SLE: differential invariants study

Random curves (produced by the Schramm–Loewner evolution SLE_κ ) of the fractal dimension d_κ = 1 + κ /8 , κ < 8 given on a surface D ⊂ℝ^3 and the conformal group ( CG ) that acts on D are considered. We study the action integral L[𝐗(t)] , 𝐗(t) ∈ SLE_κ (the fractal variation of length of a random curve (Kennedy in J Stat Phys 128(6):1263–1277, 2006)) together with the conformal group extension CGE of CG which invariant transforms SLE_κ and L[𝐗(t)] . We calculate the second-order universal differential invariant 𝐉_2 (or the multiscale representation of invariants) of the GCE and show that L[𝐗(t)] generates all second-order differential invariants of CGE by the operators of invariant differentiation. The differential invariants look like invariant quantities of different scales wherein L[𝐗(t)] plays a role of "the fractal length scale". The method of calculations of differential invariants is a kind of modern multiscale analysis (Olver and Pohjanpelto in Adv Math 222(5):1746–1792, 2009) based on the theory of symmetry group. This investigation is also motivated by Cartan’s point of view (in: Cartan, La Théoric des Groupes Finis et Continus et la Géometrie Differentielle traittée par le Méthode du Repére Mobile, Gauthier-Villars, Paris, 1937) that the local geometry properties are entirely governed by differential invariants of the group admitted.