in this short review we look at recent advances in schramm-loewner evolution (sle) theory and its application to critical phenomena. the application of sle goes beyond critical systems to other time dependent, scale invariant phenomena such as turbulence, sand-piles and watersheds. through the use of sle, the evolution of conformally invariant paths on the complex plane can be followed; hence a...

We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy’s formula. Then we prove existence, uniqueness, and conformal invarianc...

We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculate their values in the scaling limit. As a particular case we obtain conformal invariance of the crossing probabilities and Cardy’s formula. Then we prove existence, uniqueness, and conformal invarianc...

We discuss boundary conditions for conformal field theories that preserve the boundary Poincaré invariance. As in the bulk field theories, a question arises whether boundary scale invariance leads to boundary conformal invariance. With unitarity, Cardy’s condition of vanishing momentum flow is necessary for the boundary conformal invariance, but it is not sufficient in general. We show both a p...

We study scale invariant but not necessarily conformal invariant deformations of non-relativistic conformal field theories from the dual gravity viewpoint. We present the corresponding metric that solves the Einstein equation coupled with a massive vector field. We find that, within the class of metric we study, when we assume the Galilean invariance, the scale invariant deformation always pres...

Using the Wilson renormalization group, we show that if no integrated vector operator of scaling dimension -1 exists, then scale invariance implies conformal invariance. By using the Lebowitz inequalities, we prove that this necessary condition is fulfilled in all dimensions for the Ising universality class. This shows, in particular, that scale invariance implies conformal invariance for the t...

We consider classical and quantum strings in the conformally invariant background corresponding to the SL(2,R) WZWN model. This background is locally anti-de Sitter spacetime with non-vanishing torsion. Conformal invariance is expressed as the torsion being parallelized. The precise effect of the conformal invariance on the dynamics of both circular and generic classical strings is extracted. I...

We consider classical and quantum strings in the conformally invariant background corresponding to the SL(2,R) WZWN model. This background is locally anti-de Sitter spacetime with non-vanishing torsion. Conformal invariance is expressed as the torsion being parallelized. The precise effect of the conformal invariance on the dynamics of both circular and generic classical strings is extracted. I...