@MISC{(paris_theperiodicity, author = {Bernhard Keller (paris}, title = {THE PERIODICITY CONJECTURE VIA 2-CALABI-YAU CATEGORIES}, year = {} }

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Abstract

The periodicity conjecture was formulated in mathematical physics at the beginning of the 1990s, in the work of Zamolodchikov [11], Kuniba-Nakanishi [8] and Gliozzi-Tateo [4]. 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 [3] and Gliozzi-Tateo [5] for the pairs (An, A1), by Fomin-Zelevinsky [2] in the case where one of the diagrams is A1 and by Volkov [10] and Szenes [9] when both diagrams are of type A. The conjecture is about to be proved by Hernandez-Leclerc [6] in the case where one of the diagrams is of type A. We will sketch a proof [7] 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 trans-formations and by Amiot’s recent work [1] on cluster categories for algebras of global dimension 2. References 1. Claire Amiot, Cluster categories for algebras of global dimension 2 and quivers with potential, arXiv:0805.1035. 2. Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977–1018. 3. Edward Frenkel and András Szenes, Thermodynamic Bethe ansatz and diloga-