Towards a conformal field theory for Schramm-Loewner evolutions?

Eveliina Peltola

For a number of critical lattice models in \(2D\) statistical physics, it has been proven that scaling limits of interfaces (with suitable boundary conditions) are described by Schramm-Loewner evolution (SLE) curves. So-called partition functions of these SLEs (which also encode macroscopic crossing probabilities) can be regarded as specific correlation functions in the conformal field theory (CFT) associated to the lattice model in question. Although it is not clear how to define the latter mathematically, one can still make sense of many of the properties predicted for these CFTs. In particular, all of the expected CFT properties: conformal invariance, null-field equations, and fusion rules, are satisfied by the partition functions. One might then ask: Is it possible to go deeper and to construct the appropriate CFT fields as random distributions? Time permitting, I discuss some ideas to this direction.

Keywords:
Conformal field theory, correlation function, critical lattice model, Schramm-Loewner evolution.

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