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

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.


Introduction
In the variational formulation of classical mechanics, time evolution from an "initial" to a "final" state in a symplectic phase space (A, ω) is given by a relation defined by a lagrangian space L contained in the symplectic product (A ⊕ A, ω ⊕ −ω). Similarly classical field theories can be formalized rigorously in a symplectic framework. The evolution relation associates "incoming" to "outgoing" Cauchy boundary data for the case where M has incoming and outgoing boundary components, ∂M = ∂M in ∪ ∂M out . Fields are valued along the boundary altogether with their derivatives. This relation defines an isotropic space of boundary conditions that extend to solutions in the interior of M , LM ⊂ A ∂M = A ∂M in × A ∂Mout , where the symplectic From the quantum side the axiomatic setting for GBF is inspired on the axiomatic setting of Topological Quantum Field Theories (TQFT), see [At] and the formulation of G. Segal. We consider objects in the category of (n − 1)−manifolds, i.e. closed boundary components or hypersurfaces Σ, provided with additional normal structure required by germs of solutions: for instance for field theories without metric dependence we consider gluings by diffeomorphisms of tubular neighborhoods of Σ, [Mi], meanwhile for field theories depending on the metric we consider gluing by isometries of Σ, Σ ′ leaving invariant the metric tensor germ along Σ. The gluing of two regions M 1 , M 2 can be performed along hypersurfaces Σ ⊂ M 1 , Σ ′ ⊂ M 2 , both diffeomorphic and oriented manifolds, Σ ∼ = Σ ′ , and Σ ′ means reversed orientation. The precise axiomatic setting for quantum field theories along with their classical counterpart appears in [O3] and for affine theories in [O1].
Corners. This TQFT-inspired approach requires a classification of the basic regions or building blocks used to reconstruct the whole space-time region M 1 ∪ Σ M 2 , by gluing the pieces M 1 , M 2 , along the boundary hypersurface Σ ∼ = Σ ′ . This classification from the topological point of view can be achieved at least for the case of two dimensional surfaces. In higher dimensions, it would be appealing to avoid such classification issues, by considering simpler building blocks, such as n−balls. Unfortunately, the price to pay is that we should allow gluings of regions along hypersurfaces Σ with nonempty boundaries ∂Σ. For instance we can consider the gluing of two n− balls M 1 , M 2 along (n − 1)−balls contained in their boundaries Σ, Σ ′ . This means that we should allow non differentiability and lack of normal derivatives of fields along the (n − 2)−dimensional corners contained in the boundaries ∂Σ, of boundary faces, Σ ⊂ ∂M 1 . A well suited language to describe such phenomena, consists in treating regions M i as manifolds with corners. For TQFT the attempt to deal with the case of corners gives rise to Extended Topological Quantum Field Theories, one possible approach for two dimensional theories is given for instance in [LP]. There is also a specific formulation for 2−dimensional Yang-Mills with corners in [O2]. Our aim is to extend this last approach to higher dimensions.
Gauge field theories. When we consider principal connections on a principal bundle P → M, with structure compact Lie group G, they are represented by sections of the quotient affine 1−jet bundle J 1 P/G → M . In this case the space of sections K M is an affine space. Furthermore for quadratic Lagrangian densities we will have that the space of solutions, A M , is an affine space. This enable us to consider a GBF formalism for affine spaces such as is described in [O1].
The novel issue with respect to [O1], is to consider gauge symmetries, G M , acting on A M . These symmetries are vertical automorphisms of the bundle P , that in turn yield vertical automorphisms of the bundle J 1 P/G. Infinitesimal gauge symmetries should preserve the action, S M : K M → R. These can be identified with vertical G−invariant vector fields X on P , and also with sections of V P/G → M , where V P is the vertical tangent bundle of P → M . These vertical vector fields act on J 1 P/G preserving the lagrangian density.
When we consider germs of solutions of the boundary, we also have symmetriesG ∂M , and quotienting by degeneracies we obtain a gauge group action G ∂M acting by symplectomorphisms on (A ∂M , ω ∂M ). The main problem is to give sense to the quotient space A M /G M of solutions and how relate the reduced boundary conditions contained in the symplectic reduction A ∂M /G ∂M . The issues of gluing solutions need also to be clarified.
Main results. Our aim is to give an axiomatic GBF formulation for gauge field theories in the case of space-time regions with corners. For the classical theory we will consider the following simplifications: Abelian structure groups and affine structure for the space of solutions of Euler-Lagrange equations. The test example will be Yang-Mills action. The most general setting of nonabelian structure groups remains as a conjecture even in the classical case, see [CMR1]. Along this program we study the case without corners and then we focus our attention on the case with corners.
The important result for n−dimensional field theories without corners is the symplectic reduction theorem 1. For the case with corners this theorem replicates as theorem 3. One of the most important ingredients comes from establishing suitable local Fermi type coordinates, this in turn comes from a volume transport argument by Moser [Mo] in lemma 1. Dynamics as lagrangian relation is particularly interesting and it is established in theorem 2 whose statement for the case with corners remains the same.
As we were finishing writing this article we realized that Lagrangian embedding in the abelian case had been shown independently in [CMR1] for the case without corners. Nevertheless there are some differences in our approach: Whereas we used axial gauge fixing and used Hodge star in the boundary in order to describe a Hodge decomposition for the space of boundary conditions that extend to solutions the interior. On the other hand, in appendix C of [CMR1], the authors used Lorentz fixing and independently give the space of boundary conditions H 1 (M, ∂M ) this Hodge structure by considering such decompositions for any subspace of Dirichlet boundary conditions L ⊂ Ω 1 (∂M ).
Description of sections. Section 2 consists in a review of the sym-plectic formalism for classical field theories altogether with an exposition of the axiomatic setting that we propose to abelian gauge field theories. We also divide the exposition of the axioms into two cases: the case where regions are considered as manifolds and the case where regions are manifolds with corners. In section 3, we exemplify the application of the concepts and axioms described in section 2. Here we address the simpler case without corners and propose the abelian Yang-Mills theory. For local arguments Moser's argument on the transport flow for volume forms will be relevant. We describe the symplectic reduction of the space of boundary conditions and emphasize the proofs of the lagrangian embedding of solutions once reduction is achieved. The Friedrichs-Morrey-Hodge theories are the main source of our results. For the case with corners, in section 4, a complete description of the reduction can be achieved. We review dimension 2 as an example in section 4.1.
Aknowledgements. The author thanks R. Oeckl for several discussions and encouragement for writing this note. This work was partially supported by CONACYT-México postdoctoral grant at CCM-UNAM.
2 Axioms for classical abelian gauge field theories 2.1 The symplectic setting for classical lagrangian field theories For the sake of completeness, we resume the symplectic formalism for lagrangian field theories in the following paragraphs. Local descriptions for the case of the space of Dirichlet-Neumann conditions appear in [KT], on the other hand discussion on the space of germ of solutions in the axiomatic setting appears in [O], [O1], parallel developments appear in [CMR]. We adopt an abstract coordinate-free description of the (pre-)symplectic structure for boundary data, by means of a suitable a cohomological point of view for the presentation of this formalism. Classical field theories assume that over an n−dimensional space-time region M , there exist a "configuration space", K M , of fields ϕ ∈ K M . The word "space" used for referring to K M usually denote infinite dimensional Frèchet manifolds, defined as a space of sections of a smooth bundle E over M . It also assumes the existence of a Lagrangian density, Λ ∈ Ω n (J 1 M ), depending on the first-jet j 1 ϕ ∈ J 1 M , i.e. on the first order derivatives ∂ϕ and on the values of the fields ϕ. The action corresponding to the lagrangian density is then defined as On the other hand we consider the factorization of the space of k−forms over the l−jet manifold J l M as ) corresponds to horizontal (resp. vertical) k−forms. For instance, using local coordinates x i , i = 1, . . . n, for tha manifold M , take (x i ; u a ; u a i ) as local coordinates for J 1 M , horizontal forms have basis the exterior product of the basis dx i . Meanwhile for vertical forms say in J 1 M , we have as basis the exterior product of du a , du a i . The horizontal (resp. vertical) differential is where r equals the dimension of each fiber of the bundle E. Thus, vertical k−forms vanish on horizontal vector fields X such that d V ( X) = 0. This decomposition yields a variational bicomplex, see for instance [GMS]. .

Denote the space of solutions of the Euler Lagrange equations as
In the case where we are dealing with connections A M is an affine space corresponding to a linear space that we call L M .
Of course the representative θ Λ ∈ d −1 H •d V Λ depends just on the d H −cohomology class of the lagrangian density. On the other hand, by integration by parts, the differential Locally each variation δϕ is identified with a vector field, X, along the section ϕ in J 1 M . This X in turn induces a vector field j 2 X, the 2−jet prolongation of the vector field X, along j 2 ϕ, on the 2−jet manifold J 2 M , both vanishing on horizontal 1−forms. This shows that total variations consist of two contributions: One due to the variation on the bulk of the fields corresponding to Euler-Lagrange equations, but also a contribution coming from the values of the field and its normal derivatives on the boundary ∂M . Let us concentrate on the boundary term of the variation. The calculus on the 1−jet total space, J 1 M, translates onto the calculus on the infinite dimensional space, K M , so that θ Λ induces a 1−form for variations of 1−jets of solutions restricted to the boundary X ∈ T ϕ A M . This enables us to consider a 1−form dS ϕ , for variations X ∈ T ϕ A M . For a (n − 1)−dimensional boundary manifold Σ, the boundary conditions for solutions on a tubular neighborhood Σ ε ∼ = Σ × [0, ε], of the cylinder Σ × [0, 1], can be described as germs of solutions.
The affine space of germs of solutions on the boundary, and the corresponding linear space are defined as the injective limit Similarly for the linear spaces L Σ ε ′ ⊂ L Σε . We consider spaces of solutions on cylinders Σ × [0, ε], ε > 0, embedded as tubular neighborhoods Σ ε of Σ.
The submersion of variations of germsX ∈ TÃ Σ , onto variations of jets X ∈ T A Σ , leads to the definition of the 1−form onÃ ∂M , Ultimately, our purpose is to consider the presymplectic structure onÃ Σ , There are degeneracies of the presymplectic structureω Σ due to the degeneracy of the lagrangian density and the degeneracies arising from considering arbitrary order derivatives for the germs of solutions. We suppose that these degeneracies altogether can be eliminated by quotienting K ω Σ := ker ω Σ , then we obtain a symplectic space A Σ , ω Σ , we will prescribe this condition as an axiom.
Consider an action map S M (ϕ) defined for connections ϕ of a principal bundle P over M with compact abelian structure group G. We denote as A M , the space of solutions Euler-Lagrange equations in the interior of the region M . In general, we suppose that ∂M is not empty. Hence when we restrict the action functional S M , from the configuration field space K M to the space of solutions A M , it induces a non-constant map On the other hand we have the groups, G M , of gauge symmetries on regions acting on the solutions on the bulk A M that come from the Euler-Lagrange variational symmetries of lagrangian density, see definition 2.3.1 of [GMS]. Infinitesimal symmetries can be identified with G−invariant vertical vector fields on P that can be identified with vertical vector fields acting on J 1 P/G and preserving the lagrangian density.
By taking the cylinder Σ × [0, ε] as M , those symmetries by the group G Σ × [0, ε] act on germs of solutions in A Σε hence inÃ Σ . By taking the quotient by the stabilizer of theÃ Σ , we obtain a group of gauge symmetries on hypersurfaces,G acting onÃ Σ . Once we have taken the quotient of the space of germsÃ Σ , and its corresponding linear spaceL Σ , by the degeneracy space K ω Σ , we get a space A Σ , and a gauge group G Σ acting on A Σ . This groupG Σ decomposes into two kind of symmetries: those coming from the degeneracy of the presymplectic structure and those preserving the symplectic structure coming from vector fields preserving the lagrangian density. This means that there is a normal subgroup that takes into account all degeneracies altogether, K ω Σ ⊂G Σ , and whose orbits onÃ Σ consist of the integral leafs of the characteristic distribution generated by the kernel of the presymplectic structureω Σ . Meanwhile G Σ act by symplectomorphisms on A Σ with respect to the symplectic structure ω Σ .

Regions with and without corners
In the following presentation of the axiomatic for classical lagrangian field theories, we will consider regions and hypersurfaces as manifolds with corners.
Hypersurfaces are (n − 1)−dimensional topological manifolds Σ, decomposing as a union of (n − 1)−dimensional manifolds with corners, This union in turn is obtained by gluing of (n − 1)−dimensional manifolds with cornersΣ i ,Σ j , along pairs of (n−2)−faces. This can be done by means of an equivalence relation ∼ P , defined by certain set P of pairs (i, j), i = j. More precisely, non trivial equivalence identifications take place at the set This means that gluings of the faces Σ i , Σ j , take place at (n − 2)−faces A region is an n−dimensional manifold with corners M . Its boundary ∂M , is a topological manifold. Each hypersurface Σ ⊂ ∂M consists of faces Σ i ⊂ ∂M , which are manifolds with corners. For an abstract hypersurface, not necessarily related to a region M , each Σ i may be consider as a face of the n−dimensional manifold with corners given by the cylinder Σ × [0, ε]. In general ∂Σ, ∂Σ i may be nonempty.
The gluing of a region M along two nonintersecting faces Σ 0 , Σ ′ 0 , can be defined. The more general gluing along two nonintersecting hypersurfaces Σ, Σ ′ , may also be defined. Nonetheless, when we consider, for instance, the gluing of riemannian metrics, this gluing may be problematic. For if we glue faces with non intersecting boundaries ∂Σ 0 ∩ ∂Σ ′ 0 = ∅, then conic singularities of the metric along the corners may arise in the resulting spacetime region. Therefore gluings should restricted to nonintersecting faces.
For smooth hypersurfaces Σ ⊂ ∂M we consider tubular neighborhoods [Mi], Σ ε ⊂ M with diffeomorphisms We will also have regular tubular neighborhoods of faces Σ i , consisting of images Σ i ε := X(W (Σ × [0, ε])). The corners of the region M lie on the union of the (n − 2)−dimensional submanifolds, For a more detailed description of faces and corners, see for instance [AGO].

GBF axioms
Now we give a detailed description of the axiomatic setting for classical gauge field theories. Axioms (A1) to (A9) describe the kinematics of the classical theory, while axioms (A10) to (A12) describe the dynamics for gauge fields. (A2) Presymplectic structure: For every hypersurface Σ ⊂ ∂M , there is a presymplectic structureω Σ onÃ Σ invariant underL Σ actions. Equivalently we can consider aL Σ as a presymplectic vector space with presymplectic structure that we can also denote asω Σ .
(A8) Gauge action: There are groupsG Σ acting onÃ Σ preserving the affine structure and the presymplectic structureω Σ such that K ω Σ G Σ . The quotient group acts on on A Σ , preserving the symplectic structure ω Σ . There is a group, G M , of symmetries of variational symmetries S M acting on the space of solutions A M . There is a restriction map a M : A M → A ∂M and group homomorphisms h M : G M → G ∂M . There is a compatibility of gauge actions given by the commuting diagram There is also a compatible action on the corresponding linear spaces (A9) Factorization of gauge actions on hypersurfaces: For the case with corners there is a homomorphism h Σ;|Σ| n−1 : G Σ → G |Σ| n−1 form the direct product group G |Σ| n−1 := GΣ 1 ×· · ·×GΣ m onto G Σ coming from homomorphisms and similar commuting diagram for actions on linear spaces There is an involution of the gauge groups and G Σ → G Σ , compatible with the action.
We denote G Σ i as the image h |Σ| n−1 ;Σ i (G Σ ) ⊂ GΣi. The zero component of the G ∂M −orbit is isomorphic to C ⊥ ∂M , the symplectic orthogonal complement of a coisotropic subspace (A11) Locality of gauge fields: Let M 1 be the region that can be obtained by the gluing of M along the disjoint faces, Σ 0 , Then there is an injective affine maps, a M ; We consider the gluing of the actions compatible with the actions on linear spaces (A12) Gluing of gauge fields: Let M 1 , M be regions with corners M 1 is obtained by gluing M along hypersurfaces Σ, Σ ′ ⊂ ∂M . The following diagrams commute analogously for the inclusions There is also a compatibility for the gluing of the actions of the gauge groups

Further explanations on the axioms
Axioms (A1) to (A7) are just a restatement of axioms (C 1) to (C 6) for a classical setting of affine (linear) field theories as stated in [O1]. Some clarifications are added: in (A2) we consider presymplectic spaces of connections instead of symplectic spaces. We do not consider Hilbert space structures since we are not yet introducing a prequantization scenario for field theories. Some comments can be said about postulate (A4). The translation rule of the 1−form θ ∂M can be deduced from translation rule for the differential dS M of an action action map. This in turn can be deduced from (4). This last relation could be stated as a primordial property and arises from taking a quadratic lagragian density Λ. Affine structure for the space of solutions A M can also be deduced from this condition on Λ.
In (A7) we adapt the decomposition stated in (C 3) for the corners case. Here we are using the definition of stratified spaces, [AGO]. |Σ| (n−1) denote the structure of Σ as (n − 1)−dimensional stratified space This in turn is the quotient of the disjoint union of faces |Σ| n−1 :=Σ 1 ⊔ · · · ⊔Σ k by an equivalence relation ∼ P , defined by certain identification of pairs (i, j) along connected (n − 2)−dimensional facesΣ ij ⊂ ∂Σ i ,Σ ji ⊂ ∂Σ j , for certain set P of pairs (i, j). The set of corners correspond to the (n−2)−dimensional faces Σ ij := Σ i ∩Σ j , (i, j) ∈ P. The (n−2)−dimensional skeleton of corners is the stratified space The lack of surjectivity for dotted arrows in (A7) comes from the non differentiability of the hypersurface Σ along the corners |Σ| (n−2) in the inter- Axiom (A8) introduce the gauge symmetries. Axiom (A9) presents the decomposition and involution properties for gauge actions on the boundary. Finally axioms (A11) and (A12) are derivations for the locality and gluing rule of gauge fields arising from the gluing axiom (C 7).
Locality arguments for gauge fields is implicitly exploited in (A8), (A11) and (A12) deserve further clarification. For instance in (A11), the existence of the exact sequence is not trivial and it is derived from locality for connections in A M and gauge actions in G M . Thanks to the inclusions ∂M ε ⊂ M of tubular neighborhoods we get exact sequences Recall thatÃ Σ is an inductive limit, and A Σ is a quotient ofÃ Σ . For (A8) similar arguments using the following commutative diagrams The compatibility stated here as axiom (A8) arises from locality: the embedding of gauge symmetries in M as local gauge symmetries in a tubular neighborhood ∂M ε and then in G ∂Mε ⊂G ∂M , then symmetries for germs hence symmetries in the quotient group G ∂M .
The axiom (A10) encodes the dynamics a gauge fields since it is an adapted version of the lagrangian embedding onto the symplectic space A ∂M considered in (C 5). In (A10) we use the notion of reduced lagrangian space, see [We]. We could also postulate this axiom as follows. In fact this will be the approach that will be used along this work. There exists a symplectic

Simplifications in the absence of corners
As we mentioned previously for some axioms, namely (A7), (A9) and (A12), we will consider separately two cases: regions with and without corners.
We write down explicitly these axioms in the case where regions M and hypersurfaces Σ are smooth manifolds. Here ∂Σ = ∅.
(A7)' Suppose that and (n − 1)−dimensional hypersurface Σ decomposes as a disjoint union Then there are linear and affine isomorphisms respectively such that (5) holds.
(A9)' For the case without corners |Σ| n−1 ∼ = Σ and the direct product group and analogous compatibility diagrams for actions on linear spaces L ∂M , L |∂M | n−1 (A12)' Let M 1 , M be regions without corners as above with gluing along hypersurfaces Σ, Σ ′ ⊂ ∂M There is also a compatibility for the gluing of the actions of the gauge For section 3 we will consider the case without corners hence we will consider the simplified version (A7)', (A9)', (A12)'. Meanwhile for section 4 we will consider the conrers version (A7), (A9), (A12).

The n−dimensional case without corners
In this section we consider the case without corners. Another assumption is to consider linear (or affine) field theories, see the comment of axiom (A4), that are well suited for the axiomatic scenario presented in the previous section.
The examples that will be useful as a test case is Yang-Mills lagrangian density. We consider gauge principal bundles on a compact manifold M provided with a riemanian metric h, nonempty boundary ∂M and compact abelian fiber group G. For the sake of simplicity we assume the following.
We suppose that regions M are smooth manifolds dim M ≥ 2, provided with a trivial principal bundle P with abelian structure group G = U (1). Hypersurfaces Σ are also smooth manifolds, thus ∂Σ = ∅.
Under these assumptions we will prove the consistency of the axioms for the case without corners. In section 4 we will show the case of regions with corners.

Classical abelian Yang-Mills action
Since the bundle is trivial, the space of connections A M has a linear structure and can be identified with L M . We consider the Yang-Mills action where ϕ ∈ A M is a connection that is a solution of the Euler-Lagrange equations in the bulk, i.e. d ⋆ dϕ = 0. The corresponding linear space is here Ω 1 (M, g) denotes g−valued 1−forms on M that can be identified with 1− forms in M , Ω 1 (M ). This objects fulfill (A1).
The identity component of gauge symmetries can be identified with certain f ∈ Ω 0 (M ) acting by ϕ → ϕ + df , thus G 0 denote the locally constant functions on M . In addition we will suppose that there are no corners. Hence we will consider hypersurfaces as closed submanifolds Σ ⊂ ∂M . Since G 0 M preserves Yang-Mills action on A M this requirement mentioned in (A8) is satisfied.
We will describe an embedding and a normal vector field ∂ τ on Σ ε , whose flow lines are the trajectories X(·, τ ) ∈ Σ ε , 0 ≤ τ ≤ ε, that are normal to the boundary. This embedding arises from the solution of the volume preserving evolution problem on Σ, solved by Moser's trick, see [Mo].
If the volume form on Σ in local coordinates can been described as | det h ij | 1/2 dx 1 ∧ dx 2 · · · ∧ dx n−1 , then (X τ ) * (c(τ )λ τ ) = λ 0 , implies Hence the derivative of Z 0 at Σ equals Z 0 This proves assertion 1. Now, since ∂ τ is normal to Σ, Recall that the derivative of Y t the exponential map Y t at s ∈ Σ, Y 0 * : ) . This proves assertion 3. Assertion 4 is an immediate consequence of assertion 3, and assertion 6 is in turn a consequence of assertion 4.
Definition 1. The following expression corresponds to the presymplectic structure inL Σ , for the Yang Mills action, see for instance [Wo], for allξ,η ∈L Σ with representatives ξ, η ∈ L Σε .
In addition, the degeneracy subspace of the presymplectic form is From this very definition we have that the degeneracy gauge symmetries group K ω Σ is a (normal) subgroup of the identity component groupG 0 Σ ≤G Σ of the gauge symmetries, be the axial gauge fixing subspace ofL Σ . The following statement leads to a simpler expression for the presymplectic structure.
a) Every K ω Σ −orbit inL Σ intersects in just one point the subspace ΦÃ Σ .
Proof of a). Let n−1 i=1 η i dx i + η τ dτ be a local expression for a solution η ∈ L Σε . Let us apply a gauge symmetry in such a way that η τ + ∂ τ f = 0. We can solve the corresponding ODE for f (s, τ ) once we fix an initial condition f (s, 0) = g(s). If we take this initial condition g(s) as a constant, then we get a gauge symmetry in K ω Σ . The remaining part is a straightforward calculation. This proves 2. The other assertion may be inferred from lemma 2.
With this statement we satisfy (A2) and (A3). Let be the quotient by the linear space K ω Σ corresponding to degenerate gauge symmetries. And also let G 0 Σ :=G 0 Σ /K ω Σ be the quotient by the nomal subgroup. By the previous lemma, when we restrict the quotient class A Σ / / / / A Σ to ΦÃ Σ , then we get an isomorphism of affine spaces. Let ω Σ the corresponding symplectic structure on A Σ induced by the restriction ofω Σ to the subspace ΦÃ Σ ⊂L Σ . We now proceed to give a precise description of the symplectic space L Σ . Lemma 1 implies that where ⋆ Σ stands for the Hodge star on Σ. Since ι ∂τ (L ∂τ η) = ι ∂τ (ι ∂τ dη) = 0, then we have a linear map L Σε → Ω 1 (Σ) × Ω 1 (Σ), where for everyη ∈L Σ with representative ξ, η ∈ L Σε and η defined in (9). that leads to a map where we consider the identification with the tangent space T (Ω 1 (Σ)). Notice that ι ∂τ η = 0 implies that η ∈ L Σε , corresponds to a 1−form φ η on Σ. Notice also that d ⋆ dη = 0 implies We have the following expression for the symplectic structure on L Σ for every (φ ξ ,φ ξ ), (φ η ,φ η ) ∈ L Σ , with representatives ξ, η ∈ L Σε . Form this very definition we can verify (A4), i.e. translation invariance and also relation (3) where Furthermore (A6) is easily verified and the claims from (A5) can be inferred from the relation ⋆ Σ = −⋆ Σ .
Till now we have validated axions (A1) to (A8) which describe kinematic information of gauge fields. In the following paragraphs we consider gauge equivalence.

Symplectic reduction
We still need to describe the quotient for the symplectic action of the gauge group G 0 Σ on L Σ . The suitable gauge fixing space Φ Σ in A Σ for this action will be the space of divergence free 1−forms, i.e. if we define The following task is the detailed description of the symplectic quotient space According to Hodge theory [Sc], associated with the inner product we have an orthogonal decomposition where the space H 1 (Σ) of harmonic 1−forms is isomorphic to the de Rham cohomology H 1 dR (Σ). The following lemma fulfills (A8) and shows the gaugefixing space definition required in (A10).

b) Every
The existence and regularity of a solution, f (s), for this PDE on Σ is warranted precisely by Hodge theory. Since On the other hand (19) When we substitute ∂ τ η i + ∂ τ ∂ i f by the coefficients φ i τ , of a time dependent 1−form in Σ, φ τ ∈ Ω 1 (Σ), equation (19) has a solution φ τ . This leads to an ODE for g i (s, τ ) : Equation (20) can be solved, once we can fix the boundary condition . This boundary condition, in turn, has been obtained by solving (17) in Σ. We conclude that n−1 i=1 g i (s, τ )dx i is an exact form on Σ, so that there exists f (s, τ ) ∈ Ω 0 (Σ ε ) such that (18) holds.
There is extension of local gauge actions: In the particular case of trivial principal bundle local gauge symmetries in G ∂M extend via partitions of unity to symmetries in the bulk G M . This means that we can define sections σ : The corresponding axial gauge fixing space can be described with the isomorphism, given by Hodge theory where in the r.h.s. we take tangent spaces. In the abelian case holonomy hol γ (φ) = exp γ φ ∈ G of a connection φ along a closed trajectory γ, can be defined up to cohomology class of γ. Recall that for G = U (1), γ φ ∈ √ −1R. Thus by considering independent generators {γ 1 , . . . , γ b 1 } of the homology H 1 (Σ, Z), and a dual harmonic basis φ 1 h , . . . , φ b 1 h we have the exact sequence Hence a surjective map from the derivative D hol Γ : T H 1 dR (Σ) → T G b 1 . Now we consider the reduction of Φ A Σ under the action of the discrete group G Σ /G 0 Σ .
Theorem 1. We have the quotient space with reduced symplectic structure ω Σ given in (15).
With this result we end up the kinematical part of the axiomatic description, i.e. axioms (A1) to (A9).

Dynamics modulo gauge
These paragraphs are aimed to verify axioms (A10) to (A12) where dynamics of gauge fields is constructed. We discuss the behavior of the solutions near the boundary in more detail. Recall that here is a mapr M : L M →L ∂M coming from the restriction of the solutions to germs on the boundary, and composing we the quotient class map we have a map r M : L M → L ∂M . Let LM ⊂ L Σ be the image under this map. The aim is to describe the image LM ⊂ L ∂M of the space of solutions as a Lagrangian subspace once we have taken gauge quotient. The aim of this part is to verify the dynamics postulate (A10).
We recall some useful facts of Hodge-Morrey-Friedrich theory for manifolds with boundary, see for instance [Sc], [AM] and [GMS]. We can consider both Neumann and Dirichlet boundary conditions in order to define k−forms on M , i.e.
The differential d preserves the Dirichlet complex Ω k D (M ) and on the other hand, the codifferential d ⋆ preserves the Neumann complex Ω k N (M ). In addition, the space H k (M ) of harmonic fields dϕ = 0 = d ⋆ ϕ, turns out to be infinite dimensional, nevertheless finite dimensional spaces arise when we restrict to Dirichlet or Neumann boundary conditions H k N (M ), H k D (M ).

Lemma 4 ([Sc]).
1. There is an orthogonal decomposition 2. In particular there is an orthogonal decomposition for divergence-free fields 3. Each de Rham cohomology class can be represented by a unique harmonic field without normal component, i.e. there is an isomorphism 4. Each de Rham relative cohomology class can be represented by a harmonic field null at the boundary, i.e. there is an isomorphism Notice that r M Φ A M ⊂ L ∂M . If ϕ ∈ Ω k (∂M ε ) satisfies the Neumann condition X * ∂M (⋆ϕ) = 0, then it also satisfies X * ∂M (ι ∂τ ϕ) = 0 with ι ∂τ ϕ ∈ Ω k−1 (∂M ε ).
Let us consider the divergence free fields which, according to lemma 4, have an orthogonal decomposition of th axial gauge fixing space of solutions constitutes the orthogonal projection of the space of solutions L M , according to the decomposition It also coincides with the orthogonal projection ofΦ A M . From this orthogonal decomposition it can be shown that every solution ϕ ∈ L M can be transformed, modulo the bulk gauge transformation, onto a field belonging to the space Φ A M . Thus the following statement can be proven.
We have the following statement.
1. There is a well defined restriction map 2. If we adopt the identification given in theorem 1.
then the map r M coincides with the first jet of the pullback, i.e. we have a commutative diagram of linear mappings Now we are in position to prove that the image of solutions modulo gauge onto the space of boundary conditions modulo gauge, stated in proposition 1, is in fact a lagrangian space. The following statement completes the dynamical picture described in (A10). 2. LM /G 0 ∂M is a coisotropic subspace. In other words LM ∩ Φ A ∂M is a lagrangian subspace of the symplectic space Φ A ∂M .
As we mentioned in the introduction for LM /G 0 ∂M isotropy is always true, see [KT]. For the sake of completeness we give a proof that is a straightforward calculation. Take ϕ, ϕ ′ ∈ L M and consider its image where ϕ was defined in lemma 2, and where d ⋆ ∂M φ ϕ = d ⋆ ∂Mφ ϕ = 0. We also consider ϕ ∈ Φ A M . Then From a property shown in lemma 1 we have that the last expression equals Recall hat ϕ, ϕ ′ are global solutions in the interior d ⋆ dϕ = 0 = d ⋆ dϕ ′ , hence by applying Stokes' Theorem we have Proof for coisotropic embedding. Take ϕ ∈ Φ A M , as indicated in lemma 5, take φ ϕ ,φ ϕ := r M (ϕ) and suppose that ω ∂M (ϕ, ϕ ε ) = 0 for every ϕ ε ∈ L ∂Mε with φ ′ ,φ ′ ∈ L ∂M corresponding to ϕ ε with ι ∂τ ϕ ε . Recall that Then thanks to the representative (12) we have According to the orthogonal decomposition described in lemma 4, we have We calculate in more detail the first summand of the r.h.s. of equation (25). According to lemma 6,φ h = X * ∂M L ∂τ ϕ h , where we consider the orthogonal decomposition In the last line we have used the properties described for X * ∂M given in lemma 1 and dϕ h = 0. Now consider the first summand of the l.h.s. of equation (25), the extensionφ := ψ · ϕ ε ∈ Ω 1 N (M ) of ϕ ε ∈ Ω 1 (∂M ε ) , given by a partition of unity ψ : M → [0, 1], such that ∂M = ψ −1 (1).
Recall that since ϕ ∈ Φ A M , then d ⋆ dd ⋆ α = 0. From the orthogonal decomposition we have dd ⋆ α ∈ H 3 (M )∩dΩ 2 (M ). By the non-degeneracy of the Hodge inner product in M , there is a well defined exact harmonic field dβ ∈ H 3 (M )∩dΩ 2 , that is the projection of dβ, such that d ⋆ dβ = 0, and the r.h.s. of (30) reads as On the other hand consider the l.h.s. of (30). Recall that β ∈ Ω 2 (∂M ) = H 2 (∂M ) ⊕ dΩ 1 (∂M ) ⊕ d * Ω 3 (∂M ), in fact we can take On the other hand, consider extensionβ := ψβ, If we take the orthogonal projection ofβ, then X * ∂M β h = β h and X * ∂M (γ) = γ. Also for 3−forms as arguments of dd ⋆ ∂ M we have the functionals If we look more carefully the l.h.s. of expression (31), then Where in the last equality we used X * ∂M L ∂τβ = X * ∂M (L ∂τ β) =β and that L ∂τ Mβ ∧ ⋆dd ⋆ α = 0. Hence Looking back again at expression (31) and (29) we have Hence for every α we have Finally we can extend the solution ϕ ε in the cylinder ∂M ε to a solution in the interior M , by means of This ends up the validity of (A10). As we mentioned in previous section, locality follows for Yang-Mills fields and actions, in particular (A11) hold. Gluing axiom (A12) also follows from locality arguments. This completes the dynamical description for this gauge field theory.

n−dimensional case with corners
We will focus on the case of regions M , that are manifolds with corners provided with Yang-Mills fields.
Each Σ i is embedded in a regular tubular neighborhood Σ i ε , ε > 0. The analogous of lemma 1 can be established for a regular tubular neighborhood of Σ i . We can proof ibidem the existence of embedding W that have similar properties as those described for X. It needs an adaptation of Moser's argument to volume forms on open manifolds Σ i and diffeomorphisms vanishing on the boundary ∂Σ i , see [DaM].
To give a detailed description of the space of divergence-free fields on Σ, let us first consider harmonic fields.
If φ h ∈ H 1 (Σ i ), then the restriction, φ I h , over every face closure Σ I ⊂ Σ i , contained in Σ i , is harmonic as stated in the following lemma.
This finishes the description of harmonic forms on the stratified space |Σ|. Notice that when Σ i are balls then the harmonic forms φ ∈ Ω r (Σ i ) are completely defined by their Dirichlet boundary conditions on ∂Σ i .
We also have an analogous of hodge decomposition for stratified spaces, that follows also from theorems 7 and 8 in [AGO]: 1. There is an orthogonal decomposition 2. In particular there is an orthogonal decomposition for divergence-free fields ker d ⋆ : Ω r (|Σ|) → Ω r−1 (|Σ|) = Therefore the divergence free 1−forms on |Σ| are described as Thus we could define the gauge-fixing space as Form the projectionsΣ i → Σ i ⊂ Σ we obtain the linear maps This inclusion is the restriction of an inclusion referred in axiom (A7), as is described in the following commuting diagram Similarly, the inclusions in the affine spaces A Σ → m i=1 AΣi can be described. Also for the gauge symmetries we have exact sequences where GΣi ≃ Ω 0 (Σ i )/R (sinceΣ i is simply connected), and G 0 Σ stands for the identity component of the gauge symmetries group G Σ .
The analogous result of theorem 1 can be established for surfaces with corners.
According to [Du], [DuS], the space of harmonic forms on a smooth manifold M with smooth boundary ∂M , have the Betti number as rank and equals dim H 1 N (M ) = dim H 1 (M ) = dim H n−1 (M, ∂M ). Similarly for smooth closed hypersurfaces Σ, For manifolds with corners Σ, the space of harmonic forms has the same description. Take a homeomorphism F : Σ ′ → Σ, that defines a diffeomorphism, with lack of differentiability on the corners on ∂Σ ′ . If φ ∈ H 1 N (Σ) is harmonic form with null normal component, then is also well defined harmonic form on ∂Σ ′ . Hence, for stratified spaces homeomorphic to manifolds, such as |Σ| harmonic forms have also rank given by the Betti number.
Dynamics described in (A10), is also valid in the context of corners. The statement of theorem 2, remains the same as in the case without corners. Isotropic embedding of LM ∩ Φ A ∂M ⊂ Φ A M , goes ibidem as in the case without corners.
The proof of the coisotropic embedding in the case without corners depends entirely on the orthogonal decompositions and isomorphisms described in lemma 4. Explicitly we require the isomorphism which may be verified by using the results stated in [AGO] theorems 7 and 8.

Example: 2−dimensional case
For a better understanding of our model, we review our constructions in a more down to earth examples namely the 2−dimensional case. We provide this presentation as comparison tool with other known procedures for quan-be a subspace ofL Σ . As we did in lemma 2 we have that every K ω Σ −orbit iñ L Σ intersects in just one point the subspace ΦÃ Σ . The presymplectic form ω Σ restricted to the subspace ΦÃ Σ may be written as ω Σ (η,ξ) = 1 2 Σ [−η s ∂ τ ξ s + ξ s ∂ τ η s ] ds,ξ,η ∈L Σ .
Henceω Σ is non-degenerated when we restrict it to the subspace ΦÃ Σ ⊂L Σ . Let ω Σ the corresponding symplectic structure on A Σ induced by the restriction ofω Σ to the subspace ΦÃ Σ ⊂L Σ . Let us consider hypersurfaces Σ := Σ 1 ∪ · · · ∪ Σ m ⊂ ∂M where Σ i are homeomorphic to S 1 in the case without corners, or to intervals possibly identified in some pairs by their boundaries in the case with corners.
Recall that here is map,r M : L M →L ∂M , coming from the restriction of the solutions to germs on the boundary, and composing we the quotient class map we have a map r M : L M → L ∂M . Let LM be the image LM = r M (L M ) under this map. The aim is to describe the image LM = r M (L M ) ⊂ L Σ of the space of solutions as a lagrangian subspace modulo gauge.
We also have lemma 5. It follows that LM = r M (Φ A M ). The isotropic embedding described in theorem 2 is proven in (39). The corresponding coisotropic embedding in the 2−dimensional version goes as follows: Take ϕ ∈ Φ A M , φ = r M (ϕ) and suppose that ω ∂M (φ, φ ′ ) = 0 for every ϕ ′ ∈ L ∂Mε with φ ′ ∈ L ∂M corresponding to ϕ ′ . Theṅ Since ϕ ′ is a solution in a tubular neighborhood ∂M ε then ∂ τ (ϕ ′ ) s | Σ = c ϕ ′ . Henceċ global dependence of dynamics holds for the quantum version, i.e. the quantum TQFT version of Yang-Mills gauge fields. Once we have completed reduction, the picture of quantization on this finite dimensional space can be specified cfr. [DH], [Wi], [La]. For a complete description of the quantization in 2−dimensions in general non abelian case with corners see [O2].

Outlook: quantization in higher dimensions
The geometric quantization program with corners will be treated elsewhere [D]. Once the reduction -quantization procedure is completed, the next task is the formulation of the quantization -reduction process and the equivalence of both procedures. See the discusion of these issues in dimension two for instance in [Wi], [DH] and [La]. In order to administrate the geometric quantization program [Wo] for the reduced space we need to describe a suitable hermitian structure in Φ A Σ to be used as a prequantization ingredient.