I will explain and state a conjecture of Kontsevich, that relates vertex models from statistical mechanics to $E_n$-algebras (i.e., algebras for the n-dimensional little disks operad). I will also give the main ingredients of the proof of Kontsevich's conjecture, that involve discretized versions of the little disks operad. This is a work in progress with Damien Lejay.

The work presented here has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 768679).[-]