CFTs of SLEs: the radial case

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

Stochastic Schramm-Loewner evolutions (SLE) [2,3,4] are random processes adapted to a probabilistic description of fractal curves or sets growing into simply connected planar domains U ⊂ C. They are expected to provide a rigorous description of two dimensional critical clusters in their continuous limit. SLEs depend on a real parameter κ. For 2 ≤ κ < 4, they are conjecturally related to the O(n) models in their dilute phase with n = −2 cos(4π/κ), and to the Fortuin-Kasteleyn clusters of the Q-state Potts models with Q = 4 cos 2 (4π/κ) for 4 ≤ κ < 8. We refer to refs. [6,7] for an introduction to SLEs.
Two classes of SLEs, chordal or radial, have been defined. The former -chordal SLEs -describe random planar curves joining two points on the boundary of a domain U, while the later -radial SLEs -describe random curves joining a point on the boundary ∂U to a point in the bulk of U. The aim of this note is to extend to radial SLEs the relation between SLEs and conformal field theories (CFT), which we developed in the chordal case in ref. [8]. Another relation between CFTs and chordal SLEs has been presented in ref. [10].
In the first section we shall recall the definition of the radial SLEs and its basic covariance properties. The CFT formulation of radial SLEs is given in section 2 in which radial SLEs are viewed as Markov processes in a completion of the enveloping algebra of a Borel subalgebra of the Virasoro algebra. This allows us to define a stochastic evolution operator A, dual to a Fokker-Planck like operator, which acts on the CFT Hilbert space and whose geometrical meaning is given. As a consequence, we determine in section 3 a key local martingale M t and we show that the dual Fokker-Planck operator acts as the dilatation operator on spinless conformal primary fields localized in the bulk ending point of the radial SLE curves. This agrees with observations made by Cardy in ref. [11]. The last two sections illustrate the use of the martingale M t by re-deriving the derivative exponants computed in refs. [4] and by computing the restriction martingales [5] which code for the response of SLEs to deformations of the domains in which they are defined.
1-Radial SLEs. The usual description of radial SLEs is by random curves connecting the point 1 on the boundary of the unit disc to the origin. Its study involves a stochastic differential equation, whose geometric properties play an important role in what follows. If U is any simply connected domain in C, x 0 a boundary point of U and z * an interior point of U, there is a unique conformal map from the unit disc to U mapping 1 to x 0 and 0 to z * . The image of SLE curves by this map defines a statistical ensemble of random curves in U, starting at x 0 and ending at z * , which is by definition radial SLE in (U, x 0 , z * ). This ensemble is related to a new stochastic differential equation, which we now describe geometrically.
Suppose that f t (z), t ∈ [0, T ], is a familly of functions solving a stochastic differential equation of the form with ξ t a Brownian motion with covariance E[ξ t ξ s ] = κ min(t, s). By this we mean that the two functions σ and ρ are holomorphic in some domain U and that there is a non empty domain U T ⊂ U such that f t maps U T into U and solves the above differential equation for z ∈ U T . Suppose that ϕ maps U conformally to some domain V. Then Itô's formula shows that These two relations show that transfom as holomorphic vector fields under ϕ. When σ and ρ vanish at some point z * ∈ U, the equation We may view f t as a random process on N z * . Another application of Itô's formula shows that γ −1 ft ·dγ ft .·F = (dt σ +dξ t ρ)F ′ +dt κ 2 ρ 2 F ′′ , or equivalently This equation involves only intrinsic geometric objects.
To define radial SLE on any domain, we only have to choose the vector fields w −1 and w −2 appropriately: w −1 is the generator of conformal motions of U fixing z * , and w −2 is a holomorphic vector field, unique up to translation by w −1 , tangent to ∂U, fixing z * but with a pole at x 0 .
We observe that the Lie algebra of N z * is formally isomorphic to a completion of a Borel subalgebra of the Virasoro algebra. In the sequel, we shall make use of the covariance of radial SLE under conformal maps to choose (U, x 0 , z * ) in such a way that w −1 and w −2 are as simple as possible in terms of Virasoro generators, so that we can make use of its representation theory and of conformal field theory efficiently.
In the outer unit disc geometry D = {z ∈ C ; |z| ≥ 1}, w −1 = iz∂ z and 2w −2 = z z+1 z−1 ∂ z , and the Loewner equation for the SLE map g t = e −iξt f t reads [2]: with g 0 (z) = z. The SLE hulls K t are the sets of points in D which have been swallowed: K t = {z ∈ D ; τ z ≤ t} with τ z the swallowing time such that g τz (z) = U τz . The map g t is the uniformizing map of the complement of K t in D. The SLE curve γ [0,∞) , also called the SLE trace, is reconstructed using g t (γ(t)) = U t .
In the upper half plane geometry We shall mainly present the results in the case of the disc geometry but they can easily be translated to the upper half plane geometry.

2-
The stochastic evolution operator. For making contact with CFT and its operator formalism, it is useful to translate the disc by −1 so that the SLE hulls start to be created at x 0 = 0 and grows into D tr , the complement in C of the unit disc centered at −1. We define maps h t by 4 h t (z) + 1 = U −1 t g t (z + 1). Since we view the hulls as growing outside the unit disc, both maps g t and h t fix the point z * = ∞ at infinity: By the results of [9], the maps g t and h t are associated to operators G t and H t which implement them in CFT. Let L n be the generators of the Virasoro algebra vir with commutation relations: [L n , L m ] = (n − m)L n+m + c 12 n(n 2 − 1)δ n,−m . The operator G t belongs to the envoloping algebra of a Borel subalgebra of vir. From its definition and the radial Loewner equation (2), it follows that G t satisfies: The operator H t is linked to G t by H t = e −L −1 G t e iξtL 0 e L −1 . By Itô calculus, one finds that H t satisfies the stochastic equation: . Compare with eq.(1). The stochastic evolution operator A is by definition the drift term in the stochastic equation, eq. (3): It may be expressed in terms of the Virasoro generators L n , but those are associated to the vector fields ℓ n = −z n+1 ∂ z which are not adapted to the geometry of the disc. Thus, we change basis and consider the generators V n associated to the vector fields v n = − i n 2 z n+1 (z+2) n−1 ∂ z , which are the push forward of the ℓ n 's by the uniformazing map of D tr onto the upper half plane H. The V n satisfy the Virasoro algebra. The first few are: In the upper half plane geometry, the appropriate basis of vir are the L n 's and the stochastic evolution operator is The basis V n 's or L n 's correspond to two different real forms of vir.
We need to recall a few basic facts concerning the Virasoro algebra and its highest weight representations. For the disc geometry we shall consider highest weight vector representations defined with respect to the polarization of the Virasoro algebra associated to the basis V n . So highest weight vectors |v are such that V n |v = 0 for n > 0 and V 0 |v = h|v . For the upper half plane geometry, we shall consider highest weight representations with respect to the basis L n .
In particular, by projecting this local martingale on vectors v| and assuming appropriate boundedness conditions, we get that the expectations This result follows from the null vector equation (2V −2 − κ 2 V 2 −1 )|ω = 0 which selects the representation with conformal weight h 1;2 . As a consequence, A|ω = 2h 0;1/2 |ω with 2h 0;1/2 = ( κ−2 This result may alternatively be formulated in term of the boundary field Ψ 1;2 (x 0 ) creating the state |ω at the origin x 0 , the point at which the SLE trace starts. We have: Other properties emerge when testing the evolution operator against conformal primary fields. This amounts to consider correlation functions with insertions of bulk or boundary primary fields Φ ∆;∆ (z,z) · · · Ψ h (x) · · · A|ω . By commuting or deforming contours using standard rules of CFT [1,14], the action of A = −2W −2 + κ 2 W 2 −1 on |ω may be traded for an action of on the bulk and boundary fields.
Recall that for V a Virasoro generator associated to a vector field v(z), bulk primary fields Φ ∆;∆ (z,z) of dimensions (∆,∆) and boundary primary fields Ψ h (x) of dimension h satisfy: Under global conformal maps h t (z) implemented in CFT by operators H t , primary fields transform as: As a consequence 5 , the action of A T on primary fields is easy to compute using the explicit expressions of the vector fields associated to W −2 and W −1 . For bulk primary fields localized at the point z * fixed by the SLE map, we get the particularly nice result: with d = ∆ +∆ the scaling dimension and s = ∆ −∆ the spin. In other words, A T , which may be thought of as a dual of a Fokker-Planck operator, acts diagonaly on primary operators localized at the fixed point. For spinless operators, this action is simply the dilatation. This is in agreement with the points raised in ref. [11]. On boundary conformal fields, A T acts as a second order differential operator H h closely related to the Calogero hamiltonian. In the case of the translated disc geometry, with x = e iθ − 1 parametrizing the boundary of D tr , this action reads: The three properties (7,8,9) have a simple consequence: appropriate CFT correlation functions are eigenfunctions of H h . For instance, with H h defined above, eq.(9), and eigenvalue with d = ∆ +∆ and s = ∆ −∆. The simplest case is for Ψ h the identity operator. The non vanishing of Φ ∆,∆ (z * ,z * )Ψ 1;2 (x 0 ) then requires ǫ ∆,∆ = 0, or equivalently d = 2h 0;1/2 + κ 2 s 2 , which is indeed the fusion rule relation. This case also provides a simple check of the martingale M t = e −2t h 0;1/2 H t |ω . Indeed, we may compute e −2t h 0;1/2 Φ ∆,∆ (z * ,z * )H t |ω by moving H t to the left so that Φ ∆,∆ gets transformed by h t (z). Using the normalisation of the SLE map at the fixed point z * and the fusion rule d = 2h 0;1/2 + κ 2 s 2 , we get exp[isξ t + κ 2 s 2 t] which is a well known martingale for the Brownian motion.
Some of the previous results have a simple interpretation in terms of the O(n) models in the dilute phase. Using Coulomb gas techniques, the L-leg bulk operators have been identified [12,15] with the bulk operator Φ 0;L/2 , and the L-leg boundary operators with Ψ 1;L+1 . Hence the boundary operator Ψ 1;2 singled out by the SLE martingale M t corresponds to a 1-leg operator creating a single curve, while the bulk operator Φ 0;1/2 to which it couples corresponds to the termination of a single curve in the bulk as it should be.
, h ≥ 0, at large time for x on the boundary. In particular f 0 (x, t) is the probability that the point x has not been swallowed by the SLE trace up to time t. As shown in [4] using probabilistic arguments, the time evolution of f h (x, t) is governed by H h , eq. (9): t). So its large time behavior is dictated by the eigenstate of H h of largest eigenvalue. We shall identify this eigenvalue using the martingale M t = e −2t h 0;1/2 H t |ω .
Indeed, consider as above the projection of the martingale M t on the state created by primary fields localized at the fixed point and on the boundary: By construction e −2t h 0;1/2 F h (x, t), h ≥ 0, is a local martingale. It may be computed by moving H t to the left, which conformally transforms the primary fields. Hence, , 0) where we used the known asymptotic behavior of h t at the fixed point z * . As a consequence: with ǫ ∆,∆ = 2h 0;1/2 −2∆. This is of course related to the eigen-equation (10). Now, as a consequence of the null vector relation (2V −2 − κ 2 V 2 −1 )|ω = 0, the function F h (x, 0) = Φ ∆,∆ (z * ,z * )Ψ h (x)Ψ 1;2 (x 0 ) satisfies a second order differential equation [1] which depends on ∆. The primary field Φ ∆,∆ (z * ,z * ) is chosen by demanding that F h (x, 0) satisfies the same boundary condition as f h (x, t). Namely [4], it is single valued when x moves along the boundary and it vanishes when x approaches x 0 from both sides. This selects the conformal weight ∆(h), in the disc geometry. This fonction has no node, it is thus the fundamental. As a consequence, f h (x, t) decreases exponentially as e −λ(h) t with an exponant: It of course agrees with ref. [4] and with the computations of ref. [16], section 12.3, based on 2D quantum gravity. The dimension ∆(h) has a simple interpretation in the Coulomb gas representation. Let β κ = 2/κ be the charge of Ψ 1;2 creating the SLE trace and β, or 2α 0 − β, be the charge of the boundary operator Ψ h with h = 1 2 β(β − 2α 0 ) ≥ 0. Then δ + (h) = β κ β and δ − (h) = β κ (2α 0 − β) with β > α 0 > 0, reflecting the fact that the fusion relations with Ψ 1;2 are linear in terms of Coulomb charges. δ ± (h) are directly related to the dimensions of the operators produced by fusing Ψ h with Ψ 1;2 since the operator prod- x → x 0 . Hence, the vanishing of Φ ∆,∆ (z * ,z * )Ψ h (x)Ψ 1;2 (x 0 ) as x → x 0 demands to represent Ψ h with the charge β, β > α 0 and not with 2α 0 − β. Let α =ᾱ be the charges of the bulk operator Φ ∆,∆ . Demanding that there are no screening charges in the Coulomb gas representation of the correlation function Φ ∆,∆ (z * ,z * )Ψ h (x)Ψ 1;2 (x 0 ) ensures that this function has no monodromy, as we required. Since one has to compensate the background charge 2α 0 , the absence of screening charges imposes 2α + β + β κ = 2α 0 or 2α + (2α 0 − β) + β κ = 2α 0 . Demanding now that this correlation vanishes as x approaches x 0 selects the charge 2α = 2α 0 − β − β κ . The corresponding scaling dimension α(α − 2α 0 ) is 2∆(h).

5-
The restriction martingales. The martingale M t may be used to construct the restriction martingales [5] coding for the influence of deformations of domains on radial SLEs. For simplicity we present it in the case of the disc geometry. The construction is similar to the one we presented in [9], so we shall only sketch it. Let A be a hull in D tr and φ A be one of the uniformizing map of its complement onto D tr fixing x 0 . Given φ A and h t , we may write in a unique way a commutative diagram φ At • h t = h t • φ A where φ At (resp. h t ) uniformizes the complement of h t (A) (resp. φ A (K t )) onto D tr with φ At fixing x 0 and h t fixing z * = ∞. Let as above H t (resp. H t ) be the operators implementing h t (resp. h t ) in CFT. Similarly, let G A (resp. G At ) be those implementing φ A (resp. φ At ). Then [9],