Other variants of SLE

Date: 2015-10-07; view: 461.

So far we have discussed only chordal SLE, which describes curves connecting distinct points on the boundary of a simple connected domain, and radial SLE, in which the curve connects a boundary point to an interior point. Another simple variant is dipolar SLE [39], in which the curve is constrained to start at boundary point and to end on some finite segment of the boundary not containing the point. The canonical domain is an infinitely long strip, with the curve starting a point on one edge and ending on the other edge. This set-up allows the computation of several interesting physical quantities.

The study of SLE in multiply connected domains is very interesting. Their conformal classes are characterised by a set of moduli, which change as the curve grows. Friedrich and Kalkkinen [40] have argued that SLE in such a domain is characterised by diffusion in moduli space as well as diffusion on the boundary.

It is possible to rewrite the differential equations which arise from null state conditions in extended CFTs (for example super-conformal CFTs [41] and WZWN models [42]) in terms of the generators of stochastic conformal mappings which generalise that of Loewner. However, a physical interpretation in terms of the continuum limit of lattice curves appears so far to be missing.