2 00 4 SLE ( κ , ~ ρ ) and Conformal Field Theory

SLE(κ, ~ ρ ) is a generalisation of Schramm-Loewner evolution which describes planar curves which are statistically self-similar but not conformally invariant in the strict sense. We show that, in the context of boundary conformal field theory, this process arises naturally in models which contain a conserved U(1) current density Jμ, in which case it gives rise to a highest weight state |h〉 satisfying a deformation 2L−2|h〉 = (κ/2)L−1|h〉 + αJ−1L−1|h〉 of the usual level 2 null state condition. We apply this to a free field theory with piecewise constant Dirichlet boundary conditions, with a discontinuity λ at the origin, and argue that this will lead to level lines in the bulk described by SLE(4, ~ ρ ) across which there is a universal macroscopic jump ±λ∗ in the field, independent of the value of λ. We also show how SLE(κ, ~ ρ ) with κ 6= 4 may be realised in the Coulomb gas formalism of the O(n) model. ∗Address for correspondence