The periodicity conjecture via 2-Calabi-Yau categories

Bernhard Keller

The periodicity conjecture was formulated in mathematical physics at the beginning of the 1990s, in the work of Zamolodchikov, Kuniba-Nakanishi and Gliozzi-Tateo. It asserts that a certain discrete dynamical system associated with a pair of Dynkin diagrams is periodic and that its period divides the double of the sum of the Coxeter numbers of the two diagrams. The conjecture was proved by Frenkel-Szenes and Gliozzi-Tateo for the pairs (A_n, A₁), by Fomin-Zelevinsky in the case where one of the diagrams is A₁ and by Volkov and independently Szenes when both diagrams are of type A. The conjecture is about to be proved by Hernandez-Leclerc in the case where one of the diagrams is of type A. We will sketch a proof of the general case which is based on Fomin-Zelevinsky's work on cluster algebras and on the theory relating cluster algebras to triangulated 2-Calabi-Yau categories. An important role is played by Coxeter transformations and by Amiot's recent work on cluster categories for algebras of global dimension 2.