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


1 Answer 1


My name is Raoul Santachiara and I am a CNRS researcher at the Paris-Saclay university. My core expertise are the CFTs and their applications to 2D critical statistical model. I have worked (and I am still working) on the relationship between CFT and SLE. I will try to give you a (of course) partial answer to your questions, mainly concerning the 2D case (so more or less answering your questions 1,2,3).

At the beginning we were quite excited about the fact that SLE could provide an equivalent (??) formulation of CFT and therefore maybe help answering some open question in our field. We realized with time that, even if SLE represents a more natural and elegant approach to compute certain probabilities, it remains too specific to answer much more fundamental questions of CFT in which we are interested. Let me give two concrete examples:

1) BULK critical percolation clusters: the probability measures of these clusters are conformally invariant. This hints to the fact that there exist CFT solutions that provide these probabilities but are still NOT known. This is a long-standing problem: NEW CFT solutions with c<1 that go beyond the minimal models. To be clear, a CFT solution is when you can compute the four-point correlation functions, i.e. when you know the set of critical exponents AND the so-called structure constants. The physicists have long thought about that, for instance by trying to develop logarithmic CFTs (see Saleur, Jacobsen, Pearce, Rasmussen, Zuber, Vasseur, Dubail, etc). In this respect SLE is related to conformal blocks in BOUNDARY CFTs, and in particular to special functions of CFT that involve degenerate fields: BUT in this case there are some crucial SIMPLIFICATIONS, in particular the probabilities satisfy Fuchsian differential equations. However this is NOT true in general, for instance it is not true for the bulk connectivities properties of critical percolation clusters. In this respect, we obtained some nice results in arXiv:1607.07224

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).


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