Vladimir Fock: Clusters, integrable systems and tame symbol

Date: 2023-06-29

Time: 09:30 - 10:30


Vladimir Fock, Université de Strasbourg


A tame symbol is a bimultiplicative 2-cocycle on the group of nonvanishing functions a circle given by an explicit formula. The tame symbol is related to Heisenberg group, resultant, Witt ring, Gauss reciprocity and many other subjects.  We will use the tame symbol to define a homology class (with values in the multiplicative group) of a Lagrangian subvariety of a cluster A-variety. Say that a Lagrangian subvariety is Bohr-Sommerfeld if this class is trivial. We will show in the example of dimension 2 that every Bohr-Sommerfeld curve gives a solution of a difference equation, which is a quantization of the equation of the curve with quantization parameter equal to 1. The solution is given by quantum dilogarithms of algebraic functions.