# SLE applied to theoretical physics (eg. CFT), too untractable? not informative enough?

It is has been 15 years since the SLE area took off, and it helped rigourize known predictions from RG and the conformal invariance of the interface (eg. percolation and Ising) and other applications.

But besides the numerous surveys of relating CFT and SLEs, I haven't found any solid paper using SLE techniques to obtain new results in physics (in the last few years), let alone being a common place tool such as RG.

So as a person knowledgeable in the SLE area, I would like to know what the criticisms are from the physics point of view.

1)Were the physicists looking for a more tractable limiting object (eg. Gaussian free field)? Are SLEs not explicit enough, as a theoretical tool?

2)Is it that known tools can do anything that SLEs do and far more? So why bother learning about them.

3)Is it because the connection between CFT and SLE is well-established for central charge $c\leq 1$, but most models are above that (see (1))?

4)There is no 3D analogue for SLE and connection with the recent success of conformal bootstrapping in predicting critical exponents (see (2)).

(1) www.sciencedirect.com/science/article/pii/S0550321307007109

(2)"Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap" link.springer.com/article/10.1007/s10955-016-1658-x

2) The extension of SLE to theories with extended symmetries is not well known. I worked exactly on this problem. While CFT results cover a large spectrum of "conformal algebras", SLE is well-known only for the Virasoro algebra. Its extension to other algebras seems much harder than the CFT approach. If you compare with CFT, in five years after the BPZ paper, a great number of extension have been worked out. SLE has been mainly limited to the Virasoro case. Actually I was involved in some of the very few extensions of SLE to other cases. Using the ideas of Wiegmann, Gruzberg et al., I proposed a connection on SLE/Parafermionc algebras, which are conformal algebras + $\mathbb{Z}_N$ symmetries (arXiv:0705.2749). I could predict the fractal dimension of an interface of certain critical models (such as in the Ashkin-Teller model). It is quite funny that this is the only example I know where this quantity is not accessible by any of the two methods taken alone. Using BOTH I could predict it (and I numerically confirmed it).