribution. Both have conformal invariance properties in law. In the present article, some relations between the two objects are studied. We establish identities of partition functions between different versions of $\SLE$ and the free field with appropriate boundary conditions; this involves $ζ$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of $\SLE$ with the free field, show…

Gaussian free fields coupled with multiple SLEs driven by stochastic log-gases

Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and…

CFTs of SLEs: the radial case

We present a relation between conformal field theories (CFT) and radial stochastic Schramm-Loewner evolutions (SLE) similar to that we previously developed for the chordal SLEs. We construct an important local martingale using degenerate representations of the Virasoro algebra. We sketch how to compute derivative exponants and the restriction martingales in this framework. In its CFT formulation, the SLE dual Fokker-Planck operator acts as the two-particle Calogero hamiltonian on boundary primary fields and as the dilatation operator on bulk primary fields localized at the fixed point of the S…

有质量SLE$_4$与有质量调和探索子的标度极限

Massive SLE$_4$ and the scaling limit of the massive harmonic explorer

本文完整严格证明了有质量调和探索子的标度极限为共形协变、相对弦向SLE$_4$绝对连续的massive SLE$_4$，还证明合适边界条件下massive SLE$_4$可作为有质量高斯自由场的等值线耦合。