Degenerate Classical Field Theories and Boundary Theories

We introduce a framework for degenerate classical field theories in the BV formalism, which allows us to discuss many interesting examples of theories which do not admit a Lagrangian description. Further, we study phase spaces and boundary conditions for classical field theories on manifolds with boundary, and from a fixed classical field theory together with a choice of boundary condition, construct a degenerate classical field theory on the boundary. We apply these ideas to many physically interesting examples including the Kapustin-Witten twists of N=4 supersymmetric Yang-Mills, Chern-Simons theory, the chiral Wess-Zumino-Witten model, chiral Toda theory, and a new 3d classical field theory called Whittaker theory.