Motives and categorifications of quantum statistical mechanics

The talk is based on joint work with Goncalo Tabuada (arXiv:1411.3223). We show that the Bost-Connes quantum statistical mechanical system admits a natural categorification in terms of orbit categories of Tate motives. This construction is then further generalized to include a large class of Bost-Connes-type data, which have similar categorifications, obtained through the Tannakian formalism, and which give rise to interesting quantum statistical mechanical systems. These include a quantum statistical mechanical system associated to Weil numbers and the category of motives over finite fields.