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 (TQ...
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| Cite this: | General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners / H.G. Díaz-Marín // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
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Díaz-Marín, H.G. 2019-02-13T16:56:24Z 2019-02-13T16:56:24Z 2015 General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners / H.G. Díaz-Marín // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53D30; 58E15; 58E30; 81T13 DOI:10.3842/SIGMA.2015.048 https://nasplib.isofts.kiev.ua/handle/123456789/147116 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. The author thanks R. Oeckl for several discussions and encouragement for writing this note at CCM-UNAM. This work was partially supported through a CONACYT-M´exico postdoctoral grant. The author also thanks the referees for their comments and suggestions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners Article published earlier |
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General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners |
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General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners Díaz-Marín, H.G. |
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General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners |
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General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners |
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General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners |
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General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners |
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general boundary formulation for n-dimensional classical abelian theory with corners |
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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.
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General Boundary Formulation for n-Dimensional Classical Abelian Theory with Corners / H.G. Díaz-Marín // Symmetry, Integrability and Geometry: Methods and Applications. — 2015. — Т. 11. — Бібліогр.: 28 назв. — англ. |
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2025-11-26T14:35:25Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 048, 35 pages
General Boundary Formulation for n-Dimensional
Classical Abelian Theory with Corners
Homero G. DÍAZ-MARÍN †‡
† Escuela Nacional de Ingenieŕıa y Ciencias, Instituto Tecnológico y de Estudios Superiores
de Monterrey, C.P. 58350 Morelia, México
E-mail: homero.diaz@itesm.mx
‡ Centro de Ciencias Matemáticas, Universidad Nacional Autónoma de México,
C.P. 58190 Morelia, México
E-mail: homero@matmor.unam.mx
Received October 30, 2014, in final form June 04, 2015; Published online June 24, 2015
http://dx.doi.org/10.3842/SIGMA.2015.048
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 ana-
logue 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.
Key words: gauge fields; action; manifolds with corners
2010 Mathematics Subject Classification: 53D30; 58E15; 58E30; 81T13
1 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 space-time M has incoming and outgoing
boundary components, ∂M = ∂Min ∪ ∂Mout. Fields are valued along the boundary together
with their derivatives. This relation defines an isotropic space of boundary conditions that
extend to solutions in the interior of M , LM̃ ⊂ A∂M = A∂Min
× A∂Mout , where the symplectic
structure, ω∂M = ωin ⊕ ωout, is formed by certain symplectic structures ωin and ωout defined
in A∂Min
and A∂Mout , respectively. For recent progress from a categorical point of view on this
classical formalism in the case of linear symplectic spaces, see for instance [26]. In some cases,
degeneracies of the Lagrangian density yield degeneracies of a presymplectic structure ω∂M , for
the Cauchy data A∂M .
A wise observation appearing for the first time in [14], is that it is possible to formulate
a symplectic framework for field theories in general space-time regions M . Here, general boun-
daries ∂M are composed of general hypersurfaces, which do not necessarily correspond to “in”
and “out” space-like boundary components. The spaces A∂M of 1-jets arising from Cauchy data,
namely, Dirichlet and Neumann boundary data, have a presymplectic structure ωin, see [14].
A derivation of a symplectic formalism, was independently rediscovered in the general bounda-
ry formulation (GBF) for classical theories in [19, 22], this time arising from their quantum
mailto:homero.diaz@itesm.mx
mailto:homero@matmor.unam.mx
http://dx.doi.org/10.3842/SIGMA.2015.048
2 H.G. Dı́az-Maŕın
counterparts. Here, the definition of a (pre)symplectic structure is given for the space ̶M of
germs of solutions of a cylinder of the boundary ∂Mε := ∂M × [0, ε]. Axiomatic frameworks
incorporating this symplectic formalism appeared in [19, 22], for linear field theories, whereas
for the case of affine field theories they appeared in [21]. Another symplectic setting for field
theories appeared independently in [5], where it is related to the BFV and BV formalism. Here
appears explicitly the distinction for the (pre)symplectic structure for 1-jets and for germs.
The space of germs Ã∂M contains much more information than the 1-jets for fields in ∂M . As
a consequence, if we consider germs instead of 1-jets, then instead of a symplectic structure ω∂M ,
we may have a presymplectic structure ω̃∂M . Hence we need to consider the space of germs of
boundary conditions ̶M as a coisotropic space. This space of germs ̶M needs to be reduced
in order to obtain a symplectic space.
We suppose that both degeneracies, those due to germ higher order derivatives as well as
those those due to Lagrangian density, are both contained in the kernel of the presymplectic
structure ω̃∂M in Ã∂M . So the reduced space A∂M is a symplectic space.
Dynamics in the interior of the space-time region M is described as a Lagrangian immersion,
AM̃ ⊂ A∂M of the boundary data of solutions of the Euler–Lagrange equations. For infinite-
dimensional symplectic vector spaces, isotropic spaces are required to be coisotropic in order to
be Lagrangian. Isotropy is always satisfied [14], but coisotropy of the immersion AM̃ ⊂ A∂M
does not hold in general, see counterexamples in [5].
From the quantum side the axiomatic setting for the GBF is inspired by topological quantum
field theories (TQFT), see [4] and the approach of G. Segal [25]. 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 [17]. 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 M1, M2 can
be performed along hypersurfaces Σ ⊂M1, Σ′ ⊂M2, both isometric oriented manifolds, Σ ∼= Σ′.
Here Σ′ means reversed orientation. The precise axiomatic system for quantum field theories
along with their classical counterpart appears in [19] and for affine theories in [21].
Corners. This TQFT-inspired approach requires a classification of the basic regions or
building blocks used to reconstruct the whole space-time region M1 ∪Σ M2, by gluing the
pieces M1, M2, along the boundary hypersurface Σ ∼= Σ′. This classification from the topo-
logical 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 consequence is that we would have to al-
low gluings of regions along hypersurfaces Σ with nonempty boundaries ∂Σ. For instance, we
can consider the gluing of two n-balls M1, M2 along (n − 1)-balls contained in their bounda-
ries Σ, Σ′. This means that we would have to 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 ⊂ ∂M1, Σ′ ⊂ ∂M2. A well suited language for describing such phenomena,
consists in treating regions Mi as manifolds with corners. For TQFT the attempt to deal with
the case of corners gives rise to extended topological quantum field theories. A possible ap-
proach for two-dimensional theories is given for instance in [11, 16]. There is also a specific
formulation for 2-dimensional with corners in [20]. 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 J1P/G→M . In this case the space of sections KM is an affine space. Furthermore
for quadratic Lagrangian densities we will have that the space of solutions, AM , is an affine space.
This enables us to consider a GBF formalism for affine spaces such as is described in [21].
Classical Abelian Theory with Corners 3
We give a step further in relation to [21] since we consider gauge symmetries, GM , acting
on AM . Variational gauge symmetries are vertical automorphisms of the bundle P , that in
turn yield vertical automorphisms of the bundle J1P/G. Infinitesimal gauge symmetries should
preserve the action, SM : KM → R. They can be identified with vertical G-invariant vector
fields ~X on P , as well as with sections of V P/G→M , where V P is the vertical tangent bundle
of P →M . Action preservation follows from invariance of the Lagrangian density under vertical
vector fields act on J1P/G.
When we consider germs of solutions on the boundary, we have a group of variational gauge
symmetries G̃∂M . By taking the quotient by the degeneracies we obtain a gauge group ac-
tion G∂M of symplectomorphisms on (A∂M , ω∂M ). To make sense of the quotient space AM/GM
may be problematic in non-abelian gauge field theories, also taking the related reduced boundary
conditions A∂M/G∂M . The issue of gluing solutions also needs 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
to Euler–Lagrange equations. We use this axiomatic setting to construct abelian theories. The
most general setting of nonabelian structure groups for actions remains a conjecture even in
the classical case, see [6]. Along this program we study the case of smooth space-time regions
without corners as well as the case of regions with corners.
As we were writing this article we realized that Lagrangian embedding for the abelian case of
actions and other important cases such as BF and Chern–Simmons were actually proved in [6].
Here the authors use Lorenz gauge fixing and use Dirichlet boundary condition for 1-forms. Thus
by Friedrichs–Morrey–Hodge theory they describe the space of boundary conditions that extend
to solutions modulo gauge, AM̃/G∂M ⊂ A∂M/G∂M , as harmonic forms on ∂M extendable to
cocolsed forms on M . AM̃/G∂M is isomorphic to the direct sum of two spaces: On one hand
a finite-dimensional subspace of H1(M,∂M). On the other hand an infinite-dimensional space of
closed forms in ∂M , see Proposition 4 in the appendix of [6]. Independently, we use axial gauge
fixing and Neumann boundary conditions for 1-forms to describe the space AM̃/G∂M as a direct
sum of two spaces: On one and a finite-dimensional subspace of H1(∂M). On the other hand
an infinite-dimensional space consisting of coclosed 1-forms. Thus we give a complementary
view, although that was not our original aim. The proof in [6] is short and briefly describes the
main ideas. We give a more detailed proof, since our aim is to exhibit the explicit application of
an axiomatic system that seems sketched in [5]. We give explicit calculations in terms of local
coordinate decomposition. Finally our results extend to regions that are manifolds with corners.
This is essential for the physically most relevant case of gluing, where the component manifolds
as well as the composite manifold have the topology of a ball.
A related work [1], describes Killing vector field acting on 1-forms with Dirichlet and Neu-
mann boundary conditions. The author thanks the referee for pointing out the reference [23],
where gauge action is described for spin manifolds with boundary in the context of M theo-
ries.
Description of sections. Section 2 consists of a review of the symplectic formalism for
classical field theories together with an exposition of the language of abelian gauge field theories
and manifolds with corners. In Section 3 we exhibit the axioms of an abelian gauge field theory
which is divided in two cases: the case where regions are considered as smooth manifolds with
boundary and the case where regions are manifolds with corners. We construct the abelian theory
using this axiomatic system. In Section 4, we focus on the kinematics of gauge fields. This section
involves local considerations where Moser’s arguments on the transport flow for volume forms
is used. A similar argument due to Dacorogna–Moser for manifolds with boundary is crucial for
the corners case. Dynamics of gauge fields is explored in Section 5. We describe the symplectic
reduction of the space of boundary conditions and emphasize the proofs of the Lagrangian
4 H.G. Dı́az-Maŕın
embedding of solutions. This last result uses Friedrichs–Morrey–Hodge theory adapted to the
case of corners. Finally we review the special case of Yang–Mills theory in dimension 2 in
Section 6 for illustration.
2 Classical abelian gauge field theories
For the sake of completeness, we summarize 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 [14]. On the other hand the discussion of the space of germs of
solutions in the axiomatic setting appears in [21, 22]. Parallel developments appear also in [5].
We adopt an abstract coordinate-free description of the (pre)symplectic structure for boundary
data, by means of a suitable cohomological point of view.
2.1 The symplectic setting for classical Lagrangian field theories
Classical field theory assumes that over an n-dimensional space-time region M , there exists
a “configuration space”, KM , of fields ϕ ∈ KM . The word “space” used for referring to
KM usually denotes an infinite-dimensional Frèchet manifold, defined as a space of sections of
a smooth bundle E over M . It also assumes the existence of a Lagrangian density, Λ ∈ Ωn(J1M),
depending on the first-jet j1ϕ ∈ J1M , 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
SM (ϕ) =
∫
M
j1(ϕ)∗Λ.
On the other hand we consider the factorization of the space of k-forms over the l-jet mani-
fold J lM as
Ωk
(
J lM
)
=
k⊕
r=0
Ωr
H
(
J lM
)
⊗ Ωk−r
V
(
J lM
)
,
where the complex Ωk
H(J lKM ) (resp. Ωk
V (J lKM )) corresponds to horizontal (resp. vertical) k-
forms. For instance, using local coordinates xi, i = 1, . . . , n, for the manifold M , take (xi;ua;uai )
as local coordinates for J1M . Then horizontal forms have as a basis the exterior product of
the dxi. Meanwhile for vertical forms in J1M , have as basis the exterior product of dua, duai .
The horizontal (resp. vertical) differential is induced by the coordinate decomposition
dH : Ωk
H
(
J lM
)
→ Ωk+1
H
(
J l+1M
)
(resp. dV := d− dH). For instance, for horizontal 0-forms we have
dH :=
n∑
i=1
(
∂
∂xi
+
r∑
a=1
uai
∂
∂ua
)
dxi : Ω0
H
(
J0M
)
→ Ω1
H
(
J1M
)
,
where r equals the dimension of each fiber of the bundle E. Thus, vertical k-forms vanish
on horizontal vector fields ~X such that dV ( ~X) = 0. This decomposition yields a variational
Classical Abelian Theory with Corners 5
bicomplex, see for instance [12],
0 0 . . .
Ωn
H(J1M)
dV
//
dH
OO
Ωn
H(J1M)⊗ Ω1
V (J2M)
dH
OO
// . . .
Ωn−1
H (J0M)
dV //
dH
OO
Ωn−1
H (J0M)⊗ Ω1
V (J1M)
dH
OO
// . . .
...
dH
OO
...
dH
OO
...
Denote the space of Euler–Lagrange solutions as
AM =
{
ϕ ∈ KM |
(
j2ϕ
)∗
(dV Λ) = 0
}
.
In the case we are dealing with the space of connections AM is an affine space. The corresponding
linear space is denoted as LM .
Consider the image dV Λ ∈ Ωn
H(J1M)⊗Ω1
V (J2M), of the Lagrangian density, Λ ∈ Ωn
H(J1M).
Take a preimage
θΛ ∈ d−1
H ◦ dV Λ ∈ Ωn−1
H
(
J0M
)
⊗ Ω1
V
(
J1M
)
.
Of course the representative θΛ ∈ d−1
H ◦ dV Λ depends just on the dH -cohomology class of the
Lagrangian density. By integration by parts, the differential of the action dSM , evaluated on
variations δϕ = X ∈ TϕKM , may be decomposed as
dSM (δϕ) = (dSM )ϕ(X) =
∫
M
(
j2ϕ
)∗(
ι(j2 ~X)dV Λ
)
+
∫
∂M
(
j1ϕ
)∗(
ι ~XθΛ
)
.
Locally each variation δϕ is identified with a vector field, ~X, along the section j1ϕ in J1M .
This ~X in turn induces a vector field j2 ~X, the 2-jet prolongation of the vector field ~X, along
j2ϕ, on the 2-jet manifold J2M . Both ~X and j2 ~X vanish on horizontal 1-forms. This shows
that total variations consist of two contributions. One kind of variation is the one localized on
the bulk of the fields corresponding to Euler–Lagrange equations. Another kind of contribution
to the variation comes from 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, J1M , translates to the calculus on the infinite-dimensional space, KM , so that θΛ induces
a 1-form
(dSM )ϕ(X) =
∫
∂M
(
j1ϕ
)∗(
ι ~XθΛ
)
for variations X ∈ TϕAM of 1-jets of solutions restricted to the boundary. This enables us to
consider a 1-form dSM , for variations X ∈ TϕAM .
For an (n − 1)-dimensional boundary manifold Σ, the boundary conditions for solutions on
a tubular neighborhood Σε
∼= Σ× [0, ε], 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 limits
ÃΣ := lim−→AΣε , L̃Σ := lim−→LΣε ,
6 H.G. Dı́az-Maŕın
where the inclusion of tubular neighborhoods, Σε⊂Σε′ , for ε<ε′, induces an inclusion AΣε′⊂AΣε .
Similarly, there is an inclusion for the linear spaces LΣε′ ⊂ LΣε .
The submersion of variations of germs X̃ ∈ TϕÃΣ, onto variations of jets ~X ∈ TϕAΣ, leads
to the definition of the 1-form on ̶M ,(
θ̃Σ
)
ϕ
(X̃) := (dSM )ϕ(X).
Ultimately, our purpose is to consider the presymplectic structure on ÃΣ,
ω̃Σ = dθ̃Σ.
There are degeneracies of the presymplectic structure ω̃Σ 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 taking the
quotient by KωΣ := kerωΣ. Then we obtain a symplectic space (AΣ, ωΣ).
Consider an action map SM (ϕ) defined for connections ϕ of a principal bundle P over M with
compact abelian structure group G. We denote as AM , the space of Euler–Lagrange solutions
in the interior of the region M . In general, we suppose that ∂M is not empty. Hence when
we restrict the action functional SM , from the configuration field space KM to the space of
solutions AM , it induces a non-constant map
SM : AM → R.
On the other hand we have the groups, GM , of gauge symmetries on regions acting in AM the so-
lutions on the bulk that come from the Euler–Lagrange variational symmetries of the Lagrangian
density, see [12, Definition 2.3.1]. Infinitesimal symmetries can be identified with G-invariant
vertical vector fields on P , i.e., with vertical vector fields acting on J1P/G and preserving the
Lagrangian density.
By taking the tubular neighborhood, Σε as the region M , those symmetries by the group GΣε
act on germs of solutions in AΣε hence in ÃΣ. By taking the quotient by the stabilizer of the ÃΣ,
we obtain a group of gauge symmetries on hypersurfaces,
G̃Σ := lim−→GΣε
acting on ÃΣ.
Once we have taken the quotient of the space of germs ÃΣ, and its corresponding linear
space L̃Σ, by the degeneracy space KωΣ , we get a space AΣ, and a gauge group GΣ acting on AΣ.
The group G̃Σ 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 KωΣ ⊂ G̃Σ,
that takes into account all degeneracies. The KωΣ-orbits on ÃΣ consist of the integral leafs
of the characteristic distribution generated by the kernel of the presymplectic structure ω̃Σ.
Meanwhile, the quotient group GΣ acts by symplectomorphisms on AΣ with respect to the
symplectic structure ωΣ.
2.2 Regions with and without corners
In the following presentation of the axiomatic system for classical Lagrangian field theories, we
will consider regions and hypersurfaces as manifolds with corners. We adopt the definition of
stratified spaces in [2]. In order to establish the notation that will be used along this work we
sketch the definitions of manifolds with corners, for a more detailed description see the cited
reference.
Classical Abelian Theory with Corners 7
Hypersurfaces are (n− 1)-dimensional topological manifolds Σ which decompose as a union
of (n− 1)-dimensional manifolds with corners,
Σ = ∪mi=1Σi ∼= tmi=1Σ̌i/ ∼P .
This union in turn is obtained by gluing of (n− 1)-dimensional manifolds with corners: Σ̌i, Σ̌j ,
along pairs of (n− 2)-faces. This can be done by means of an equivalence relation ∼P , defined
by a certain set P of pairs (i, j), i 6= j. More precisely, non trivial equivalence identifications
take place for the set
∪(i,j)∈PΣij := ∪(i,j)∈PΣi ∩ Σj .
This means that gluings of the faces Σi, Σj , take place at (n − 2)-faces Σij ⊂ ∂Σi, Σji ⊂ ∂Σj ,
Σji ∼= Σij .
Consider a hypersurface Σ as a stratified space consisting of a union ∪mi=1Σi of manifolds
with corners Σ̌i identified along their their faces ∂Σ̌i. Denote the structure of stratified spaces,
as |Σ| respectively. For a stratified space |Σ| we denote the k-dimensional skeleton as |Σ|(k),
k = 0, 1, 2, . . . , n− 1, notice that |Σ|(n−1) ∼= Σ and
|Σ|(n−2) = ∪(i,j)∈PΣij ⊂ Σ
corresponds to the corners set. We adopt the notation for the set of k-dimensional faces as |Σ|k.
Thus
|Σ|n−1 =
{
Σ̌1, . . . , Σ̌m
}
is the set of (n− 1)-dimensional faces and
|Σ|n−2 =
{
Σ̌ij | (i, j) ∈ P
}
is the set of (n − 2)-faces. Here Σ̌ij ⊂ Σ̌i is the preimage of the corner Σij = Σi ∩ Σj ⊂ Σ,
(i, j) ∈ P.
A region is an n-dimensional manifold with corners M . Its boundary ∂M , is a topological
manifold. The corresponding stratified space structures are |M |, |∂M |. Each hypersurface
Σ ⊂ ∂M consists of the union of faces Σi ⊂ ∂M , which are manifolds with corners.
An abstract closed smooth hypersurface Σ, not necessarily related to a region M , may be
considered as a component of the boundary of a cylinder Σ× [0, ε], ∂Σ = ∅ [17].
The notion of a cylinder can be generalized for a manifold with corners Σ, ∂Σ 6= ∅. A regular
cylinder consists of
Σ̂ε :=
{
(s, t) ∈ Σ× [0, ε] | t ∈ [0, ε(s)ε], s ∈ Σ
}
⊂ Σ× [0, ε], (2.1)
where ε : Σ→ [0, 1] is an increasing smooth function such that ε−1(0) = ∂Σ and Σε := ε−1(1) ⊂ Σ
is a smooth retract deformation of Σ.
Thus Σ corresponds to one face of the n-dimensional manifold with corners given by the
regular cylinder Σ̂ε. In general ∂Σ, ∂Σi may be nonempty.
For smooth hypersurfaces Σ ⊂ ∂M we consider tubular neighborhoods [17], Σε ⊂ M with
diffeomorphisms
X : Σ× [0, ε]→ Σε.
On the other hand, a regular tubular neighborhood for a face Σ ⊂ ∂M , consists of a homeomor-
phism that becomes a diffeomorphism outside the corners ∂Σ ⊂ Σ̂ε
X : Σ̂ε → Σε
8 H.G. Dı́az-Maŕın
Recall that the corners of the region M lie in the union of the (n− 2)-dimensional submani-
folds, ∪(i,j)∈PΣij .
The gluing of a region M along two nonintersecting faces Σ0, 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 along general
hypersurfaces may be problematic. For if we glue faces with non intersecting boundaries ∂Σ0 ∩
∂Σ′0 = ∅, then conic singularities of the metric along the corners may arise in the resulting
space-time region.
We consider gluings along nonintersecting faces and do not consider gluings along hypersur-
faces.
3 Axiomatic system proposal
Now we give a detailed description of the axiomatic framework for classical gauge field theories.
Axioms A1–A9 describe the kinematics of the classical theory, while Axioms A10–A12 describe
the dynamics for gauge fields.
3.1 GBF Axioms
We consider space-time regions M that are manifolds with corners of dimension n, as well as
hypersurfaces Σ that are topological (n − 1)-dimensional topological manifolds with stratified
space structure |Σ|.
A1 Affine structure. For space-time regions M we have the affine spaces AM with the associa-
ted linear spaces LM of Euler–Lagrange solutions. On the other hand, for hypersurfaces Σ
we have affine spaces AΣ with associated linear spaces L̃Σ, of boundary conditions. There
are also affine maps ãM : AM → Ã∂M , as well as linear maps r̃M : LM → L̃∂M .
A2 Presymplectic structure. For every hypersurface Σ ⊂ ∂M , there is a presymplectic struc-
ture ω̃Σ on ÃΣ invariant under L̃Σ actions. Equivalently we can consider L̃Σ as a presym-
plectic vector space with presymplectic structure denoted also as ω̃Σ.
A3 Symplectic structure. There is a group KωΣ acting freely by translations on ÃΣ, such
that KωΣ is isomorphic to the closed linear subspace ker ω̃Σ ⊂ L̃Σ. So ω̃Σ induces a sym-
plectic structure, ωΣ, on the orbit space
AΣ := ÃΣ/KωΣ .
This space is an affine space modeled on the linear space LΣ := L̃Σ/KωΣ . By taking
the quotients, the maps ãM and r̃M induce affine and linear maps aM : AM → A∂M ,
rM : AM → A∂M , respectively.
A4 Symplectic potential. There is a symplectic potential, i.e., an LΣ-valued 1-form θΣ(ϕ, ·) for
each ϕ ∈ AΣ, identified with a linear map θΣ(ϕ, ·) : LΣ → R. There is also a bilinear map
[·, ·]Σ : LΣ × LΣ → R such that
[φ, φ′]Σ + θΣ(η, φ′) = θΣ(φ+ η, φ′), η ∈ AΣ, φ, φ
′ ∈ LΣ
and
ωΣ(φ, φ′) =
1
2
[φ, φ′]Σ −
1
2
[φ′, φ]Σ, φ, φ′ ∈ LΣ. (3.1)
Classical Abelian Theory with Corners 9
There exists an action map SM : AM → R, such that
SM (η) = SM (η′)− 1
2
θ∂M (η, η − η′)− 1
2
θ∂M (η′, η − η′) (3.2)
and also SM (η) = SM (η′) for aM (η) = aM (η′).
A5 Involution. For each hypersurface Σ there exists an involution AΣ → AΣ, where Σ is the
hypersurface with reversed orientation. There is also a linear involution LΣ → LΣ. We
have: θΣ(η, φ) = −θΣ(η, φ) and [φ, φ′]Σ = −[φ, φ′]Σ.
A6 Disjoint regions. For a disjoint union, M = M1tM2, there is a bijection AM1×AM2 → AM
and compatible linear isomorphism LM1×LM2 → LM , such that aM = aM1×aM2 and rM =
rM1 × rM2 , satisfy associative conditions. For the action map we have SM = SM1 + SM2 .
A7 Factorization of fields on hypersurfaces. For a hypersurface Σ obtained as the quotient Σ̌1t
· · ·tΣ̌k by an equivalence relation ∼P , define A|Σ|n−1 := AΣ̌1×· · ·×AΣ̌m , L|Σ|n−1 := LΣ̌1⊕
· · · ⊕ LΣ̌m . Then there are affine gluing maps aΣ,|Σ|n−1 : AΣ → A|Σ|n−1 , and compatible
linear maps rΣ,|Σ|n−1 : LΣ → L|Σ|n−1 with commuting diagrams
A|Σ|n−1
��
AΣ
//
aΣ,|Σ|n−1
;;
AΣ̌i
L|Σ|n−1
��
LΣ
//
rΣ,|Σ|n−1
;;
LΣ̌i
We also have the relation
[·, ·]Σ = r∗Σ;|Σ|n−1
(
[·, ·]Σ̌1 + · · ·+ [·, ·]Σ̌m
)
, θΣ = r∗Σ;|Σ|n−1
(
θΣ̌1 + · · ·+ θΣ̌m
)
(3.3)
we denote AΣi as the image of AΣ into AΣ̌i , and similarly LΣi .
A8 Gauge action. There are groups G̃Σ acting on ÃΣ preserving the affine structure and the
presymplectic structure ω̃Σ such that KωΣ E G̃Σ. The quotient group
GΣ := G̃Σ/KωΣ
acts on AΣ, preserving the symplectic structure ωΣ. There is a group, GM , of gauge
variational symmetries for SM acting on the space of solutions AM . There is a group
homomorphism hM : GM → G∂M . For the map aM : AM → A∂M the compatibility of
gauge group actions is given by the commuting diagram
AM ×GM //
��
A∂M ×G∂M
��
AM // A∂M
There is also a compatible action on the corresponding linear spaces rM : LM → L∂M
LM ×GM //
��
L∂M ×G∂M
��
LM // L∂M
A9 Factorization of gauge actions on hypersurfaces. For the case of regions with corners there
is a homomorphism hΣ;|Σ|n−1 : GΣ → G|Σ|n−1 from the direct product group G|Σ|n−1 :=
GΣ̌1 × · · · ×GΣ̌m onto GΣ coming from homomorphisms
h|Σ|n−1;Σ̌i : G|Σ|n−1 → GΣ̌i
10 H.G. Dı́az-Maŕın
and commuting diagrams
G|Σ|n−1
��
GΣ
//
;;
GΣ̌i
and
A|Σ|n−1
��
A|Σ|n−1 ×G|Σ|n−1
��
OO
AΣ
88
33AΣ ×GΣ
oo //
66
AΣ̌i ×GΣ̌i
// AΣ̌i
and similar commuting diagrams for actions on linear spaces
L|Σ|n−1
��
L|Σ|n−1 ×G|Σ|n−1
��
OO
LΣ
88
33LΣ ×GΣ
oo //
66
LΣ̌i ×GΣ̌i
// LΣ̌i
.
There is an involution of the gauge groups GΣ → GΣ, compatible with the action.
We denote the image h|Σ|n−1;Σ̌i(GΣ) ⊂ GΣ̌i as GΣi .
A10 Lagrangian relation modulo gauge. Let AM̃ be the image aM (AM ) ⊂ A∂M of boundary
conditions on ∂M that extend to solutions on the bulk M . Let LM̃ be the corresponding
linear subspace that is the image rM (LM ) ⊂ L∂M . The subspace LM̃ ⊂ L∂M is Lagrangian.
The zero component of the G∂M -orbit is isomorphic to C⊥∂M , the symplectic orthogonal
complement of a coisotropic subspace C∂M ⊂ L∂M . There is a Lagrangian reduced sub-
space isomorphic to
LM̃ ∩ C∂M/LM̃ ∩ C
⊥
∂M
of the symplectic reduced space C∂M/C
⊥
∂M .
A11 Locality of gauge fields. Let M1 be the region that can be obtained by the gluing of M
along the disjoint faces, Σ0,Σ′0 ⊂ ∂M , where Σ′0
∼= Σ0. Then there is an injective affine
map, a
M ;Σ0,Σ′0
: AM1 ↪→ AM , a compatible linear map, r
M ;Σ0,Σ′0
: LM1 ↪→ LM , and a ho-
momorphism h
M ;Σ0,Σ′0
: GM1 ↪→ GM , with exact sequences
AM1 ↪→ AM ⇒ AΣ0 , LM1 ↪→ LM ⇒ LΣ0 , GM1 ↪→ GM ⇒ GΣ0 ,
where we consider the involution A
Σ′0
→ AΣ0 , for the second arrow on the double map.
Recall that AΣ0 is the image in AΣ̌0
. We consider the gluing of the actions
AM1 ×GM1
//
��
AM ×GM
..
00
��
AΣ0 ×GΣ0
��
AM1
// AM
,,
22 AΣ0
Classical Abelian Theory with Corners 11
compatible with the actions on linear spaces
LM1 ×GM1
//
��
LM ×GM
..
00
��
LΣ0 ×GΣ0
��
LM1
// LM
,,
22 LΣ0
and also SM1 = SM ◦ aM ;Σ0;Σ′0
.
A12 Gluing of gauge fields. Let M1, M be regions with corners M1 is obtained by gluing M
along hypersurfaces Σ0,Σ′0 ⊂ ∂M . The following diagrams commute
AM1
//
��
AM
��
A∂M1
��
A∂M
��
A|∂M1|n−1 A|∂M |n−1oo
GM1
//
��
GM
��
G∂M1
��
G∂M
��
G|∂M1|n−1 G|∂M |n−1oo
LM1
//
��
LM
��
L∂M1
��
L∂M
��
L|∂M1|n−1 L|∂M |n−1oo
where if |∂M |n−1 = Σ̌0 t Σ̌′0 t (Σ̌1 t · · · t Σ̌r), and |∂M1|n−1 = Σ̌1 t · · · t Σ̌r, then the
map a|∂M |n−1,|∂M1|n−1 : A|∂M |n−1 → A|∂M1|n−1 equals the canonical inclusion
AΣ̌1 × · · · ×AΣ̌r ⊂ AΣ̌0
×A
Σ̌′0
×
(
AΣ̌1 × · · · ×AΣ̌r
)
.
We have similar inclusions
r|∂M |n−1,|∂M1|n−1 : L|∂M |n−1 → L|∂M1|n−1 ,
h|∂M |n−1,|∂M1|n−1 : G|∂M |n−1 → G|∂M1|n−1 .
Compatibility for the gluing of the actions of the gauge groups is described by the com-
muting diagrams:
AM1
//
��
AM
��
A∂M1 ×GM1
//
��
ii
AM ×GM
��
55
A∂M1
��
A∂M1 ×G∂M1
��
oo A∂M ×G∂M
��
// A∂M
��
A|∂M1|n−1 ×G|∂M1|n−1
uu
A|∂M |n−1 ×G|∂M |n−1oo
((
A|∂M1|n−1 A|∂M |n−1oo
12 H.G. Dı́az-Maŕın
LM1
//
��
LM
��
LM1 ×GM1
//
��
ii
LM ×GM
��
66
L∂M1
��
L∂M1 ×G∂M1
��
oo L∂M ×G∂M
��
// L∂M
��
L|∂M1|n−1 ×G|∂M1|n−1
uu
L|∂M |n−1 ×G|∂M |n−1oo
((
L|∂M1|n−1 L|∂M |n−1oo
3.2 Further discussion of the axioms
Axioms A1–A7 are just a restatement of Axioms C1 to C6 for a classical setting of affine (linear)
field theories in [21]. Some clarifications are added: In Axiom A2 we consider presymplectic
spaces of connections instead of symplectic spaces. We do not consider Hilbert space structures
since we are not introducing yet prequantization.
Some comments can be made about postulate Axiom A4. The translation rule of the 1-form
θ∂M can be deduced from the translation rule for the differential dSM of the action map. This in
turn can be deduced from (3.2). This last relation could be stated as a primordial property and
arises from considering a quadratic Lagrangian density Λ. The affine structure for the space of
solutions AM can also be deduced from this condition on Λ.
In Axiom A7 we adapt the decomposition stated in Axiom C3 for the corners case.
The set of corners correspond to the (n−2)-dimensional faces Σij := Σi∩Σj , (i, j) ∈ P. The
lack of surjectivity for dotted arrows in Axiom A7 comes from the non differentiability of the
hypersurface Σ along the corners |Σ|(n−2) in the intersections Σi ∩ Σj , (i, j) ∈ P.
Axiom A8 introduces 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 C7.
Locality arguments for gauge fields is used in Axioms A8, A11 and A12. They deserve further
clarification. For instance in Axiom A11, the existence of the exact sequence is not trivial and
it is derived from locality for connections in AM and gauge actions in GM . From the inclusions
∂Mε ⊂M of regular tubular neighborhoods we get the following exact sequences
AΣε
AM1−(Σε∪Σ′ε)
� � // AM
88 88
&& &&
AΣ′ε
If we consider the maps
AΣε
� � // ÃΣ
// // AΣ,
then we can induce the sequence proposed in the axiom, when ε → 0. Recall that ÃΣ is an
inductive limit, and AΣ is a quotient of ÃΣ. For Axiom A8 similar arguments using the following
Classical Abelian Theory with Corners 13
commutative diagrams
AM ×GM //
��
A∂Mε ×G∂Mε
��
AM // A∂Mε
Axiom A8 arises from locality : there is an embedding of gauge symmetries in M as local
gauge symmetries in a tubular neighborhood ∂Mε, and then there is an inclusion G∂Mε ⊂ G̃∂M .
Finally symmetries from GM acting on germs yield symmetries in the quotient group G∂M .
Axiom A10 encodes the dynamics of gauge fields since it is an adapted version of the Lag-
rangian embedding to the symplectic space A∂M considered in Axiom C5. In Axiom A10 we use
the notion of reduced Lagrangian space, see [26]. We could also postulate this dynamics axiom
as follows.
There exists a symplectic closed subspace ΦA∂M ⊂ L∂M , such that LM̃ intersects transversally
the space ΦA∂M . Hence LM̃ ∩ΦA∂M ⊂ ΦA∂M is a Lagrangian subspace. Furthermore every G∂M -
orbit intersects ΦA∂M is a discrete set. We also call ΦA∂M a gauge-fixing space for the gauge
symmetries G∂M .
3.3 Simplifications in the absence of corners
As we mentioned previously for some axioms, namely Axioms A7, A9 and A12, we will consider
separately two cases:
Smooth case. Regions M and hypersurfaces are smooth manifolds of dimension n and n− 1
respectively, Σ is closed.
Corners case. Regions M are n-dimensional manifolds with corners, and hypersurfaces are
(n− 1)-dimensional topological manifolds Σ with stratified space structure |Σ|.
We write down explicitly these axioms in the smooth case, where regions M and hypersur-
faces Σ are smooth manifolds.
A7′ Suppose that an (n− 1)-dimensional hypersurface Σ decomposes as a disjoint union
Σ := Σ1 t · · · t Σm
of connected components Σ1, . . . ,Σm. Define A|Σ|n−1 := AΣ1 × · · · × AΣm , L|Σ|n−1 :=
LΣ1 ⊕ · · · ⊕ LΣm . Then there are linear and affine isomorphisms respectively
rΣ;|Σ|n−1 : LΣ → L|Σ|n−1 , aΣ;|Σ|n−1 : AΣ → A|Σ|n−1
such that (3.3) holds.
A9′ For the case without corners |Σ|n−1 ∼= Σ and the direct product group G|Σ|n−1 := GΣ1 ×
· · · × GΣm is isomorphic to GΣ with a gluing homomorphisms hΣ;|Σ|n−1 : GΣ → G|Σ|n−1
with compatibility commuting diagrams
AΣ ×GΣ
oo //
��
A|Σ|n−1 ×G|Σ|n−1
��
AΣ
oo // A|Σ|n−1
and analogous compatibility diagrams for actions on linear spaces L∂M , L|∂M |n−1 .
14 H.G. Dı́az-Maŕın
A12′ Let M1, M be regions without corners as above with gluing along hypersurfaces Σ, Σ′ ⊂
∂M
AM1
//
��
AM
��
A∂M1 A∂Moo
GM1
//
��
GM
��
G∂M1 G∂Moo
LM1
//
��
LM
��
L∂M1 L∂Moo
There is also a compatibility for the gluing of the actions of the gauge groups
AM1
//
��
AM
��
AM1 ×GM1
//
��
gg
AM ×GM
��
88
A∂M1 ×G∂M1
xx
A∂M ×G∂Moo
&&
A∂M1 A∂Moo
LM1
//
��
LM
��
LM1 ×GM1
//
��
ff
LM ×GM
��
88
L∂M1 ×G∂M1
xx
L∂M ×G∂Moo
&&
L∂M1 A∂Moo
4 Kinematics of gauge fields
In this section we consider affine field theories (comment of Axiom A4). The action that is
used as a test case comes from the Lagrangian density. We consider gauge principal bundles
on a compact manifold M provided with a Riemannian metric h, nonempty boundary ∂M
and compact abelian fiber group G. We suppose that regions M are manifolds of dimension
n = dimM ≥ 2, provided with a trivial principal bundle P with abelian structure group
G = U(1).
4.1 Classical abelian action
Along this subsection we assume the following two descriptions of a face Σ of hypersurface.
A. Smooth case: Σ is a smooth closed (n− 1)-dimensional manifold, or
B. Corners case: Σ is an (n− 1)-dimensional topological manifold with corners.
Since the bundle is trivial, the space of connections AM has a linear structure and can be
identified with LM . We consider the action
SM (ϕ) =
∫
M
dϕ ∧ ?dϕ,
Classical Abelian Theory with Corners 15
where ϕ ∈ AM 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
LM =
{
ϕ ∈ Ω1(M) | d?dϕ = 0
}
.
Here Ω1(M, g) ' Ω1(M) denotes g-valued 1-forms on M . These objects fulfill Axiom A1.
The identity component of gauge symmetries can be identified with certain f ∈ Ω0(M)
acting by ϕ 7→ ϕ+ df , thus G0
M
∼= Ω0(M)/Rb0 , where Rb0 denote the locally constant functions
on M . We consider hypersurfaces as closed submanifolds Σ ⊂ ∂M . Since G0
M preserves the
action on AM the requirement mentioned in Axiom A8 is satisfied.
We will describe an embedding, that in the smooth case is
X : Σ× [0, ε]→ Σε.
We also consider a normal vector field ∂τ on Σε, whose flow lines are the trajectoriesX(·, τ) ∈ Σε,
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 [18].
Lemma 4.1. A. Smooth case. Let Σ be a compact closed (n− 1)-manifold that is a component
of the boundary of a Riemannian manifold Σε diffeomorphic to a cylinder Σ × [0, ε], provided
with a Riemannian metric h. Then there exists an embedding X : Σ× [0, ε]→ Σε such that:
1. The vector field ∂τ is normal to Σ. The flow lines through s ∈ Σ correspond to trajectories
X(s, τ) ∈ Σε, 0 ≤ τ ≤ ε, transverse to Σ.
2. If ?Σ denotes the Hodge operator defined in Σ, and if XΣ : Σ→ Σε stands for the inclusion
XΣ(·) := X(·, 0) then
?ΣX
∗
Σ(ϕ) = X∗Σ(?ϕ), ∀ϕ ∈ Ωk(Σε).
3. If L· denotes the Lie derivative, then
X∗Σ (L∂τ ? ·) = X∗Σ (?L∂τ ·) = ?ΣX
∗
Σ (L∂τ ·) .
4. X∗Σ(L∂τ (d?ϕ)) = X∗Σ(d?(L∂τϕ)), for any ϕ ∈ Ωk(Σε).
5. Suppose that ϕ ∈ Ω1(Σε) satisf ies ι∂τϕ = 0 then
?ΣX
?
Σ(L∂τϕ) = X?
Σ(?dϕ).
6. If X∗Σ(d?ϕ) = 0, then X∗Σ(d?L∂τϕ) = 0, for any ϕ ∈ Ωk(Σε).
B. Corners case: Let Σ be a compact (n−1)-manifold with corners that is a component of the
boundary of a Riemannian manifold with corners Σε diffeomorphic to a regular cylinder Σ̂ε, (2.1),
provided with a Riemannian metric h. Then there exists an embedding
X : Σ̂ε → Σε
such that all previous assertions hold.
Proof of case A. Consider the exponential map Y : Σ × [0, ε] → Σε, Y
t(·) := Y (·, t). on
a tubular neighborhood Σε of Σ (see for instance [17]). This means that for every initial condition
s ∈ Σ and t ∈ [0, ε], Y t(s) ∈ Σε, is a geodesic passing trough s = Y 0(s) whose arc-length is t.
The initial velocity vector field ∂Y t(s)
∂t |t=0 = ∂Y 0(s)
∂t , s ∈ Σ, is a vector field ∂τ , normal to Σ ⊂ Σε.
16 H.G. Dı́az-Maŕın
Let λ ∈ Ωn−1(Σε) be the (n − 1)-volume form associated to the Riemannian metric in Σε,
recall that dim Σε = n. Define λt := (Y t)∗λ as the form induced by the restriction of the (n−1)-
volume form on the embedded (n − 1)-hypersurface Y t(Σ) ⊂ Σε. Now take the differentiable
function c(t) :=
∫
Σ λ
0/
∫
Σ λ
t ∈ R+, ∀ t ∈ [0, ε]. Notice that c(0) = 1. Then by the compactness
of Σ ⊂ ∂M , [c(τ)λτ ] = [λ0] ∈ Hn−1
dR (Σ), for every fixed τ ∈ [0, ε]. Hence by Moser’s trick,
see [18], there exists an isotopy of the identity, Z : Σ× [0, τ ]→ Σ such that (Zτ )∗(c(τ) ·λτ ) = λ0,
Z0(s) = s, ∀ s ∈ Σ, where Zt(s) := Z(s, t).
We define
X(s, τ) := Zτ ◦ Y τ (s), ∀ (s, τ) ∈ Σ× [0, ε]
also Xt(·) := X(·, t), XΣ(·) := X(·, 0) = X0(·).
Consider the explicit form of the Hodge star operator, ?, for the Riemannian metric h on Σε,
and the star operator, ?Σ, for the induced metric h := X∗Σh on Σ. Take a k-form ϕ ∈ Ωk(Σε).
If we consider a coordinate chart (x1, . . . , xn−1) in Σ. Then locally X∗Σ(?ϕ) equals the pullback
of the k-form
?
(∑
I
aIdx
i1 ∧ · · · ∧ dxik +
∑
I′
bI′dx
i′1 ∧ · · · ∧ dxi
′
k−1 ∧ dτ
)
=
√
|det(hij)|
(∑
J
hi1j1 · · ·hikjkaIdxj1 ∧ · · · ∧ dxjn−k−1 ∧ dτ
)
+
√
|det(hij)|
(∑
J ′
hi
′
1j
′
1 · · ·hi
′
kj
′
n−kbI′dx
j′1 ∧ · · · ∧ dxj
′
n−k
)
.
Here the indexes denote ordered sets I = {i1 < · · · < ik}, J = {j1 < · · · < jn−k−1} such that
their union I ∪ J , as an ordered set, corresponds to a basis (dx1, . . . , dxn−1) of 1-forms on Σ.
The ordered sets I ′ = {i′1 < · · · < i′k−1}, J = {j′1 < · · · < j′n−k} are constructed in a similar way.
Thus
X∗Σ(?ϕ) =
√
| det(hij)|
(∑
J ′
hi
′
1j
′
1 · · ·hi
′
kj
′
n−kbI′dx
j′1 ∧ · · · ∧ dxj
′
n−k
)
.
Meanwhile
?ΣX
∗
Σ(ϕ) =
√∣∣ det
(
hij
)∣∣ ∑
{j′1<···<j′n−k}
h
i′1j
′
1 · · ·hi
′
kj
′
kbI′dx
j′1 ∧ · · · ∧ dxj′k
.
But h
ij
= hij for i, j ∈ {1, . . . , n−1}, and also hin = δi,n, the Kronecker delta, since ∂τ is normal
to Σ. Hence
√
|det(hij)| =
√
|det(hij)|, and ?ΣX
∗
Σ(ϕ) = X∗Σ(?ϕ). This proves assertion 2.
If the volume form on Σ in local coordinates can been described as |det(hij)|1/2dx1 ∧ dx2 ∧
· · · ∧ dxn−1, then (Xτ )∗(c(τ)λτ ) = λ0, implies
c(τ)
√∣∣det
(
h ◦Xτ
)
ij
∣∣dx1 ∧ · · · ∧ dxn−1 =
∣∣ det
(
hij
)∣∣1/2dx1 ∧ dx2 ∧ · · · ∧ dxn−1.
Furthermore (Xτ )∗[L∂τ (c(τ)λ)] = ∂
∂τ (Xτ )∗(c(τ)λ), then
X∗Σ [L∂τ (c(τ)λ)] =
∂
∂τ
(Xτ )∗ (c(τ)λ) |τ=0 =
∂
∂τ
(
λ0
)
|τ=0 = 0.
Classical Abelian Theory with Corners 17
Hence
∂
∂τ
(
c(τ)
√∣∣det
(
h ◦Xτ
)
ij
∣∣dx1 ∧ · · · ∧ dxn−1
) ∣∣
τ=0
=
[
∂
∂τ
∣∣det
(
h ◦Xτ
)
ij
∣∣1/2 · c(t) +
∣∣ det
(
h ◦Xτ
)
ij
∣∣1/2∂c(τ)
∂τ
] ∣∣
τ=0
dx1 ∧ · · · ∧ dxn−1 = 0.
Recall that c(τ) = | det(h
ij
)|1/2/|det(h ◦Xτ )|1/2; hence
∂c(τ)
∂τ
∣∣∣
τ=0
=
−3
2
∣∣det
(
hij
)∣∣1/2∣∣det
(
h ◦Xτ
)
ij
∣∣−3/2 ∂
∂τ
∣∣ det
(
h ◦Xτ
)
ij
∣∣∣∣
τ=0
.
Therefore
∂
∂τ
∣∣det
(
h ◦Xτ
)
ij
∣∣1/2∣∣
τ=0
=
∂c(τ)
∂τ
∣∣∣
τ=0
= 0.
Hence the derivative of Z0 at Σ equals Z0
? = Id, since ∂c(τ)
∂τ |τ=0 = 0. Therefore
∂Xτ
∂τ
|τ=0 = Z0
∗
(
∂Y 0
∂τ
)
= ∂τ .
This proves assertion 1.
Now, since ∂τ is normal to Σ,
∂
∂τ
∣∣det(h)ij
∣∣1/2∣∣
τ=0
=
∂
∂τ
∣∣det
(
h
)
ij
∣∣1/2∣∣
τ=0
= 0,
then X∗Σ(L∂τ (?ϕ)) equals√∣∣ det
(
hij
)∣∣ ∑
{j′1<···<j′n−k}
∂
∂τ
(
hi
′
1j
′
1 · · ·hi
′
n−kj
′
n−kbI′
)∣∣
τ=0
dxj
′
1 ∧ · · · ∧ dxj
′
n−k .
Recall that the derivative of the exponential map Y t at s ∈ Σ, Y 0
∗ : T0(TsΣε) ' TsΣε → TsΣε,
equals the identity, Y 0
∗ = Id. This in turn implies that
∂
∂τ
(
hi
′
1j
′
1 · · ·hi
′
n−kj
′
n−k
)∣∣
τ=0
= 0.
Therefore X∗ΣL∂τ (?ϕ) equals√∣∣ det
(
hij
)∣∣ ∑
{j′1<···<j′n−k}
hi
′
1j
′
1 · · ·hi
′
n−kj
′
n−k
∂
∂τ
(bI′)|τ=0dx
j′1 ∧ · · · ∧ dxj
′
n−k
= X∗Σ(?L∂τ (ϕ)).
This proves assertion 3. Assertion 4 is an immediate consequence of assertion 3, and assertion 6
is in turn a consequence of assertion 4.
Part 5 is a direct calculation for if ι∂τϕ = 0, ϕ ∈ Ω1(Σε), then locally ϕ =
n−1∑
i=1
fi(x, τ)dxi,
thus X∗Σ(?dϕ) equals
X∗Σ ?
n−1∑
i=1
∑
j 6=i
∂jfi(x, τ)dxj ∧ dxi
+
n−1∑
i=1
∂
∂τ
fi(x, τ)dτ ∧ dxi
= |dethij |1/2
n−1∑
i=1
(−1)i+1h1,i · · · ĥi,i · · ·hn,i ∂
∂τ
fi(x, τ)dx1 ∧ · · · ∧ d̂xi ∧ · · · ∧ dxn−1,
where ĥi,i, d̂x
i
denote missing terms. This last expression corresponds to ?ΣX
∗
Σ(L∂τϕ), therefore
assertion 5 holds. �
18 H.G. Dı́az-Maŕın
Proof of case B. Now we consider Σ as a manifold with corners and ∂Σ 6= ∅. We consider the
exponential map Y t(s) for all 0 ≤ t ≤ ε(s), where ε−1(0) = Σ, recall the definition of a regular
cylinder Σ̂ε in (2.1).
Then we use Dacorogna–Moser’s argument for manifolds with boundary. This result is proved
in [7, Theorem 7] for domains Σ ⊂ Rn−1 with smooth boundary ∂Σ, and for Lipschitz bounda-
ries ∂Σ. In the case of general manifolds with corners Σ, the same results holds, see Remark 2.4
and Theorem 2.3 in [3]. This proves the case of regions with corners.
Since ε : Σ→ [0, 1] in equation (2.1) is a smooth increasing function, then the (n− 1)-volume
form c(τ)·λτ coincides with the volume form λ0 of Σ along ∂Σ. They define the same cohomology
form [λ0] ∈ Hn−1
dR (Σ, ∂Σ). Now by Dacorogna–Moser, there exists Z : Σ× [0, τ ]→ Σ, such that
(Zτ )∗(λτ ) = λ0. Now we consider
Xτ = Zτ ◦ Y τ (s), ∀ (s, τ) ∈ Σ× [0, ε(s)ε]
Recall that Σε := ε−1(1) ⊂ Σ is a smooth deformation retract Σε ⊂ Σ. All statements for the
case A remain valid in Σε, since they depend on local coordinate arguments.
Take a sequence ε→ 0 and the corresponding regular cylinders Σ̂ε, (2.1). There are smooth
increasing functions εε : Σ → [0, 1], such that for every ε1 < ε2, the corresponding deformation
retracts, Σεεi = ε−1
εi (1), i = 1, 2, may be contained one in another Σεε2 ⊂ Σεε1 . Recall that
ε−1
εi (0) = ∂Σ. Therefore, Σ can be obtained as the closure of
⋃
ε>0 Σεε . Hence the statements
for case A remain valid for the whole domain Σ. �
Definition 4.2. The following expression corresponds to the presymplectic structure in L̃Σ, for
the Yang–Mills action, see for instance [28],
ω̃Σ
(
η̃, ξ̃
)
=
1
2
∫
Σ
X∗Σ
(
η ∧ d?ξ − ξ ∧ d?η
)
, (4.1)
for all ξ̃, η̃ ∈ L̃Σ with representatives ξ, η ∈ LΣε .
In addition, the degeneracy subspace of the presymplectic form is
KωΣ :=
{
η̃ ∈ L̃Σ | η = df, f(s, 0) = 0, f ∈ Ω0(Σε), ∀ s ∈ Σ
}
.
From this very definition we have that the degeneracy gauge symmetries group KωΣ is a (normal)
subgroup of the identity component group G̃0
Σ ≤ G̃Σ of the gauge symmetries,
KωΣ E G̃
0
Σ ≤ G̃Σ.
Let
ΦÃΣ
:=
{
η̃ ∈ L̃Σ | ι∂τ η = 0, η ∈ LΣε reprentative of η̃
}
(4.2)
be the axial gauge fixing subspace of L̃Σ. The following statement leads to a simpler expression
for the presymplectic structure.
Lemma 4.3. For every ϕ ∈ LΣε corresponding to a solution, there is the gauge orbit represen-
tative
ϕ = ϕ+ df, (4.3)
such that ι∂τϕ = 0, and f |Σ = 0.
a) Every KωΣ-orbit in L̃Σ intersects in just one point the subspace ΦÃΣ
.
Classical Abelian Theory with Corners 19
b) The presymplectic form ω̃Σ restricted to the subspace ΦÃΣ
may be written as
ω̃Σ
(
η̃, ξ̃
)
=
1
2
∫
Σ
X∗Σ
(
η ∧ ?L∂τ ξ − ξ ∧ ?L∂τ η
)
,
for every ξ̃, η̃ ∈ L̃Σ with representatives ξ, η ∈ LΣε. Hence ω̃Σ is a non-degenerate 2-form
when restricted to the gauge fixing subspace ΦÃΣ
⊂ L̃Σ.
Proof of a). Let
( n−1∑
i=1
ηidxi
)
+ ητdτ be a local expression for a solution η ∈ LΣε . Let us apply
a gauge symmetry
X∗Σ (η + df) =
n−1∑
i=1
(
ηi + ∂if
)
dxi +
(
ητ + ∂τf
)
dτ
in such a way that ητ +∂τf = 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 KωΣ . The remaining part is a straightforward calculation. This proves a).
The other assertion may be inferred from Lemma 4.3. �
This result shows that Axioms A2 and A3 are satisfied. Let
AΣ := ÃΣ/KωΣ , LΣ = L̃Σ/KωΣ
be the quotients by the linear space KωΣ corresponding to degenerate gauge symmetries. And
also let G0
Σ := G̃0
Σ/KωΣ be the quotient by the normal subgroup. By Lemma 4.3, when we
restrict the quotient class ÃΣ
// // AΣ to ΦÃΣ
, then we get an isomorphism of affine spaces.
Let ωΣ be the corresponding symplectic structure on AΣ induced by the restriction of ω̃Σ to the
subspace ΦÃΣ
⊂ L̃Σ.
We now proceed to give a precise description of the symplectic space LΣ.
Lemma 4.1 implies that
X∗Σ
(
ξ ∧ ?L∂τ η
)
= X∗Σ
(
ξ
)
∧ ?ΣX
∗
Σ
(
L∂τ η
)
, (4.4)
where ?Σ stands for the Hodge star on Σ. Since ι∂τ (L∂τ η) = ι∂τ (ι∂τdη) = 0, then we have
a linear map LΣε → Ω1(Σ)× Ω1(Σ), where
η 7→
(
φη, φ̇η
)
:=
(
X∗Σ(η), X∗Σ
(
L∂τ η
))
, (4.5)
for every η̃ ∈ L̃Σ with representative ξ, η ∈ LΣε and η defined in (4.3). that leads to a map
LΣ → Ω1(Σ)× Ω1(Σ) ' T
(
Ω1(Σ)
)
, (4.6)
where we consider the identification with the tangent space T (Ω1(Σ)).
Notice that ι∂τ η = 0 implies that η ∈ LΣε corresponds to a 1-form φη on Σ. Notice also that
d?dη = 0 implies
d?(ι∂τdη) = d?(L∂τ η) = 0.
Hence d?Σ(X∗Σ(L∂τ η)) = 0. Therefore φ̇η ∈ ker d?Σ .
We have the following expression for the symplectic structure on LΣ
ωΣ
((
φη, φ̇η
)
,
(
φξ, φ̇ξ
))
=
1
2
∫
Σ
(
φη ∧ ?Σφ̇ξ − φξ ∧ ?Σφ̇η
)
, (4.7)
20 H.G. Dı́az-Maŕın
for every (φξ, φ̇ξ), (φη, φ̇η) ∈ LΣ, with representatives ξ, η ∈ LΣε . From this very definition we
can verify Axiom A4, i.e., translation invariance and also relation (3.1) where[(
φη, φ̇η
)
,
(
φξ, φ̇ξ
)]
Σ
:=
∫
Σ
φη ∧ ?Σφ̇ξ.
Furthermore Axiom A6 is easily verified and the claims from Axiom A5 can be inferred from
the relation ?Σ = −?Σ.
With this result we finish the kinematical part of the axiomatic description, i.e., Axioms A1–
A6.
4.2 Symplectic reduction
In this subsection we assume also two cases as in the previous subsection, either:
A. Σ is a smooth closed (n− 1)-dimensional manifold, or
B. Σ is an (n− 1)-dimensional manifold with corners.
We still need to describe the quotient for the symplectic action of the gauge group G0
Σ on LΣ.
The suitable gauge fixing space ΦΣ in LΣ for this action will be the space of divergence free
1-forms, i.e., we define
ΦAΣ
:=
{(
φ, φ̇
)
∈ LΣ | d?Σφ = 0 = d?Σ φ̇
}
.
The following task is the detailed description of the symplectic quotient space
LΣ/G
0
Σ ' ΦAΣ
.
A. Smooth case. We recall some useful facts of Hodge–Morrey–Friedrich theory for manifolds
with boundary, see for instance [1, 9, 10, 24]. We can consider both Neumann and Dirichlet
boundary conditions in order to define k-forms on a manifold V , i.e.,
Ωk
N (V ) :=
{
ϕ ∈ Ωk(V ) |X∗∂V (?ϕ) = 0
}
, Ωk
D(V ) :=
{
ϕ ∈ Ωk(V ) |X∗∂V (ϕ) = 0
}
.
The differential d preserves the Dirichlet complex Ωk
D(V ) and on the other hand, the codifferen-
tial d? preserves the Neumann complex Ωk
N (V ). In addition, the space Hk(V ) 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 HkN (V ), HkD(V ).
According to Hodge theory [24], associated with the inner product∫
Σ
φ ∧ ?Σφ
′, φ, φ′ ∈ Ω1(Σ)
we have an orthogonal decomposition φ = φh + d?Σαφ,{
φ ∈ Ω1(Σ) | d?φ = 0
}
= H1(Σ)⊕ d?ΣΩ2(Σ),
where the space H1(Σ) of harmonic 1-forms has rank b = dimH1(Σ).
According to [9, 10], the space of harmonic forms on a smooth manifold Σ with smooth
boundary ∂Σ, has rank
b = dimH1
N (Σ) = dimH1(Σ) = dimHn−2(Σ, ∂Σ).
B. Corners case. 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
Classical Abelian Theory with Corners 21
differentiability on ∂Σ′. Here Σ′ is a smooth manifold with smooth boundary homeomorphic
to Σ. If φ ∈ H1
N (Σ) is a harmonic form with null normal component, then F ∗(φ)|∂Σ′ ∈ H1
N (Σ′) is
also a well defined harmonic form on ∂Σ′. Hence for manifolds with corners Σ, harmonic forms
have also rank given by the Betti number.
The following lemma fulfills Axiom A8 and provides the gauge-fixing space definition required
in Axiom A10.
Lemma 4.4. Let Σ be a closed smooth manifold or a manifold with corners of dimension n−1.
For η ∈ LΣε, take (φη, φ̇η) ∈ TΩ1(Σ) as defined in (4.5), with gauge transformation group G0
Σ.
a) The gauge group action of G0
Σ on LΣ is induced in the tangent space TΩ1(Σ) by the
translation action φ 7→ φ+ df, f ∈ Ω0(Σ) on Ω1(Σ).
b) Every G0
Σ-orbit in LΣ intersects the subspace ΦAΣ
in just one point .
c) The symplectic form ωΣ is preserved under the G0
Σ-action.
Proof of b). Consider X∗Ση =
n−1∑
i=1
ηidxi, a local expression for a solution η ∈ LΣε ∩ΦÃΣ
⊂ LΣε .
Consider f : Σ→ R, then d?Σ(X∗Σ(η) + df) = 0 implies
n−1∑
i=1
∂i
[∣∣det
(
h
)∣∣1/2 n−1∑
i=1
(
ηi + ∂if
)
(−1)ih1,i · · · ĥi,i · · ·hi,n−1
]
= 0. (4.8)
The existence and regularity of a solution, f(s), for this PDE on Σ is warranted precisely by
Hodge theory. Since
X∗Σ(η) ∈ Ω1(Σ) ' dΩ0(Σ)⊕ H1(Σ)⊕ d?ΣΩ2(Σ),
there exists f ∈ Ω0(Σ), such that X∗Σ(η) + df is the orthogonal projection of X∗Σ(η) onto
ker d?Σ ' H1(Σ)⊕ d?ΣΩ2(Σ). Define
φη := X∗Σ(η) + df ∈ ker d?Σ .
On the other hand
d?L∂τ (η + df) = 0 (4.9)
implies
n−1∑
i=1
∂i
[
|det(h)|1/2
n−1∑
i=1
(
∂τη
i + ∂τ∂if
)
(−1)ih1,i · · · ĥi,i · · ·hi,n−1
]
= 0. (4.10)
When we substitute ∂τη
i + ∂τ∂if by the coefficients φiτ , of a time dependent 1-form in Σ,
φτ ∈ Ω1(Σ), equation (4.10) has a solution φτ . This leads to an ODE for gi(s, τ) := ∂if ,
∂τη
i + ∂τ∂if = φiτ . (4.11)
Equation (4.11) can be solved once we fix the boundary condition ∂if(xi, 0) = ∂if(xi). This
boundary condition, in turn, has been obtained by solving (4.8) in Σ.
We conclude that
n−1∑
i=1
gi(s, τ)dxi is an exact form on Σ, so that there exists f(s, τ) ∈ Ω0(Σε)
such that (4.9) holds.
To conclude define φ̇η := X∗Σ(L∂τ (η + df)), notice that (φη, φ̇η) ∈ ΦAΣ
.
Remark that from the very form of the solution φτ = X∗Σ(ητ ) + df τ , φτ and ∂τη have the
same integrals along closed cycles, hence they have the same cohomology class in H1(Σ). �
22 H.G. Dı́az-Maŕın
The axial Gauge fixing space ΦΣ can be described with
T
[
H1(Σ)⊕ d?ΣΩ2(Σ)
]
'
[
TH1(Σ)
]
×
[
T (d?ΣΩ2(Σ))
]
,
where we take tangent spaces. Recall that according to Hodge theory the space of coclosed
1-forms can be described as H1(Σ) ⊕ d?ΣΩ2(Σ). In the abelian case the holonomy holγ(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
{γ1, . . . , γb} of the homology H1(Σ), and a dual harmonic basis φ1
h, . . . , φ
b
h we have the exact
sequence
0 // ⊕bi=1Z ·
[
φih
]
// H1(Σ)
holγi // Gb // 1.
Hence there is a surjective map by the differential DholΓ : TH1(Σ)→ TGb.
Now we consider the reduction of ΦAΣ
under the action of the discrete group GΣ/G
0
Σ.
Lemma 4.5. Let Σ be a (n− 1)-dimensional smooth manifold or a manifold with corners. We
have the quotient space
AΣ/GΣ = ΦAΣ
/
(
GΣ/G
0
Σ
)
' T
(
Gb
)
× T
(
d?ΣΩ2(Σ)
)
with reduced symplectic structure ωΣ given in (4.7).
4.3 Factorization on hypersurfaces
Now we complete the symplectic reduction picture for hypersurfaces described in Axioms A7–
A9. First we consider the factorization given in Axiom A7. In this section we have denoted
alternatively smooth closed manifolds or manifolds with corners as Σ. For the results stated in
this particular subsection, the convention will be different:
In this subsection Σ will denote a hypersurface, i.e., a topological manifold with a stratified
space structure |Σ|.
The r-forms in the k-skeleton, r ≤ k, k = 0, 1, 2, . . . , n− 1 is the set of restrictions of these r-
forms to its faces:
Ωr
(
|Σ|(n−1)
)
= ∪mi=1
{
ϕi ∈ Ωk
(
Σi
)
|ϕi|Σj = ϕj ∀ (i, j) ∈ P
}
,
Ωr
(
|Σ|(n−2)
)
= ∪
{
ϕIΩkΣI |ϕI |ΣJ = ϕJ , I, J ∈ P
}
.
Notice that we have the inclusion of k-forms on stratified spaces, for r = 0, 1, given by the
pullbacks
Ωr(Σ) = Ωr
(
|Σ|(n−1)
)
⊂ Ωr
(
|Σ|(n−2)
)
.
To give a detailed description of the space of divergence-free fields on Σ, let us first consider
harmonic fields.
If φh ∈ H1(Σi), then the restriction, φIh, over every face closure ΣI ⊂ Σi, contained in Σi, is
harmonic as stated in the following lemma, which consists of two parts one for regular cylinders
and another one for (n− 1)-dimensional stratified spaces.
Lemma 4.6 (Theorems 7 and 8 in [2]). Let Σε be a Riemannian manifold with corners home-
omorphic to the regular cylinder Σ̂ε, (2.1). For every harmonic form ϕ ∈ Hr(Σε), the following
are true:
1. ϕ is closed and coclosed, that is dϕ = 0 = d?ϕ, i.e., ϕ ∈ Hr(Σε).
Classical Abelian Theory with Corners 23
2. ϕi := ϕ|Σi ∈ Hr(Σi), where Σi ⊂ |Σ|(n−1) are the (n− 1)-dimensional faces.
3. The boundary and coboundary operators satisfy,((
dϕ
)i)
=
(
d(ϕi)
)
,
((
d?ϕ
)i)
=
(
d?Σi
(
ϕi
))
,
so that they define complexes (Ω(|Σ|(n−2)), d), (Ω(|Σ|(n−2))), d?).
4. ϕ|Σi ∈ HrN (Σi), if and only if ι∂τϕ = 0, where ∂τ is a vector field normal to Σi.
Let |Σ|(n−1) = |Σ| be an (n − 1)-dimensional stratified space homeomorphic to an (n − 1)-
dimensional manifold. For every harmonic r-form φ ∈ Hr(|Σ|)
1. ϕ is closed and coclosed, that is dφ = 0 = d?Σiφ, i.e., φi := φ|Σi ∈ Hr(Σi), for each
(n− 1)-dimensional closed stratum Σi.
2. φI := φ|ΣI ∈ Hr(ΣI), where ΣI ⊂ |Σ|(n−2) are the (n− 2)-dimensional faces.
3. The boundary and coboundary operators satisfy,((
dφ
)I)
=
(
d
(
φI
))
,
((
d?φ
)I)
=
(
d?ΣI
(
φI
))
,
so that they define complexes (Ω(|Σ|(n−2)), d), (Ω(|Σ|(n−2))), d?).
This finishes the description of harmonic forms on the stratified space |Σ|. Notice that
when Σi are balls then the harmonic forms φ ∈ Ωr(Σi) are completely defined by their Dirichlet
boundary conditions on ∂Σi.
We also have an analogous of Hodge–Morrey–Friedrichs decomposition for stratified spaces,
that follows also from Theorems 7 and 8 in [2]:
Corollary 4.7.
1. There is an orthogonal decomposition
Ωr(|Σ|) = HrN (|Σ|)⊕
(
Hr(|Σ|) ∩ dΩr−1(|Σ|)
)
⊕ dΩr−1
D (|Σ|)⊕ d?Ωr+1
N (|Σ|).
2. In particular there is an orthogonal decomposition for divergence-free fields
ker
[
d? : Ωr(|Σ|)→ Ωr−1(|Σ|)
]
= HrN (|Σ|)⊕ d?Ωr+1
N (|Σ|).
Therefore the divergence free 1-forms on |Σ| are described as
H1(|Σ|)⊕ d?ΣΩ2(|Σ|),
thus we could define the gauge-fixing space as
ΦAΣ
:= T
(
H1(|Σ|)
)
× T
(
?Σ dΩ0(|Σ|)
)
.
From the projections Σ̌i → Σi ⊂ Σ we obtain the linear maps
H1(|Σ|) ⊂ ⊕ri=1H
1
(
Σ̌i
)
and
Ω0(|Σ|) ⊂ ⊕mi=1Ω0
(
Σ̌i
)
.
24 H.G. Dı́az-Maŕın
If ΦAΣ̌i
:= T (H1(Σ̌i))⊕ T (?ΣidΩ0(Σ̌i)), then we get the injective maps
ΦAΣ
→
m∏
i=1
ΦAΣ̌i
=: ΦA|Σ|2
.
This inclusion is the restriction of an inclusion referred to in Axiom A7, as is described in the
following commuting diagram
T
(
Ω1(|Σ|)
)
//
m∏
i=1
T
(
Ω1
(
Σ̌i
))
= T
(
⊕mi=1 Ω1
(
Σ̌i
))
LΣ
//?�
OO
⊕mi=1LΣ̌i = L|Σ|2
?�
OO
ΦAΣ
?�
OO
//
m∏
i=1
ΦAΣ̌i
=: ΦA|Σ|2
?�
OO
Similarly, the inclusions in the affine spaces AΣ →
m∏
i=1
AΣ̌i can be described. Also for the
gauge symmetries we have exact sequences
H0
dR(|Σ|) //
��
Rm
��
Ω0(|Σ|)
��
// ⊕mi=1Ω0
(
Σ̌i
)
��
G0
Σ
// ⊕mi=1GΣ̌i =: G|Σ|2
where GΣ̌i ' Ω0(Σ̌i)/R (since Σ̌i is simply connected), and G0
Σ stands for the identity component
of the gauge symmetries group GΣ.
We use Lemma 4.5 to establish the following claim.
Theorem 4.8. Let Σ be a hypersurface with a stratified space structure |Σ|. We have the gauge
fixing space
ΦAΣ
= ΦAΣ
/G0
Σ ' T
(
H1(|Σ|)
)
× T
(
?Σ dΩ0(|Σ|)
)
⊂
m∏
i=1
ΦAΣ̌i
with symplectic structure ωΣ induced by the pullback of ωΣ̌1 ⊕ · · · ⊕ ωΣ̌m. We also have the
quotient space
AΣ/GΣ = T
(
Gb
)
× T
(
d?ΣΩ2(|Σ|)
)
.
Where b = dimH1(|Σ|) is the Betti number of Σ,
b = dimHn−2(Σ; ∂Σ) = dimH1(Σ), n− 1 = dim Σ.
This proves the validity of the factorization Axioms A7, A9.
Classical Abelian Theory with Corners 25
For the smooth case let us consider a hypersurface Σ as a disjoint union of oriented hyper-
surfaces Σ = Σ1 t · · · t Σm. Then there is a linear map
Ω1(Σ)→ Ω1(Σ1)⊕ · · · ⊕ Ω1(Σm)
given by η 7→ X∗Σ1(η) ⊕ · · · ⊕ X∗Σm(η). This map induces the isomorphism rΣ;|Σ|n−1 : LΣ →
L|Σ|n−1 = LΣ1 ⊕ · · · ⊕ LΣm . Furthermore, the chain decomposition∫
Σ
· =
∫
Σ1
·+ · · ·+
∫
Σm
·
verifies Axiom A7′. The proof of Axiom A9′ is similar.
Let us now look at the content of Axiom A8. Let G0
M = {df | f ∈ Ω0(M)} be the identity
component of the bulk gauge symmetry group GM . Each bulk symmetry df ∈ G0
M induces
a symmetry X∗Σ(df) ∈ G0
∂Mε
in the boundary cylinder ∂Mε := ∪iΣi
ε, and also a symmetry
hM (df) ∈ G0
∂M in the boundary conditions. This was mentioned in the locality arguments in
Subsection 3.2.
There is an 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 GM .
This means that we can define sections σ : G0
∂M → GM of the homomorphism GM → G∂M .
Hence there is a well defined (set-theoretic) orbit map
rM : LM/G
0
M → L∂M/G
0
∂M .
Furthermore, by linearity of the actions LM/G
0
M , AM/G
0
M have linear and affine structures
respectively. This proves Axiom A8.
The kinematic description of gauge fields is now completed from Axioms A1–A9.
5 Dynamics modulo gauge
These paragraphs are aimed to verify Axioms A10–A12 where dynamics of gauge fields is con-
structed. We discuss the behavior of the solutions near the boundary in more detail. Recall
that here is a map r̃M : LM → L̃∂M coming from the restriction of the solutions to germs on
the boundary. Composing with the quotient class map by the space Kω∂M , we have a map
rM : LM → L∂M . Let LM̃ ⊂ LΣ be the image under this map. The aim is to describe the image
LM̃ ⊂ L∂M of the space of solutions as a Lagrangian subspace once we take the gauge quotient.
We recall some useful facts of Hodge–Morrey–Friedrich theory for manifolds with boundary
for k-forms on M .
Lemma 5.1 (A. Smooth case [24]). Suppose that M is a smooth Riemannian manifold with
boundary
1. There is an orthogonal decomposition
Ωk(M) = HkN (M)⊕
(
Hk(M) ∩ dΩk−1(M)
)
⊕ dΩk−1
D (M)⊕ d?NΩk+1
N (M).
2. In particular there is an orthogonal decomposition for divergence-free fields
ker
[
d? : Ωk
N (M)→ Ωk−1
N (M)
]
= HkN (M)⊕ d?Ωk+1
N (M).
3. Each de Rham cohomology class can be represented by a unique harmonic field without
normal component, i.e., there is an isomorphism
Hk
dR(M) ' HkN (M).
26 H.G. Dı́az-Maŕın
4. Each de Rham relative cohomology class can be represented by a harmonic field null at the
boundary, i.e., there is an isomorphism
Hk
dR(M,∂M) ' HkD(M).
For the corners case we consider the stratified space structure |M | of the manifold with
corners M in order to give a precise definition of the spaces of Neumann and Dirichlet boundary
conditions on k-forms
Ωk
D(|M |) :=
{
ϕ ∈ Ωk(|M |) |ϕ||∂M | = 0
}
, Ωk
N (|M |) :=
{
ϕ ∈ Ωk(|M |) | ? ϕ||∂M | = 0
}
.
The decomposition given in Corollary 4.7, has a corners counterpart given by the following
result.
Lemma 5.2 (B. Corners case). Suppose that M is a Riemannian manifold with corners with
stratified space structure |M |.
1. There is an orthogonal decomposition
Ωk(|M |) = dΩk−1
D (|M |)⊕ HkN (|M |)⊕
(
Hk(|M |) ∩ dΩk−1(|M |)
)
⊕ d?Ωk+1
N (|M |).
2. There is an isomorphism
H1(|M |) ' H1
N (|M |).
3. And also an orthogonal decomposition
Ω2(|∂M |) = H2(|∂M |)⊕ dΩ1(|∂M |)⊕ d?Ω3(|∂M |).
The proof follows from restating arguments used in [2, Theorems 7 and 8].
Define
ΦAM := LM ∩ Ω1
N (|M |) ⊂ LM .
Notice that rM (ΦAM ) ⊂ L∂M . If ϕ ∈ Ωk(|∂Mε|) satisfies the Neumann condition X∗∂M (?ϕ) = 0,
then it also satisfies X∗∂M (ι∂τϕ) = 0 with ι∂τϕ ∈ Ωk−1(|∂Mε|).
Let us consider coclosed fields which, according to Lemma 5.1, have an orthogonal decom-
position of the axial gauge fixing space of solutions
ΦAM =
{
ϕh + d?α ∈ LM |ϕh ∈ H1
N (|M |), α ∈ Ω2
N (|M |), d?dd?α = 0 = dd?dα
}
.
If ϕ ∈ ΦAM , then ϕ satisfies the equation d?ϕ = 0, it also satisfies the Euler–Lagrange equation
d?dϕ = 0, since d?dd?α = 0 = dd?dα. This space
ΦAM ⊂ H1
N (|M |)⊕ d?Ω2
N (|M |),
constitutes the orthogonal projection of the space of solutions LM , according to the decompo-
sition
Ω1(|M |) = H1
N (|M |)⊕ d?Ω2
N (|M |)⊕ dΩ0
D(|M |)⊕
(
H1(|M |) ∩ dΩ0(|M |)
)
.
From this orthogonal decomposition it can be shown that every solution ϕ ∈ LM can be trans-
formed, modulo the bulk gauge transformation,
ϕ 7→ ϕ = ϕ+ df
onto a field belonging to the space ΦAM . Thus the following statement can be proven.
Classical Abelian Theory with Corners 27
Lemma 5.3.
1. Every G0
M -orbit intersects ΦAM ⊂ LM in exactly one point, i.e., for every ϕ ∈ LM there
exists f ∈ Ω0(|M |), such that ϕ = ϕ+ df ∈ ΦAM .
2. rM : ΦAM → LM̃ ∩ ΦA∂M is a linear surjection.
Corollary 5.4. LM̃/G
0
∂M = rM (ΦAM )/G0
∂M ⊂ L∂M/G0
∂M .
Consider the identification H1
N (|M |) ' H1(|∂M |), and d?NΩ2(|M |) ' d?∂MΩ2(|∂M |). We have
the following statement.
Lemma 5.5.
1. There is a well defined restriction map
X∗∂M : ΦAM → H1(|∂M |)⊕ d?∂MΩ2(|∂M |)
2. If we adopt the identification given in Lemma 4.5,
L∂M/G
0
∂M ' T
(
H1(|∂M |)
)
× T
(
d?∂MΩ2(|∂M |)
)
,
then the map rM coincides with the first jet of the pullback, i.e., we have a commutative
diagram of linear mappings
ΦAM� _
��
rM // L∂M/G
0
∂MOO
��
H1
N (|M |)⊕ d?Ω2
N (|M |)
j1X∗∂M
// T
(
H1(|∂M |)
)
× T
(
d?∂MΩ2(|∂M |)
)
As a matter of fact de Rham theorems also apply in stratified spaces. There is an isomorphism
for the singular cohomology
H1(|∂M |,R) ' H1
dR(|∂M |)
only in the case when the stratification is complete, see definition in [2, Theorem 18]. This
occurs for instance in the case where the faces Σi ⊂ ∂M are homeomorphic to balls. In general
there is just a morphism H1
dR(|∂M |)→ H1(|∂M |).
Now we are in a position to prove the result that encodes dynamics in a Lagrangian context.
We prove that the image of solutions modulo gauge onto the space of boundary conditions
modulo gauge, stated in Proposition 5.4, is in fact a Lagrangian space. The following statement
completes the dynamical picture described in Axiom A10.
Theorem 5.6. Let LM̃ = rM (LM ) be the boundary conditions that can be extended to solutions
in the interior LM . Then for the symplectic vector space L∂M/G
0
∂M
1. LM̃/G
0
∂M is an isotropic subspace.
2. LM̃/G
0
∂M is a coisotropic subspace.
In other words LM̃ ∩ ΦA∂M is a Lagrangian subspace of the symplectic space ΦA∂M .
28 H.G. Dı́az-Maŕın
As we mentioned in the introduction for LM̃/G
0
∂M isotropy is always true, see [14]. For the
sake of completeness we give a proof that is a straightforward calculation. Take ϕ,ϕ′ ∈ LM and
consider its image(
j1X∗∂M
)
(ϕ) =
(
X∗∂Mϕ,X
∗
∂M (L∂τϕ)
)
=
(
φϕ, φ̇ϕ
)
∈ L∂M ∩ ΦA∂M
where ϕ was defined in Lemma 4.3, and where d?∂Mφϕ = d?∂M φ̇ϕ = 0. We also consider
ϕ ∈ ΦAM . Then
ω∂M
(
ϕ,ϕ′
)
=
1
2
∫
∂M
φϕ ∧ ?∂M φ̇ϕ
′ − φϕ′ ∧ ?∂M φ̇ϕ
=
1
2
∫
∂M
X∗∂M (ϕ) ∧ ?∂MX∗∂M
(
L∂τϕ′
)
−X∗∂M
(
ϕ′
)
∧ ?∂MX∗∂M (L∂τϕ).
From a property shown in Lemma 4.1 we have that the last expression equals
1
2
∫
∂M
X∗∂M
(
ϕ ∧ ?dϕ′ − ϕ′ ∧ ?dϕ
)
.
Recall hat ϕ, ϕ′ are global solutions in the interior d?dϕ = 0 = d?dϕ′, hence by applying Stokes’
theorem we have
ω∂M
(
ϕ,ϕ′
)
=
∫
M
dϕ ∧ ?dϕ′ − dϕ′ ∧ ?dϕ = 0.
Proof for coisotropic embedding. Take ϕ ∈ ΦAM , as indicated in Lemma 5.3, take
(φϕ, φ̇ϕ) := rM (ϕ) and suppose that ω∂M (ϕ,ϕε) = 0 for every ϕε ∈ L∂Mε with (φ′, φ̇′) ∈ L∂M
corresponding to ϕε with ι∂τϕ
ε. Recall that d?∂Mφ′ = 0 = d?∂M φ̇′. Then thanks to the repre-
sentative (4.4) we have∫
∂M
φϕ ∧ ?∂M φ̇′ =
∫
∂M
φ′ ∧ ?∂M φ̇ϕ, ∀
(
φ′, φ̇′
)
∈ L∂M . (5.1)
According to the orthogonal decomposition described in Lemma 5.1, we have
φϕ = φh + d?∂MX∗∂M (α), φ̇ϕ = φ̇h + d?∂MX∗∂M (α̇),
where
φh, φ̇h ∈ H1(|∂M |), α, α̇ ∈ Ω2(|∂Mε|), α̇ = L∂τα, d∗dd∗α = 0
Hence equation (5.1) implies ∀ (φ′, φ̇′) ∈ L∂M∫
∂M
φh ∧ ?∂M φ̇′ +
∫
∂M
d?∂MX∗∂M (α) ∧ ?∂M φ̇′
=
∫
∂M
φ′ ∧ ?∂M φ̇h +
∫
∂M
φ′ ∧ ?∂Md?∂MX∗∂M (α̇). (5.2)
We calculate in more detail the first summand of the r.h.s. of equation (5.2). According to
Lemma 5.5, φ̇h = X∗∂M (L∂τϕh), where we consider the orthogonal decomposition
ϕ = ϕh + d∗α ∈ ΦAM ⊂ H1
N (|M |)⊕ d∗Ω2
N (|M |)
with ϕh ∈ H1
N (|M |), α ∈ Ω2
N (|M |). Hence∫
∂M
φ′ ∧ ?∂M φ̇h =
∫
∂M
φ′ ∧ ?∂MX∗∂M
(
L∂τϕh
)
=
∫
∂M
φ′ ∧ ?∂MX∗∂M
(
dϕh
)
= 0.
In the last line we have used the properties described for X∗∂M given in Lemma 4.1 and dϕh = 0.
Classical Abelian Theory with Corners 29
Now consider the first summand of the l.h.s. of equation (5.2), the extension ϕ̃ := ψ · ϕε ∈
Ω1
N (|M |) of ϕε ∈ Ω1(|∂Mε|), given by a partition of unity
ψ : M → [0, 1], such that ∂M = ψ−1(1). (5.3)
Then ∫
∂M
φh ∧ ?∂M φ̇′ =
∫
∂M
φh ∧ ?∂MX∗∂M (L∂τ ϕ̃) =
∫
∂M
φh ∧X∗∂M (?dϕ̃).
Furthermore, Lemma 5.5 claims that there exists ϕh ∈ H1
N (|M |) such that φh = X∗∂M (ϕh).
Therefore by Stokes’ theorem∫
∂M
φh ∧ ?∂M φ̇′ =
∫
∂M
X∗∂M (ϕh ∧ ?dϕ̃) =
∫
M
d(ϕh ∧ ?dϕ̃) = 0.
Therefore equation (5.2) yields∫
∂M
d?∂MX∗∂M (α) ∧ ?∂M φ̇′ =
∫
∂M
φ′ ∧ ?∂Md?∂MX∗∂M (α̇). (5.4)
According to the isomorphism H1(|M |) ' H1
N (|M |) in Lemma 5.1, [ϕ̃] ∈ H1(|M |) corresponds
to a harmonic field
[ϕ̃]↔ ϕ′h ∈ H1
N (|M |), (5.5)
and this in turn corresponds to the harmonic component φ′h ∈ H1(|∂M |) of
X∗∂M (ϕ′) = φ′ = φ′h + d?∂Mβ
with β ∈ Ω2(|∂M |), as is stated in Lemma 5.5. Thus, for the r.h.s. of equation (5.4) we have∫
∂M
φ′h ∧ ?∂Md?∂MX∗∂M (α̇) +
∫
∂M
d?∂Mβ ∧ ?∂Md?∂MX∗∂M (α̇).
Notice that ∂∂M = 0, therefore the last expression equals∫
∂M
dφ′h ∧ ?∂MX∗∂M (α̇) +
∫
∂M
d?∂Mβ ∧ ?∂Md?∂MX∗∂M (α̇)
=
∫
∂M
d?∂Mβ ∧ ?∂Md?∂MX∗∂M (α̇).
Similarly for φ̇′ = φ̇h + d?∂M β̇ with φ̇h ∈ H1(|∂M |), β̇ ∈ Ω2(|∂M |) and therefore the l.h.s. of
equation (5.4) equals
∫
∂M d?∂M β̇ ∧ ?∂Md?∂MX∗∂M (α).∫
∂M
β ∧ ?∂Mdd?∂MX∗∂M (α̇) =
∫
∂M
β̇ ∧ ?∂Mdd?∂MX∗∂M (α). (5.6)
Finally this eqution describes a condition on pairs β, β̇ ∈ Ω2(|∂M |), ∀α, α̇ ∈ Ω2
N (|∂Mε|) ⊂
Ω2
N (|M |). Again by Stokes’ theorem applied to the r.h.s. of the previous expression (5.6) we
have ∫
∂M
β ∧ ?∂Mdd?∂MX∗∂M (α̇) =
∫
M
d
˜̇
β ∧ ?dd?(α), ∀α ∈ Ω2
N (|M |), (5.7)
where
˜̇
β = ψ · β̇ ∈ Ω2
N (|M |) is an extension of a 2-form in the cylinder β̇ ∈ Ω2(|∂M |) to the
interior of M , given by a partition of unity ψ, (5.3).
30 H.G. Dı́az-Maŕın
Recall that since ϕ ∈ ΦAM , then d?dd?α = 0. From the orthogonal decomposition
Ω3(|M |) = dΩ2
D(|M |)⊕ H3
N (|M |)⊕
(
H3(|M |) ∩ dΩ2(|M |)
)
⊕ d?Ω4
N (|N |)
we have dd?α ∈ H3(|M |)∩ dΩ2(|M |). By the non-degeneracy of the Hodge inner product in M ,
there is a well defined exact harmonic field dβ̌ ∈ H3(|M |) ∩ dΩ2, that is the projection of d
˜̇
β,
such that d?dβ̌ = 0, and the r.h.s. of (5.7) reads as∫
M
dβ̌ ∧ ?dd?(α),
therefore∫
∂M
β ∧ ?∂Mdd?∂MX∗∂M (α̇) =
∫
∂M
X∗∂M
(
β̌
)
∧ ?∂Mdd?∂MX∗∂M (α). (5.8)
On the other hand consider the l.h.s. of (5.7). Recall that β ∈ Ω2(|∂M |) = H2(|∂M |) ⊕
dΩ1(|∂M |)⊕ d∗Ω3(|∂M |), in fact we can take
β = βh + dγ ∈ H2(|∂M |)⊕ dΩ1(|∂M |).
Consider the extension β̃ := ψβ,
β̃ = β̃h + d(γ̌ + γD) + d∗θ ∈ Ω2(|M |)
= H2
N (|M |)⊕
(
H2(|M |) ∩ dΩ1(|M |)
)
⊕ dΩ1
D(|M |)⊕ d∗Ω3
N (|M |).
If we take the orthogonal projection of β̃,
β̂ := β̃h + dγ̂ ∈ H2
N (|M |)⊕
(
H2(|M |) ∩ dΩ1(|M |)
)
⊕ dΩ1
D(|M |), γ̂ = γ̌ + γD, (5.9)
then X∗∂M (β̃h) = βh and X∗∂M (γ̂) = γ. Also for 3-forms as arguments of dd?∂M we have the
functionals∫
∂M
X∗∂M
(
β̃
)
∧ ?∂Mdd?∂M · =
∫
∂M
X∗∂M
(
β̃h + dγ̂
)
∧ ?∂Mdd?∂M ·,∫
∂M
(βh + dγ) ∧ ?∂Mdd?∂M · =
∫
∂M
X∗∂M
(
β̂
)
∧ ?∂Mdd?∂M · .
If we look more carefully the l.h.s. of expression (5.8), then∫
∂M
X∗∂M (β̂) ∧ ?∂Mdd?∂MX∗∂M (L∂τ (α)) =
∫
∂M
X∗∂M (β̂) ∧X∗∂M
(
L∂τ ? (dd?α)
)
=
∫
∂M
X∗∂ML∂τ
(
β̂ ∧ ?(dd?α)
)
+
∫
∂M
X∗∂M
(
L∂τ β̂
)
∧X∗∂M
(
? (dd?α)
)
= L∂τ
(∫
M
d
(
β̂ ∧ ?(dd?α)
))
+
∫
∂M
X∗∂M
(
L∂τ β̂
)
∧X∗∂M
(
? (dd?α)
)
=
∫
∂M
β̇ ∧ ?∂Mdd?∂MX∗∂M (α),
where in the last equality we used X∗∂M (L∂τ β̂) = X∗∂M (L∂τβ) = β̇ and that L∂τ
∫
M β̂∧?dd?α = 0.
Hence∫
∂M
X∗∂M (β̂) ∧ ?∂Mdd?∂MX∗∂M (L∂τ (α)) =
∫
∂M
β̇ ∧ ?∂Mdd?∂MX∗∂M (α).
Classical Abelian Theory with Corners 31
Looking back again at expression (5.8) and (5.6) we have∫
∂M
β̇ ∧ ?∂Mdd?∂MX∗∂M (α) =
∫
∂M
β ∧ ?∂Mdd?∂MX∗∂M (α̇)
=
∫
∂M
X∗∂M
(
β̌
)
∧ ?∂Mdd?∂MX∗∂M (α).
Hence for every α we have∫
∂M
X∗∂M
(
L∂τ β̂
)
∧X∗∂M
(
? (dd?α)
)
=
∫
∂M
X∗∂M
(
β̌
)
∧ ?∂Mdd?∂MX∗∂M (α).
This implies that X∗∂M (L∂τ β̂) = X∗∂M (β̌).
Finally we can extend the solution ϕε in the cylinder ∂Mε to a solution in the interior M , by
means of
ϕ′ := ϕ′h + d?
(
β̂
)
,
where ϕ′h was defined in (5.5) and β̂ is defined in (5.9). Notice that d∗dd∗β̂ = d∗dd∗dγ = 0,
therefore ϕ′ ∈ ΦAM . Furthermore,
φ′ = X∗∂M
(
ϕ′
)
, φ̇′ = X∗∂M
(
L∂τϕ′
)
. �
This finishes the proof of the validity of Axiom A10.
As we mentioned in Subsection 3.2, locality follows for fields and actions, in particular
Axiom A11 hold. The gluing Axiom A12 also follows from locality arguments. This completes
the dynamical description for this gauge field theory.
Thus abelian theory is fully constructed within this axiomatic framework.
6 Example: 2-dimensional case
For a better understanding of our model, we review our constructions in a more down to earth
example, namely the 2-dimensional case. We provide this presentation as a comparison tool with
some quantizations of two-dimensional theories, see for instance [8, 15, 27]. This also suggest
the steps that are necessary in quantization for general dimensions in further research.
Recall that we are supposing that we have a trivial gauge principal bundles on a compact
surface M , with structure group G = U(1). The following lemma will lead to a description of
the presymplectic structure ω̃Σ, on ÃΣ, for a proof see [13]. Lemma 4.1 in this case can be
simplified as the following statement.
Lemma 6.1 (Fermi). Given a cylinder Σ× [0, 1], there exists an embedding
X : Σ× [0, ε]→M
of the cylinder Σ× [0, ε] into a tubular neighborhood Σε of Σ, such that if (s, τ) are local coordi-
nates, then ∂/∂s, ∂/∂τ are orthonormal vector fields along Σ. Here s corresponds to arc length
along Σ with respect to the Riemannian metric h on M . Furthermore h|Σ is locally described as
the identity matrix.
The presymplectic structure can be written by using these local coordinate as in (4.1),
ω̃Σ
(
η̃, ξ̃
)
=
1
2
∫
Σ
[
ηs
(
∂sξ
τ − ∂τξs
)
− ξs
(
∂sη
τ − ∂τηs
)]
ds,
32 H.G. Dı́az-Maŕın
where X∗Σ(η) = ηsds+ ητdτ,X∗Σ(ξ) = ξsds+ ξτdτ are 1-forms corresponding to solutions in the
cylinder, i.e., ξ, η ∈ Ω1(Σε) satisfying Euler–Lagrange equations. We can also describe the gauge
group GΣ on AΣ, by considering the action of the identity component gauge group of germs:
G̃0
Σ := lim−→G0
Σε
. Here G0
Σε
:= Ω0(Σε)/Rb0 is acting by translations η 7→ η + df and inducing the
corresponding action η̃ 7→ η̃ + df̃ on germs η̃ ∈ L̃Σ.
The degeneracy subspace of the symplectic form is
KωΣ :=
{
η̃ ∈ L̃Σ | η = ∂τfdτ, ∂sf(s, 0) = 0, f ∈ Ω0(Σε)
}
.
From this very definition we have that the degeneracy gauge symmetry group KωΣ is a (normal)
subgroup of the abelian group G̃0
Σ.
By considering an axial gauge fixing, as in (4.2), let
ΦÃΣ
:=
{
η ∈ L̃Σ | ι∂τ η = 0
}
be a subspace of L̃Σ. As we did in Lemma 4.3 we have that every KωΣ-orbit in L̃Σ intersects
in just one point the subspace ΦÃΣ
. The presymplectic form ω̃Σ restricted to the subspace ΦÃΣ
may be written as
ω̃Σ(η̃, ξ̃) =
1
2
∫
Σ
[
−ηs∂τξs + ξs∂τη
s
]
ds, ξ̃, η̃ ∈ L̃Σ. (6.1)
Hence ω̃Σ is non-degenerate when we restrict it to the subspace ΦÃΣ
⊂ L̃Σ.
Let ωΣ the corresponding symplectic structure on AΣ induced by the restriction of ω̃Σ to the
subspace ΦÃΣ
⊂ L̃Σ.
Hypersurfaces are Σ := Σ1∪· · ·∪Σm ⊂ ∂M . In the smooth case Σi are homeomorphic to S1.
In the case with corners, Σi are intervals identified in some pairs by their boundaries.
Then there is a linear map
Ω1
∂M → Ω1
(
Σ1
)
⊕ · · · ⊕ Ω1
(
Σm
)
,
where η 7→
(
XΣ1
0
)∗
(η)⊕ · · · ⊕
(
XΣm
0
)∗
(η). This map induces r∂M ;Σ : L∂M →= LΣ1 ⊕ · · · ⊕LΣm .
Furthermore, the chain decomposition
∫
∂M · =
∫
Σ1 ·+ · · ·+
∫
Σm · induces Axioms A7′ and A7.
Recall that here is a map, r̃M : LM → L̃∂M , coming from the restriction of the solutions to
germs on the boundary. Composing with the quotient class map, we have a map rM : LM →
L∂M . Let LM̃ be the image rM (LM ) under this map.
Our aim is to describe the image LM̃ = rM (LM ) ⊂ LΣ of the space of solutions as a Lag-
rangian subspace modulo gauge.
Take ϕ ∈ LM so that d?dϕ = 0, then dϕ is constant a scalar multiple of the h-area form µ,
i.e., dϕ = ċϕµ, for a constant ċϕ. Suppose that ϕ ∈ LM is such that ι∂τϕ = 0, then ϕτ = 0.
Hence ∂sϕ
τ − ∂τϕs = ċϕ is constant. That is, −∂τϕs = ċϕ. Therefore if φ := rM (ϕ), φ′ :=
rM (ϕ′) ∈ L∂M , then by substituting in (6.1) we obtain
ω∂M
(
ϕ,ϕ′
)
=
∫
∂M
(
ϕsċϕ′ − (ϕ′)sċϕ
)
ds, ∀ϕ,ϕ′ ∈ L∂M .
Recall that ϕ,ϕ′ ∈ L∂Mε . By Stokes’ theorem
ω∂M
(
ϕ,ϕ′
)
= ċϕ′
∫
M
dϕ− ċϕ
∫
M
dϕ′ = (ċϕ′ ċϕ − ċϕċϕ′) · area(M) = 0, (6.2)
where area(M) :=
∫
M µ.
Classical Abelian Theory with Corners 33
We now consider the orbit space for gauge orbits. We consider the unit component subgroup
G0
Σ E GΣ. Recall the map (4.6). Take the gauge fixing subspace
ΦAΣ
:= {(η0, η̇0) ∈ AΣ | ∂sη0 = 0 = ∂sη̇0} =
{
(cds, ċds) ∈ AΣ | (c, ċ) ∈ R2
}
.
We can see the proof of Lemma 4.4 for this context. Let (cds, ċds) be a point in ΦAΣ
∼= R2.
Consider X∗Σ(η) = ηsds + ητdτ = ηsds, a local expression for a solution η ∈ LΣε ∩ ΦÃΣ
. By
considering a gauge symmetry we can get an ODE for f : Σε → R,
ηs + ∂sf = c, (6.3)
∂τη
s + ∂τ∂sf = ċ (6.4)
Equation (6.4) can be solved for g(s, τ) := ∂sf , once we can fix the boundary condition
∂sf(s, 0) = g(s, 0). This boundary condition in turn can be obtained by solving (6.3) in Σ.
The holonomy along Σ,
holΣ(η) = exp
√
−1
∫
Σ
η ∈ G = U(1)
remains the same for c and for η, furthermore since they are in the same component,
∫
Σ cds
equals
∫
Σ η
sds mod 2πZ. Here η belongs to the G0
Σ-orbit of c, therefore there is a homotopy
between both evaluations. Hence
c · length(Σ) =
∫
Σ
cds =
∫
Σ
ηsds
this implies that equation (6.3) can be solved.
Lemma 5.3 is also satisfied. It follows that LM̃∩ΦA∂M = rM (ΦAM ). The isotropic embedding
described in Theorem 5.6 is proved in (6.2). The corresponding coisotropic embedding in the
2-dimensional version goes as follows:
Take ϕ ∈ ΦAM , φ = rM (ϕ) and suppose that ω∂M (φ, φ′) = 0 for every ϕ′ ∈ L∂Mε , with
φ′ ∈ L∂M corresponding to ϕ′. Then
ċϕ
∫
∂M
(ϕ′)sds =
∫
∂M
(
ϕs∂τ (ϕ′)s
)
ds.
Since ϕ′ is a solution in a tubular neighborhood ∂Mε then ∂τ (ϕ′)s|Σ = ċϕ′ . Thus
ċϕ
∫
∂M
(ϕ′)sds = ċϕ′
∫
∂M
ϕsds = ċϕċϕ′
∫
M
µ
therefore∫
∂M
ϕ′ = ċϕ′ · area(M).
We claim that this is a sufficient condition, so that ϕ′ ∈ L∂Mε can be extended to the interior
of M . There exists a solution ϕ̌ ∈ LM such that ϕ′ = rM (ϕ̌). This will be an exercise of calculus
of differential forms.
The first step is to construct an extension θ = ψϕ ∈ Ω1(M), where we take a partition of
unity ψ whose value on ∂Mε is 1 and is 0 outside an open neighborhood V ⊂M of ∂Mε. We see
that ċϕ′dθ is closed and also has the same relative de Rham cohomology class in H2
dR(M,∂M ;R)
as ċϕ′µ. Thus ċϕ′(dθ − µ) = ċϕ′dβ, for a 1-form β such that β|∂M = 0. Therefore we can
define ϕ̌ := θ − β such that it is a solution. Therefore dϕ̌ = ċϕ′µ and it is also an extension,
ϕ̌|Mε = ϕ′|Mε .
34 H.G. Dı́az-Maŕın
This proves in the 2-dimensional case the Lagrangian embedding of Theorem 5.6. Nevertheless
we should notice that the proof of coisotropy is rather obvious in this case, since the reduced
symplectic space AΣ/GΣ is finite-dimensional.
Notice that in this case the bilinear form [·, ·]Σ used in Axiom A4, corresponds to[
φη, φξ
]
Σ
:= −
∫
Σ
(
ηs∂τξ
s
)
ds.
Here (cds, ċds) ∈ ΦAΣ
can be identified with c +
√
−1ċ ∈ C, provided with the Kähler
structure: length(Σ) · dc ∧ dċ. The holonomy holΣ : Ω1(Σ) → U(1) induces the derivative map
DholΣ : ΦAΣ
→ TU(1). We have the following commutative diagram
ΦAΣ D holΣ
//
OO
��
TU(1)
OO
��
C exp // C×
We can finally define the reduced space as the topological cylinder
AΣ/GΣ := ΦAΣ
/GΣ ↔ C×,
where GΣ := GΣ/G
0
Σ ' Z. We can get the symplectic structure ωΣ on AΣ/GΣ. This ωΣ is
length(Σ) times the area form on the cylinder TU(1). The reduced symplectic structure: ωΣ
on AΣ/GΣ, is length(Σ) times the area form on the cylinder TU(1). For ∂M = Σ1 ∪ · · · ∪ Σm:
AM/GM → A∂M/G∂M = TU(1)× · · · × TU(1).
The space LM has Lagrangian image which in each factor is the quotiented line:{
(c, ċ) ∈ R2 | c · length(∂M) = ċ · area(M)
}
/Z ⊂ TU(1).
As a consequence the map AM/GM → A∂M/G∂M does depend on global data of the metric,
such as area(M) and length(∂M). Recall that the same global dependence of dynamics holds
for the quantum version, i.e., the quantum TQFT version of gauge fields.
Once we have completed reduction, the picture of quantization on this finite-dimensional
space can be specified, cf. [8, 15, 27]. For a complete description of quantization in 2-dimensions
in general non abelian case with corners see [20].
7 Outlook: quantization in higher dimensions
The geometric quantization program with corners will be treated elsewhere. 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 discussion of these issues
in dimension two for instance in [8, 15, 27]. In order to administer the geometric quantization
program [28] for the reduced space we need to describe a suitable hermitian structure in ΦAΣ
.
Another work in progress with more physical applications is the formulation corresponding to
Lorentzian manifolds rather than the Riemannian case.
Acknowledgements
The author thanks R. Oeckl for several discussions and encouragement for writing this note at
CCM-UNAM. This work was partially supported through a CONACYT-México postdoctoral
grant. The author also thanks the referees for their comments and suggestions.
Classical Abelian Theory with Corners 35
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1 Introduction
2 Classical abelian gauge field theories
2.1 The symplectic setting for classical Lagrangian field theories
2.2 Regions with and without corners
3 Axiomatic system proposal
3.1 GBF Axioms
3.2 Further discussion of the axioms
3.3 Simplifications in the absence of corners
4 Kinematics of gauge fields
4.1 Classical abelian action
4.2 Symplectic reduction
4.3 Factorization on hypersurfaces
5 Dynamics modulo gauge
6 Example: 2-dimensional case
7 Outlook: quantization in higher dimensions
References
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