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 $\zeta$-regularization and the Polyakov-Alvarez conformal anomaly formula. We proceed with a construction of couplings of $\SLE$ with the free field, showing that, in a precise sense, chordal $\SLE$ is the solution of a stochastic"differential"equation driven by the free field. Existence, uniqueness in law, and pathwise uniqueness for these SDEs are proved for general $\kappa>0$.


Introduction
In 2d statistical mechanics, various important models such as percolation or the Ising model are expected (or proved) to have, at criticality, a conformally invariant scaling limit. The general notion of conformal invariance underlaid the development of Conformal Field Theory. In 1999, Schramm ([32]) proposed a precise version of the notion of conformal invariance in distribution. This consists in considering the distribution of an isolated path (typically, an interface in the model) connecting two boundary points of a simply connected planar domain (in the chordal case). One obtains a collection of distributions (µ SLE c ) on simple paths indexed by configurations, viz. domains with two marked boundary points; the conformal invariance requirement reads ϕ * µ SLE c = µ SLE ϕ(c) for a conformal equivalence ϕ : c → ϕ(c) (in other words, µ SLE is a covariant functor on the groupoid of configurations). Under the conformal invariance requirement and an additional domain Markov property, the collection of measures is classified by a positive parameter κ > 0 ( [32]).
Another type of conformally invariant scaling limits involves distributions. In the case of dimers, a height function is associated to a configuration, following the definition of Thurston; Kenyon ([16,17]) proved that in the case of the square lattice, with appropriate boundary conditions, this height converges in distribution to the massless free field. This is the Gaussian measure with covariance operator given by the Green kernel with Dirichlet boundary conditions. It can be seen as a random distribution (element of C ∞ 0 (D) ′ ) and is a basic object in constructive Field Theory ( [37,14]).
Temperley's bijection (see eg [19]) relates dimer configurations (tilings) to uniform spanning trees; branches of these trees are distributed as loop-erased random walks. In this discrete setting, two types of invariance principle may be considered: a branch converges to SLE 2 , as proved by Lawler,Schramm,Werner ([24]); the height function (at least in closely related set-ups) converges to a free field. Moreover, in the discrete setting, the height function determines the branches. The relation between the height of the tiling and the branches can be understood in terms of winding (of a curve running along the branches on the medial lattice, [19]), as first conjectured by Benjamini. It was then proved that the scaling limit of (the Peano path of) the tree is SLE 8 ( [24]). A question raised in [17] is whether the reconstruction of the tree from the height function, which is possible in the discrete set-up, can be carried out in the continuum. This will be answered affirmatively in Section 8.
In [33], Schramm and Sheffield prove that the zero level line of a discrete Gaussian free field on the triangular lattice (with appropriate boundary conditions) converges in distribution to chordal SLE 4 , as the mesh goes to zero. Trivially, the discrete free field converges to the continuous massless free field. The relation between chordal SLE 4 and the free field in the scaling limit, in particular in terms of couplings, is studied in details in the forthcoming [34]. A closely related situation is that of double domino tilings, that was conjectured by Kenyon to lie in the same universality class.
Work in progress relating the free field and SLE κ for κ = 4 has been reported by Scott Sheffield, based partly on the "winding" of SLE curves, seen as "flow lines of e ich ", h a free field, c a parameter. A notion of "local sets" of the free field, that applies to and extends the case of contour lines, has also been advanced.
In the examples of spanning trees and double domino tilings/discrete free field, two types of boundary conditions for fields appear: piecewise constant, with jumps at prescribed points; and a multiple of the winding of the boundary curve, again with jumps at prescribed points.
In the present article, we study relations between different variants of SLE and the free field with appropriate boundary conditions. The first main result concerns partition functions of SLE and the free field. For the free field, the partition function is defined in a natural way from its Gaussian structure: where m is the mean of the field. The Laplacian (in a bounded domain with Dirichlet condition on its smooth boundary) has a discrete spectrum going to infinity. In this situation, it is customary to resort to the ζ-regularized det ζ (∆). Partition functions of SLE are defined in a way compatible with its absolute continuity properties; the form of the partition function is Z SLE = det(∆) − c 2 times a conformally invariant tensor, where c is the central charge, that depends on κ. For many variants of SLE (chordal, radial, multiple chordal and radial), we match the boundary condition (involving the winding of the boundary) of the free field and the SLE variant in such a way that the identity of partition functions (see Theorem 5.3): holds. These are functions on the configuration space (a configuration being now equipped with a Riemannian metric, not merely a complex structure), that are well defined up to a multiplicative constant. The apparent mismatch of exponents of the Laplacian determinant is resolved via the Polyakov-Alvarez conformal anomaly formula. We note that these partition functions are also relevant to Conformal Field Theory (as correlators of primary fields) and Virasoro representations, as detailed in the forthcoming [11]. For earlier considerations on partition functions/CFT correlators in relation with SLE, see [12,2,21] and references therein.
When a field and an SLE are matched through their partition functions, one gets easily a "local" coupling restricted to the SLE and field seen in disjoint subdomains. This plays a rôle closely analogous to that of local commutation of SLEs considered in [9] (here, the two "commuting" objects are an SLE and a field, instead of two SLEs). In the context of SLE reversibility, Zhan showed in [42] how to lift local couplings to global couplings. In [10], it is shown how to extend this to the framework of local commutation, in which partition functions intervene naturally. We use similar techniques here to couple in a domain one SLE strand with a free field, in conjunction with Gaussian arguments and free field properties. One can also couple systems of commuting SLEs with a free field, in such a way that the different SLE strands are independent conditionally on the field; the identity of partition functions is instrumental at this point.
In order to elucidate the nature of these results, we introduce a notion of stochastic "differential" equation driven by the free field, by analogy with the classical framework of SDEs driven by linear Brownian motion. The relation between the SLE path and the free field does not involve a stochastic calculus, but is a condition that can be checked pathwise by an explicit construction. Informally, the field near the SLE trace converges to its boundary value given the position of the trace; some care has to be given to the fact that this boundary value is not defined on the trace (for κ = 4), which is rough. The equation reads: where φ is the field, K the Loewner chain and h an harmonic function depending on the chain (playing the rôle of SDE coefficients (σ, b)). As in the case of SDEs, there are conditions of adaptness w.r.t. a filtration; the filtration is indexed here by a partially ordered set (for inclusion) of open subsets of the domain.
In this context, we prove that chordal SLE κ , for κ > 0, is a solution of a stochastic equation driven by the free field, for which uniqueness in law holds (Theorem 7.3). The compatibility of the construction with various duality identities (reversibility for κ = 4) leads to a proof of pathwise uniqueness for general κ > 0 (Theorem 7.7). Both existence and pathwise uniqueness are local properties, hence hold in a variety of set-ups.
The article is organized as follows. In Section 1, we discuss discrete couplings. Section 2 contains results on the Brownian loop measure, in particular in relation with functional (ζ and Fredholm) determinants. Schramm-Loewner Evolutions are discussed in Section 3, with emphasis on partition functions. Section 4 gathers material on the free field. Relevant boundary conditions are introduced in Section 5, before establishing identities of partition function. Section 6 is concerned with local and global couplings. Stochastic equations driven by the free field are discussed in Section 7, where uniqueness results are proved. Some consequences (Temperley's bijection in the continuum and strong duality identities) are discussed in Section 8.

Discrete couplings
In this section, we discuss some examples of discrete couplings between a path converging to some SLE and a field converging to the free field, for motivation and intuition.

The Temperley coupling
A complete discussion would involve introducing a lot of material that will not be used later in the article, so we shall only sketch the construction, in the case of the square lattice. For a detailed treatment, see eg [19] and references therein.
Consider a portion of the square lattice approximating a simply connected domain: this gives a finite graph Γ. The outer boundary is seen as a single extended vertex. A spanning tree on the graph rooted at the extended vertex determines by planar duality a spanning tree on the dual graph Γ † . The two graphs may be oriented towards their root (this involves picking a root on the dual graph). From each vertex of the graph starts an outgoing edge in the tree. A square lattice with a twice smaller mesh can be constructed by superimposing the original graph Γ and its dual Γ † : this gives a graph DΓ, which is bipartite (black vertices of DΓ correspond to vertices and faces of Γ, white vertices to edges of Γ). To an oriented edge in the original graph Γ or the dual Γ † , one can associate an edge of the new graph DΓ: its initial half. Thus a spanning tree on Γ determines first a dual tree on Γ † and this two trees yield a collection of edges in DΓ. Being careful with the treatment of the boundary, this collection of edges is a perfect matching on the bipartite graph DΓ. This describes Temperley's bijection between uniform spanning trees on Γ and perfect matching of a related graph DΓ; see Figure 1, panel 2.
To the dimer configuration is associated an integer valued height function on vertices of the medial lattice (DΓ) † , as defined by Thurston. The variation of the height along an edge of (DΓ) † with a black vertex of DΓ to its left is (−3) if it crosses a matched edge of DΓ and 1 otherwise. The data of the original tree, the dimer configuration, and the (admissible) height function are equivalent. Kenyon proved ( [16,17]) that, as the mesh of the lattice goes to zero, the height converges in distribution (in a weak topology) to the free field, with boundary value given by a multiple of the winding of the boundary, measured from the root.
In order to make the connection with SLE, it seems convenient to modify the situation as follows. Picking a point y inside the domain, one can consider the branch of the tree from y to the boundary. The branch hits the boundary at x, it is also convenient to condition on x. It is known that the branch is distributed as a Loop Erased Random Walk and that its scaling limit is radial SLE 2 from x to y ( [24]). Given the tree, one obtains a dimer configuration by Temperley's bijection; one can create a hole by sliding the dimers along the LERW, a construction introduced by Kenyon; see Figure 1. The height function has now an additive monodromy around the puncture y (ie is additively multiply valued, picking an additive constant 4 when traced counterclockwise along a simple loop around y). We stress however that the dimer configuration is not uniform on tilings of the punctured graph (it is uniform on a set of admissible matchings).
In this context, one also gets a natural definition of the partition function of the LERW from y to x: the number of spanning trees of Γ such that the branch starting from y exits at x. The total number of spanning trees in Γ is, by the well known Matrix Tree Theorem, det(∆ Γ ) where ∆ Γ is the combinatorial Laplacian on Γ with Dirichlet conditions on the boundary (so that it is invertible). The probability that the LERW exits at x is the probability that the underlying random walk exits at x, ie Harm Γ (y, {x}). This gives a partition function: This is coherent with the partition functions of SLE in the continuum that we will consider later on; in the case of radial SLE 2 , it is written: An important point is that the partition function accounts for an ambient "environment", through det(∆). Justification for including this contribution will be given later, in particular in terms of absolute continuity properties.

Discrete free field and double dominos
For continuity of the discussion, we start with the case of double domino dimers ( Figure 2). One possible setup is as follows. Consider a portion of the square lattice Γ, from which a boundary square is deleted (say a rectangle with odd sides minus a corner). One can sample uniformly a domino tiling of this domain. It is associated to a height function, that converges to the free field, with boundary condition given by a multiple of the winding of the boundary, jumping at the excised corner. One can proceed likewise by deleting another corner, sampling independently the domino tiling. The superposition of the two tilings consists of doubled dimers, closed loops (nested), and one open path connecting the two excised corners. Kenyon conjectured that this converges to chordal SLE 4 .
Each tiling is associated to an height function. The difference h of the two height functions is such that h jumps by ±4 when crossing a loop or the open path (and is constant along these paths). Hence one can think of the superposition of tilings in terms of contour lines of h. A double application of the invariance principle in [16,17] shows that h converges to a free field with piecewise constant boundary conditions, jumping at the excised corners (the winding contributions cancelling out when taking the difference of the two height functions).
A closely related situation is completely analyzed in [33]. Consider a portion Γ of the triangular lattice, approximating a simply connected domain. One can project orthogonally the continuous free field on the space of functions that are piecewise constant on the triangulation. As an alternative description of the discrete field, one can take the Gaussian measure (on functions on the triangulation) with covariance given by the discrete Green kernel with Dirichlet boundary conditions; the mean of the field is the harmonic extension of a piecewise constant function on the boundary, with jumps at two marked points. From the field, one gets a coloring of vertices (black for positive values of the field, white for negative values). There is an interface running in the domain between black and white vertices, connecting the two marked boundary points. As the mesh of the lattice goes to zero, this interface converges in distribution to chordal SLE 4 , for a precisely tuned boundary condition. The discrete field trivially converges to the continuous free field with piecewise constant boundary conditions.
In the double domino model, the field jumps by ±4 at the marked point and, according to [17], converges to 4 √ 2 √ π times a standard free field. Normalizing the field, we find a jump of π 2 . In the case of the discrete Gaussian free field, it is mentioned in [33] that the jump is 2 π 8 for a normalized field. In the continuum, we will consider a jump of π 2 πκ , κ = 4. All these expressions are thus coherent. In these situations, it is quite natural to consider multiple paths created by, say, excising an even number of boundary squares in the double domino model or flipping the boundary condition an even number of times for the discrete free field. This is analyzed combinatorially in [18]; see also [9,7]. In the context of the discrete free field, the Gaussian structure gives a natural definition for the partition function: where m, m ∇ is the (discrete) Dirichlet energy of the mean of the field (which is also the state of minimal energy, under the boundary condition constraints). We will consider regularized versions of this in the continuum.

Loop measure, determinants
In order to relate quantities arising from SLE and free field densities, we need to introduce the loop measure [22,26] and relate the masses of some sets under this measure to functional determinants of two types: Fredholm and ζ-regularized. Some relations between loop measures, free fields and functional determinants are discussed in [27]. Most of the discussion here can be carried out at the level of Markov chains ( [25,27]) or diffusions on manifolds; only the conformal anomaly formula is specific to the two-dimensional case.
Consider the (positive) Laplacian ∆ on a compact manifold M with boundary, with Dirichlet condition on the boundary (more generally, the negative generator of a diffusion). Following Ray-Singer, one attaches to ∆ a ζ-function: where P t = e −t∆ is the transition kernel for Brownian motion (running at speed 2), trace class in L 2 (M ) for t > 0, and Tr(P t ) = M P t (x, x)dvol(x). This is absolutely convergent for ℜs > (dim M )/2. If λ 1 ≤ λ 2 ≤ · · · is the spectrum of ∆, ζ ∆ (s) = n≥1 λ −s n . Under regularity assumptions on the boundary, ζ ∆ has a meromorphic extension to C, in particular regular at 0, so that one can define the spectral invariant: Note that P t (x, y)dvol(y) is the disintegration of the measure on paths starting from x ′ , stopped at t, w.r.t. the endpoint y. Let us denote W t x→y this subprobability measure (killing on the boundary). This gives another justification to introduce the (rooted) loop measure ( [26]) (and also to normalize it this way): It is a measure on rooted loops, ie on functions δ : for ℜ(s) > 1. Again formally, we have: The divergence of the RHS comes from small loops. We will be able to phrase identities by various inclusionexclusion arguments that cancel the small loops.
Of particular interest is the following quantity: if D is a domain, K 1 , K 2 are disjoint subsets of D (say connected closed sets), then define: (D); ψ(K 1 ), ψ(K 2 )) = m l (D; K 1 , K 2 )). Typically, K i is a crosscut or a hull attached to ∂D. We have: Proof. Under these assumptions, the ζ functions have a meromorphic extension to C and are regular at 0. Then: using the restriction property. It is easy to see that the measure µ loop D restricted to loops intersecting both K 1 and K 2 is finite (with mass m l (D; K 1 , K 2 )), and that the RHS is an entire function in s (the mass of loops connecting K 1 to K 2 in a short time t is of order exp(−dist(K 1 , K 2 ) 2 /t), from the Varadhan large deviation estimate for the heat kernel). So taking the derivative at 0 gives the result.
Note that the LHS is defined under more general assumptions (K 1 , K 2 have positive capacity) than the RHS.
Another expression in terms of Fredholm determinants det F is also useful. Again D is a domain; K 1 , K 2 are smooth curves (typically, crosscuts). The metric on D induces a length on K 1 , K 2 . Let us define a map T 12 : and T 21 : L 2 (K 2 ) → L 2 (K 1 ) is defined similarly. These operators have smooth kernels and T = T 12 T 21 is trace class on L 2 (K 1 ).

Proposition 2.2. Under the above assumptions:
Proof. We have the expansion (see eg [38], Chapters 2-3): More precisely, T 12 , T 21 are strictly uniformly subMarkov kernel, ie: where p > 0 is the infimum of probabilities that a particle starting from K i hits ∂D before K 2−i , i = 1, 2. It follows that: sup so that Tr((T 12 T 21 ) n ) = O((1 − p) 2n ), and the determinant expansion is legitimate.
This expansion counts loops intersecting both K 1 and K 2 (T 12 corresponds to paths starting from x ∈ K 2 and stopped when they hit K 1 ). We just have to check that the count is correct.
Let us go back to the rooted loop measure. For a rooted loop δ, consider the sequence of successive hits of K 1 and K 2 by δ; this can be represented by an alternating sequence σ of 1's and 2's. We consider the parabolic harmonic measure P H D\K1 (x, t, y)dtdl(y) seen for x ∈ D \ K 1 , for t > 0, y ∈ K 1 . The mass under W t x of loops that start from x, hit K 1 , then K 2 , and return to dA(x) at time t without returning to K 1 is: By the semigroup property: So integrating out x, one gets: which can be rewritten as (integrating t 1 ): If we add the symmetrized term (obtained by interchanging K 1 , K 2 ), we get: Proceeding as above, one gets the following expression for µ loop r {length(σ) = 2n, 2n + 1}, after having integrated out the root x: with cyclical indexing (y n+1 = y 1 ). Plainly, taking a cyclic permutations of indices does not change the value of that expression. So averaging over the n cyclic permutations, one gets: which concludes the proof.
To illustrate the result and to fix normalization, we embark on a sample computation, similar to the one in [26]. Let D = H, K 1 a small hull near 0 (with half-plane capacity 2t, see Section 3.1), K 2 the unit semicircle. The harmonic measure in the semidisk D + = D ∩ H (on the semicircle) can be obtained from the harmonic measure in the disk by a reflection principle argument: and classically Harm D (z, y) = 1 2π ℜ y+z y−z . So for z close to 0, On the other hand, starting from y ∈ U, if X τ is Brownian motion stopped on exiting H \ K 1 , E(ℑ(X τ )) ≃ −2tℑ(y −1 ). It follows that on L 2 (K 2 ), the operator T 21 T 12 has kernel: It follows that: Let x ∈ (0, 1). Consider the homography: ϕ(z) = 1−xz x−z . It permutes −1, 1, hence preserves K 2 (an hyperbolic geodesic); moreover ϕ(0) = x −1 , ϕ ′ (0) = x −2 − 1. Besides, it is easy to see that ψ(z) = z + z −1 is the conformal equivalence H \ D → H with hydrodynamic normalization at ∞. Then: Hence: We now discuss the Polyakov-Alvarez ([29, 1, 28]) conformal anomaly formula, that describes the transformation of det ζ (∆) under a conformal change of metric. The key point is that under a change of metric g 0 → g = e 2σ g 0 , the Laplacian transforms as ∆ → e −2σ ∆ 0 (this is particular to dimension 2). It is then a matter of short time heat kernel asymptotics (Pleijel-Minakshisundaram in the bulk, McKean-Singer near the boundary). We also give a simplified version in the case of planar domains; this expression is used in [28] to prove that among simply connected Riemannian surfaces with given boundary length, flat disks have extremal Laplacian determinants. 1. Let (M, g 0 ) be a compact surface with boundary, g = e 2σ g 0 . Then: where K 0 is the Gauss curvature and k 0 is the geodesic curvature of the boundary for g 0 .
2. Let D be a planar simply connected domain with smooth boundary and Euclidean metric, D the unit disk. Let ϕ : D → D be a conformal equivalence, σ = log |ϕ ′ |. Then: Proof. One deduces 2 from 1 as follows. It is equivalent to consider the Euclidean Laplacian in D and to consider the Laplacian in D with pulled back metric g = |ϕ ′ | 2 g 0 , so that the conformal factor is σ = log |ϕ ′ |.

Chordal SLE
First we recall some definitions and fix notations. We briefly discuss here chordal SLE in the upper half-plane H, from a real point to ∞. Chordal SLE in other (simply connected) domains are obtained by conformal equivalence. We will use chordal SLE both in itself and as a reference measure. For general background on SLE, see [31,40,23].
Consider the family of ODEs, indexed by z in H: with initial conditions g 0 (z) = z, where W t is some real-valued (continuous) function. These chordal Loewner equations are defined up to explosion time τ z (possibly infinite). Define: Then (K t ) t≥0 is an increasing family of compact subsets of H; moreover, g t is the unique conformal equivalence H \ K t → H such that (hydrodynamic normalization at ∞): The coefficient of 1/z in the Laurent expansion of g t at ∞ is by definition the half-plane capacity of K t at infinity; this capacity equals (2t).
is a standard Brownian motion, then the Loewner chain (K t ) (or the family (g t )) defines the chordal Schramm-Loewner Evolution with parameter κ in (H, x, ∞). The chain K t is generated by the trace γ, a continuous process taking values in H, in the following sense: The trace is a continuous non self-traversing curve. It is a.s. simple if κ ≤ 4 and a.s. space-filling if κ ≥ 8 ( [31]). The boundary of a nonsimple SLE κ (κ > 4) is locally absolutely continuous w.r.t. SLEκ,κ = 16/κ (SLE duality, [10,41]).
Note that chordal SLE depends only on two boundary points, and radial SLE depends on one boundary and one bulk point. In several natural instances, one needs to track additional points on the boundary. This has prompted the introduction of SLE κ (ρ) processes in [22], generalized in [6]. The driving Brownian motion is replaced by a semimartingale which has local Girsanov density w.r.t. the original Brownian motion. These turn out to be technically useful processes (eg [10]).
In the chordal case, let ρ be a multi-index, i.e. : Let k be the length of ρ; if k = 0, one simply defines SLE κ (∅) as a standard SLE κ . If k > 0, assume the existence of processes (W t ) t≥0 and (Z and such that the processes (W t − Z (i) t ) do not change sign. Then we define the chordal SLE κ (ρ) process starting from (w, z 1 , . . . z k ) as a chordal Schramm-Loewner evolution the driving process of which has the same law as (W t ) as defined above, with W 0 = w, Z

Partition functions
In this subsection we introduce partition functions of SLE (a predefinition is in [10]), and give some basic properties. These partition functions are null vectors of some canonical Virasoro representations ( [11]). They correspond to some correlators in Conformal Field Theory.
We begin with an informal discussion to motivate the definition (see eg [2] and references therein for related topics). Consider the Ising model on, say, the triangular lattice. Let D be a (simply connected) portion of the triangular lattice with boundary vertices partitioned in two arcs ∂ − , ∂ + . A spin configuration ε consists of an assignment of ± spins to vertices of D, the spins being fixed on the boundary (± on ∂ ± ). The energy of a configuration is H(ε) = −β i∼j δ εi,εj . The partition function Z is defined as: Except for exceptional cases (torus), there is no explicit asymptotic expansion (as the mesh of the lattice goes to 0) of this.
In this situation, one can define an interface γ running between the connected clusters of negative spins attached to ∂ − and positive spins attached to ∂ + ; it connects the two marked boundary points x, y separating ∂ − , ∂ + . Consider the following relative situation: D ′ is another configuration which is identical to D in a neighbourhood U of x. The two models induce measures µ, µ ′ on γ U , the interface γ started from x stopped upon exiting U . This defines two new configurations, denoted simply by D \ γ U , D ′ \ γ U , in which the spins neighbouring γ U (which are fixed by construction) are taken as part of the boundary, and the marked point x is moved to the tip of γ U . Then it is easy to see that: This is only using the local form of the interaction (and the existence of a "Markovian" set of boundary conditions). If this converges to SLE (for critical β), the LHS is well-defined; this suggests looking for continuous analogues of Z compatible with Radon-Nikodỳm derivatives. This is achieved by the Here (Poisson excursion kernel, relative to the local coordinates z x , z y ). This can also be seen as a tensor. The local coordinate z x maps a neighbourhood of x in D conformally to a neighbourhood of 0 in H.
The following situation will be typical. Let c 1 = (D 1 , x 1 , y 1 ) be a configuration; δ a crosscut separating x 1 , y 1 , C a collar neighbourhood of δ at positive distance of x 1 , y 1 . Let c 2 = (D 2 , x 2 , y 2 ) be another configuration that agrees with D 1 in the collar C. One can generate hybrid configurations c ij = (D ij , x i , y j ), such that c ij agrees with D i left of δ and with D j right of δ, i, j ∈ {1, 2}. The local coordinates at x i of D i1 , D i2 are the same, symmetrically at y i . The metrics of D i1 , D i2 agree to the left of δ (and a bit further), symmetrically for D 1j , D 2j . Then one can form the ratio: The point is that this is independent of choices of local coordinates (as tensor dependences cancel out) and of metrics (due to the local form of the Polyakov-Alvarez formula, Proposition 2.3). This is an analogue with boundary of the "train track" argument of [21].
The definition of the partition function is now justified by the following result.
Proof. This is proved in Proposition 3 in [10], based on results in [22]. The loop measure term is identified via Proposition 2.1.
There is a number of variants of SLE. A configuration c can consist of a Riemannian bordered surface (oriented, otherwise general topology) D with marked points x 1 , . . . , x n on the boundary and y 1 , . . . , y m in the bulk. Analytic coordinates (or merely 1-jets of local coordinates) at the marked points are given. A partition function Z is a positive function of such configurations. It has a tensor dependence on analytic coordinates (ie it transforms as i (dz i ) hi j |dw j | 2hj , z i local coordinate at x i , w j local coordinate at y j ), and depends on the metric as det ζ (∆ D ) −c/2 . The partition function can be seen as a section of a line bundle over a moduli space, as exposed in [12,20].
where γ τ is the SLE trace stopped upon exiting V .
This does not depend on the choice of ϕ, from the previous result. We proceed to show how some variants of SLE fit in this construction. An important situation is when the same partition function Z generates SLE's starting from different seeds : the two SLE's then satisfy local commutation ([9]). This imposes precise conditions on Z. Definitions of SLE's in general configurations from CFT correlators are considered in [12].
In order to express partition functions invariantly, we need to introduce some harmonic constructions: • If D is a domain, y ∈ D, x ∈ ∂D, the Poisson kernel P D (y, x) is a 1-form in x given by P D (y, x)dx = Harm D (y, dx).
• If D is a domain, x, y ∈ ∂D, the Poisson excursion kernel H D (x, y) is a 1-form in x, y given by: • If D is a simply connected domain, y ∈ D, ϕ : D → D, y → 0, a conformal equivalence, let H D (y) = |dϕ| |y (ϕ is unique up to a phase); this is a version of the conformal radius.
Let us use chordal SLE κ in H as reference measure (normalized at infinity as usual). Let z i 's be marked points (initially) on the real line), and Z i t = g t (z i ) − W t . Then a simple computation (eg Section 6.1 in [10]) shows that: Using this as a density produces an SLE κ (ρ) process, ρ = κβ 1 , . . . , κβ n . This process is invariant in distribution under homographies if ρ 1 + · · · + ρ n = κ − 6 (Lemma 3.2 in [9]). In terms of partition functions, this can expressed by: Z in a configuration c = (D, x 0 , x 1 , . . . , x n ) with the seed at x 0 and the convention ρ 0 = 2.
To treat the radial case, we use chordal results, together with a reflection argument.
The map g t : H \ K t → H can be extended by Schwarz reflection: g t (z) = g t (z). This is compatible with Loewner evolution. In the result above, one can take z i on the real line or a pair of conjugate z i , z ′ i = z i , pairing terms so as to get a real process. For instance, take two marked points y, y ′ = y; set ρ = ρ ′ = (κ−6)/2 (to get invariance under homographies). Then the resulting SLE is simply radial SLE κ aiming at y ∈ H.

Massless Euclidean free field
In this section, we gather a few facts on the massless (Euclidean) free field that will be needed later. We consider here only fields with Dirichlet boundary conditions. See [37,14] for background on the free field, [15] for Gaussian Hilbert spaces; also the survey [36].

Discrete free field
To illustrate some of the notions while avoiding technicalities, we consider first a discrete analogue of the situation (leading to finite dimensional Gaussian vectors).
Let Γ be a connected graph with some vertices marked as the boundary ∂Γ. Fields φ with Dirichlet boundary conditions are elements of H 1 0 (Γ), that is functions on vertices that vanish on the boundary, with centered Gaussian distribution relative to the Dirichlet inner product: where ∆ is the (positive) combinatorial Laplacian: Hence (φ(x)) x∈Γ is a Gaussian vector with distribution: where the normalization constant Z Γ is given by The free field with boundary conditions φ ∂ ∈ R ∂Γ is the Gaussian variable on the affine space {φ ∈ R Γ , φ |∂Γ = φ ∂ }, with covariance operator ∆ −1 . It is easy to see that the mean m of the field is the harmonic extension of φ ∂ to Γ. Furthermore, φ is distributed as φ = m + φ 0 , where φ 0 is a free field with (zero) Dirichlet boundary conditions. Finally, the partition function can be expressed as: One can also put weights on (unoriented) edges and vertices. Assume that the vertex set is partitioned in connected subsets V l , δ, V r , in such a way that no vertex of V l is adjacent to V r (one may also require: no vertex of δ has all its neighbours in δ). It is easy to see that: that are harmonic except on δ, Γ l is the graph with inner vertices V l , so that δ is part of its boundary. Functions in W are in bijection with functions on δ vanishing on δ ∩ ∂Γ (unique harmonic extension to Γ l , Γ d ). Thus there is an inner product on functions on δ induced by the inclusion W ֒→ H 1 0 (Γ). Let P r (resp. P l ) denotes the operator from W to functions on Γ r (resp. Γ l ) that associates to w ∈ W its unique harmonic extension to Γ r (with Dirichlet boundary condition on ∂Γ). Define a Neumann jump operator in End(W ) as follows: If (P w) ∈ H 1 0 (Γ) is the function equal to P l w (resp. P r w, w) on Γ l (resp. Γ r , δ), then This shows that a free field φ on Γ is the sum of three independent components: is a free field on Γ l (resp. Γ r ) with Dirichlet boundary conditions on ∂Γ ∪ δ and w = φ |δ is a Gaussian variable taking values in W with covariance operator N −1 (which is the restriction of ∆ −1 to δ). The restrictions of φ to Γ l , Γ r (not to confuse with φ l , φ r ) are independent conditionally on w: Chasing normalizing constants in Gaussian integrals, one also get the identity: Consider now the following situation: Γ 1 , Γ 2 are graphs as above that agree in a neighbourhood of Γ r . Let µ i be the discrete free field measure on H 1 0 (Γ i ), i = 1, 2; let R be the restriction φ → φ |Γr . We are interested in the Radon-Nikodỳm derivative: From the decomposition φ |Γr = φ r + P r w where φ r is independent of w = T φ and its distribution is the same for Γ 1 , Γ 2 , it is clear that: Looking at the marginal distribution T φ, we may as well assume that Γ 1 , Γ 2 only agree in a collar neighbourhood of δ. Since these distributions are Gaussian, we have: where N i , i = 1, 2, is the jump operator in each situation. We note that: which is better suited to scaling limits. Also, while N 1 , N 2 will converge to first order pseudodifferential operators, N 1 − N 2 will converge to a smoothing kernel operator.
We conclude with a computation of partition functions, that is an elementary discrete analogue of Lemma 6.3. Let Γ 1 , Γ 2 be graphs that agree in a neighbourhood of a cut δ, with boundary conditions φ ∂1 , φ ∂2 . Let Γ ij , i, j ∈ {1, 2}, be the graph that agrees with Γ i (resp. Γ j ) left (resp. right) of δ, with induced boundary conditions φ ∂ij . Consider the measure on φ |δ : and the one-sided partition function Z Γ l i ,φ ∂ i ,φ δ for the field left of δ with boundary conditions φ ∂i on the part of ∂ i left of δ and φ δ on the cut δ; Z Γ r i ,φ ∂ i ,φ δ is defined similarly. Chasing definitions, one gets the decomposition It is then immediate that (T denotes the restriction to δ): where µ ij is the measure of the discrete free field in Γ ij with boundary conditions ∂ ij . One deduces the identity: which is better suited to scaling limits (see Lemma 6.3). Without additional difficulty, one gets a similar identity when δ is a region with m connected components in its complement.

Continuous free field
Let D be a bounded planar domain with Jordan boundary (allowing the bounding Jordan arc to have double points). The massless (Euclidean) free field is a random distribution φ, ie a random element of It has a Gaussian distribution, with mean 0 and covariance operator G D (the Green kernel with Dirichlet boundary conditions G D ). As in the case of Brownian motion, it is sometimes convenient (if only for psychological reasons) to take as model of the underlying probability space a "path space". The space C ∞ 0 (D) ′ is usually taken as Wiener space; we will also use a tighter H −s (D) for some s > 0. Let where dA is the Lebesgue measure. We have a Poincaré inequality ||f || 2 L 2 ≤ (λ 1 ) −1 ||f || 2 H 1 , λ 1 the lowest eigenvalue of ∆ in D (Dirichlet boundary conditions), so that ||.|| H 1 is equivalent to the usual Sobolev norm. One feature of this norm is the conformal invariance: for ψ : D ′ → D a conformal equivalence.
By duality, one defines H −1 (D), a space of distribution with norm: where , is the evaluation of the distribution f against the test function g.
, it follows that: where ∆ −1 is given by convolution with the Green kernel G D .
One can give a first definition of the free field. Let (Ω, F , P ) be a probability space carrying a sequence (ε n ) n of iid centered, unit variance Gaussian variables, F the Borel algebra generated by cylinder events. Let e n be a Hilbert basis of H −1 (D). Denote φ(e n ) = ε n ∈ L 2 (Ω, P ). This maps the e n 's isometrically from H −1 (D) to L 2 (Ω, P ), so this can be extended to an isometric embedding H −1 (D) ֒→ L 2 (Ω, P ), denoted by φ(.). In the language of Gaussian processes, L 2 (Ω, P ) is a Gaussian Hilbert space; it is indexed by the Hilbert space H −1 ; it is also a special case of a Gaussian stochastic process, with index set H −1 and covariance function ρ(f, g) = f, g H −1 . Plainly, φ(f ) is a Gaussian variable for any f ∈ H −1 , and E(φ(f )φ(g)) = ρ(f, g) = f, g H −1 .
By duality, one can also think of A more explicit construction goes as follows. Let (e n ) be an orthonormal basis of H 1 0 (D) consisting of smooth functions. Formally, φ = n ε n e n , where (ε n ) n is a sequence of iid random variables; so that for f a test function, We have to determine a space in which this is a.s. convergent. For a smooth function f on D vanishing on the boundary, consider the Fourier decomposition: 2a jk sin(πjx) sin(πky).
Then ||f || 2 L 2 = j,k>0 |a jk | 2 and ||f || 2 let ||f || H −s be the norm of this bounded operator. The completion of H s 0 (D) for this norm is H −s (D). In terms of Fourier coefficients, , in such a way that e jk is an orthonormal basis of H 1 0 (D). It is easy to see that h converges a.s. in any Hence we can take Ω = H −s (D), F its (countably generated) Borel algebra, P the measure described above. As before, for any f ∈ H −1 , φ(f ) is defined as an element of L 2 (Ω, P ). It is easy to see that F is generated by these random variables. This reconstructs the Gaussian Hilbert space indexed by H −1 (D) from a Radon measure on H −s (D). Moreover, for a fixed φ ∈ Ω, f → φ, f

Field decompositions, trace
To prepare the description of the spatial Markov property, we describe decompositions of a free field in different areas of a domain D. More specifically, δ is a smooth crosscut separating D into two open subdomains D l , D r . Most of what is discussed there works for more general topologies (and also in higher dimension).
We have already seen a discrete analogue of the situation. We describe here first a Gaussian space approach, and then a pathwise construction. The main result is that one can define a trace of the free field on δ, a Gaussian variable in H −s (δ).

Decomposition in Gaussian spaces
We consider decompositions of free fields, from the point of view of Gaussian spaces. The index set H 1 0 (D) splits as: where W is the closure of functions that are harmonic on D l ⊔ D r , and as above can be identified as a function space on δ.
The Neumann jump operator is defined as: where: P l is the Poisson operator extending w to a harmonic function on D l with Dirichlet boundary condition on ∂D l ∩ ∂D; similarly P r is the Poisson operator on D r ; the crosscut ∂ is oriented (with D l to its left) and ∂ n is the normal derivative pointing to D l . We also denote by P lr the harmonic extension of w to D l ⊔ D r . The Green's formula readily shows that δ wN wdl = D |∇P lr w| 2 dA (dl is the length element on δ induced by the metric on D).
The Neumann operator N is a first order pseudodifferential operator.
(ψ(y)−ψ(x)) 2 by conformal invariance of the Green's function and explicit computations in H (this expression is Moebius invariant). This shows that We note that in the unit disk D = D(0, 1), say, the Dirichlet energy of P D w (harmonic extension of w, w a continuous function on the circle) can be expressed in terms of w as follows: and this is a conformally invariant expression. Hence we have: The last asymptotic follows from the local characterization of Sobolev spaces: for 0 < s < 1, in dimension n, an element f of H s is an element of L 2 such that: and this quantity gives an equivalent norm (modulo constant functions).
We can construct w from φ as follows. As noted earlier, the φ, f H 1 are elements of L 2 (Ω, P ). Take (e n ) a Hilbert basis of W , which is isometrically embedded in H 1 0 (D); one can choose the e n 's with smooth restriction on δ. Then consider: This converges a.s. in any H −s (δ) = H 1/2−n/2−s (δ) where s > 0, n = 1 (dimension of δ); it follows from the equivalence of norms ., N. L 2 and ||.|| 2 H 1/2 . We will give a more explicit construction later on. This can also be seen from the point of view of Wiener chaos decomposition (isometrically, the Fock space, [37], I.4). We have: (complexified, symmetrized tensor algebra). For the probability space associated with T δ ψ: W ⊙n and the isometric embedding L 2 (Ω δ , P δ ) ֒→ L 2 (Ω, P ) is induced by the isometric embedding W ֒→ H 1 0 (D) (second quantization). This embedding is also positive and preserves 1. It follows that it is induced by Note also that φ(f ) is defined for any f ∈ H −1 (D). There is a bounded operator, the Sobolev trace, Applying the transpose of the Sobolev trace operator to an element of H − 1 2 (δ) yields a distribution with support on δ that belongs to H −1 (D), hence can be evaluated against φ.

A pathwise construction
We discuss here a pathwise construction of the trace of the field on the crosscut δ. The issue is that an instance φ of the free field lies in H −s (D); the Sobolev trace theorem defines a bounded linear map from H s (D) to H s− 1 2 (δ) for s > 1 2 , thus cannot be applied here. But we are only concerned with defining a trace almost everywhere on H −s (D). For lightness of notation, we take D bounded, with smooth boundary, and so drop the loc subscripts.
Starting from φ ∈ H −s (D), we want to define a trace T φ in some function (or distribution) space on δ, almost everywhere in φ. For simplicity, we will define T φ in H −1 (δ). The topological condition that δ is a crosscut plays no role here, we can simply assume that δ is a smooth curve in D, possibly intersecting ∂D only at its endpoints. It is parameterized by t Let us consider a kernel operator K : where the kernel K is smooth, Markov (K ≥ 0, K1 = 1), and with finite range ε > 0 (K(t, z) = 0 when |δ t − z| ≥ ε). It follows that Kh is a smooth function on δ. We want to take ε ց 0. For this we need to estimate ||Kh|| H −1 (δ) .
Let ψ be a smooth test function on δ with compact support (ie vanishing in a neighbourhood of the endpoints), and f be a continuous function, F = t 0 f (s)ds. Then: , and after some manipulations: which we now specialise to f = Kφ.
First we estimate: given that G D (z 1 , z 2 ) = O(log |z 1 − z 2 |). One can think of the RHS as drawing z 1 (resp. z 2 ) from the distribution K(δ u , .)dA (resp. K(δ v , .)dA) and taking the expectation of G D (z 1 , z 2 ). Given z 1 , the probability that |z 2 − z 1 | < ηε is at worse of order η 2 (if δ u , δ v are very close). This gives a contribution of order ηε 0 ε −2 rdr = O(| log ε|) (fixing η = 1/10, say). If |z 1 − z 2 | ≥ ηε, one also get a contribution of order | log ε|. This gives the (crude) uniform estimate: . If K 1 , K 2 are two smooth Markov kernels as above with range ε, we have the following estimate if (We use here the fact that K i 1 = 1). Combining with the uniform estimate above, we get: Consider now a sequence of Markov kernels K n with range ε n = O(n −1−η ) for some η > 0, say. Then: which is summable. It follows that: For another choice of kernel sequence, one gets a.s. the same element. Since the K n 's are bounded linear operators, T δ is Borel measurable.

Remark 4.1. One can proceed similarly if δ is a boundary arc. Estimates of the Green kernel near the boundary show that the trace of the field on the boundary vanishes a.s., as it should (Dirichlet boundary conditions).
Since the K n 's are linear, it appears readily that T δ φ is also Gaussian, with covariance operator given by the restriction of the covariance operator of the free field (that is, G D ). Note that while the inverse of G is the Laplacian (a differential operator), the inverse of its restriction to δ is the Neumann jump operator (a first order pseudodifferential operator, which is nonlocal). This can be checked directly: if ψ is a smooth function on δ, say with compact support, then: δ G D (x, .)ψ(x)dl(x) is a continuous function on D that vanishes on ∂D, is harmonic on D \ δ, and its normal derivative across δ jumps by f (x) at x ∈ δ.
The Poisson kernel is smooth, so that P T δ φ defines a harmonic function away from δ. One can recover φ l (resp. φ r ) by φ l = φ |D l − P l T δ φ.

Markov property
Let us consider a free field in a domain D with Dirichlet boundary condition (probability space (Ω, F , P ), Ω = H −s (D)), and δ a crosscut that splits D into two subdomains D l , D r . (This works for more general topology). We describe here a spatial Markov property of the free field first pointed out by Nelson and Symanzik; we essentially follow [37] here. We also describe some marginal and conditional distributions that will be needed later on.
In Wiener chaos decomposition, the conditional expectation operator: is generated by the projection of H 1 0 (D) onto its closed subspace H 1 0 (U ) (this is a contraction). Similarly, E(.|F K ) corresponds to the projection H 1 0 (D) → H 1 0 (D \ K) ⊥ . Consider the trace T δ φ of φ on δ and the decomposition: φ = φ l + P T δ h + φ r , φ |D l = φ l + P l T δ φ.
We have the following description: 2. F D l and F Dr are independent conditionally on F δ .
In turn this follows from the locality of the inverse covariance (viz. the Laplacian). More precisely, if f has support in D r , we have to prove that p D l f = p ∂ f . It is enough to see that for g ∈ C ∞ 0 (D l ), p D l f, g L 2 = 0. Now: 3. This is an expression of the orthogonal decomposition :

Absolute continuity
We begin by recalling some general results (following here [5], see also [37], I.6). Let H be a Hilbert space, Q a trace class (symmetric, positive) covariance operator. Denote by dN Q the centered Gaussian measure on H with covariance Q (it exists since Q is trace class) and by dN m,Q the Gaussian measure with mean m, covariance Q. Then: 1. Let Q be a positive, trace class operator; M a symmetric operator such that Q 1/2 M Q 1/2 < 1; and m ∈ H. Then: Indeed, if f ∈ Q 1/2 (H), Q −1/2 f, h H is defined, and this mapping Q 1/2 (H) → L 2 (Ω, N Q ) can be completed given the isometry property: Also the properties 2,3 can be combined to give a more general expression: Let us now specialize this to the free field case. Let us take H = H −s (D), s > 0, with inner product: The covariance operator of the free field w.r.t. ., . L 2 is G D . In terms of ., . H −s : We can rewrite the expressions above in terms of , . L 2 given: which is simply saying that while the choice of H −s is arbitrary, the Cameron-Martin space H 1 0 (D) is canonical.
Let us consider the following situation. Two domains D 1 , D 2 agree in a subdomain containing a crosscut δ. The crosscut splits D 1 in D l,1 , D r and D 2 in D l,1 , D r (so that the two domains agree in a neighbourhood of D r . We can define the massless free field in D 1 , D 2 , and then restrict it to D r ; in this way, we get absolutely continuous measures. We will need an expression for the Radon-Nikodỳm derivative. As in the discrete case, the Markov property shows that the derivative factors through the trace of the field on δ: where µ i are the free field measures, R is the restriction to D r , T the trace on δ. So we have only to consider dT * µ2 dT * µ1 ; for this purpose it is enough to assume that D 1 , D 2 agree in a collar neighbourhood C of δ. Let N i be the Neumann jump operator on δ ⊂ D i , i = 1, 2. While N 1 , N 2 are first order pseudodifferential operators (on functions on δ), the difference N 2 − N 1 is a smoothing kernel. Indeed, This follows from the fact that G . (x, y) + 1 2π log |x − y| is smooth. Alternatively, G D (x, y) counts Brownian paths from x to y in D; decomposing w.r.t. the first exit of the collar C, one eliminates the contribution of paths staying in C (viz. G C ) which accounts for the singularity. In this fashion, one can represent (G D2 − G D1 ) |δ in terms of the Poisson kernel in C. It follows that w, (N 2 − N 1 )w L 2 is defined for all w ∈ H −s (δ).
It will also be convenient to identify the normalization constant in (4.4); here , a smooth kernel operator on δ. Recall that m l (D; K 1 , K 2 ) denotes the mass of loops in the loop measure in D that intersect both K 1 and K 2 .
The RHS does not depend on the choice of collar C, due to the restriction property of the loop measure.
Proof. We have to prove that: Given the multiplicative structure of the result, it is enough to prove it for D 2 = C ⊂ D 1 ; one may even assume that D = D 1 and C = D 2 agree on one side of δ. Taking K 1 = δ and K 2 = ∂C a crosscut "parallel" to δ, we have to prove: det F (N 1 N −1 2 ) = exp(−m l (D; K 1 , K 2 )) Let T 12 , T 21 be as in Proposition 2.2. Given the result there, we need only prove: In the reduced case, D is a domain, K 1 , K 2 are two disjoint crosscuts and where X is a Brownian motion (running at speed 2) started at x, killed when it hits ∂D at time τ ; σ 2 is the first time it hits K 2 ; and σ 1 is the first time it hits K 1 after σ 2 . Disintegrating w.r.t. X σ2 , X σ1 , we get: When x, y are on K 1 , this can be phrased more tersely as: which is what we needed.

Boundary conditions and partition functions
Quoting from [37], "While we will not use Gaussian variables of mean different from zero, they may well play a role in the future development of the theory". A free field in D with boundary conditions φ |∂D = φ ∂ is written as φ = m + φ 0 where φ 0 is a free field with Dirichlet boundary conditions and m, the mean of the field, is the harmonic extension of φ ∂ to D.
We will define here appropriate sets of boundary conditions that are continuous in Carathéodory-type topologies, and study partition functions of associated free fields.

Domain continuity
In what follows, we will be primarily interested in the chordal case, in which a configuration c = (D, x, y) consists of a simply connected domain D with two points x, y marked on the boundary. We begin with the case of smooth domains.
From the examples of the Temperley coupling and the discrete Gaussian Free Field, it is natural to consider free fields with the following boundary conditions in a configuration c = (D, x, y): where wind(y → w) is the winding of the boundary arc from y to w contained in (xy), (yx) respectively. We refer to this set of boundary conditions as (a, b) boundary conditions. Note that this is in general asymmetric in x, y (jump +πa at x, −πa − 2πb at y). In a configuration, there is a unique harmonic function satisfying the (a, b) boundary conditions. If c = (H, x, ∞), then h 0 (z) = a arg(z − x) and in a general smooth domain D, if ϕ is a conformal equivalence (D, x, y) → (H, 0, ∞), then Note that ϕ ′ does not vanish in the simply connected domain D, so that there is a single valued branch of arg(ϕ ′ ) in D.
One can generalize this to configurations (D, x 1 , . . . , x n , y), with jump πa i at x i , i ≤ n, and −π i a i −2πb at y; we call those (chordal) (a, b) boundary conditions.
It will be convenient to consider fields with (additive) monodromy. For a domain D with a marked point y in the bulk (puncture), we shall consider the affine space of additively multiply valued functions on D \ {y} that augment by a fixed quantity (the monodromy) along a counterclockwise circle around y and are locally bounded near y. • f has monodromy π(a 1 + · · · + a n ) + 2πb around y, There is a unique harmonic function h 0 satisfying these conditions, that can be expressed in the unit disk D, y = 0, as: In the SLE context, it is necessary to consider domains with rough boundaries. Then the winding of the boundary is no longer defined. However, boundary conditions for the free field intervene only through their harmonic extension. Hence one can use the covariance formula: where ϕ : D → D is a conformal equivalence preserving marked points, to define h D in general simply connected domains. This is up to an additive constant. If the boundary is rough everywhere and b = 0, there is no very natural way to fix the constant. On the other hand, it is enough for the boundary to be regular enough in a neighbourhood of, say, y to get an unambiguous definition.
We will mostly concerned with the behaviour of the boundary conditions under deformation of the domain. Proof. Let ϕ n be the unique conformal equivalence (D, 0, 1) → (D n , y, x).
, where h D depends implicitly on the φ −1 n (x n i ). Carathéodory convergence implies that ϕ n converges to ϕ n uniformly on compact sets of D; consequently ϕ ′ n also converges uniformly on compact sets. Given the hypothesis on the boundary around x, it is not hard to see (using eg the Loewner equations) that ϕ and its derivative converge uniformly in a neighbourhood of 1. This yields local uniform convergence of h cn .
The weak convergence of (R U ) * µ FF cn follows from the form of the characteristic functional: where f runs over C ∞ 0 (U ).

Dirichlet energy
We study here regularised Dirichlet energy of the harmonic extension of boundary conditions described above. One can think of this as a ground state energy. From the discrete situation, it is natural to define the partition function for the free field in D with boundary condition φ |∂D = φ ∂ , for φ ∂ a smooth (for now) function on ∂D as: The use of the regularized det ζ (∆) is customary in the physics literature, see eg [13]; notice that this introduces a metric in addition of the complex structure.
When φ ∂ is piecewise smooth (with jumps), the Dirichlet energy m, m H 1 diverges. We will use another (also customary, see eg [39]) regularization method, that requires introducing local coordinates (or rather 1-jets) at the marked points where φ ∂ jumps.
Consider a domain D with smooth boundary, φ ∂ a piecewise smooth function on ∂D with jumps δ i at x i (say in counterclockwise order, i = 1 . . . n), m its harmonic extension to D. Let z i be an analytic local coordinate at x i (i.e. z i (x i ) = 0, z i maps a neighbourhood of x i in D to a neighbourhood of 0 in H). Define: It is easy to see that this limit exists. There is a simple dependence on the choice of coordinates, that can be expressed by saying that the tensor where h i = δ 2 i /2π, is well defined. Similarly, for functions with monodromy 2πα around a bulk point y, one can use a local coordinate w at y and define: {x∈D,|w(x)|≥ε} |∇m| 2 dA(x) + 2πα 2 log(ε) so that setting h 0 = πα 2 2 , one defines a tensor: If c = (D, x, y) is a configuration with smooth boundary, one can define: the partition function for (a, b) boundary conditions. We proceed to evaluating this partition function.
2. Consider the case of (a, b) boundary conditions on a configuration (D, x ′ i , y ′ ). Let us consider first the case D = D, with marked point 0. Let ϕ : (D, x i , 0) → (D, x ′ i , y ′ ) be a conformal equivalence. Then it is easy to see that: We have to compute the regularized Dirichlet energy, up to an additive constant. The only new term is: We get the following expression: and we conclude by identifying the conformal invariants P D , H D in the unit disk.
By comparing with the partition functions of SLE, one obtains the following:

Variations of harmonic quantities
We are considering here a local boundary perturbation of a domain D (growth of a hull at a boundary point) and its effect on various harmonic quantities.
Let (D t ) t≥0 be a decreasing sequence of domains, x ∈ ∂D 0 , so that for any neighbourhood U of x, for t small enough, the domains D t agree outside of U . The domains are assumed to be Jordan (the boundary can be parameterized as a continuous, not necessarily simple function); by x ∈ ∂D, we mean a prime end that is a point.
Let G t be the Green kernel of D t . For any z, z ′ in D, z, z ′ ∈ D t for small t, and G t (z, z ′ ) decreases. It follows that G 0 (z, z ′ ) − G t (z, z ′ ) is positive and harmonic in the two variables in D t . Let t n ց 0 and a n ր ∞ such that a n (G 0 (z, z ′ ) − G n (z, z ′ )) has a positive limit for some x, y ∈ D. Then (Harnack principle) a n (G 0 (., z ′ ) − G n (., z ′ )) converges to a positive harmonic function in D; moreover this function extends continuously to 0 on the boundary except at x. It thus has to be proportional to the Poisson kernel P D,x as a function of z; by symmetry, the same is true for the z ′ variable. Hence for an appropriate choice of a n : lim a n (G(z, This argument carries to more general topologies. Let us compute in coordinates for the rest, with the usual SLE conventions. In the upper half-plane H, G(z, z ′ ) = − 1 2π log z−z ′ z−z ′ and P H,x (z) = − 1 π ℑ 1 z−x . For a family (H \ K t ) corresponding to conformal equivalences (g t ), we get G t (z, z ′ ) = G(g t (z), g t (z ′ )) and at t = 0, for a hull growing at x, Let m t (z) be the harmonic function in (H \ K t ) with boundary conditions: − π 2 a + b.(π + wind(∞ → .)) on (∞, γ t ) and h = π 2 a + b.(π − wind(. → ∞)) on (γ t , ∞). Then: where P t = P H\Kt,γt and P ′ t = ∂ ∂Wt P t (this is somewhat dependent on the Loewner convention).
6 Couplings of SLEs and free fields 6

.1 Local invariance of the free field under SLE dynamics
We have obtained partition function identities between (versions of) SLE on the one hand and the free field (with corresponding boundary conditions) on the other hand. We now show that this implies local identities in distribution between SLE and the free field, in a way closely analogous to local commutation statements (between two SLE's with the same partition function) in [9].
Let c = (D, x 1 , . . . , x n , y) be a configuration (y in the bulk), (a, b) a set of boundary conditions for the free field corresponding to SLE κ (ρ), as in Theorem 5.3. (This covers the chordal case, when the monodromy around y is 0). Among the marked points on the boundary, x 1 , . . . , x m are seeds of SLE (ρ i = 2 for i = 1, . . . , m). We denote by µ SLE c , µ FF c the respective distributions of the SLE system and the free field in c.
Up to now, we have considered boundary conditions for the free field up to an additive constant. We now need to fix this constant (in order to compare fields in different domains). For instance, one can require that φ(x + n ) = 0, where x n is not a seed. Let U be a connected open subset of D, not having any seed x i on its boundary. Let τ i be stopping times for each SLE such that γ τi i is a.s. at distance at least η > 0 of U . Let s 1 , . . . , s m be time parameters for each SLE (these are somewhat arbitrary, up to bicontinuous time change). Then c s is the configuration (D \ ∪ i γ τi i , γ 1,τ1 , . . . , γ m,τm , x m+1 , . . . , y). We have the following: Lemma 6.1. In the above situation, the following identity of distributions on C ∞ 0 (U ) ′ holds: where R U denotes the restriction from a subdomain of D (containing U ) to U .
In words: run the i-th SLE strand to time τ i ; this generates a random configuration c τ ; sample the free field in c τ , conditionally independently; restrict this field from the random domain D \ (∪ i γ τi i ) to the fixed domain U . Then the resulting mixture of Gaussian fields is again Gaussian, identical in distribution to the restriction of the free field in D to U .
Proof. A distribution ν on C ∞ 0 (U ) ′ is determined by the characteristic functional: where f runs over C ∞ 0 (U ). By general Gaussian properties, where µ FF cs has mean m s and covariance G s . Therefore we have to prove that: Since this has to hold also for all stopping times lesser than τ 1 , . . . , τ m , we have to prove that: is a martingale in s j (stopped at τ j ), the other times being fixed. Due to the (joint) Markov property, we can assume that the other times are 0; we have a single time parameter t = s j . Hence we are left with a stochastic calculus problem.
At this point it is rather convenient to compute in coordinates. So one can assume that D = H, the marked point x n (where the field is 0) is at infinity. In this situation, the mean m t of the field in c t is given by: and this has to be martingale (stopped at positive distance of z), under the SLE κ (ρ) evolution (conventionally, g t (x j ) = W t , the driving process). This can be checked directly; we give another argument that avoid computations.
We introduced radial SLE κ (ρ) by means of the local martingale (w.r.t. the reference measure, that is chordal SLE in (H, 0, ∞)): where two of the marked points are conjugates y, y, for appropriate coefficients. We can perturb the above situation (radial SLE κ (ρ)) by adding a marked point z with weight ρ z = κε, say. We compute: which thus a (local) martingale for radial SLE κ (ρ)'s. Taking the imaginary part, one gets a process proportional to m t (z).
This proves that t → m t , f L 2 is a martingale (stopped away from the support of f ). There is only one term in m t with quadratic variation, so one computes easily: where P is the Poisson kernel. Besides, we have computed that: which is thus a martingale, given that a 2 j = 2 πκ . Remark 6.2. The (pointwise) first moment martingale m t (z) was pointed out by Sheffield in the context of the free field, in the chordal case. The proof shows that E is, up to a multiplicative constant, the exponential martingale of m (see e.g. [30]).
As in [10], this can used to construct "local couplings". Define a function in γ s = γ s1 1 , . . . , γ sm m (SLE strands) and φ (a field). Then: The first line is obvious while the second one is a rephrasing of the lemma. This shows that ℓ.µ SLE c ⊗ µ FF c is a coupling of µ SLE c , µ FF c (this builds on the fact that we have a coupling restricted to γ τ , φ |U , that extends to a coupling of γ, φ using the Markov property of the SLE system and of the free field).
Let us analyze the density ℓ. Let δ i be crosscuts around x i , i = 1, . . . , m, that are at positive distance of each other and of all marked points. Let δ = ⊔ i δ i be the union of crosscuts and U the connected component of D \ δ with no seed x i , i = 1, . . . , m, on its boundary. From the Markov property of the free field, it readily appears that: i.e. the density factors through the trace on δ, which we now denote simply by T . Given the multiplicative identity: Then the Cameron-Martin formula yields: The middle term was analyzed in Lemma 4.4: so that: We denote by N s the Neumann jump operator on δ relative to the configuration c s , in such a way that w, N w L 2 (δ) is the Dirichlet energy of the harmonic extension of w to D (with zero boundary condition on ∂D). Observe that when different time indices evolve, the configuration varies in distinct connected components of D \ δ; thus where N = N 0 , N si = N 0,...,0,si,0,... , and w = (w 1 , . . . , w m ) identifies L 2 (δ) to ⊥ i L 2 (δ i ). So N s has no cross dependence in the s parameters. We proceed to show that this is also the case for N s T m s .
Observe that N w = i ∂ n P i w, where P i is the harmonic extension to the i-th connected component of D \ δ and ∂ n is outward pointing (on δ). Considerm the harmonic function on D \ δ that agrees with m on ∂D and vanishes on δ. Then m −m is harmonic on D \ δ, agrees with m on δ and vanishes on ∂D. Thus m−m = P T m. Now i ∂ n m = 0, since ∂ n m on δ i is counted once in each direction (and m is smooth across δ). Thus N T m = − i ∂ nm . In the varying situation,m s depends only on s i in the connected component having γ i in its boundary; this is due to the local character of the boundary condition. Hence N s T m s has no cross dependence in the s parameters. More precisely, (N T m) si − (N T m) vanishes on δ j , j = i.
Since N s has no cross dependence in the s parameters, one gets: N s1,...,sn − N = (N s1 − N ) + (N s1,s2 − N s1 ) + · · · + (N s1,...,sn − N s1,...,sn−1 ) and a similar identity holds for (N T m) s . This shows that: Putting things together, we get: where Z FF si = Z FF 0,...,0,si,0,... . Using the identity Z FF = Z SLE (Theorem 5.3), this translates into: It follows that at the stopping times τ i : The measure in the denominator is the measure induced on the stopped paths γ τi i by the SLE system; the measure in the numerator has the same m marginals, but these are independent.
is a coupling of µ SLE c , µ FF c such that γ τ1 1 , . . . , γ τm m are jointly independent conditionally on T φ. More precisely: Informally, all the interaction between the different SLE strands is carried by the field. This depends on the fact that the SLE strands are occulted from each other by the crosscuts δ i . Note that this conditional independence property is trivially satisfied in a coupling where the SLE's are determined by the field. The lemma hinges on and contains the following Gaussian integral evaluation: In the discrete setting, we gave an elementary version of the computation above, thinking of partition functions as (matrix elements of) transfer operators.
We have established Lemmas 6.1, 6.3 in the case where the SLE strand is absolutely continuous w.r.t. chordal SLE in a neighbourhood of its starting point x. However, it will be also useful to consider versions where it is absolutely continuous w.r.t. SLE κ (ρ − , ρ + ) starting from x, x − , x + . In the case ρ − , ρ + > −2 (which is the only one of use here), the SLE κ (ρ − , ρ + ) is defined for all times and is driven by a semimartingale ( [22], Section 4 in [33]). The absolute continuity properties of these processes can be expressed as in Section 3.2; one may note there that the Radon-Nikodỳm derivative is still well defined in the case where x − or x + are displaced under the evolution, which happens when ρ ± < κ 2 − 2. Consider a configuration (D, x 1 , . . . , x m , y) as above, x − = x = x + = x 1 . The jump of the field at x is ± ρ − +2+ρ + √ 2πκ . The crosscuts δ i are defined as before. Then Lemma 6.1 holds, with the same proof. Lemma 6.3 also follows.

Global coupling
We have constructed a "local coupling" between a system of SLE's and a free field, in the case where the partition functions coincide, for a choice of crosscuts. We now use a limiting argument to construct a global coupling that will enjoy the same properties for any choice of crosscuts. This requires compatibility of the construction with the respective Markov properties of SLE systems and the free field. A "local to global" argument is introduced in [42] in the context of SLE reversibility. We broadly follow here the presentation in [10], one difference being in the nature of the Markov properties; there is also here additional structure (conditional independence).
Again, c = (D, x 1 , . . . , x n , y) is a configuration (y in the bulk), (a, b) a set of boundary conditions for the free field corresponding to SLE κ (ρ); x 1 , . . . , x m are seeds of SLE (ρ i = 2 for i = 1, . . . , m). We denote by µ SLE c , µ FF c the respective distributions of the SLE system and the free field in c.
The goal is to construct a coupling of µ SLE and µ FF with natural compatibility with Markov properties, and such that the different SLE strands are independent conditionally on the field. It is unclear whether it is possible to do this "in one go". So we shall consider first the coupling of two objects: one SLE strand (case m = 1) and a free field.
Let η > 0 be a small parameter, say much smaller than the diameter of D and the distance between marked points. We define a sequence of stopping times for γ by τ 0 = 0, It is easy to see that if D is bounded, there is a fixed N = N (D) such that a.s. τ n = ∞ for n > N .
Let ∂ be the union of {y} and the smallest connected boundary arc containing all marked points except x = x 1 . Let p 0 be the random integer: where K p denotes the hull of the SLE stopped at τ p . Let δ p be the boundary component of (K p ) 2η that disconnects K p from other marked points. (One can consider variants that ensure that the crosscut δ p is smooth. The important point is that is no closer than, say, 3 2 η of K p and no farther than, say, 2η, and is F τp measurable). We define: where c p is the configuration sampled at time τ p and T p = T δp (trace on δ p ). This quantity depends on the SLE strand up to time τ p+1 .
Consider the measure L.µ SLE c ⊗ µ FF c , where the density L is given by: First we have to check that this is a coupling of µ SLE c , µ FF c . For fixed φ, p → p−1 q=0 ℓ q (γ, φ) is a discrete time martingale (it is bounded when stopped at p 0 ). This boils down to: which follows from the local case (Lemma 6.1) in the configuration c p (note that δ p is determined by γ [0,τp] ). For the other marginal, we need another expression of the density, thinking now of γ as fixed. This follows from ℓ q (γ, φ) = q≤r≤p0 ℓ q,r (γ, φ) where for q ≤ r < p 0 and This density factors through T q φ because of the Markov property of the free field (the conditional distributions of T q+i φ given T q φ do not depend on what is on the other side of the crosscut δ q ). Thus: so that integrating T 0 φ, then T 1 φ, . . . (with the convention that the conditioning by T p0+1 φ is empty), one gets: for fixed γ. This shows that we have indeed a coupling of µ SLE c , µ FF c . It is easy to see that in this coupling, γ τp is independent of φ conditionally on φ inside (K p ) 2η (more precisely, T 0 φ, . . . , T p φ). This holds for fixed p or a stopping time for the discrete time filtration (σ(γ τp )) p≥0 .
In the general case, one can proceed in different ways. For simplicity, we can first use a common (discrete) time scale for the SLE system: at each step, each strand moves at distance η, synchronously; this yields a sequence of configurations c p = c τ 1 p ,...,τ m p . Consider δ p = δ 1 p ⊔ · · · ⊔ δ m p . Then we can define similarly to the m = 1 case: where γ is now an m-uplet of paths. As before, the local study ensures that p → p−1 q=0 ℓ q is a martingale for fixed φ (say stopped at p 0 , the first p such that (p, . . . , p) / ∈ G). Moreover, due to the nested structure of the δ p 's and the Markov property of the field, we can write: so that for fixed γ, Ldµ FF c0 (φ) = 1, integrating first T 0 φ, then defines a coupling of µ FF c0 , µ SLE c0 . Let us go back to the m = 1 case. We saw that the measure L.µ SLE c ⊗ µ FF c , where the density L is given by: is a coupling of µ SLE c , µ FF c . Let n be a stopping time in the discrete filtration (σ(γ τp )) p≥0 . Then the measure induced on (γ τn , φ |(K 2η n ) c ) can be described as follows. The first marginal is just the SLE strand stopped at τ n . Conditionally on γ τn , the distribution of T n−1 φ is that induced by the free field in c n ; consequently, the conditional distribution of φ |(K 2η n ) c is that of the free field in c n restricted to (K 2η n ) c . Indeed, the measure induced on (γ τn , φ) is simply: ) is a discrete time martingale for fixed φ). Then, reasoning as above (with n − 1 replacing p 0 ), one gets the expression: and then one integrates out successively T 0 φ, . . . , T n−2 φ.
The construction depends on a small parameter η, which we now take to 0. The sequence of paired measures L η .µ SLE c ⊗ µ FF c has fixed marginals, hence is tight. Thus there exists a subsequential limit Θ, which is again a coupling of µ SLE c , µ FF c . To phrase properties of the coupling, we need to introduce filtrations. First, (F SLE t ) t≥0 is the filtration generated by the SLE strand. The time scale is arbitrary and the discussion here is invariant under bicontinuous progressive time change (under which the class of stopping times is invariant). A possible time scale is the half-plane capacity of ψ(γ . ), where ψ is some conformal equivalence D → H.
Recall that for the free field, we defined F FF The set of open subsets of D is partially ordered for inclusion, so we can think of (F FF U ) U as a filtration with partially ordered index set (plainly, U ⊂ V implies F FF U ⊂ F FF V ). We can phrase now:

2.
For all open set U having the seed x of the SLE on its boundary (a continuous arc), the field restricted to D \ U is independent of the SLE stopped upon exiting U conditionally on F FF ∂U . Equivalently, the SLE stopped upon exiting U is independent of the field conditionally on the field restricted to U .
Proof. The limiting arguments here are similar to those in Theorem 6 in [10]. Let η k ց 0 be a sequence along which L η .µ SLE c ⊗ µ FF c has a limit Θ. For the first statement, one can consider a probability space with sample (γ, φ, φ 1 , . . . , φ k , . . . ) such that φ k → φ a.s. (e.g. in the Fréchet topology of C ∞ 0 (D) ′ ) and the marginal (γ, φ k ) has distribution L η k µ SLE c ⊗ µ FF c . Let us assume first that γ τ is at uniformly bounded below distance of ∂. Consider n(k) = inf{n : τ k n ≥ τ } where the sequence of stopping times (τ k n ) n is from the definition of L k = L η k . Then τ k = τ k n(k) is a stopping time and τ k ց τ a.s. Let ε > 0 be fixed. For k large enough (viz. 3η k ≤ ε), conditionally on F FF τ k , the field φ k restricted to the connected component of D \ (γ τ ) ε having ∂ on its boundary has the distribution of the free field in c τ k restricted to that set. One concludes by taking k → ∞ and then ε ց 0.
One obtains the second statement by applying the first statement to τ U = inf{t : γ t / ∈ U } in conjunction with the Markov property of the free field.
In the situation with several strands, one can rely on the local computation in Lemma 6.3 to reduce the problem to one strand.  (γ 1 , φ), . . . , (γ m , φ) are maximal couplings; 2. the SLE strands γ 1 , . . . , γ m are independent conditionally on the field φ.
One can obtain more general stopping statements (involving e.g. sequences of stopping times for the different SLE strands), which are a bit heavy to formulate and of no direct use here.
Proof. For η > 0, consider a coupling where L i η (γ i , φ) is the density we considered above. The marginal distributions are dµ SLE c (γ i ) (SLE system restricted to the i-th strand, i = 1, . . . , m) and dµ FF c (φ). Moreover, the γ i 's are independent conditionally on φ, due to the split form of the density for fixed φ. As η ց 0, the family of measures is tight.
Consider a sequence η k ց 0 along which these couplings converge to a measure Θ on (γ 1 , . . . , γ m , φ). In particular, the distributions of the marginals (γ 1 , φ), . . . , (γ m , φ) converge. Then the limiting distributions of these paired marginals are maximal couplings, as in the proof of the previous theorem. It is also clear that the conditional independence of the SLE strands given the field is preserved in the limit.
What remains to check is that under Θ, (γ 1 , . . . , γ m ) is (jointly) distributed according to µ SLE c . Consider disjoint crosscuts δ i , i = 1 . . . m separating x i (the seed of the i-th SLE) from all other marked points; more precisely, D \ δ i = L i ⊔ R i , with x i ∈ ∂L i . The i-th SLE is stopped at time τ i , when it comes within distance ε > 0 of δ i . For i = j, γ i is independent from γ j conditionally on φ; besides, γ τi i is depends on φ only through its restriction to L i . Moreover, the restrictions of the field in the L i 's are independent conditionally on the trace of the field on δ = ⊔ i δ i . Hence the γ τi i 's are independent given the trace T δ φ. Since (γ i , φ) is a maximal coupling, the joint distribution of (γ τi i , T δ φ) is that of the SLE stopped at τ i , and conditionally on γ τi i , φ is distributed as the field in c τi . This shows that the distribution under Θ of (γ τ1 1 , . . . , γ τm m , T δ φ) is the same as the one in the local coupling of Lemma 6.3. In particular, the joint distribution of (γ τ1 1 , . . . , γ τm m ) is that of the SLE system µ SLE c with the i-th strand stopped at τ i . Since this is valid for all crosscuts (δ i ) and all ε > 0, the joint distribution of (γ 1 , . . . , γ m ) under Θ is indeed µ SLE c . 7 Stochastic "differential" equations driven by the free field In order to build some intuition on the nature of the relationship between SLE and the free field studied here, it appears rather convenient to draw an analogy with the standard theory of stochastic differential equations (SDEs) driven by (real) Brownian motion (see eg [30]). However the free field/SLE situation does not involve a stochastic calculus w.r.t. free field.

Definitions
Let us briefly recall the set-up for stochastic differential equations. Let (X, B) be a pair of adapted processes in a probability space (Ω, F , P), σ t ((X t . )), b t ((X t . )) progressively measurable functions of the process X. The SDE reads: A pair (X, B) is a solution of the SDE if B is an F -Brownian motion and the relation is satisfied (given B and X, the RHS is defined as a stochastic integral). It is a strong solution if moreover F is generated by B.
There is uniqueness in law if in all solutions (X, B), the marginal X has the same distribution, and pathwise uniqueness if for any pair of solutions (X, B), (X ′ , B) defined on a common filtered space (with common driving BM's), the processes X, X ′ are undistinguishable (a.s. equal).
For simplicity, we restrict ourselves to the chordal case: a configuration c = (D, x, y) consists in a simply connected domain D with two marked points x, y on the boundary.

Consider a filtration (F U ) U indexed by open neighbourhoods of x in D.
An F -free field is a free field such that F FF U ⊂ F U and φ restricted to D \ U is independent of F U conditionally on F FF ∂U . A stochastic Loewner chain K . starting from x is F -adapted if K stopped at first exit of U is F U -measurable. We only consider Loewner chains with continuous driving functions. Assume given an assignment: where h is a harmonic function in D t = D \ K t . (One may also consider the situation where h is defined in D \ γ [0,t] , for a Loewner chain generated by a trace γ).
We are interested in comparing the boundary values of h = h((K s ) 0≤s≤t ) and φ in D t ; the issue is that neither need be defined pointwise on ∂D t . One may proceed as follows: consider a sequence δ n of closed, smooth curves converging to ∂D t (eg equipotentials seen from a bulk point). The δ n 's depend on the chain but not on the field. Then one requires that the harmonic extension of the trace of the field on δ n inside δ n (this is a.s. well defined) converges to h uniformly on compact sets of D t . Plainly, this can be checked pathwise. A more compact (if less explicit) formulation in terms of conditional expectation of the field is possible: ) measurable sequence of nested closed curves approximating ∂K t . Then a.s. conditionally on K t . , the harmonic extension of T δn φ inside δ n converges uniformly on compact sets to E(φ |Dt |F FF ∂Dt ), the conditional expectation of the field restricted to D t given F FF ∂Dt .
Proof. The curve δ n splits D t into L n (that has ∂K t . on its boundary) and R n . A compact set C of D t is contained in R n for n large enough. From the properties of the field trace, ) and the conditional expectation is constant, since δ m is contained in R n . So we can compute it for the free field in R n with Dirichlet boundary conditions (so that h n = 0). Thus: This can be exactly evaluated by general Gaussian arguments. Under µ FF Rn , w = T δm φ has covariance (G Rn ) |δm . For a symmetric kernel B(x, y) = i,j f i (x)f j (y) on (δ m ) 2 , one gets: This is valid for finite rank kernels, and by approximation applies to the trace class kernel: It follows that: In terms of path decompositions, this corresponds to a Brownian loop in R n starting and ending at z and decomposed w.r.t. its first and last visit to δ m . Given that the transition kernels in R n have uniform exponential decay and that as n → ∞, the transition kernel in R m converges uniformly to that of D t uniformly on C × C × [0, t], it is easy to see that E(||h m − h n || 2 L 2 (C) ) converges to 0 as m → ∞, uniformly in n ≥ m. One can refine this (using eg the Harnack inequality to control derivatives of the Poisson kernel) to get that for any k ≥ 0, E(||h m − h n || 2 H k (C) ) converges to 0 as m → ∞, uniformly in n ≥ m. It follows that (h m ) converges a.s. in any H k (C), and consequently (Sobolev imbedding) converges uniformly in C (ie in (C 0 (C), ||.|| ∞ )).
From the free field Markov property, we have h = E(φ |C |F FF ∂Dt ) = E(φ |C |F FF Kt∪∂D ). By definition of F FF on closed sets, we have F FF Kt∪∂D = ∩ n>0 F FF Ln . For any m > 0, we have: Since lim m h m exists a.s. and is consequently (∩ n F FF Ln )-measurable, we get that h = lim m h m , which concludes the proof.
Consider the following problem, given the data of h = h((K s ) 0≤s≤t ): find a probability space with filtration (F U ) U on which are defined a field φ and a stochastic Loewner chain (K t ) t≥0 such that: This imposes some compatibility conditions on h under stopping of the Loewner chain: if s ≤ t, h s = h((K s . )) and h t = h((K t . )) agree on ∂D s in the sense that if (δ n ) is a sequence of closed curves approaching ∂D s , the harmonic extension of the restriction of h t to δ n converges to h s locally uniformly in D s . Under continuity assumptions (as in Lemma 5.1), it is enough to check the condition E(φ |Dt |F FF ∂Dt ) = h((K t . )) for a countable dense set of times {t i }.
By analogy with the SDE framework, one can phrase: There is pathwise uniqueness if for any filtered space on which are defined a field φ and two chains K, K such that (φ, K) and (φ,K) are solutions, the Loewner chains are a.s. equal.

Existence and uniqueness in law
We have considered different types of boundary conditions, in particular chordal (a, b) boundary conditions in a configuration c = (D, x, y). This defines an assignment h a,b = h a,b (K t . ), provided that the domain D is regular enough (C 1 ) around y and stopped chains K t . stay away from y. Proof. Existence. Take a maximal coupling of a free field with (a, b) boundary conditions and a chordal SLE κ in c, which exists by Theorem 6.4. Define F U = F SLE τU ∨ F FF U , where τ U is the time of first exit of U by the SLE. By definition, the SLE is F -adapted; and φ is an F -free field by Theorem 6.4. Besides, for a time t, φ |Dt is distributed as an (a, b) free field in D t conditionally on F ∂Dt . It follows that E(φ |Dt |F FF ∂Dt ) = h a,b ((K t . )) is a.s. satisfied at t; consequently it is a.s. satisfied for t in a dense countable set of times {t i }. It is then easy to see that the equation is satisfied for all times, a.s.
Uniqueness. We reason as in Lemma 6.1, in reverse (a standard argument, see eg [24]). Consider a solution (φ, K . ), with filtration F ; denote G t = F Kt . By the Markov property of the field, the distribution of φ |Dt conditionally on G t is that of a free field in D t with mean h t = h a,b (K t . ). Consequently, if f ∈ C ∞ 0 (D) is a test function and τ = inf{t ≥ 0 : dist(K t . , supp(f )) ≤ ε}, we have: and M is by construction a bounded G-martingale, and is continuous (the Loewner chain is also assumed to be generated by a continuous process).
, we have: Zs is also a semimartingale. Thus one can write dW t = σ t dB t + db t ; plugging this back in t dt evaluated at two distinct z points, one gets σ t ≡ √ κ and db t = 0. Thus the Loewner chain is distributed as chordal SLE κ .
We restricted to the chordal case for simplicity; however it is clear that the result applies whenever an identity of partition functions as in Theorem 5.3 holds. Following the discussion at the end of Section 6.1, it also applies when the SLE strand is absolutely continuous w.r.t. an SLE κ (ρ − , ρ + ), ρ ± > −2, near its start at x − = x = x + .

Pathwise uniqueness
In this subsection, we are considering the question of pathwise uniqueness in the chordal case for general κ > 0. Pathwise uniqueness combined with the already established existence of weak solutions implies existence of strong solutions (in which the SLE path is a function of the field).
The general strategy consists in starting from a weak solution (φ, K) to construct a triplet (K, φ,K) whereK is a dual SLE path (or collection of such paths) such that K,K are independent conditionally on the field andK determines K. This implies that K is actually a strong solution. Moreover, if K,K are two weak solutions defined on the same probability space (common field), then K,K are equal since they are determined by the common auxiliaryK; this yields pathwise uniqueness.

Case κ = 4
In the cases κ = 4, 8, the corresponding free field boundary conditions have symmetries compatible with reversibility. We now exploit this fact, in conjunction with the simplicity of the trace, for κ = 4.
We have already proved the existence of a solution. It is enough to prove that if (φ, K), (φ,K) are two solutions defined on the same filtered space, then K =K. It implies in particular that all solutions are strong (as in the case of SDEs).
To be able to use densities, we prove a different version. Namely, consider a solution (φ, K) of the problem in (D, x, y) ((a, 0) boundary conditions) and (φ,K) a solution in (D, y, x) ((−a, 0) boundary conditions), coupled so that the fields agree (they have the same boundary conditions) and the chains are independent conditionally on the field. Then we claim that K,K have the same range. As these are simple paths, this determines the chain completely. Applying twice this result (take (φ,K) a solution of the problem in (D, y, x), independent of K,K conditionally on φ; then K andK are the reverse ofK), one gets pathwise uniqueness.
Hence we consider a triplet (K, φ,K) such that (K, φ), (φ,K) are solutions in (D, x, y), (D, y, x) respectively, and K,K are independent conditionally on φ. We reason now as in Theorem 6.5. Consider a crosscut δ splitting D in L, R (x on the boundary of L, y on the boundary of R). The chains K,K are stopped at τ,τ when they come within distance ε > 0 of the crosscut δ. Then K τ is independent of F FF R conditionally on F FF δ ;Kτ is independent of F FF L conditionally on F FF δ ; K τ is independent ofKτ conditionally on the field; F FF L , F FF R are independent conditionally on F FF δ . It follows that K τ ,Kτ are independent conditionally on F FF δ . Besides, the marginal distributions of (K τ , T δ φ), (Kτ , T δ φ) are fixed: and symmetrically for (Kτ , T δ φ). The joint distribution of (K, T δ φ,Kτ ) is thus: To obtain the joint distribution of (K τ ,Kτ ), one integrates out T δ φ; as in Lemma 6.3, this yields the joint distribution: ie the same distribution as whenK is the reverse of K. Since this holds for all crosscuts δ and all ε > 0, it is easy to see that in this couplingK is the reverse of K.
Case κ ∈ (0, 4) When κ / ∈ {4, 8}, the coupling of the free field and chordal SLE is not compatible with SLE reversibility (at least, not in an obvious way). But it is still compatible with some duality identities (eg [10], in particular Proposition 10), which will be enough for our purposes.
The parameters are chosen so that there is a dual chainK which is an SLEκ(ρ 1 ,ρ 2 ) from y to x,κ = 16/κ, ρ 1 =κ − 4,ρ 2 = (κ − 4)/2, with the same partition function. For instance so that the fields associated to K,K share the same boundary conditions, up to a global sign.
Given a weak solution (K, φ), one can thus construct a triplet (K, φ,K) whereK is independent of K conditionally on the field and (K, φ) is a solution of the dual equation. We study the joint distribution (K,K). Consider two disjoint crosscuts δ 1 , δ 2 disconnecting x (resp. y) from other marked points; the chains K,K are stopped at τ,τ when they come within distance ε > 0 of δ = δ 1 ⊔ δ 2 . Arguing as in the κ = 4 case, we see that K τ ,Kτ are independent conditionally on T δ φ, and consequently the distribution of the triplet (K τ , T δ φ,Kτ ) is as in Lemma 6.3. Integrating out T δ φ shows that (K,K) is a maximal coupling of K,K. In such a coupling ( [10], Proposition 10), K (stopped when it hits the boundary arc (z 2 , z 1 )) is the (say, left) boundary of the range ofK (stopped when it hits (z 1 , z 2 ) at x). Thus K is determined byK; one concludes as in the κ = 4 case.
The argument here is best understood in terms of Uniform Spanning Trees (UST). Chordal SLE 8 is the scaling limit of the Peano path of a UST with Dirichlet/Neumann boundary conditions, [24]. The auxiliary objectK we are using is an arbitrarily fine subtree (and dual subtree) with finitely many branches, that are SLE 2 -type curves.
The following lemma provides path decompositions for some versions of SLE κ , κ ≥ 8 (notice that the statements are simpler in the case κ = 8).
Lemma 7.4. In a configuration c = (D, x, z 1 , y, z 2 ), consider an SLE κ (ρ) chain K, κ ≥ 8, ρ = κ 2 − 4, 2, κ 2 − 4 at z 1 , y, z 2 (x, z 1 , y, z 2 in this order on the boundary), coupled with a free field φ; let µ c be the law of that SLE. Let z be a point on the boundary arc (xy); D l the random subdomain swallowed when the trace hits z, K the boundary arc ∂D l ∩ D; D r = D \ D l ; the endpoints ofK are z and a random point z ′ on (yx). The dual pathK is determined by the field and the restrictions of (K, φ) to D l , D r respectively are independent conditionally onK. The marginal distributions are (κ = 16/κ): Proof. Given a solution (K, φ) in the configuration c, consider a solution (K, φ) of the dual problem, as summarized in Table 1, 2 depending on the position of z;K is taken independent of K conditionally on the field. In the tables, entries in a row are ρ parameters, except [κ] that designates the starting point of the Table 1: z ∈ (x, z 1 ) Table 2: z ∈ (z 1 , y) Reasoning as in the case κ = 4 (see Lemma 6.3), we see that (K,K) is a maximal coupling. The ρ coefficients at z ± are chosen so thatK is the boundary of K stopped when it hits z; this is a duality identity of the type considered in [10], [41]. SinceK is determined by K and independent of it conditionally on φ, it is determined by φ. The situation in c r is the same as in c, by the Markov property and the fact that (K, φ) is a solution.
The chain K stays in D l until it reaches z at time τ z . We have to determine the distribution of K up to τ z conditionally onK. By construction, (K, φ) and (K, φ) are solutions of dual problems in D. By Lemma 5.1 (asK gets closer to its random endpoint z), conditionally onK, φ restricted to D l is a free field with (a, b) boundary conditions, with jumps at x, z 1 ∧ z, z 2 ∧ z ′ , z ′ (the jumps at z + , y in D agglomerate in a jump at z ′ ).
The fact that (K, φ) is a solution in D is a pathwise, local condition (Lemma 7.1). It follows that (K τz , φ |D l ) is a solution in D l . Uniqueness in law then determines the distribution of K τz conditionally on K.
One can use the previous lemma to reconstruct a chordal SLE κ , κ ≥ 8 from its dual branches as follows (see also [35] for related considerations).
Start from a chordal SLE κ in a domain (D, x, y) coupled with a free field; pick a point z on the boundary. This is a particular case of Lemma 7.4 with z 1 = y = z 2 . The branchK starting from z is determined by the field; it splits D into D l , D r . In D l , the branchK l starting from z ′ is determined by the restricted field; similarly,K r is the branch starting from z ′ in D r . The branchK l splits D l into D ll and D lr . Recursively, every subdomain is dissected in two by a branch determined by the field (see Figure 3). All the branches are boundary arcs of the chain K at some time; and the cells are visited in a prescribed order (eg at level 2, D ll , D lr , D rl , D rr ).
We can extract information on K from the field in this form of the branchesK. We need to prove that all the information on K can be obtained by this countable set of branches; informally, the splitting procedure yields information on K at arbitrary small scales, everywhere in D.
Lemma 7.5. Let w 1 , w 2 be distinct interior points of D. In the iterative splitting of (D, x, y), w 1 and w 2 are eventually in distinct cells a.s.
Proof. Without loss of generality, assume that D is bounded with, say, Jordan boundary. Assume by contradiction that w 1 , w 2 are in the same cell at any level of the splitting. Note that a.s. they are not on any branchK (that have zero Lebesgue measure). The splitting start from z = z 0 in (xy) (distinct from x, y). The endpoint of the first branch is z 1 ; at the next level, the cell containing w 1 , w 2 is dissected by a branch from z 1 to some z 2 , and so on. The branchesK n from z n to z n+1 are simple and disjoint except at their endpoints, and of "alternating colors" (ie alternately left and right boundary arcs of K).
There are two possibilities: either the concatenation of theK n contains infinitely many simple disjoint cycles circling around w 1 , w 2 , or eventually the successive cells containing w 1 , w 2 have a common boundary point, and theK n are arranged in a zigzag. In the first case, the diameter of each simple cycle circling around w 1 , w 2 is bounded away from 0. All the points on these cycles are visited by the trace in prescribed order. This contradicts the continuity of the trace of K ( [31,24]).
In the second case, consider the harmonic measure h n of the branchK n in the cell D n , seen from w 1 or w 2 . Then (h 2n ) n and (h 2n+1 ) n are eventually increasing (and never zero), hence bounded away from 0. Since the harmonic measure of the connected setK n is bounded away from 0 seen from two distinct points w 1 , w 2 , the diameter ofK n is also bounded away from 0; one concludes as in the first case.
One can now conclude that in a solution (K, φ), the chain K is determined by the field. Indeed, the splitting of the domain is determined by the field. Enumerate a dense sequence of points w n in D; they are hit by K at times τ n , which constitute a dense sequence of stopping times. Enumerate also the cells of the splitting at all levels; let σ m be the random time at which the trace enters the m-th cell, which it does at a point x m determined by the field. By the previous lemma, for i = j, the times τ i , τ j are a.s. separated by a random time σ m . Hence the family of times σ m is a.s. dense. The position of the continuous trace of K is prescribed on an a.s. dense set of times. Thus we get pathwise uniqueness of the solution K.
Case κ ∈ (4,8) The argument is similar to the case κ ≥ 8, however a bit more involved. Again, the SLE trace can be recovered from a tree of dual arcs which is independent conditionally on the field.
We begin with a path decomposition, analogous to Lemma 7.4; the formal commutation identities are the same, but the geometry of paths is different. Recall in particular that for κ ∈ (4, 8), SLE develops cutpoints ( [3,8]). Hence the complement of the boundary of the SLE hull stopped at a finite time has countably many connected components (rather than just two in the κ ≥ 8 case).
Lemma 7.6. In a configuration c = (D, x, z 1 , y, z 2 ), consider an SLE κ (ρ) chain K, κ ∈ (4, 8), ρ = κ 2 − 4, 2, κ 2 − 4 at z 1 , y, z 2 (x, z 1 , y, z 2 in this order on the boundary), coupled with a free field φ; let µ c be the law of that SLE. Let z be a point on the boundary arc (xy); D r = D \ K τz a random simply connected subdomain, LetK be a solution of the dual problem starting from z (see Tables 1,2), independent of K conditionally on the field, stopped when it hits (yx) at z ′ ; its last visit on (xy) before hitting (yx) is at z ′′ . Conditionally onK, the restriction of K to different connected components of D \K are independent.
1. If z ∈ (x, z 1 ),K is the first excursion ofK from (xy) to (yx). Let D l be the component of D \K with x on its boundary. Conditionally onK, K τz in D l has distribution µ c l where c l = (D l , x, z, z, z 2 ∧ z ′ ), stopped when it hits z ′′ . After τ z ′′ , the distribution of K is µ cr , c r = (D r , z ′′ , z 1 , y, z ′ ∨ z 2 ).
2. If z ∈ (z 1 , y),K =K. Let D l = D \ D r , D ′ l the connected component of D l with x on its boundary. Conditionally onK, the distribution of K τ z ′′ is µ c ′ l in c ′ l = (D ′ l , x, z 1 , z ′′ , z 2 ∧ z ′ ); the distribution of K in another connected component D ′′ l of D l corresponding to an excursion ofK from y ′′ to x ′′ on (xy) is µ c ′′ l , c ′′ l = (D ′′ l , x ′′ , x ′′ , y ′′ , x ′′ ). After τ z , the distribution of K is µ cr , c r = (D r , z, z, y, z ′ ∨ z 2 ) Proof. The general argument is as in Lemma 7.4, based on the same commutation relations (Tables 1, 2), the difference being in the geometric interpretation.
In the case z ∈ (x, z 1 ), considerK a solution of the dual problem (Table 1), independent of K conditionally on the field. The pathK hits (xy) between z and z 1 (and not in (xz) or (z 1 y)) before it first hits (yx) at z ′ , where it is stopped. Then Lemma 6.3 shows that (K,K) is a maximal coupling in the sense of [10], for K stopped at τ z andK stopped atτ z ′ .
For any stopping timeτ forK lesser thanτ z ′ , stop K the first time it hitsKτ or disconnect it from y. Given the values of the ρ parameters, it can hitKτ only at the tipKτ ; if it does not, the pathK goes back to (xy) at the point hit by K at disconnection time. Hence any point ofK on the first excursionK from (xy) to (yx) is on K, while points on excursions ofK from (xy) to (xy) are not. Moreover,K can hit K only on its right boundary. Reasoning as eg in [10], one concludes that the right boundary of K τz is the first excursion ofK from (xy) to (yx).
Given that (K, φ) is a solution, conditionally onK, φ restricted to different connected components of D \K is a free field with prescribed (a, b)-type boundary conditions. Since (K, φ) is also a solution, and this is a local property (Lemma 7.1), this determines the distribution of K in the connected components of D \K it visits.
The case z ∈ (z 1 , x) is similar (and simpler). There (see Table 2),K hits (xy) between z 1 and z (and not in (xz) or (zy)) before it first hits (yx) at z ′ , where it is stopped. The chain hits z a.s.; the right boundarỹ K of K τz intersects (xy) between z 1 and z. Reasoning as before, we see that (K,K) is a maximal coupling for K stopped at τ z andK stopped atτ z ′ ; given the choice of parameters, this implies thatK =K.
As in the κ ≥ 8 case, this path decomposition result can be used recursively to describe a chordal SLE κ by a collection of dual paths determined by the field. We note that in this context it is rather natural to consider not simply chordal SLE κ , but a fuller version, such as branching SLE κ ( [4,35]). Let us start with a chordal SLE κ in (D, x, y), 4 < κ < 8, coupled with a free field φ; pick a point z on (xy). This is the situation of Lemma 7.6 with z 1 = y = z 2 . The original domain D is split in subdomains byK; the trace of K is contained in D l , D r . The distribution of (K, φ) in these subdomains is of the general type considered in Lemma 7.6, and we can take z ′ to play the rôle of z. Cells intersecting K are split recursively. Note that when case 2 of Lemma 7.6 applies, one gets countably many subcells (and new marked points have to be picked in these cells). The collection of dual branches thus produced is independent of K conditionally on the field.
There remains to check that points separated by the trace of K are also separated by the dual branches. For this purpose, it seems practical to use a slightly different subdivision scheme. One proceeds as described above; the difference being that when case 1 in Lemma 7.6 applies, one takes z ′′ (rather than z ′ ) as the new marked point in D l . Also, one does not further divide cells the interior of which is not visited by the trace of K (these are determined by the tree structure). This ensures that the successive arcsK are boundary arcs of the original SLE κ (rather than some branching version of it).
Consider w 1 , w 2 distinct points of D. If at some level n, there is a cell containing both w 1 , w 2 which is not further divided, this means that w 1 , w 2 are swallowed at the same time by K. Reasoning as in Lemma 7.5 shows that w 1 , w 2 cannot belong both to cells in an infinite strictly decreasing sequence, as that would violate the continuity of the trace of K. Therefore points w 1 , w 2 that are swallowed at distinct times by the trace are separated by the collection of dual arcs constructed above. For (w n ) a dense sequence of points in D, the sequence of stopping times (τ wn ) is a.s. dense, and one concludes as in the case κ ≥ 8.
Proof. This follows immediately from pathwise uniqueness and the fact that both members of the equation in Theorem 7.7 have the same covariance rule.
The use of the chordal model is essentially conventional and for simplicity. However we observe that existence of strong solutions and pathwise uniqueness are local properties, hence they hold in more general settings.
To illustrate this, consider the following situation. Let D be a planar simply connected domain. Points x 1 , . . . , x m are marked on the boundary and a point y is marked in the bulk. (One could mark more points in the bulk). Consider a field with (a, b) boundary conditions, where κ > 0, a 1 = ± 2 πκ , b = a 1 (1 − κ 4 ). Let ∂ be the smallest boundary arc containing all marked points except x 1 . Then: The partition function Z a,b is as in Theorem 5.3. The resulting SLE is a radial SLE κ (ρ) (one can also omit y to get a chordal SLE κ (ρ)).
Proof. Let U be a subdomain of D having on its boundary a boundary an arc of ∂D containing x 1 ; assume also that U is simply connected and at positive distance of other marked points.
Consider a solution (K, φ) of the stochastic equation with (a, b) boundary conditions and restrict it to (K τU , φ |U ) (τ U time of first exit of U by K). One can construct a solution (K,φ) of the chordal problem in common boundary arc. This fixes the offset of φ n in cells with rough boundaries.
This defines from the continuous tree a sequence of a.e. harmonic functions φ n . The point is to prove that this converges to a free field. Let φ be a free field coupled with the tree as above, so that φ n is the expected value of the field given the branches at level ≤ n. Let F n be the σ-algebra generated by the tree at level n and G n be the Green kernel in the complement of the tree at level n, with Dirichlet boundary conditions. Then if f ∈ C ∞ 0 (D), f (x)G n (x, y)f (y)dA(x)dA(y)).
Notice that G n (x, y) is nonnegative and decreasing in n for fixed x, y (domain monotonicity). Moreover, by Lemma 7.5, G n (x, y) = 0 eventually for fixed x = y. It follows that φ − φ n , f converges to 0 in L 2 . This implies that φ, f is F ∞ = σ(F 1 , F 2 , . . . )-measurable. As this holds for all f ∈ C ∞ 0 (D), φ itself is F ∞ -measurable.
Note that there is a similar correspondence between fields and chordal SLE for any κ ≥ 8. For κ ∈ (4,8), the data of one chordal SLE path is not sufficient to reconstruct the field; it can be expected that reconstruction is possible from a "fuller" version, such as branching SLE κ ( [4,35]), with similar arguments.

Strong duality identities
Duality for SLE, conjectured by Duplantier, states that boundary arcs of SLE κ , κ > 4, can be described as (versions of) SLEκ,κ = 16/κ. Various such identities are established in [10,41]. In [6], "strong" duality identities are conjectured; these bear on the joint distribution of an SLE κ and its boundary (rather than just the marginal distribution of the boundary) and are based on computations that can be understood in terms of partition function identities.
Proof. The case κ ≥ 8 is part of the pathwise uniqueness proof. Let us briefly discuss the modification for the case κ ∈ (4, 8). Conditionally on d, K is an SLE κ (ρ), ρ = κ − 4, −4 at x, d. The relevant parameters are summarized in Table 3. One can couple K, under the conditional measure and stopped at τ x , with a free field. TakingK a solution of the dual problem with the same field, starting from d and stopped when it hits (0, x), we see reasoning as before thatK is the boundary arc of K straddling x (see Theorem 1 in [10]). This gives the conditional distribution of the field in D 0 . Considering K τx in D 0 shows that it is a solution of a stochastic equation there (since this is a local condition, see Lemma 7.1), which determines its distribution by weak uniqueness.
Similarly, one can consider two-sided situations, ie versions of SLE κ conditioned on both left and right boundary arcs. In [8], properties of SLE κ (ρ) in (D, x, y), ρ = κ − 4, κ − 4 at x − , x + are studied; in particular, for κ ∈ (4, 8), it is a chain of iid "beads". We will now briefly discuss how to identify the distribution of these beads conditionally on their boundary.
The relevant system of commuting SLE's is summarized in Table 4. Consider three chains K,K l ,K r Table 4: strong duality -two sided corresponding to the lines of Table 4 coupled with a common field φ. Reasoning on K,K l shows thatK l is the left boundary of K; symmetrically,K r is its right boundary. This entails pathwise uniqueness for K l ,K r and the fact that they do not cross (however they intersect at the cutpoints of K if κ ∈ (4, 8)). This determines the distribution of the field right ofK l and also the distribution ofK r limited to the domain right ofK l . Consequently, one gets the distribution of the field betweenK l andK r , and finally the distribution of K in the domain (or chain of domains if κ ∈ (4, 8)) delimited byK l ,K r . The conclusion is that the distribution of K restricted to a bead D (between consecutive cutpoints X, Y of K) is that of an SLE κ (ρ), from X to Y in D, ρ = κ 2 − 4, κ 2 − 4 at X − , X + .