# Phase Space for Gravity with Boundaries

Alberto S. Cattaneo  Institut für Mathematik, Universität Zürich
Winterthurerstrasse 190, CH-8057 Zürich, Switzerland<EMAIL_ADDRESS>
###### Abstract.

This explanatory note, based on the geometrical method by Kijovski and
Tulczyjew, describes the construction of the reduced phase space of Lagrangian
field theories, i.e., the correct space of initial conditions with its
symplectic structure. Several examples and, in particular, the case of four-
dimensional gravity in the coframe formalism (Palatini–Cartan theory) are
analyzed.

###### Key words and phrases:

Gravity. Kijovski–Tulczyjew method. Lagrangian field theory. Reduced phase
space. Symplectic reduction.

I acknowledge partial support of the SNF Grant No. 200020_192080 and of the
Simons Collaboration on Global Categorical Symmetries. This research was
(partly) supported by the NCCR SwissMAP, funded by the Swiss National Science
Foundation. This article is based upon work from COST Action 21109 CaLISTA,
supported by COST (European Cooperation in Science and Technology)
(www.cost.eu). This article is commissioned by the Encyclopedia of
Mathematical Physics, edited by M. Bojowald and R. J. Szabo, to be published
by Elsevier [8].

Key points

  * •

Definition of Lagrangian field theories

  * •

Euler–Lagrange equations and Noether $1$-form

  * •

Boundary structure

  * •

Evolution relations

  * •

Cauchy data and the reduced phase space

  * •

Some examples: scalar field, electromagnetism, Palatini–Cartan theory.

###### Contents

  1. 1 Introduction
  2. 2 Some preliminary examples
    1. 2.1 Mechanics
    2. 2.2 A degenerate example
    3. 2.3 The scalar field
    4. 2.4 Electromagnetism
    5. 2.5 Conclusions
  3. 3 Lagrangian field theory
    1. 3.1 Euler–Lagrange spaces
    2. 3.2 Reduction
  4. 4 Preboundary fields
    1. 4.1 Composition of evolution relations
    2. 4.2 Cauchy data
  5. 5 Boundary fields
    1. 5.1 Boundary conditions
  6. 6 Examples
  7. 7 Gravity in the coframe formalism
  8. 8 Conclusion

##  1\. Introduction

One useful description of a mechanical system is in terms of hamiltonian
evolution. This is given by a symplectic manifold and a hamiltonian
function—the evolution being given by the flow of its hamiltonian vector
field. This description is particulary interesting because it is, in
principle, ready-made for quantization, with the symplectic manifold being
turned to a Hilbert space and the hamiltonian function to a hamiltonian
operator (how this is done, if anyway possible, is another story).

Very often a fundamental system is however introduced via an action functional
and its associated Euler–Lagrange (EL) equations. Recasting it as a
hamiltonian system—in particular in the case of gravity—is the goal of this
note.

The (reduced) phase space of the system refers to the symplectic manifold on
which one formulates the hamiltonian evolution of the system, whenever
possible, once one chooses a time axis. Roughly speaking, it is the space of
Cauchy data for the EL equations. The main issue is that the latter may have
also a nonevolutionary content, which then gives rise to constraints. This is
typically the case of gauge theories and of gravity.

The traditional way of proceding is due to Dirac [17]. This is an algebraic
method with the goal of extending the Legendre transform to the case of
degenerate Lagrangians. The method is iterative, with the construction of
primary and a sequence of secondary constraints, and can in principle break
down at every step.

In this note we will focus on a more geometric approach due to Kijovski and
Tulczyjew (KT) [22] (we also rely on the presentation in [14]).111A related
construction, which we will not discuss here, goes under the name of
“covariant phase space” (see [29, 4, 20, 23, 24], and also [25] for an
overview). In this approach, the space of solutions, equipped with its
symplectic structure, is usually constructed via the bivariational complex
(see, e.g., [9] and references therein). The central idea is to reformulate
the space of solutions to the EL equations as an isotropic relation between
the spaces of initial and final field configurations naturally endowed with a
closed $2$-form. The flow property is here replaced by the set-theoretic
composition of relations, which only in particular cases are graphs of a flow.
The initial (or final) space is then naturally endowed with a subset of Cauchy
data—the points that can belong to the relation for an arbitrary small
interval of time. As the restriction222Following the common terminology in
symplectic geometry, by “restriction” we mean the “pullback by the inclusion
map.” of the $2$-form to the space of Cauchy data is usually degenerate, one
has to consider an appropriate quotient (by the null directions of the
$2$-form) to recover the reduced phase space as a symplectic manifold. It may
of course happen that the quotient is singular. However, there is a natural
way of doing this reduction in stages. In most physical theories, the only
problematic quotient turns out to be the last one: the symplectic reduction of
a coisotropic (i.e., defined by first-class constraints) submanifold of a
symplectic manifold of (appropriately defined) boundary fields. To deal with
the last reduction, there are appropriate techniques (e.g., the
Batalin–Fradkin–Vilkovisky formalism (BFV) [19, 18], which will not be
discussed in this note).

One advantage of the KT method is that it is rather flexible. One does not
really have to consider evolution between an initial and a final time, but one
can focus on the initial time only. One can even drop the condition that the
initial submanifold should be space-like—the formalism working in more general
situations—and one does not have to consider cylinders (a product of a
manifold by an interval), but one can consider more general manifolds with
boundary. One further advantage is that this method can be readily applied to
the extension of the BV formalism [6, 7] to manifolds with boundary (and
possibly lower-dimensional strata), producing the BV-BFV formalism discussed
in [15].

We will start with a warm up dealing with simples cases. Next we will
formulate the general theory and will apply it to the already presented
examples. Finally, we will discuss gravity in the coframe (a.k.a.
Palatini–Cartan) formulation.

######  Acknowledgment.

I thank J. Huerta, M. Schiavina, S. Speziale, and M. Tecchiolli for useful
comments on a first draft.

##  2\. Some preliminary examples

We start reviewing some simple examples that will motivate the general theory.
We will review in Section 6 how they are described within the general KT
method.

###  2.1. Mechanics

Let us start with the familiar example of mechanics in one dimension. The
action functional is

$S[q]=\int_{a}^{b}\left(\frac{1}{2}m\dot{q}^{2}-V(q)\right)dt,$

where $q$ (the “field”) is a path $[a,b]\to{\mathbb{R}}$, $m$ is the mass, and
$V$ is the potential energy. We want to compute a variation of $S$ isolating
in the integral variations of $q$ but not of its derivative(s). We can do this
integrating by parts:

$\delta S=-\int_{a}^{b}\left(m\ddot{q}+V^{\prime}(q)\right)\delta
q\,dt+m\dot{q}\,\delta q\Big{|}_{a}^{b}.$

The bulk term—i.e., the integral over the interval—contains the EL equations
as the coefficient of $\delta q$:

$m\ddot{q}+V^{\prime}(q)=0.$

The boundary term can be reinterpreted as the difference, at times $b$ and
$a$, of a $1$-form, on the space of positions $q$ and velocities $v=\dot{q}$,
that we denote by $\alpha$ and call the Noether $1$-form:333More precisely, we
regard this $1$-form as defined on the tangent bundle
$T{\mathbb{R}}={\mathbb{R}}\times{\mathbb{R}}\ni(v,q)$ of the space of fields
at $a$ (or $b$), so $\delta$ becomes the usual de Rham differential. The
boundary term in the variation of the action is more explicitly written as
$m\dot{q}\,\delta q\Big{|}_{a}^{b}=\pi_{b}^{*}\alpha-\pi_{a}^{*}\alpha,$ where
$\pi_{a}$ and $\pi_{b}$ are the maps from the space of paths to
${\mathbb{R}}\times{\mathbb{R}}$ that send the path $q$ to $(\dot{q}(a),q(a))$
and to $(\dot{q}(b),q(b))$, respectively.

$\alpha=mv\,\delta q.$

We then take its differential

$\omega\coloneqq\delta\alpha=m\,\delta v\,\delta q,$

which is an example of a symplectic form. This simply means that $\omega$ is
closed (i.e., $\delta\omega=0$) and nondegenerate (i.e., hamiltonian vector
fields are uniquely determined). We call the space $T{\mathbb{R}}$ of
positions and velocities together with $\omega$ the phase space of the
theory.444Note that the theory produces two copies of the phase space, one for
the initial time $a$ and the other for the final time $b$. (Usually, one calls
phase space the space $T^{*}{\mathbb{R}}$ of positions $q$ and momenta $p=mv$
with $\omega=\delta p\,\delta q$, but this is just a change of variables.) As
the EL equations are of second order, the phase space $T{\mathbb{R}}$ is the
space of possible initial conditions, a.k.a. Cauchy data. Assuming the
existence of global solutions, this may also be regarded as the space of
solutions to the EL equations.

The dynamics is then given by the Hamiltonian $H=\frac{1}{2}mv^{2}+V(q)$ on
$T{\mathbb{R}}$ (or, as usual, $H=\frac{p^{2}}{2m}+V(q)$ on
$T^{*}{\mathbb{R}}$) in the sense that the time evolution is governed by the
first-order ODE $(\dot{v},\dot{q})=X$ where $X=(-\frac{V^{\prime}}{m},v)$ is
the unique vector field satisfying

$\iota_{X}\omega+{\mathrm{d}}H=0$

with $\iota$ denoting contraction. The vector field $X$ is called the
hamiltonian vector field of $H$. The graph $L_{[a,b]}$ of the flow of $X$ can
be shown to be a lagrangian submanifold of $T{\mathbb{R}}\times
T{\mathbb{R}}$, upon changing the sign of the symplectic form on one of the
two factors. This is an example of what we will in general call the evolution
relation of the system.

###  2.2. A degenerate example

Consider now, as action functional, the euclidean length of a curve:

$S[q]=\int_{a}^{b}\|\dot{q}\|\;dt=\int_{a}^{b}\sqrt{\dot{q}\cdot\dot{q}}\;dt,$

where $q\colon[a,b]\to{\mathbb{R}}^{n}$ is a path. The first, minor, remark
concerning this example is that, to avoid singularities, we have to restrict
the space of paths to the regular ones, i.e., those with nonvanishing speed
everywhere (this is an open condition, not a constraint). If we compute the
variation of the action, we get

$\delta
S=-\int_{a}^{b}\frac{{\mathrm{d}}}{{\mathrm{d}}t}\frac{\dot{q}}{\|\dot{q}\|}\,\delta
q\;dt+\frac{\dot{q}\cdot\delta q}{\|\dot{q}\|}\Bigg{|}_{a}^{b}.$

The EL equations are then

$\frac{{\mathrm{d}}}{{\mathrm{d}}t}\frac{\dot{q}}{\|\dot{q}\|}=0,$

which are solved by straight regular paths, with an unspecified
parametrization. The Noether $1$-form is

$\alpha=u\cdot\delta q,$

where we have introduced the notation $u$ for the normalized velocity
$\frac{\dot{q}}{\|\dot{q}\|}$. The corresponding $2$-form

$\omega=\delta u\cdot\delta q$

is degenerate. To look for its kernel, consider a vector field

$X=X_{q}\cdot\frac{\partial}{\partial q}+X_{u}\cdot\frac{\partial}{\partial
u}$

and compute the contraction

$\iota_{X}\omega=X_{u}\cdot\delta q-X_{q}\cdot\delta u.$

The first term may vanish only for $X_{u}=0$ because the components $\delta
q^{i}$ of $\delta q$ are independent. On the other hand, since $u$ is
normalized, we have $u\cdot u=1$ which, differentiated, yields the relation
$u\cdot\delta u=0$. Since $\delta u$ is orthogonal to $u$, the last term in
$\iota_{X}\omega$ vanishes exactly when $X_{q}$ is parallel to $u$.

To formulate evolution in hamiltonian terms, we have to mod out the kernel of
the $2$-form $\omega$. This reduction then consists in modding $q$ out along
the $u$ direction. The reduced space is the union over $u\in S^{n-1}$, the
$(n-1)$-dimensional sphere, of the spaces

$F_{u}=\\{q\in{\mathbb{R}}^{n}\\}/[q\sim q+\lambda u,\
\lambda\in{\mathbb{R}}].$

As a representative for each equivalence class, one can choose a vector
orthogonal to $u$. Therefore, we may identify $F_{u}$ with

$T_{u}\coloneqq\\{q\in{\mathbb{R}}^{n}:q\cdot u=0\\}.$

But this is just the tangent space of the sphere (viewed as submanifold of
${\mathbb{R}}^{n}$) at the point $u$. We then have that the reduced phase
space is $\cup_{u\in S^{n-1}}T_{u}=TS^{n-1}$, the tangent bundle of the
sphere. The symplectic structure is $[\delta u]\cdot\delta q$. Using the round
metric, one can actually identify $TS^{n-1}$ with $T^{*}S^{n-1}$. Under this
transformation, the symplectic form is mapped to the canonical one, $\delta
p\cdot\delta q$.

Finally note that the symplectic manifold $TS^{n-1}$ constructed above can
also be regarded as the space of solutions to the EL equations, i.e., the
space of oriented straight lines in ${\mathbb{R}}^{n}$: an oriented line is in
fact specified by its direction (the unit tangent vector $u\in S^{n-1}$) and
by its position in space (a point $q\in{\mathbb{R}}^{n}$ modulo translations
along $u$). Also note that these data do not change along the line, so the
evolution on the reduced phase space $TS^{n-1}$ is given by the identity map,
and the Hamiltonian vanishes.

###  2.3. The scalar field

Let us now consider one example from field theory in $d$ dimensions. To a
$d$-dimensional Lorentzian manifold $(M,g)$ we associate the space of fields
$C^{\infty}(M)$ and the action functional

$S_{M}[\phi]=\frac{1}{2}\int_{M}g^{\mu\nu}\,\partial_{\mu}\phi\,\partial_{\nu}\phi\,\sqrt{|\det
g|}\;{\mathrm{d}}^{d}x,$

where $(g^{\mu\nu})$ are the components of the inverse of $g$. The EL equation
is the wave equation on $(M,g)$:

$\partial_{\mu}(g^{\mu\nu}\partial_{\nu}\phi\sqrt{|\det g|})=0.$

We consider $M=\Sigma\times I$, where $\Sigma$ is space-like and $I=[a,b]$ is
a time interval. We also assume for simplicity that near the boundary the
metric splits as

$g=-({\mathrm{d}}x^{0})^{2}+h_{ij}{\mathrm{d}}x^{i}{\mathrm{d}}x^{j},$

where $0$ denotes the time direction and $(h_{ij})$ is a euclidean metric.
When we compute the variation of $S_{M}$, we get a boundary term—the Noether
$1$-form—on each of the two boundary copies of $\Sigma$:

$\alpha^{\partial}_{\Sigma}=\int_{\Sigma}\phi_{0}\,\delta\phi\,\sqrt{\det
h}\;{\mathrm{d}}^{d-1}x,$

where $\phi_{0}$ denotes $\partial_{0}\phi$.

In the symplectic setting, we consider $\phi$ and $\phi_{0}$ as two
independent functions on $\Sigma$: the initial configuration and the initial
velocity. The space $F^{\partial}_{\Sigma}$, with coordinates $\phi$ and
$\phi_{0}$, can be identified with the space of solutions to the wave equation
on $\Sigma\times[a,b]$ with $\phi$ and $\phi_{0}$ as initial conditions at
time $a$. The space $F^{\partial}_{\Sigma}$ with symplectic form

(1)
$\omega^{\partial}_{\Sigma}\coloneqq\delta\alpha^{\partial}_{\Sigma}=\int_{\Sigma}\delta\phi_{0}\,\delta\phi\,\sqrt{\det
h}\;{\mathrm{d}}^{d-1}x$

is then the phase space of the theory.

###  2.4. Electromagnetism

Electromagnetism on a $d$-dimensional lorentzian manifold $(M,g)$ is described
by a $1$-form555More generally, a $U(1)$ connection.
$A=A_{\mu}{\mathrm{d}}x^{\mu}$ on $M$ (called the potential) and the action

$S_{M}[A]=\frac{1}{4}\int_{M}g^{\mu\nu}g^{\rho\sigma}\,F_{\mu\rho}\,F_{\nu\sigma}\,\sqrt{|\det
g|}\;{\mathrm{d}}^{d}x,$

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the
electromagnetic field. The EL equations are in this case the Maxwell equations

$\partial_{\mu}(g^{\mu\nu}g^{\rho\sigma}\,F_{\nu\sigma}\,\sqrt{|\det
g|})=0\quad\forall\rho.$

Again we consider $M=\Sigma\times I$, where $\Sigma$ is space-like and
$I=[a,b]$ is a time interval with split metric near the boundary. In this
case, the boundary contribution—the Noether $1$-form—on each copy of $\Sigma$
is

$\alpha^{\partial}_{\Sigma}=\int_{\Sigma}h^{ij}\,F_{0i}\,\delta
A_{j}\,\sqrt{\det h}\;{\mathrm{d}}^{d-1}x,$

where $h_{ij}$ is the Euclidean metric on $\Sigma$ obtained by restricting
$g$, and $0$ denotes the time component. Again, in the symplectic formalism,
we consider $F_{0i}$ and $A_{j}$ as independent fields. We denote by
$F^{\partial}_{\Sigma}$ the space with coordinates $F_{0i}$ and $A_{j}$ and
with symplectic structure

$\omega^{\partial}_{\Sigma}=\delta\alpha^{\partial}_{\Sigma}=\int_{\Sigma}h^{ij}\,\delta
F_{0i}\,\delta A_{j}\,\sqrt{\det h}\;{\mathrm{d}}^{d-1}x.$

######  Remark 2.1.

If we introduce the electric field $E^{j}\coloneqq h^{ij}F_{0i}$, then we get
the usual formula

$\omega^{\partial}_{\Sigma}=\int_{\Sigma}\delta E^{j}\,\delta A_{j}\sqrt{\det
h}\;{\mathrm{d}}^{d-1}x,$

which simply says that the electric field and the vector potential are
canonically conjugate variables.

The crucial point now is that, unlike in the case of the scalar field, the
space $F^{\partial}_{\Sigma}$ is not the space of solutions. In fact, near the
boundary, the EL equations read

$\partial_{0}(g^{\rho j}F_{0j}\sqrt{\det
h})+\partial_{i}(h^{ij}g^{\rho\sigma}F_{j\sigma}\sqrt{\det
h})=0\quad\forall\rho.$

For $\rho=k$ a space index, we get the evolution equation

$h^{kj}\dot{F}_{0j}\sqrt{\det h}=\partial_{i}(h^{ij}h^{kl}F_{jl}\sqrt{\det
h}),$

where we denoted the time derivative by a dot (assuming for simplicity that
$h$ is time-independent). The equation for $\rho=0$,

$\partial_{i}(h^{ij}F_{0j}\sqrt{\det h})=0,$

is however not an evolution equation and instead defines a constraint on the
variables on $F^{\partial}_{\Sigma}$. Note that, with the notations in Remark
2.1, this equation corresponds to the Gauss law ${\mathrm{div}}E=0$. We then
define $C$ as the set of solutions to the Gauss law or, equivalently, as the
common zero set for all scalars $\lambda$ of the functions

$J_{\lambda}\coloneqq-\int_{\Sigma}\lambda\,\partial_{i}(h^{ij}F_{0j}\sqrt{\det
h})\;{\mathrm{d}}^{d-1}x=\int_{\Sigma}\partial_{i}\lambda\,h^{ij}F_{0j}\sqrt{\det
h}\;{\mathrm{d}}^{d-1}x$

The restriction of $\omega^{\partial}_{\Sigma}$ to $C$ is degenerate. Its
kernel is generated by the hamiltonian vector fields $X_{\lambda}$ for all
$J_{\lambda}$s. We immediately get

$X_{\lambda}(A_{i})=\partial_{i}\lambda,\qquad X_{\lambda}(F_{0j})=0,$

which can be interpreted as infinitesimal gauge transformations. The reduced
phase space $\underline{C}$ is then defined as the quotient of the space $C$
of solutions to the Gauss law by gauge transformations.

###  2.5. Conclusions

All the above examples have something in common: the evolution can eventually
be described in hamiltonian terms, and the symplectic form is ultimately
derived from the Noether $1$-form that arises as boundary term of a variation
of the action.

On the other hand, there are also several differences. In the case of
mechanics or of the scalar field this is the end of the story. In the
degenerate case of the length functional, the $2$-form arising from the the
differential of the Noether $1$-form is degenerate, so one has to reduce by
its kernel in order to get a symplectic form. In the case of electromagnetism
(and more generally of Yang–Mills theory), the construction we have presented
actually yields a symplectic form on an appropriate space of boundary fields.
However, some of the EL equations (the Gauss law) put constraints on them.
Imposing the constraints leads however to a degeneracy of the restriction of
the symplectic form, so one has now to reduce by its kernel.

In order to put more uniformity to these examples, but also to get ready to
deal with more intricated examples, like gravity, we need a more conceptual
perspective, which will be developed in the next sections.

##  3\. Lagrangian field theory

In this and in the next two sections, we discuss the general method. We
encourage the reader to apply each new construction to the examples of Section
2 (the results will be briefly reviewed in Section 6).

A Lagrangian field theory on a manifold $M$, possibly with boundary, is
specified by a space of fields $F_{M}$ and a local action functional $S_{M}$.
The fields are local on $M$ (sections of some sheaf); typically:

  * •

maps from $M$ to some fixed target manifold,

  * •

sections of some vector (or, more generally, fiber) bundle over $M$,

  * •

connections on $M$ for some principal bundle.

The action functional is of the form $S_{M}=\int_{M}L$, where the lagrangian
density $L$ is a function of the fields and some of their derivatives (jets)
at a point. In this note, we assume $M$ to be compact.666If $M$ is not
compact, the fields are assumed to vanish fast enough to ensure convergence of
the integral. Alternatively, one considers the Lagrangian as a distribution to
be applied to test functions with compact support. To start with, we fix no
boundary conditions for the fields.

######  Remark 3.1.

The lagrangian density $L$ may require some additional structure on $M$. For
example, to define a theory for spinors we need $M$ to allow for spin bundles.
To define gravity we need $M$ to allow for lorentzian metrics. For a given
theory, we call a manifold that admits the required structures a space–time
manifold.

The EL equations are encoded in the variation $\delta S_{M}$ of the action. We
think of $\delta$ as the de Rham differential on $F_{M}$. For this, we have to
assume that $F_{M}$ has a suitable structure of infinite-dimensional manifold
that allows defining differential forms and the de Rham differential.

######  Remark 3.2.

In this note, we prefer to consider smooth fields and give $F_{M}$ the
structure of a Fréchet manifold. One may also prescribe a different regularity
on the fields and give $F_{M}$ the structure of a Banach manifold.

######  Digression 3.3 (A categorical approach).

A different option, which is actually closer to the physical viewpoint but
which we will not explore in this note, consists in regarding $F_{M}$ as an
internal hom space in the category of smooth manifolds. Let us explain this in
the case of a sigma model where $F_{M}$ is the space of maps from $M$ to a
fixed target manifold $N$ (possibly ${\mathbb{R}}$, as in scalar field
theory). We then regard $F_{M}$ as the functor that to a finite-dimensional
manifold $Z$ associates the set of smooth maps from $Z\times M$ to $N$. We
think of this as a way of exploring $F_{M}$ by manifolds $Z$ (it is the notion
of Grothendiek’s functor of points, but it is also the usual textbook way of
parametrizing the fields when computing functional derivatives). The natural
context is that of “presheaves” over the category $\mathbf{Mfld}$ of finite-
dimensional smooth manifolds, namely, of functors from
$\mathbf{Mfld}^{\text{op}}$ to the category $\mathbf{Set}$ of sets. A finite-
dimensional manifold $X$ can also be regarded as a presheaf $\underline{X}$,
namely, the functor that to a finite-dimensional manifold $Z$ associates the
set of smooth maps from $Z$ to $X$. We regard presheaves, e.g., $F_{M}$ or
$\underline{X}$, as generalized manifolds. A natural transformation among the
corresponding functors is regarded as a map between the generalized manifolds.
Particularly useful is the generalized manifold $\Omega^{k}$, namely, the
functor that to a finite-dimensional manifold $Z$ associates the set
$\Omega^{k}(Z)$ of smooth $k$-forms on $Z$. A $k$-form on $X$ is now
equivalently described as a map from $\underline{X}$ to $\Omega^{k}$. One then
defines $k$-forms on a generalized manifold, e.g., $F_{M}$, as maps to
$\Omega^{k}$. For every $k$, we have the universal de Rham differential
$\delta$, the natural transformation from $\Omega^{k}$ to $\Omega^{k+1}$,
which, when instantiated on a given $Z$, is the usual de Rham differential
${\mathrm{d}}\colon\Omega^{k}(Z)\to\Omega^{k+1}(Z)$. The action functional
$S_{M}$ is now a map from $F_{M}$ to $\Omega^{0}$, namely, the natural
tranformation, which, when instantiated on a given $Z$, is the smooth function
on $Z$ given by $\int_{M}L$ (recall that the fields are now parametrized by
$Z$). The variational derivative $\delta S_{M}$ is now just the composition of
the universal de Rham differential $\delta$ with the map $S_{M}$, a
generalized map from $F_{M}$ to $\Omega^{1}$, so a $1$-form on $F_{M}$.

###  3.1. Euler–Lagrange spaces

If $M$ has no boundary, solutions to the EL equations are, by definition, the
critical points of $S_{M}$, viz., the fields at which $\delta S_{M}=0$. If we
introduce the Euler–Lagrange $1$-form $el_{M}\coloneqq\delta S_{M}$, then the
solutions to the EL equations are the zeros of $el_{M}$. We denote by $EL_{M}$
the space of solutions.

Note that there is an ambiguity in writing $el_{M}$, as there was already in
writing $S_{M}$, because of total derivatives, whose integral is zero. If $M$
has nonempty boundary, the different ways are no longer equivalent but differ
by boundary terms.

We take the definition of $S_{M}$ as fixed (we will return to this later on),
but in $el_{M}$ we only want variations of the fields, and not of their
derivatives, to appear. This requires integrating by parts (several times if
necessary). Finally, we may write (in a unique way)

(2) $\delta S_{M}=el_{M}+\alpha_{M},$

where $\alpha_{M}$, also a $1$-form on $F_{M}$, collects the boundary terms.
It follows that

(3) $\alpha_{M}=\int_{\partial M}a_{M},$

where $a_{M}$ is a density on the boundary $\partial M$ depending on the
fields and their jets (including transversal ones) at a point on $\partial M$.

We keep denoting by $EL_{M}$ the zero locus of $el_{M}$. In this note, for
simplicity, we will assume that $EL_{M}$ is a submanifold of $F_{M}$. Note
that (2) has the following consequences:

  1. (i)

The restriction of $\alpha_{M}$ to $EL_{M}$ is an exact $1$-form.

  2. (ii)

If we change the definition of $S_{M}$ to $\widetilde{S}_{M}$ via the addition
of total derivatives to the Lagrangian $L_{M}$, we have
$\widetilde{S}_{M}=S_{M}+\phi_{M}$, where $\phi_{M}=\int_{\partial M}f_{M}$
collects the boundary terms. Since our convention for $el_{M}$ is fixed
(namely, we write $\delta\widetilde{S}_{M}=el_{M}+\widetilde{\alpha}_{M}$), we
get

$\widetilde{\alpha}_{M}=\alpha_{M}+\delta\phi_{M}.$

Changing the action as in (ii) should be regarded as an equivalence (anyway,
the space $EL_{M}$ is does not change).777In the quantum theory, what matters
is the Gibbs weight $\mathrm{e}^{\frac{{\mathrm{i}}}{\hbar}S_{M}}$, which, in
view of this remark, should be considered not as a function but as a section
of a line bundle over $F_{M}$: the change in the definition of $S_{M}$ is the
action of the gauge transformation
$\mathrm{e}^{\frac{{\mathrm{i}}}{\hbar}\phi_{M}}$. From this point of view,
$\alpha_{M}$ should be considered as a $1$-form connection. We then define the
$2$-form

$\omega_{M}\coloneqq\delta\alpha_{M}$

which has invariant meaning ($\delta\alpha_{M}=\delta\widetilde{\alpha}_{M}$).
Point (i) above then becomes:888Explicitly, this means that $\omega_{M}$
evaluated on a solution and contracted with a solution of the linearized
equations vanishes. See footnote 2.

> The restriction of $\omega_{M}$ to $EL_{M}$ is zero.

Borrowing the terminology of symplectic geometry, we say that $EL_{M}$ is an
isotropic submanifold of $(F_{M},\omega_{M})$.

###  3.2. Reduction

The $2$-form $\omega_{M}$ is closed by construction but in general is
degenerate, so it is not a symplectic form. We can remedy for this quotienting
$F_{M}$ by the kernel of $\omega_{M}$, i.e., by the distribution of vector
fields $X$ on $F_{M}$ satisfying $\iota_{X}\omega=0$.999Here $\iota$ denotes
contraction. Namely, $\iota_{X}\omega$ is the $1$-form that satisfies
$\iota_{X}\omega(Y)=\omega(X,Y)$ for every vector field $Y$.

We will assume that the quotient $F^{\partial}_{\partial M}$ has a smooth
manifold structure such that the canonical projection $\pi\colon F_{M}\to
F^{\partial}_{\partial M}$ is a surjective submersion.101010We first have to
assume that this distribution is regular, i.e., that the kernel distribution
consists of sections of a vector bundle. Since $\omega$ is closed, this
distribution is automatically involutive (viz., $\iota_{X}\omega=0$ and
$\iota_{Y}\omega=0$ imply $\iota_{[X,Y]}\omega=0$), but, in the Fréchet
context, the Frobenius theorem does not hold, so we have to assume explicitly
that the distribution is integrable. Finally, we have to assume that the leaf
space $F^{\partial}_{\partial M}$ has smooth structure as in the text. All
this turns out to work in the case of usual field theories.

The reason for the notation using the boundary symbol $\partial$ is that,
because of (3), a vector field $X$ that changes the bulk fields preserving
their boundary values (including their transversal jets) is necessarily in the
kernel of $\omega_{M}$. As a result, $F^{\partial}_{\partial M}$ only depends
on “boundary data.” We call an element of $F^{\partial}_{\partial M}$ a
boundary field.

By construction, there is a unique symplectic form
$\omega^{\partial}_{\partial M}$ on $F^{\partial}_{\partial M}$ such that
$\pi^{*}\omega^{\partial}_{\partial M}=\omega_{M}$.

A central concept is the space of boundary fields that arise from solutions to
EL equations:

(4) $L_{M}\coloneqq\pi(EL_{M}).$

Note that $L_{M}$ is automatically isotropic with respect to
$\omega^{\partial}_{\partial M}$. We will later require further structure on
$L_{M}$ (viz., that it is a split lagrangian submanifold; see Definition 5.2).

To make further structures clearer, but also to present a version that still
works when $F^{\partial}_{\partial M}$ is singular, it is convenient to
perform the reduction in stages, with the first reduction, which we will
denote by $\widetilde{F}^{\partial}_{\partial M}$, always possible. If also
$F^{\partial}_{\partial M}$ is smooth, we will then have the following
commutative diagram of surjective submersions:

(5) ${F_{M}}$${\widetilde{F}^{\partial}_{\partial M}}$${F^{\partial}_{\partial
M}}$$\scriptstyle{\widetilde{\pi}}$$\scriptstyle{\pi}$$\scriptstyle{p}$

We will call points of $\widetilde{F}^{\partial}_{\partial M}$ preboundary
fields. We study them in the next section.

##  4\. Preboundary fields

We already observed that a vector field $X$ on $F_{M}$ that changes the bulk
fields preserving their boundary values (including their transversal jets,
which encode the inward-pointing derivative data of the fields at the
boundary) is necessarily in the kernel of $\omega_{M}$ (actually, also in that
of $\alpha_{M}$).111111Alternatively, we may consider vector fields that
preserve germs of boundary values. Namely, we require that, for every field
$\phi$, $X$ evaluated at $\phi$ is compactly supported in the interior of $M$.
This construction produces a different version of the intermediate space
$\widetilde{F}^{\partial}_{\partial M}$, but of course the final reduction
$F^{\partial}_{\partial M}$ is the same. We now consider the distribution
consisting of only such vector fields and denote by
$\widetilde{F}^{\partial}_{\partial M}$ its leaf space and by

$\widetilde{\pi}\colon F_{M}\to\widetilde{F}^{\partial}_{\partial M}$

the canonical projection. This is a space of fields on $\partial M$ (with the
transversal jets now being new fields). As already mentioned, we call an
element of $\widetilde{F}^{\partial}_{\partial M}$ a preboundary field.

Note that $\widetilde{\pi}$ is a surjective submersion (corresponding to
restricting fields and their transversal jets to the boundary) and that there
is a unique closed local $2$-form $\widetilde{\omega}^{\partial}_{\partial M}$
on $\widetilde{F}^{\partial}_{\partial M}$ such that
$\widetilde{\pi}^{*}\widetilde{\omega}^{\partial}_{\partial M}=\omega_{M}$.

The $2$-form $\widetilde{\omega}^{\partial}_{\partial M}$ is in general still
degenerate, so a further reduction will be needed, leading again to
$F^{\partial}_{\partial M}$ with canonical projection $p\colon F_{M}\to
F^{\partial}_{\partial M}$.

There is also a unique local $1$-form $\widetilde{\alpha}^{\partial}_{\partial
M}$ on $\widetilde{F}^{\partial}_{\partial M}$ such that
$\widetilde{\pi}^{*}\widetilde{\alpha}^{\partial}_{\partial M}=\alpha_{M}$.
Moreover, we have $\widetilde{\omega}^{\partial}_{\partial
M}=\delta\widetilde{\alpha}^{\partial}_{\partial M}$ and

(6) $\delta
S_{M}=el_{M}+\widetilde{\pi}^{*}\widetilde{\alpha}^{\partial}_{\partial M}.$

We now define the analogue of (4), namely, the space of preboundary fields
that can be extended to a, not necessarily unique, solution to the EL
equations:

$\widetilde{L}_{M}\coloneqq\widetilde{\pi}(EL_{M}).$

Note that $\widetilde{L}_{M}$ is automatically isotropic with respect to
$\widetilde{\omega}^{\partial}_{\partial M}$ and that
$L_{M}=p(\widetilde{L}_{M})$. It is convenient to assume that
$\widetilde{L}_{M}$ is a submanifold of $\widetilde{F}^{\partial}_{\partial
M}$.

###  4.1. Composition of evolution relations

Note that $\widetilde{F}^{\partial}_{\Sigma}$ is actually defined for every
manifold $\Sigma$ that can arise as a boundary component of a space–time
manifold $M$. Moreover, if we denote by $\\{\partial_{s}M\\}_{s=1,\dots,k}$
the connected components of $\partial M$, we have121212Here $\prod$ denotes
the cartesian product of the given spaces. In the second formula, the pulback
of $\widetilde{\alpha}^{\partial}_{\partial_{s}M}$ by the projection to the
corresponding factor is understood.

$\widetilde{F}^{\partial}_{\partial
M}=\prod_{s=1}^{k}\widetilde{F}^{\partial}_{\partial_{s}M}$

and

$\widetilde{\alpha}^{\partial}_{\partial
M}=\sum_{s=1}^{k}\widetilde{\alpha}^{\partial}_{\partial_{s}M}.$

Suppose now we cut the manifold $M$ along a hypersurface $\Sigma$ into two
pieces $M_{1}$ and $M_{2}$, so we can recover $M$ as the gluing of $M_{1}$ and
$M_{2}$ along their common boundary component $\Sigma$ (we assume here that
$\Sigma$ does not cut the boundary of $M$). We have

$\partial M=\partial_{1}M\sqcup\partial_{2}M,\quad\partial
M_{1}=\partial_{1}M\sqcup\Sigma,\quad\partial
M_{2}=\partial_{2}M\sqcup\Sigma,$

where $\partial_{i}M$ denotes the component of the boundary of $M$ that is
also a component of the boundary of $M_{i}$.

A solution to the EL equations on $M$ is then uniquely given by a pair of
solutions to the EL equations on $M_{1}$ and $M_{2}$ that match on $\Sigma$.
More precisely, denote by $\widetilde{\pi}_{i}\colon
F_{M_{i}}\to\widetilde{F}^{\partial}_{\Sigma}$, $i=1,2$, the map that
corresponds to restricting fields on $M_{i}$ and their transversal jets to the
boundary component $\Sigma$. We then have the fiber product formula

(7)
$EL_{M}=EL_{M_{1}}\times_{\widetilde{F}^{\partial}_{\Sigma}}EL_{M_{2}}=\\{(\phi_{1},\phi_{2})\in
EL_{M_{1}}\times
EL_{M_{2}}:\widetilde{\pi}_{1}(\phi_{1})=\widetilde{\pi}_{2}(\phi_{2})\\}.$

Analogously, we have

(8)
$\widetilde{L}_{M}=\\{(\psi_{1},\psi_{2})\in\widetilde{L}_{M_{1}}\times\widetilde{L}_{M_{2}}:\pi_{\Sigma,1}(\psi_{1})=\pi_{\Sigma,2}(\psi_{2})\\},$

where $\pi_{\Sigma,i}$ is the canonical projection from
$\widetilde{F}^{\partial}_{\partial M_{i}}$ to
$\widetilde{F}^{\partial}_{\Sigma}$.

It is more instructive to view the above formulae using the language of
relations and correspndences.

######  Definition 4.1.

A relation from the set $A$ to the set $B$ is a subset of $A\times B$. If
$R_{AB}$ is a relation from $A$ to $B$ and $R_{BC}$ is a relation from $B$ to
$C$, one defines the composition

$R_{BC}\circ R_{AB}\coloneqq\\{(a,c)\in A\times C:\exists b\in B\ (a,b)\in
R_{AB}\ (b,c)\in R_{BC}\\}$

as a relation from $A$ to $C$.131313A particular example of a relation is the
graph of a map. The composition of graphs, as relations, is the same as the
graph of the usual composition of maps. A correspondence from the set $A$ to
the set $B$ is a map $C\to A\times B$ where $C$ is some set. The composition
of correspondences is defined as their fibered product. The image of a
correspondence is a relation, and the composition of such images, as
relations, is the image of the composition of the correspondences.

In this setting we may view $\widetilde{L}_{M_{1}}$ ($EL_{M_{1}}$) as a
relation (correspondence) from $\widetilde{F}^{\partial}_{\partial_{1}M}$ to
$\widetilde{F}^{\partial}_{\Sigma}$ and $\widetilde{L}_{M_{2}}$ ($EL_{M_{2}}$)
as a relation (correspondence) from $\widetilde{F}^{\partial}_{\Sigma}$ to
$\widetilde{F}^{\partial}_{\partial_{2}M}$. In this language, (8) is the
composition of relations, whereas (7) is the composition of correspondences.

For this reason, given a space–time manifold $M$ and a decomposition of its
boundary into two components, we call $\widetilde{L}_{M}$ ($EL_{M}$) the
evolution relation (evolution correspondence).

######  Remark 4.2 (Evolutionary flow).

A particular case is $M=\Sigma\times[t_{1},t_{3}]$, with $\Sigma$ having empty
boundary and $t_{1}<t_{3}$. We now pick $t_{2}\in(t_{1},t_{3})$ and cut $M$
along $\Sigma\times\\{t_{2}\\}$. In this case,
$M_{i}=\Sigma\times[t_{i},t_{i+1}]$, and we have

$\widetilde{L}_{\Sigma\times[t_{1},t_{3}]}=\widetilde{L}_{\Sigma\times[t_{2},t_{3}]}\circ\widetilde{L}_{\Sigma\times[t_{1},t_{2}]}.$

This law may be thought of as the general version of time evolution (we will
see that in many cases there is a further reduction in which the evolution
relation actually becomes the graph of a flow).

######  Remark 4.3 (Relative structures).

Suppose again we cut the manifold $M$ along a hypersurface $\Sigma$ into two
pieces $M_{1}$ and $M_{2}$. Since $S_{M}=S_{M_{1}}+S_{M_{2}}$, equation (6)
for $M$, $M_{1}$, and $M_{2}$ implies

$\widetilde{\alpha}^{\partial}_{\Sigma,1}=-\widetilde{\alpha}^{\partial}_{\Sigma,2},$

where $\widetilde{\alpha}^{\partial}_{\Sigma,i}$ denotes the $1$-form on
$\widetilde{F}^{\partial}_{\Sigma}$ viewed as a boundary component of $M_{i}$.
Therefore, with analogue notation, we have

$\widetilde{\omega}^{\partial}_{\Sigma,1}=-\widetilde{\omega}^{\partial}_{\Sigma,2}.$

This shows that choosing a boundary component $\Sigma$ as the target or the
source of the evolution relation yields opposite sign to the associated
$2$-form.141414If our theory is defined on oriented space–time manifolds and
the Lagrangian is viewed as a top form, then we view $a_{M}$ in (3) also as a
top form (now on the boundary). In this setting, $\alpha_{M}$ depends on a
choice of orientation, and we choose the orientation of $\Sigma$ in opposite
ways depending on its being viewed as the source or the target of the
evolution relation.

###  4.2. Cauchy data

Let $\Sigma$ be a manifold that can appear as a boundary of a space–time
manifold. We then consider the theory on
$M_{\epsilon}=\Sigma\times[0,\epsilon]$ with $\epsilon$ a positive real
number. We view $\widetilde{L}_{M_{\epsilon}}$ as a relation from
$\widetilde{F}^{\partial}_{\Sigma}=\widetilde{F}^{\partial}_{\Sigma\times\\{0\\}}$
to $\widetilde{F}^{\partial}_{\Sigma\times\\{\epsilon\\}}$. We define

$\widetilde{C}_{\Sigma}\coloneqq\\{c\in\widetilde{F}^{\partial}_{\Sigma}:\exists\epsilon>0\
\exists u\in F^{\partial}_{\Sigma\times\\{\epsilon\\}}\
(c,u)\in\widetilde{L}_{M_{\epsilon}}\\}.$

The space $\widetilde{C}_{\Sigma}$ consists of all the preboundary fields that
can be extended to solutions on some cylinder. Note that in general
$\widetilde{C}_{\Sigma}$ is not the whole of
$\widetilde{F}^{\partial}_{\Sigma}$ (in the examples we will see that
$\widetilde{C}_{\Sigma}$ is determined by the constraints of the theory, viz.,
by those EL equations that do not describe an evolution in the transverse
direction). We call $\widetilde{C}_{\Sigma}$ the space of Cauchy data.

More generally, we denote by $\widetilde{C}_{\partial M}$ the product of the
spaces of Cauchy data associated to each boundary component of $M$, viewed as
a subset of $\widetilde{F}^{\partial}_{\partial M}$.

We may restrict the closed $2$-form $\widetilde{\omega}^{\partial}_{\Sigma}$
to $\widetilde{C}_{\Sigma}$ and denote this restriction by
$\widetilde{\omega}^{C}_{\Sigma}$. In general, it will be degenerate. In some
good cases, $\widetilde{C}_{\Sigma}$ is a submanifold and the leaf space
$\underline{C_{\Sigma}}$ of the distribution given by the vector fields in the
kernel of $\widetilde{\omega}^{C}_{\Sigma}$ has a smooth structure such the
the canonical projection
$\widetilde{p}_{C}\colon\widetilde{C}_{\Sigma}\to\underline{C_{\Sigma}}$ is a
surjective submersion. In this case, there is a unique symplectic structure
$\underline{\omega}^{C}_{\Sigma}$ on $\underline{C_{\Sigma}}$ satisfying
$\widetilde{\omega}^{C}_{\Sigma}=\widetilde{p}_{C}^{*}\underline{\omega}^{C}_{\Sigma}$.
We call the symplectic manifold
$(\underline{C_{\Sigma}},\underline{\omega}^{C}_{\Sigma})$ the reduced phase
space associated to $\Sigma$. If this constuction works for every boundary
component of $M$, we write $\underline{C_{\partial M}}$ for the product of the
symplectic manifolds associated to each component. The evolution relation
$\widetilde{L}_{M}$ may be intersected with $\widetilde{C}_{\partial M}$ and
projected to $\underline{C_{\partial M}}$ giving rise to the reduced evolution
relation $\underline{L}_{M}$. This describes the correct evolution of the
system. (In some particularly good cases, $\underline{L}_{\Sigma\times[a,b]}$
will be the graph of a hamiltonian flow.)

##  5\. Boundary fields

We now return to the full reduction $F_{M}\to F^{\partial}_{\partial M}$
mentioned in Section 3.2; see the diagram (17) for reference.

We assume here that the quotient $F^{\partial}_{\partial M}$ has a smooth
manifold structure such that the canonical projection $\pi\colon F_{M}\to
F^{\partial}_{\partial M}$ is a surjective submersion and recall that an
element of $F^{\partial}_{\partial M}$ is called a boundary field.

The excursus through the space of preboundary fields in Section 4 and the fact
that we can obtain $F^{\partial}_{\partial M}$ from
$\widetilde{F}^{\partial}_{\partial M}$ by a further reduction allow us to see
more structure.

The first consequence is that, for every manifold $\Sigma$ that can arise as a
boundary component of a space–time manifold $M$, we may define
$F^{\partial}_{\Sigma}\coloneqq p(\widetilde{F}^{\partial}_{\Sigma})$. This
implies that, if we denote by $\\{\partial_{s}M\\}_{s=1,\dots,k}$ the
connected components of $\partial M$, we have

$F^{\partial}_{\partial M}=\prod_{s=1}^{k}F^{\partial}_{\partial_{s}M}$

and

$\omega^{\partial}_{\partial
M}=\sum_{s=1}^{k}\omega^{\partial}_{\partial_{s}M}.$

The second consequence is that $L_{M}=\pi(EL_{M})=p(\widetilde{L}_{M})$ is
also a relation, which we keep calling the evolution relation, once we choose
a decomposition of $\partial M$ into two components. Moreover, the evolution
relation of a manifold cut along a hypersurface is the composition of the
evolution relations of its parts. Remark 4.3 now implies that

$\omega^{\partial}_{\Sigma,1}=-\omega^{\partial}_{\Sigma,2},$

so $F^{\partial}_{\Sigma}$ has opposite symplectic structure if it is regarded
as target instead of source space.

######  Digression 5.1 (Composition of isotropic relations).

Following [30, 31], we can give a better description of the composition of
evolution relations, which we now assume to be submanifolds, in terms of
symplectic geometry. Observe that $L_{M}$ can be obtained as follows: first,
we consider $L_{M_{1}}\times L_{M_{2}}$ as a submanifold of
$F^{\partial}_{\partial M_{1}}\times F^{\partial}_{\partial
M_{2}}=F^{\partial}_{\partial_{1}M}\times F^{\partial}_{\Sigma}\times
F^{\partial}_{\Sigma}\times F^{\partial}_{\partial_{2}M}$; then we intersect
it with
$F^{\partial}_{\partial_{1}M}\times\Delta_{F^{\partial}_{\Sigma}}\times
F^{\partial}_{\partial_{2}M}$, where $\Delta_{F^{\partial}_{\Sigma}}$ denotes
the diagonal in $F^{\partial}_{\Sigma}\times F^{\partial}_{\Sigma}$; finally
we project to $F^{\partial}_{\partial M}$ by the map $\pi_{\Delta}\colon
F^{\partial}_{\partial_{1}M}\times\Delta_{F^{\partial}_{\Sigma}}\times
F^{\partial}_{\partial_{2}M}\to F^{\partial}_{\partial M}$ that forgets the
middle factor. Therefore, we have

$L_{M}=\pi_{\Delta}\left(L_{M_{1}}\times L_{M_{2}}\cap
F^{\partial}_{\partial_{1}M}\times\Delta_{F^{\partial}_{\Sigma}}\times
F^{\partial}_{\partial_{2}M}\right).$

The point of this construction is that the map $\pi_{\Delta}$ performs the
symplectic reduction of the restriction of
$\omega^{\partial}_{F^{\partial}_{\partial M_{1}}\times F^{\partial}_{\partial
M_{2}}}$ to
$F^{\partial}_{\partial_{1}M}\times\Delta_{F^{\partial}_{\Sigma}}\times
F^{\partial}_{\partial_{2}M}$. The crucial point here is that
$\Delta_{F^{\partial}_{\Sigma}}$ is a lagrangian submanifold of
$F^{\partial}_{\Sigma}\times F^{\partial}_{\Sigma}$ when the two factors are
given, as is the case here, opposite symplectic structures.

###  5.1. Boundary conditions

In this note we assume that there is a local $1$-form
$\alpha^{\partial}_{\partial M}$ such that $\pi^{*}\alpha^{\partial}_{\partial
M}=\alpha_{M}$.151515This condition actually fails in some interesting
examples (like a charged particle in a magnetic field or the WZW model). What
still happens in these examples however is that $\alpha^{\partial}_{\partial
M}$ exists as a $1$-form connection instead of a global $1$-form. It follows
that

  1. (i)

$\alpha^{\partial}_{\partial M}$ is uniquely determined,

  2. (ii)

$\omega^{\partial}_{\partial M}=\delta\alpha^{\partial}_{\partial M}$, and

  3. (iii)

equation (2) yields

(9) $\delta S_{M}=el_{M}+\pi^{*}\alpha^{\partial}_{\partial M}.$

In order to select solutions to the EL equations that could also be regarded
as critical points of $S_{M}$, one may then choose a submanifold $B$ of
$F^{\partial}_{\partial M}$ with the following two properties:

  1. (i)

The restriction of $\alpha^{\partial}_{\partial M}$ to $B$ vanishes, so
$\delta S_{M}=el_{M}$ on $\pi^{-1}(B)$.

  2. (ii)

$B$ intersects $L_{M}$ transversally, so each solution is isolated.

We will see that such submanifolds $B$ are actually related to possible
boundary conditions of the theory.

The first condition implies that $B$ should also be isotropic. The second
condition in particular entails that $T_{u}B$ and $T_{u}L_{M}$ are
complementary subspaces of $T_{u}F^{\partial}_{\partial M}$, for every
intersection point $u$ (i.e., the boundary value in $B$ of a solution). Note
that $\omega^{\partial}_{\partial M}$ at $u$ defines a symplectic form on the
vector space $T_{u}F^{\partial}_{\partial M}$, and that $T_{u}B$ and
$T_{u}L_{M}$ are isotropic with respect to it. We say that $T_{u}B$ and
$T_{u}L_{M}$ are split lagrangian subspaces, according to the following

######  Definition 5.2.

[13] An isotropic subspace of a symplectic vector space is called split
lagrangian if it admits an isotropic complement.

The motivation for this terminology is that an isotropic subspace of a finite-
dimensional symplectic space is split lagrangian precisely when it is half-
dimensional, so lagrangian according to the usual definition. In the infinite-
dimensional symplectic case, there are several definitions, of different
strength, of what lagrangian should be, split lagrangian being the
strongest.161616Note that, if $L$ is split lagrangian in a symplectic vector
space $V$, then its symplectic orthogonal $L^{\perp}$ is equal to $L$, so the
symplectic reduction of $L$ is a point. The condition $L=L^{\perp}$ is usually
taken to define lagrangian subspaces. In particular, split lagrangian implies
lagrangian.

If we want that every point $u$ of $L_{M}$ could be obtained by such an
intersection, we have to require that $T_{u}L_{M}$ should be split lagrangian
for every $u\in L_{M}$: we say, in this case, that $L_{M}$ is a split
lagrangian submanifold.171717The submanifolds $B$ that select isolated
solutions have also to be split lagrangian, but in principle it is enough to
require this in a neighborhood of the intersection points.

By construction, $L_{M}$ is always isotropic. On the other hand, the condition
that $L_{M}$ should be split lagrangian is required for the theory to behave
well when boundary conditions are imposed and does not necessarily hold in
general. Also note that this may depend on the choice of space–time manifold
$M$. For a Lagrangian theory to be good, we have to require that there is a
class of manifolds with boundary for which the condition holds (this class
should at least contain manifolds of the form $\Sigma\times I$, with $I$ a
time interval and $\Sigma$ belonging to a physically interesting class, e.g.,
being space-like). The theories encountered in physics (scalar or spinor field
theories and Yang–Mills theories, pure or coupled to the former) are good in
this sense.

In the case of gravity, we will see that one has to be a bit more careful in
the definition of $F_{M}$ in order for the theory to be well behaved on a
large class of space–time manifolds with boundary: namely, one has to impose
some open condition for the fields in $F_{M}$ on the boundary of $M$.181818We
call this the boundary metric nondegenerate PC theory in Section 7. We do not
view this as a boundary condition (such as the one imposed by the intersection
with an isotropic $B$) which would instead correspond to a closed condition.

######  Remark 5.3 (Coisotropic Cauchy spaces).

One can show that, if $L_{\Sigma\times[0,\epsilon]}$ is lagrangian for all
$\epsilon>0$, then $C_{\Sigma}$ is coisotropic (i.e., the symplectic
orthogonal of its tangent bundle is contained in the tangent bundle itself).
In this case, $C_{\Sigma}$ is locally given by first-class constraints (using
Dirac’s terminology). Moreover, the reduced phase space
$\underline{C}_{\Sigma}$ is obtained modding out by the hamiltonian vector
fields of the constraints (which one can regard as gauge transformations).
This is the typical case we will analyze in the rest of this note.

##  6\. Examples

We now briefly return to the examples discussed in Section 2, in light of the
general KT method.

In the case of mechanics, with target ${\mathbb{R}}$ as in Section 2.1, we
have $\underline{C}=C=F^{\partial}=T{\mathbb{R}}$, with coordinates
interpreted as initial position $q$ and initial velocity $v$. The symplectic
form is $m\,\delta v\,\delta q$. If we set $p\coloneqq mv$, we identify these
spaces with $T^{*}{\mathbb{R}}$ with canonical symplectic form $\delta
p\,\delta q$. The evolution relation
$\underline{L}_{[a,b]}\subset\underline{C}\times\underline{C}=T^{*}{\mathbb{R}}\times
T^{*}{\mathbb{R}}$ is the graph of the hamiltonian flow from time $a$ to time
$b$.

In the degenerate example of Section 2.2, we have
$\underline{C}=C=F^{\partial}=T^{*}S^{n-1}$ with canonical symplectic
structure. The evolution relation $\underline{L}_{[a,b]}$ is just the graph of
the identity.

In the case of the scalar field of Section 2.3, we have
$\underline{C}_{\Sigma}=C_{\Sigma}=F^{\partial}_{\Sigma}=C^{\infty}(\Sigma)\oplus
C^{\infty}(\Sigma)\ni(\phi,\phi_{0})$ and symplectic form as in (1). The
evolution relation $\underline{L}_{\Sigma\times[a,b]}$ consists of pairs
$((\phi^{a},\phi^{a}_{0}),(\phi^{b},\phi^{b}_{0}))$ with
$(\phi^{b},\phi^{b}_{0})$ the evaluation at time $b$ of the solution $\phi$
and its time derivative $\partial_{0}\phi$ of the Cauchy problem with initial
condition $(\phi^{a},\phi^{a}_{0})$ at time $a$.

In the case of electromagnetism of Section 2.4, we have
$F^{\partial}_{\Sigma}=\mathfrak{X}(\Sigma)\oplus\mathfrak{X}(\Sigma)\ni(A,E)$
and symplectic form as in Remark 2.1.191919$\mathfrak{X}(\Sigma)$ denotes the
vector fields on $\Sigma$. The Cauchy space $C_{\Sigma}$ consists of the pairs
$(A,E)$ with $E$ satisfying the Gauss law ${\mathrm{div}}E=0$. The reduced
phase space $\underline{C}_{\Sigma}$ is the quotient of $C_{\Sigma}$ by gauge
transformations $A_{i}\mapsto A_{i}+\partial_{i}\lambda$. The evolution
relation $\underline{L}_{\Sigma\times[a,b]}$ consists of pairs
$(([A]^{a},E^{a}),([A]^{b},E^{b}))$ with $([A]^{b},E^{b})$ the evaluation at
time $b$ of the solution of the Cauchy problem for the time-dependent Maxwell
equations with initial condition $([A]^{a},E^{a})$ at time $a$.202020$[A]$
denotes the gauge equivalence class of $A$.

##  7\. Gravity in the coframe formalism

We now come to the example of the reduced phase space we want to discuss in
this note: gravity. In its usual formulation, gravity is described by a
Lorentzian metric $g$ and by the Einstein–Hilbert action. The construction of
the reduced phase space turns out to be more convenient and elegant in its
formulation via (co)frames (using ideas mainly stemming from Cartan and Weyl),
usually called the Palatini–Cartan (PC) theory or the first-order formulation
of gravity (see, e.g., the reviews [21] and [28] and references therein). We
will mainly follow [11].212121For the treatment with Dirac’s method, see,
e.g., [2, 5]. For the covariant phase space formalism, see [26, 3].

The first dynamical field in this formulation is a smooth field of frames on
our space–time manifold $M$. The dynamical metric $g$ is recovered using the
frame starting from some reference metric (whose choice is irrelevant). A
frame field may also be viewed as an isomorphism between a reference vector
bundle $\mathcal{V}$ on $M$ and the tangent bundle $TM$. It turns out that the
computations are handier if one uses the inverse of a frame field, which we
call a coframe (field). It turns out that to construct the action one also
needs a second dynamical field, a connection for the orthogonal frame bundle.
This construction renders gravity closer in spirit to gauge theories. More
precisely, given a $d$-dimensional manifold $M$, the data are

  * •

a vector bundle $\mathcal{V}$ over $M$ isomorphic to $TM$ (over the identity
map on $M$), and

  * •

a smooth family $\eta$ of inner products with signature $(1,d-1)$ on the
fibers of $\mathcal{V}$.222222Note that this is possible if and only if $M$
admits a Lorentzian structure.

The first dynamical field is an isomorphism $e\colon TM\to\mathcal{V}$ (over
the identity map on $M$), called the coframe. The dynamical gravity field $g$
is recovered pulling $\eta$ back by $e$. Namely, if we pick coordinates on $M$
(we use Greek indices for them) and a local basis for the fiber of
$\mathcal{V}$ (we use the first lower case letters of the Latin alphabet for
its components), then the coframe $e$ is locally specified by its components
$e_{\mu}^{a}$ (an invertible $d\times d$ matrix at each point of $M$).232323We
denote by $\bar{e}\colon\mathcal{V}\to TM$, with components
$\bar{e}^{\mu}_{a}$, the inverse of $e$. At each point $x\in M$, for each $a$
we have a tangent vactor $\bar{e}_{a}(x)$ with components
$(\bar{e}^{1}_{a}(x),\dots,\bar{e}^{d}_{a}(x))$. The vectors
$\bar{e}_{1}(x),\dots,\bar{e}_{d}(x)$ are linearly independent, so they yield
a frame for the tangent space $T_{x}M$. For this reason, $\bar{e}$ is called a
frame and $e$ a coframe. If we denote the components of $\eta$ by $\eta_{ab}$,
we get the components of the space–time metric $g$ as

(10) $g_{\mu\nu}=e_{\mu}^{a}\,e_{\nu}^{b}\,\eta_{ab}.$

Note that $g$ has only $d(d+1)/2$ independent components, so the $d^{2}$
components of $e$ are $d(d-1)/2$ too many. Essentially, we have to consider
the internal orthogonal group to mod out the extra components.

In particular, we have to introduce a connection $A$ to compute the covariant
derivative ${\boldsymbol{D}}$ of sections of $\mathcal{V}$. If we denote by
$\phi^{a}$ the components of a section $\phi$ of $\mathcal{V}$, we have

$({\boldsymbol{D}}_{\mu}\phi)^{a}=\partial_{\mu}\phi^{a}+A^{a}_{\mu
b}\phi^{b}.$

In particular we want this connection to be compatible with the metric $\eta$
or, equivalently, its inverse, whose components we denote by $\eta^{ab}$.
Namely,

$0=({\boldsymbol{D}}_{\mu}\eta)^{ab}=\partial_{\mu}\eta^{ab}+A^{a}_{\mu
c}\eta^{cb}+A^{b}_{\mu c}\eta^{ac}.$

It is convenient to decompose

$A^{a}_{\mu b}=\eta_{bc}(\alpha_{\mu}^{ac}+\omega_{\mu}^{ac})$

with $\alpha$ symmetric and $\omega$ antisymmetric in the upper indices. The
above equations then read

$0=\partial_{\mu}\eta^{ab}+\alpha_{\mu}^{ab}+\alpha_{\mu}^{ba},$

which can be solved as

$\alpha_{\mu}^{ab}=-\frac{1}{2}\partial_{\mu}\eta^{ab}.$

Note that there are no further conditions on $\omega$. The covariant
derivative can then be written as

$({\boldsymbol{D}}_{\mu}\phi)^{a}=(D_{\mu}\phi)^{a}+\omega_{\mu}^{ab}\eta_{bc}\phi^{c}$

with

$(D_{\mu}\phi)^{a}=\partial_{\mu}\phi^{a}-\frac{1}{2}\partial_{\mu}\eta^{ab}\,\eta_{bc}\phi^{c}.$

The second field of the theory is $\omega$. If we pick a reference
$\omega_{0}$, then $\omega-\omega_{0}$ is a $1$-form taking values in
$\Lambda^{2}\mathcal{V}$.

It is convenient to introduce the following piece of terminology. A $k$-form
on $M$ taking values in $\Lambda^{l}\mathcal{V}$ will be called a $(k,l)$-form
or a form of type $(k,l)$. In particular, $e$ is a $(1,1)$-form and
$\omega-\omega_{0}$ is a $(1,2)$-form. The operator ${\boldsymbol{D}}$ acts on
$(k,l)$-forms producing $(k+1,l)$-forms by skew-symmetrizing the lower
space–time indices. We call this operation the covariant differential and
denote it ${\mathrm{d}}_{\omega}$.

The curvature $2$-form $F_{\omega}$ is the $(2,2)$-form defined via
${\mathrm{d}}_{\omega}^{2}\phi=F_{\omega}\cdot\phi$, for any section $\phi$ of
$\mathcal{V}$, with
$(F_{\omega}\cdot\phi)^{a}=\eta_{bc}F_{\omega}^{ac}\phi^{b}$ in local
coordinates.

From now on, we focus on the four-dimensional case, $d=4$, with cosmological
constant $\Lambda$. For a space–time manifold $M$, admitting $\mathcal{V}$ as
above, we define the action functional (the Palatini–Cartan action)

$S_{M}[e,\omega]\coloneqq\int_{M}\left(\frac{1}{2}e^{2}F_{\omega}+\frac{\Lambda}{24}e^{4}\right),$

where $e^{k}$ denotes the $(k,k)$-form obtained by taking the wedge product of
$k$ factors $e$. Here the wedge product is done both with respect to the form
and to the $\mathcal{V}$-components.242424In local coordinates, the integrand
is then given, up to a normalization factor, by
$\epsilon_{abcd}\epsilon^{\mu\nu\rho\sigma}\left(\frac{1}{2}e^{a}_{\mu}e^{b}_{\nu}(F_{\omega})^{cd}_{\rho\sigma}+\frac{\Lambda}{24}e^{a}_{\mu}e^{b}_{\nu}e^{c}_{\rho}e^{d}_{\sigma}\right).$
It can be checked [11] that $(4,4)$-forms, in particular the integrand in the
action, are canonically identified with densities. By taking a variation of
$S_{M}$, as in (2), we get

$\alpha_{M}=\int_{\partial M}\frac{1}{2}e^{2}\,\delta\omega$

and the EL equations

(11a) $\displaystyle eF_{\omega}+\frac{\Lambda}{6}e^{3}$ $\displaystyle=0,$
(11b) $\displaystyle e{\mathrm{d}}_{\omega}e$ $\displaystyle=0.$ One can check
that the second equation is actually equivalent to (11c)
$\displaystyle{\mathrm{d}}_{\omega}e$ $\displaystyle=0,$

as a consequence of the fact that $e$ is nondegenerate. Moreover, for a given
$e$, there is a unique connection $\omega(e)$ satisfying this equation. The
connection $\omega(e)$ corresponds, via $e$, to the Levi-Civita connection for
$g$ as in (10). Finally, the equation
$eF_{\omega(e)}+\frac{\Lambda}{6}e^{3}=0$ corresponds to the Einstein
equation, with cosmological constant $\Lambda$, for $g$.

######  Notation 7.1.

From now on we will put a tilde on the name of the bulk fields, so we will
write $\widetilde{e}=\widetilde{e}_{\mu}{\mathrm{d}}x^{\mu}$ and
$\widetilde{\omega}=\widetilde{\omega}_{\mu}{\mathrm{d}}x^{\mu}$ (with $\mu$
running from $1$ to $4$). We will reserve the notation without the tilde for
the fields on the boundary: $e=e_{i}{\mathrm{d}}x^{i}$ and
$\omega=\omega_{i}{\mathrm{d}}x^{i}$ (with $i$ running from $1$ to $3$ and
$x\in\partial M$). Note that the fields on the boundary take values in the
restriction $\mathcal{V}|_{\partial M}$ of $\mathcal{V}$ to the boundary. The
nondegeneracy condition now says that the three components $e_{1}(x)$,
$e_{2}(x)$, and $e_{3}(x)$ are linearly independent in $\mathcal{V}_{x}$ for
every $x\in\partial M$.

From the form of $\alpha_{M}$, it might seem that the space of boundary fields
consists of $e$ and $\omega$ as 1-forms on the boundary (taking values in
$\mathcal{V}|_{\partial M}$ and $\Lambda^{2}\mathcal{V}|_{\partial M}$,
respectively). Let us denote this space by $\widehat{F}^{\partial}_{\partial
M}$. It turns out however that the $2$-form

(12) $\omega_{M}=\delta\alpha_{M}=\int_{\partial M}e\,\delta e\,\delta\omega$

is degenerate, so some reduction is still needed in order to get the space
$F^{\partial}_{\partial M}$ of boundary fields. More precisely, if $X$ is a
vector field on $\widehat{F}^{\partial}_{\partial M}$ with component $u$ along
$e$ and $v$ along $\omega$, we get that $\iota_{X}\omega_{M}=0$ iff $eu=0$ and
$ev=0$. One can check that nondegeneracy of $e$ implies $u=0$. However, $v$
may be nonzero.252525One can check that, for each $x\in\partial M$, the space
of $v(x)$s satisfying $e(x)\,v(x)=0$ is six-dimensional. In conclusion, the
space $F^{\partial}_{\partial M}$ of boundary fields consists of coframes $e$
on the boundary and equivalence classes $[\omega]$ of connections under the
equivalence $\omega\sim\omega+v$ with $ev=0$. We denote by $\widehat{p}$ the
canonical projection $\widehat{F}^{\partial}_{\partial M}\to
F^{\partial}_{\partial M}$.

We now want to discuss the Cauchy data. We then assume
$M=\Sigma\times[0,\epsilon]$ and consider the first boundary component
$\Sigma=\Sigma\times\\{0\\}$. We have to split the EL equations (11a) and
(11c) into constraints and evolution equations. The constraints are just their
restrictions, as differential forms, to the boundary; this way, using Notation
7.1, we can define the space of Cauchy data $\widehat{C}_{\Sigma}$ inside
$\widehat{F}^{\partial}_{\Sigma}$ as

$\widehat{C}_{\Sigma}=\left\\{(e,\omega)\in\widehat{F}^{\partial}_{\Sigma}:eF_{\omega}+\frac{\Lambda}{6}e^{3}=0,\
{\mathrm{d}}_{\omega}e=0\right\\}.$

The space of Cauchy data $C_{\Sigma}$ is then obtained as
$\widehat{p}(\widehat{C}_{\Sigma})$.

Proceeding this way is however a bit difficult, since the constraint function
${\mathrm{d}}_{\omega}e$ is not invariant under the equivalence
relation.262626The other constraint function,
$eF_{\omega}+\frac{\Lambda}{6}e^{3}$, is instead invariant upon using
${\mathrm{d}}_{\omega}e=0$. In fact, upon sending $\omega$ to $\omega+v$, it
becomes $e{\mathrm{d}}_{\omega}v$ which is equal to
${\mathrm{d}}_{\omega}(ev)-{\mathrm{d}}_{\omega}e\,v$. The first term vanishes
by the condition on $v$ and the second by the other constraint. It would
indeed be simpler to consider instead the space

(13)
$\widehat{C}^{\prime}_{\Sigma}=\left\\{(e,\omega)\in\widehat{F}^{\partial}_{\Sigma}:eF_{\omega}+\frac{\Lambda}{6}e^{3}=0,\
e{\mathrm{d}}_{\omega}e=0\right\\}\supsetneq\widehat{C}_{\Sigma}.$

Note that we have replaced the condition ${\mathrm{d}}_{\omega}e=0$ with the
strictly weaker condition

(14) $e{\mathrm{d}}_{\omega}e=0,$

and it is not difficult to show that this is invariant under the equivalence
relation. It turns out that replacing $\widehat{C}_{\Sigma}$ with
$\widehat{C}^{\prime}_{\Sigma}$ is indeed possible under a simple assumption.

######  Definition 7.2.

We say that $e$ is metric nondegenerate if $g^{\partial}_{ij}\coloneqq
e_{i}^{a}\,e_{i}^{b}\,\eta_{ab}$ is nondegenerate.

In order to get to this situation, we have to define the space of fields
$F_{M}$ accordingly. Namely, we have to put the condition that the coframe
$\widetilde{e}$ becomes metric nondegenerate when restricted to the boundary.
A stricter condition would be that the coframe $\widetilde{e}$ produces a
riemannian metric on $\partial M$ (i.e., $\partial M$ is space-like). These
are open conditions on the space of fields, so they are part of the definition
of the theory.272727Reference [12] studies the case when $\partial M$ is
light-like. For the treatment of this case with Dirac’s method, see [1]. We
call the PC theory with coframe field required to be metric nondegenerate on
the boundary the boundary metric nondegenerate PC theory. We then have [16,
11] the

######  Theorem 7.3.

In the boundary metric nondegenerate PC theory,
$C_{\Sigma}=\widehat{p}(\widehat{C}^{\prime}_{\Sigma})$.

This means that in studying $C_{\Sigma}$ we can instead use
$\widehat{C}^{\prime}_{\Sigma}$ and work up to the equivalence relation. This
turns out to be very convenient, e.g., in proving that $C_{\Sigma}$ is
coisotropic (see Theorem 7.4 below).

###### Proof.

Instead of restricting (11c) to the boundary, it is more convenient to go back
to (11b) and separate the constraints and the evolution equations there. It is
clear that (14) is a constraint. On the other hand, there are more constraints
hidden in the transversal part of the equation. We get, using Notation 7.1 and
denoting the transversal index by $n$,

(15)
$\widetilde{e}_{n}({\mathrm{d}}_{\widetilde{\omega}}\widetilde{e})+\widetilde{e}({\mathrm{d}}_{\widetilde{\omega}}\widetilde{e})_{n}=0.$

The transversal derivative of $\widetilde{e}$ in the last term shows that this
equation also describes evolution. The problem is that it is not in normal
form. In particular, it might happen that the first term is not of the form
$\widetilde{e}$ times something, which would prevent the evolution equation to
be solved. The condition for this not to happen is also a constraint.

From now on we focus for simplicity on the case when the space of fields
$F_{M}$ is defined by the condition that the coframe $\widetilde{e}$ produces
a riemannian metric $g^{\partial}$ on $\partial M$ (i.e., $\partial M$ is
space-like).282828The general case is treated in [11]. This condition in
particular implies that the restriction $\mathcal{V}|_{\partial M}$ of
$\mathcal{V}$ admits a global time-like section $\varepsilon$. We now fix such
an $\varepsilon$. By nondegeneracy, the three components of $e$ together with
$\varepsilon$ form a basis of $\mathcal{V}_{x}$ at every point $x$ of the
boundary. This means that on the boundary we can expand
$\widetilde{e}_{n}=\rho\epsilon+\xi^{i}e_{i}=\rho\varepsilon+\iota_{\xi}e$,
where $\rho$ is a (nowhere vanishing) function and $\xi$ is interpreted as a
vector field on $\Sigma$ ($\rho$ and $\xi$ are uniquely determined). Inserting
this into (15) evaluated on points on the boundary yields, after some algebra,

$\iota_{\xi}(e{\mathrm{d}}_{\omega}e)+\rho\varepsilon{\mathrm{d}}_{\omega}e+e(({\mathrm{d}}_{\widetilde{\omega}}\widetilde{e})_{n}-\iota_{\xi}{\mathrm{d}}_{\omega}e)=0.$

The first term vanishes upon using the constraint (14), so we can ignore it.
We then see that the additional constraint, callled the structural constraint
in [11], is

(16) $\varepsilon{\mathrm{d}}_{\omega}e=e\sigma$

for some (1,1)-form $\sigma$.292929The evolution equation at boundary points
can now be written in normal form as
$({\mathrm{d}}_{\widetilde{\omega}}\widetilde{e})_{n}=\iota_{\xi}{\mathrm{d}}_{\omega}e-\rho\sigma$.
We then have, for whatever $\varepsilon$ we chose,

$\widehat{C}_{\Sigma}=\left\\{(e,\omega)\in\widehat{F}^{\partial}_{\Sigma}:eF_{\omega}+\frac{\Lambda}{6}e^{3}=0,\
e{\mathrm{d}}_{\omega}e=0,\ \exists\sigma\
\varepsilon{\mathrm{d}}_{\omega}e=e\sigma\right\\}.$

The crucial observation now is Theorem 17 in [11] which asserts that, for
every metric nondegenerate $e$, there is a unique $\omega$ in each equivalence
class satisfying the structural constraint (16).303030In [12] it is shown how
to modify the result in the light-like case. This means that we can identify
the quotient space $F^{\partial}_{\Sigma}$ with the subspace

$F^{\varepsilon}_{\Sigma}\coloneqq\left\\{(e,\omega)\in\widehat{F}^{\partial}_{\Sigma}:\exists\sigma\
\varepsilon{\mathrm{d}}_{\omega}e=e\sigma\right\\}$

of $\widehat{F}^{\partial}_{\Sigma}$. We therefore get the diagram

(17)
${F^{\varepsilon}_{\Sigma}}$${\widehat{F}^{\partial}_{\Sigma}}$${F^{\partial}_{\Sigma}}$$\scriptstyle{i}$$\scriptstyle{\tau^{\varepsilon}}$$\scriptstyle{\widehat{p}}$

where $i$ is the inclusion map and $\tau^{\varepsilon}$ is a diffeomorphism.
We then obtain the space of Cauchy data
$C_{\Sigma}=\widehat{p}(\widehat{C}_{\Sigma})$ as
$\tau^{\varepsilon}(C_{\Sigma}^{\varepsilon})$ with

$C_{\Sigma}^{\varepsilon}\coloneqq
i(\widehat{C}_{\Sigma})=\left\\{(e,\omega)\in
F^{\varepsilon}_{\Sigma}:eF_{\omega}+\frac{\Lambda}{6}e^{3}=0,\
e{\mathrm{d}}_{\omega}e=0\right\\}.$

However, it is clear that
$C_{\Sigma}^{\varepsilon}=i(\widehat{C}^{\prime}_{\Sigma})$, with
$\widehat{C}^{\prime}_{\Sigma}$ defined in (13), so
$C_{\Sigma}=\tau^{\varepsilon}(C_{\Sigma}^{\varepsilon})=\tau^{\varepsilon}(i(\widehat{C}^{\prime}_{\Sigma}))=\widehat{p}(\widehat{C}^{\prime}_{\Sigma})$.
∎

We conclude this section with the following important

######  Theorem 7.4.

In the boundary metric nondegenerate PC theory, $C_{\Sigma}$ is coisotropic.

###### Proof.

Thanks to Theorem 7.3, we can work with $\widehat{C}^{\prime}_{\Sigma}$ which
we can rewrite as the common zero locus of the functions

$\displaystyle P_{c}$ $\displaystyle=\int_{\Sigma}c\,e{\mathrm{d}}_{\omega}e,$
$\displaystyle T_{\mu}$
$\displaystyle=\int_{\Sigma}\mu\,\left(eF_{\omega}+\frac{\Lambda}{6}e^{3}\right),$

with $c$ a $(0,2)$-form and $\mu$ a $(0,1)$-form. We now compute their
hamiltonian vector fields ($X_{c}$ and $Y_{\mu}$) with respect to the
degenerate $2$-form $\omega_{M}$ defined in (12):
$\iota_{X_{c}}\omega_{M}=\delta P_{c}$, $\iota_{Y_{\mu}}\omega_{M}=\delta
T_{\mu}$. The nontrivial fact—a consequence of Theorem 7.3—is that they exist.
They are not unique, but their ambiguity lies in the direction of the
equivalence relation, along which the functions are invariant. As a
consequence, their Poisson brackets are well-defined. We will only compute
their restrictions to $C_{\Sigma}^{\varepsilon}$ and show that they vanish.
This is enough to prove the theorem.

We present the hamiltonian vector fields and leave the rest of the computation
to the reader:

$\displaystyle X_{c}(e)$ $\displaystyle=c\cdot e,$ $\displaystyle
X_{c}(\omega)$ $\displaystyle={\mathrm{d}}_{\omega}c+v,$ $\displaystyle
Y_{\mu}(e)$ $\displaystyle\approx{\mathrm{d}}_{\omega}\mu$ $\displaystyle
eY_{\mu}(\omega)$
$\displaystyle=\mu\,\left(F_{\omega}+\frac{\Lambda}{2}e^{2}\right),$

where $(c\cdot e)_{i}^{a}=\eta_{rs}c^{ar}e_{i}^{s}$, $v$ satisfies $ev=0$, and
$\approx$ means upon restriction to $C_{\Sigma}^{\varepsilon}$. ∎

######  Remark 7.5.

Looking into the proof, we see that $X_{c}$ generates the internal gauge
transformations. On the other hand, one can see that $Y_{\mu}$ generates
diffeomorphisms, both tangential and transversal to $\Sigma$. To see this, it
is better to expand $\mu=\lambda\varepsilon+\iota_{\xi}e$, with $\varepsilon$
as in the proof to Theorem 7.3, where $\lambda$ is a function and $\xi$ a
vector field on $\Sigma$. We then have $T_{\mu}=H_{\lambda}+P_{\xi}$ with

$\displaystyle H_{\lambda}$
$\displaystyle=\int_{\Sigma}\lambda\,\varepsilon\,\left(eF_{\omega}+\frac{\Lambda}{6}e^{3}\right),$
$\displaystyle P_{\xi}$
$\displaystyle=\int_{\Sigma}\iota_{\xi}e\,eF_{\omega}.$

Computing the hamiltonian vector fields separately, one can see that that of
$P_{\xi}$ generates the tangential vector fields and that of $H_{\lambda}$ the
transversal ones. One can also recognize $H_{\lambda}$ and $P_{\xi}$ as the
hamiltonian and momentum constraints. The full structure of the Poisson
brackets among $P_{C}$, $H_{\lambda}$, and $P_{\xi}$ is described in [11].

######  Remark 7.6.

In the computations in the proof of Theorem 7.4, several integration by parts
are involved. If one wants to extend this construction to the case when
$\Sigma$ has a boundary, the boundary terms generate an interesting “corner”
structure. This is better studied using the BFV formalism [19, 18]; see [10].
An alternative approach would consist in extending to the context of gravity
the results of [27].

##  8\. Conclusion

In this note, we have described the construction of the reduced phase space
for Lagrangian fields theories following the geometrical method by Kijowski
and W. M. Tulczyjew [22] and, in particular, we have applied it to the case of
coframe gravity in four dimensions.

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