Lagrangian and Hamiltonian structures in an integrable hierarchy

par Vincent Caudrelier (University of Leeds)

jeudi 10 nov. 2016, 11:00 → 12:30 Europe/Paris

Auditorium (Annecy-le-Vieux)

The classical and quantum versions of the $R$ matrix are the cornerstones in classical and quantum integrable systems, typically formulated in $1+1$ dimensions. They are the heart of the theory developed by the Fields medallist V.~Drinfeld. However, they traditionally concentrate all the attention on only one of the independent variables: the space one while time evolution is encoded more or less trivially. The latter point is in fact deeply related to the boundary conditions imposed on the system. A big success of the theory of classical integrable systems is the systematic Hamiltonian formulation of the corresponding PDEs. The essential object capturing the Hamiltonian properties (infinite number of conserved quantities, etc.) is the so-called classical $r$-matrix. Motivated originally by the question of integrability of certain field theories in the presence of defects, we will show how a dual Hamiltonian structure naturally emerges which gives a fully fledged $r$-matrix structure to the time variable. This is inspired and related to the notion of covariant field theory. The interplay between the standard classical $r$-matrix structure and the dual one raises many questions and begs for a ''multisymplectic $r$-matrix theory''.