Quantum Abelian Yang-Mills Theory on Riemannian Manifolds with Boundary

Abstract

We quantize abelian Yang-Mills theory on Riemannian manifolds with boundaries in any dimension. The quantization proceeds in two steps. First, the classical theory is encoded into an axiomatic form describing solution spaces associated to manifolds. Second, the quantum theory is constructed from the classical axiomatic data in a functorial manner. The target is general boundary quantum field theory, a TQFT-type axiomatic formulation of quantum field theory.Combining (3.11) and (3.10) then yieldsThe relevant integral in (3.12) is

A Poisson Algebra for Abelian Yang-Mills Fields on Riemannian Manifolds with Boundary

“…The following assertion related to Proposition 3.7 explains how the relative cohomology codifies the description of A U /G U with respect to the boundary conditions, see also [6].…”

“…There are gauge translations X = X ψ , X ′ = X ψ ′ ∈ G U , ψ, ψ ′ : U → R such that the gauge translations V, V ′ are divergence-free, see for instance the Appendix [6]. Recall that V, V ′ are defined by ϕ − dψ, ϕ ′ − dψ ′ , respectively.…”

“…There is a commutative diagram of linear maps as follows. Recall the gluing procedure for abelian YM, see [6]. The doted arrow is a Lie algebra morphism.…”

“…From the Lagrangian embedding of LŨ with respect to the symplectic structure, ω ∂U,L , it follows that the Dirichlet conditions along Σ 1 and Σ 2 completely determine the Neumann conditions in U 1 and U 2 , respectively. Here we consider an axial gauge fixing for solutions in ∂U satisfying also the Lorentz gauge fixing condition in ∂U , see Appendix in [6]. This means that the continuous gluing condition (26) will suffice to reconstruct modulo gauge the first variation V ϕ = V ϕ1 #V ϕ2 for V ϕi ∈ F Ui , i = 1, 2 disregarding the normal derivatives along Σ.…”

“…d h is the horizontal differential, see the notation of the variational bicomplex formalism in Appendix A. We adapt helicity for hypersurfaces embedded properly in general compact regions U , rather than considering cylinder regions with space-like slices, Σ × [t 1 , t 2 ], this is related to the General Boundary Formalism for field theories, see [6] and references therein.…”