General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners

Abstract

Abstract. We propose a general reduction procedure for classical field theories provided with abelian gauge symmetries in a Lagrangian setting. These ideas come from an axiomatic presentation of the general boundary formulation (GBF) of field theories, mostly inspired by topological quantum field theories (TQFT). We construct abelian Yang-Mills theories using this framework. We treat the case for space-time manifolds with smooth boundary components as well as the case of manifolds with corners. This treatment is the GBF analogue of extended TQFTs. The aim for developing this classical formalism is to accomplish, in a future work, geometric quantization at least for the abelian case.

Quantum Abelian Yang-Mills Theory on Riemannian Manifolds with Boundary

“…This amounts to requiring L M,∂M = L ∂M and A M,∂M = A ∂M , resulting in a considerable simplification of the axioms. The more general version of the axioms presented here is motivated precisely by the discovery in [7] that for abelian Yang-Mills theory the stricter Lagrangian subspace condition is not satisfied in general. We shall demonstrate in Section 4, however, that the present generalized axioms are satisfied.…”

“…We work on smooth manifolds equipped with a Riemannian metric. For the reader's convenience we review some of the geometric facts for YM fields in Appendix C, see also [7]. We rely on the results obtained in [8].…”

Dirichlet to Neumann operator for Abelian Yang–Mills gauge fields

“…The affine and linear maps from the space of solutions to the corresponding boundary conditions a M : A M → A ∂M and r M : L M → L ∂M are compatible with the corresponding gauge group actions in the bulk and in the boundary respectively, see [17] axiom (A8). There is also a section G 0 ∂M → G 0 M , see (29).…”

“…Part 1 has already been shown. Part 3 is proved independently in [7] and [17], see also Theorem 3 below. We prove part 2.…”

“…In the linear and affine case, by taking the quotient by ker ω ∂M gauge reduction can be achieved. This was explored in a previous work [17] where we exposed an axiomatic approach in the GBFT formalism for abelian gauge fields applying it to the Yang-Mills case, see also [6,7].…”