Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

In Mechanics, the cotangent space to the configuration space, understood as the space of initial data for (regular) Lagrangian theories, is equipped with a natural symplectic structure. Similarly, in field theory, one gets an infinite-dimensional symplectic space of boundary fields, containing the space of initial data as a Lagrangian submanifold: such spaces are at the heart of the so-called BV-BFV theories with boundary, which are enjoying a renewed interest. In this talk, I will review a recent work with J. Kijowski, where the symplectic structures behind higher-order fields theories have been studied in detail. The so-obtained framework, based on an obvious jet-theoretic analogy with Mechanics, represents probably the simplest geometric description of the dynamics of a Lagrangian field theory. The symplectic structures involved are defined ‘fibre-by-fibre’ and, in this sense, they can be treated as finite-dimensional.