  We review some aspects of the cutting and gluing law in local quantum field
theory. In particular, we emphasize the description of gluing by a path
integral over a space of polarized boundary conditions, which are given by
leaves of some Lagrangian foliation in the phase space. We think of this path
integral as a non-local $(d-1)$-dimensional gluing theory associated to the
parent local $d$-dimensional theory. We describe various properties of this
procedure and spell out conditions under which symmetries of the parent theory
lead to symmetries of the gluing theory. The purpose of this paper is to set up
a playground for the companion paper where these techniques are applied to
obtain new results in supersymmetric theories.
