Q_{b}},Q_{b}\right)+\frac{\mathrm{i}\hbar}{2}f_{\alpha\beta}^{\gamma}\,c^{\alpha}_{b}c^{\beta}_{b}\frac{\partial}{\partial
c^{\gamma}_{b}}.$

The first consequence of Assumption 12.1 is that $\Omega_{a}$ and $\Omega_{b}$
square to zero, so $\Omega^{2}=0$. In fact, since the hamiltonians are linear
in the momentum variables, their $\star$-commutators, in the induced
$\star$-products, are the same as $\mathrm{i}\hbar$ times their Poisson
brackets, so (89) is automatically satisfied (for the $H$s and for the
$\tilde{H}$s).

###  12.2. Gluing

As announced, we will compute $Z_{\text{INL}}$ via a gluing procedure,
similarly to what we did in Section 11.4. Namely, we will compose a partition
function from $(q_{a},c_{a})$ to $(p^{1},c_{1})$ with a partition function
from $(p^{1},c_{1})$ to $(q_{2},b^{2})$ with a partition function from
$(q_{2},b^{2})$ to $(Q_{b},c_{b})$, where $(p^{1},c_{1})$ and $(q_{2},b^{2})$
are some intermediate boundary fields. See Figure 9. The first partition
function belongs to Case I. The second, which we will denote as Case II’, can
be computed along the lines of Case II with the only difference that we now
fix $p$ at the initial endpoint and $q$ at the final one (instead of the other
way around). The analogue of (94) is

$Z_{\text{II'}}(p^{a},c_{a},q_{b},b^{b})=\mathrm{e}^{\frac{\mathrm{i}}{\hbar}(p^{a}_{i}q_{b}^{i}+b^{b}_{\alpha}c_{a}^{\alpha})}.$

The third partition function belongs to the analogue of Case III with $Q_{b}$
instead of $p^{b}$. We will call it Case IIINL and compute it in Section 12.3.
Equipped with all this, we have

(114) $Z_{\text{INL}}(q_{a},c_{a},Q_{b},c_{b};T,T^{+})=\int
Z_{\text{I}}(q_{a},c_{a},p^{1},c_{1};T,T^{+})\\\
Z_{\text{II'}}(p^{1},c_{1},q_{2},b^{2})\,Z_{\text{IIINL}}(q_{2},b^{2},Q_{b},c_{b})\;\frac{d^{n}p^{1}\,d^{n}q_{2}}{(2\pi\hbar)^{n}}\
{d^{k}c_{1}\,d^{k}b^{2}}(-\mathrm{i}\hbar)^{k},$

where again $n$ is the dimension of the target configuration space, $k$ is the
dimension of the Lie algebra, and we have conveniently normalized the measure.

$(q_{a},c_{a})$$(Q^{b},c^{b})$$Z_{\text{INL}}$$=$$(q_{a},c_{a})$$(p^{1},c_{1})$$Z_{\text{I}}$$(q_{2},b^{2})$$Z_{\text{II'}}$$(Q_{b},c_{b})$$Z_{\text{IIINL}}$
Figure 9. Composition of the partition function $Z_{\text{INL}}$

As a composition of partition functions that satisfy the mQME,
$Z_{\text{INL}}$ satisfies it as well.

###  12.3. The Case IIINL

In this case, we have to use the function
$\check{f}(b^{a},c_{a},p^{b},q_{b})=f(q_{b},Q(p^{b},q_{b}))-c_{a}^{\alpha}b^{a}_{\alpha}$
to modify the potential

$\alpha_{\text{BFV}}^{\check{f}}=\alpha_{\text{BFV}}+\mathrm{d}\check{f}(q_{b},Q_{b})=p_{i}^{a}\mathrm{d}q^{i}_{a}+c^{\alpha}_{a}\mathrm{d}b_{\alpha}^{a}-P^{b}_{i}\mathrm{d}Q_{b}^{i}-b_{\alpha}^{b}\mathrm{d}c^{\alpha}_{b}.$

Following again the recipe (87), we get

(115a) $\displaystyle\Omega_{a}$ $\displaystyle=\mathrm{i}\hbar
H_{\alpha}\left(\mathrm{i}\hbar\frac{\partial}{\partial
q_{a}},q_{a}\right)\frac{\partial}{\partial
b_{\alpha}^{a}}+\frac{\hbar^{2}}{2}f_{\alpha\beta}^{\gamma}\,b_{\gamma}^{a}\frac{\partial^{2}}{\partial
b_{\alpha}^{a}\partial b_{\beta}^{a}},$ (115b) $\displaystyle\Omega_{b}$
$\displaystyle=c^{\alpha}_{b}\tilde{H}_{\alpha}\left(-\mathrm{i}\hbar\frac{\partial}{\partial
Q_{b}},Q_{b}\right)+\frac{\mathrm{i}\hbar}{2}f_{\alpha\beta}^{\gamma}\,c^{\alpha}_{b}c^{\beta}_{b}\frac{\partial}{\partial
c^{\gamma}_{b}},$

and we clearly have $\Omega^{2}=0$ (again by Assumption 12.1). The BV action
for the given boundary polarizations is then

$S_{\text{BV}}^{\check{f}}=S_{\text{BV}}-f(q(t_{b}),Q(p(t_{b}),q(t_{b})))-c^{\alpha}(t_{a})e^{+}_{\alpha}(t_{a}).$

Next we choose the discontinuous splitting of the fields $e^{+}$, $c$ as in
(96) with boundary conditions (97). For the $p$, $q$ fields we choose instead

$q(t)=\begin{cases}q_{a}&t=t_{a}\\\ \hat{q}(t)&t>t_{a}\end{cases},\qquad
p(t)=\begin{cases}\hat{p}(t)&t<t_{b}\\\ p^{0}&t=t_{b}\end{cases},$

with

(116) $p^{0}\mathrel{\mathop{\mathchar 58\relax}}=\frac{\partial f}{\partial
q}(\hat{q}(t_{b}),Q_{b})$

and boundary conditions

$\hat{q}(t_{a})=0,\quad\hat{p}(t_{b})=0.$

The BV action adapted to this splitting is then the same as (98) with only the
first term changed:

$\begin{split}\tilde{S}_{\text{BV}}^{\check{f}}=&-f(\hat{q}(t_{b}),Q_{b})-\hat{e}^{+}_{\alpha}(t_{b})c^{\alpha}_{b}-\hat{p}_{i}(t_{a})q_{a}^{i}+\hat{c}^{\alpha}(t_{a})b^{a}_{\alpha}\\\
&+\int_{t_{a}}^{t_{b}}\bigg{(}\hat{p}_{i}\mathrm{d}\hat{q}^{i}-\hat{e}^{+}_{\alpha}\mathrm{d}\hat{c}^{\alpha}-\hat{e}^{\alpha}H_{\alpha}(\hat{p},\hat{q})\\\
&+\hat{c}^{\alpha}\left(\hat{q}^{+}_{i}\frac{\partial H_{\alpha}}{\partial
p_{i}}(\hat{p},\hat{q})-\hat{p}_{+}^{i}\frac{\partial H_{\alpha}}{\partial
q^{i}}(\hat{p},\hat{q})\right)\\\
&-\frac{1}{2}f_{\alpha\beta}^{\gamma}\,\hat{c}^{+}_{\gamma}\hat{c}^{\alpha}\hat{c}^{\beta}+f_{\alpha\beta}^{\gamma}\,\hat{e}^{+}_{\gamma}\hat{c}^{\alpha}\hat{e}^{\beta}\bigg{)}.\end{split}$

We pick again the gauge-fixing lagrangian (92). This produces the same
$\hat{p}\hat{q}$ and $\hat{e}^{+}\hat{c}$ propagators as in Case III (since
the hat fields here and there have exactly the same boundary conditions), and
we can then easily compute

$Z_{\text{IIINL}}(q_{a},b^{a},Q_{b},c_{b})=\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}(f(q_{a},Q_{b})-c_{b}^{\alpha}b^{a}_{\alpha})}.$

One can immediately check that $\Omega Z_{\text{IIINL}}=0$. Assumption 12.1
plays here a fundamental role.

######  Remark 12.3 (mQME vs unitarity).

The physical part of the partition function,
$\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(q_{a},Q_{b})}$, is exactly the
exponential of the HJ action (6) as in (10). This is not unexpected because,
since the physical part decouples from the ghost part, in the physical part we
can go back to the continuous splitting, as in Example 2.12. Note that the
partition function we get is not unitary, as it misses the corrections
discussed in Digression 2.13. In principle, we might add such
corrections—e.g., as in (12)—if possible, but this would spoil the mQME unless
we were able to find corrections to the $H_{\alpha}$s and
$\tilde{H}_{\alpha}$s that satisfy (113) with the corrected $f$. There is no
guarantee that such corrections exist. This is the price we have to pay in the
quantum BV-BFV formalism in order to get a solution to the mQME. The mQME is
fundamental to ensure that the result of a composition, as in the procedure of
Section 12.2, also solves the mQME. As a consequence, if there is a conflict,
we prefer to have $Z_{\text{INL}}$ solve the mQME at the expense of not being
unitary. We wil return to this in Remark 12.7.

###  12.4. The computation of $Z_{\text{INL}}$

We can now use (114) to compute

$\begin{split}Z_{\text{INL}}(q_{a},c_{a},Q_{b},c_{b};T,T^{+})&=\int\mathrm{e}^{\frac{\mathrm{i}}{\hbar}(p^{1}_{i}(q_{2}^{i}-q_{a}^{i})-f(q_{2},Q_{b}))}\mathrm{e}_{\star}^{-\frac{\mathrm{i}}{\hbar}\langle
H,T\rangle}(p^{1},q_{a})\;\frac{d^{n}p^{1}\,d^{n}q_{2}}{(2\pi\hbar)^{n}}\\\
&\phantom{=\int}\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\langle
T^{+},F_{-}(\mathrm{ad}_{T})c_{b}+F_{+}(\mathrm{ad}_{T})c_{a}\rangle}\;\mathrm{e}^{\mathbb{W}(T)}.\end{split}$

######  Remark 12.4.

If $f(q,Q)=Q_{i}q^{i}$, then one easily sees that
$Z_{\text{INL}}=Z_{\text{I}}$ with $p^{b}=Q_{b}$.

######  Remark 12.5.

At the leading order in $\hbar$, the physical part of $Z_{\text{I}}$ is the
exponential of $\mathrm{i}/\hbar$ times the HJ action, see (101). The integral
in the $p^{1},q_{2}$ variables then produces, also at the leading order, the
composition of the generating functions $f$ and $\hat{S}_{\text{HJ}}$.
Therefore, we have

(117)
$Z_{\text{INL}}^{\text{phys}}(q_{a},Q_{b};T)=\mathrm{e}^{\frac{\mathrm{i}}{\hbar}(\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b},\mathrm{e}^{T})+O(\hbar))}.$

######  Example 12.6 (Linear case).

In the case of linear constraints of Example 4.3, there are no corrections in
the physical part to the HJ action, see Example 11.7. Moreover, since the HJ
action (24) is at most linear in the momenta, the integral in the
$p^{1},q_{2}$ variables produces exactly the composition of the generating
functions $f$ and $\hat{S}_{\text{HJ}}$. Finally, the theory is abelian in
this case. Therefore, we get

(118)
$\begin{split}Z_{\text{INL}}(q_{a},c_{a},Q_{b},c_{b};T,T^{+})&=\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}\left(f(q_{a}+T^{\alpha}v_{\alpha},Q_{b})+T^{\alpha}w_{\alpha,i}q_{a}^{i}+\frac{1}{2}T^{\alpha}T^{\beta}A_{\alpha\beta}+T^{+}_{\alpha}\,(c_{a}^{\alpha}-c_{b}^{\alpha})\right)}\\\
&=\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\hat{S}^{f}_{\text{HJ}}(q_{a},Q_{b};T)}\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}T^{+}_{\alpha}\,(c_{a}^{\alpha}-c_{b}^{\alpha})},\end{split}$

with $\hat{S}^{f}_{\text{HJ}}$ as in (25). This result is the prototype for
abelian 7D Chern–Simons theory with nonlinear polarization, see [21, Sections
6], i.e., for the quantization of [34].

######  Remark 12.7.

As a consequence of Remark 12.3, the partition function $Z_{\text{INL}}$ is in
general nonunitary if we consider a nonlinear change of polarization. On the
other hand, it satisfies the mQME. An application of this, following Section
9.2.8, consists in composing $Z_{\text{INL}}$ with an $\Omega$-closed state
$\psi$ in the final variables $(Q_{b},c_{b})$. The result is a state
$\psi^{\prime}$ in the initial variables $(q_{a},c_{a})$ and in the residual
fields $(T,T^{+})$ which satisfies the mQME. The state $\psi^{\prime}$ will
not be unitarily equivalent to the state $\psi$ in general, but the idea is
that we introduce the state $\psi$ just to produce the solution
$\psi^{\prime}$ of the mQME. In the abelian case, for $\psi$ we may take an
arbitrary function of $Q_{b}$ times a delta function of all the ghost
variables. This is automatically $\Omega$ closed. As an application, in [21]
we will use the state (68) (times the delta function of the ghosts) as the
final state $\psi$. This will show that the result of [34], which connects the
Kodaira–Spencer [39] action [8] to 7D Chern–Simons theory (with the nonlinear
Hitchin polarization [35] at the final endmanifold), survives at the quantum
level with no corrections.

##  Appendix A Symplectic geometry and generating functions

In this appendix we recall some basic facts on symplectic geometry (also in
the infinite-dimensional case) and on (generalized) generating functions for
lagrangian submanifolds (and canonical relations). Our main reference will be
[9]. For the infinite-dimensional case, we will also follow [13].

We will discuss first the linear case, where actually most of the subtleties
are already present, and then briefly review the extension to manifolds.

###  A.1. Symplectic spaces

A symplectic form on a vector space $V$ is a skew-symmetric bilinear form
$\omega$ with the property of being nondegenerate:

(119) $\omega(v,w)=0\ \forall w\in V\Rightarrow v=0.$

The pair $(V,\omega)$ is called a symplectic space.

A symplectomorphism between symplectic spaces $(V,\omega_{V})$ and
$(W,\omega_{W})$ is an isomorphism $\phi\colon V\to W$ such that

(120)
$\omega_{W}(\phi(v_{1}),\phi(v_{2}))=\omega_{V}(v_{1},v_{2}),\quad\forall
v_{1},v_{2}\in V.$

######  Remark A.1.

Note that to a bilinear form $\omega$ on $V$ one can associate the linear map
$\omega^{\sharp}\colon V\to V^{*}$,

$\omega^{\sharp}(v)(w)\mathrel{\mathop{\mathchar 58\relax}}=\omega(v,w),\qquad
v,w\in V.$

The nondegeneracy of $\omega$ is equivalent to the injectivity of
$\omega^{\sharp}$. If $V$ is finite-dimensional, then this implies that
$\omega^{\sharp}$ is actually a bijection, but this is not guaranteed in the
infinite-dimensional case.

######  Remark A.2.

In the infinite-dimensional case, the above condition is often referred to as
weak nondegeneracy and $\omega$ is called a weak symplectic form, reserving
the term symplectic form to the case when $\omega^{\sharp}$ is a bijection. In
this paper we will only consider weak symplectic forms, and we will just call
them symplectic forms.

######  Example A.3.

One of the infinite-dimensional examples for this paper is the symplectic
space associated to abelian Chern–Simons theory. Let $\Sigma$ be a closed,
oriented $2(2k+1)$-manifold. Let $V=\Omega^{2k+1}(\Sigma)$. Then the bilinear
form

$\omega(\alpha,\beta)=\int_{\Sigma}\alpha\wedge\beta,\qquad\alpha,\beta\in V,$

is skew-symmetric and nondegenerate, hence symplectic.

####  A.1.1. Special subspaces

Given a subspace $W$ of $V$, one defines its orthogonal space as

$W^{\perp}\mathrel{\mathop{\mathchar 58\relax}}=\\{v\in V\colon\omega(v,w)=0,\
\forall w\in W\\}.$

Note that $V^{\perp}=\\{0\\}$.

######  Lemma A.4.

Let $W$ and $Z$ be two subspaces. Then

  1. (1)

$W\subseteq Z\Rightarrow Z^{\perp}\subseteq W^{\perp}$;

  2. (2)

$(W+Z)^{\perp}=W^{\perp}\cap Z^{\perp}$.

  3. (3)

$W\subseteq W^{\perp\perp}$ and $W^{\perp}=W^{\perp\perp\perp}$.

  4. (4)

If $V$ is finite-dimensional, then $\dim V=\dim W+\dim W^{\perp}$ and
$W=W^{\perp\perp}$.

###### Proof.

For $(1)$, let $v\in Z^{\perp}$. By definition $\omega(v,z)=0\ \forall z\in
Z$. A fortiori $\omega(v,w)=0\ \forall w\in W$. So $v\in W^{\perp}$.

As for $(2)$, observe that $v\in(W+Z)^{\perp}$ if and only if
$\omega(v,w+z)=0\ \forall w\in W,\ \forall z\in Z$. In particular, taking
either $w$ or $z$ equal to zero, we see that $v$ belongs to both $W^{\perp}$
and $Z^{\perp}$. On the other hand, if $v\in W^{\perp}\cap Z^{\perp}$, by
linearity we get $\omega(v,w+z)=\omega(v,w)+\omega(v,z)=0\ \forall w\in W,\
\forall z\in Z$; so $v\in(W+Z)^{\perp}$.

As for $(3)$, let $w\in W$. Then $\omega(w,v)=-\omega(v,w)=0\ \forall v\in
W^{\perp}$. Hence, $w\in W^{\perp\perp}$. For the second equation, taking
orthogonals in the first yields $W^{\perp\perp\perp}\subseteq W^{\perp}$. On
the other hand, applying the first equation to the subspace $W^{\perp}$ yields
$W^{\perp}\subseteq W^{\perp\perp\perp}$.

In finite dimensions, $\omega$ yields an isomorphism $V/W^{\perp}\to W^{*}$,
so $\dim V-\dim W^{\perp}=\dim W^{*}=\dim W$. Moreover,

$\dim W^{\perp\perp}=\dim V-\dim W^{\perp}=\dim V-(\dim V-\dim W)=\dim W.$

∎

A subspace $W$ is called isotropic if $W\subseteq W^{\perp}$, coisotropic if
$W^{\perp}\subseteq W$ and lagrangian if $W=W^{\perp}$.

Note that a subspace $W$ is isotropic if and only if $\omega(w_{1},w_{2})=0$
for all $w_{1},w_{2}\in W$ and that a subspace is lagrangian if and only if it
is at the same time isotropic and coisotropic.

In the finite-dimensional case, $L$ is lagrangian if and only if $L$ is
isotropic and $2\dim L=\dim V$. Finally, an isotropic subspace $L$ is called
split lagrangian if it admits an isotropic complement. (For more on lagrangian
and split lagrangian subspaces, we refer to [13]).

######  Lemma A.5.

$\text{split lagrangian}\Longrightarrow\text{lagrangian}$

In the finite-dimensional case, the two concepts are equivalent.

###### Proof.

Suppose that $L$ is split lagrangian. Then we can find an isotropic subspace
$W$ such that $V=L\oplus W$. In particular, $V=L+W$. By $(2)$ in Lemma A.4, we
then have $L^{\perp}\cap W^{\perp}=\\{0\\}$. Since $W\subseteq W^{\perp}$, we
also have $L^{\perp}\cap W=\\{0\\}$, which implies $L^{\perp}\subseteq L$.
Therefore, $L$ is lagrangian.

Suppose on the other hand that $L$ is lagrangian and $V$ is finite-
dimensional. Suppose $W$ is isotropic but not lagrangian and $L\cap
W=\\{0\\}$. Observe that by dimensional reasons, $L+W$ cannot be the whole
$V$. Therefore $(L+W)^{\perp}$ cannot be $\\{0\\}$. Using $(2)$ in Lemma A.4,
we then have $W^{\perp}\cap L\not=\\{0\\}$.

We now want to show that there is a nonzero vector $v\in W^{\perp}$ that does
not belong to $L$ nor to $W$. In fact, if this were not the case, we would
have $W^{\perp}\subseteq W+L$. By $(1)$ and $(2)$ in Lemma A.4, we would then
have $W\supseteq W^{\perp}\cap L^{\perp}=W^{\perp}\cap L$. Intersecting with
$L$, we would conclude $\\{0\\}=W^{\perp}\cap L$, which is a contradiction.

Let finally $\mathcal{I}$ be the set of isotropic subspaces of $V$ whose
intersection with $L$ is $\\{0\\}$. Note that $\\{0\\}\in\mathcal{I}$. If
$W\in\mathcal{I}$ is not lagrangian, by the above argument we can find a
nonzero vector $v\in W^{\perp}$ that does not belong to $L$ nor to $W$. But
then $W\oplus\operatorname{span}{v}$ is also in $\mathcal{I}$ and of dimension
strictly higher than that of $W$. After finitely many steps, we end up with a
$W\in\mathcal{I}$ which is lagrangian. But since $\dim L=\dim W=\dim V/2$, we
see that $W$ is a complement to $L$. ∎

######  Remark A.6 (Split symplectic spaces).

For applications in field theory, we will only consider symplectic spaces that
admit split lagrangian subspaces. In particular, we fix a lagrangian subspace
$W$ and a lagrangian complement $W^{\prime}$ and write

(121) $V=W\oplus W^{\prime}.$

We will call this a split symplectic space.

Note that in the finite-dimensional case, $\omega^{\sharp}$ establishes an
isomorphism between $W^{\prime}$ and $W^{*}$, which in turns establishes a
symplectomorphism between $V$ and $W\oplus W^{*}$ with the canonical
symplectic form

(122) $\omega(v\oplus a,w\oplus b)=b(v)-a(w),\qquad v,w\in W,\ a,b\in W^{*}.$

In the infinite-dimensional case, $\omega^{\sharp}|_{W^{\prime}}\colon
W^{\prime}\to W^{*}$ is only injective. Still, we may regard $W\oplus
W^{\prime}$ as a symplectic subspace of $W\oplus W^{*}$, which is what we will
do in the following.

######  Example A.7.

Consider Example A.3 with $k=0$. If we work with complex valued forms, then we
have the splitting of $V=\Omega^{1}(\Sigma)\otimes\mathbb{C}$ into
$W=\Omega^{1,0}(\Sigma)$ and $W^{\prime}=\Omega^{0,1}(\Sigma)$, with respect
to some complex structure on $\Sigma$.

####  A.1.2. Generating functions

Assume we have a split symplectic space as in (121). Consider the graph of a
linear map $\phi\colon W\to W^{\prime}$:

$L\mathrel{\mathop{\mathchar 58\relax}}=\\{w\oplus\phi(w),\ w\in W\\}\subset
V.$

Since

$\omega(w_{1}\oplus\phi(w_{1}),w_{2}\oplus\phi(w_{2}))=\omega(w_{1},\phi(w_{2}))+\omega(\phi(w_{1}),w_{2}),$

we see that $L$ is isotropic, hence split lagrangian because $W^{\prime}$ is
an isotropic complement to it, if and only if

$\omega(w_{1},\phi(w_{2}))=-\omega(\phi(w_{1}),w_{2}),\quad\forall
w_{1},w_{2}\in W.$

In this case, we can use $\phi$ and $\omega$ to define a quadratic form on
$W$:

$\psi(w)\mathrel{\mathop{\mathchar 58\relax}}=\frac{1}{2}\omega(w,\phi(w)).$

Note then that

$\delta\psi(w)=\omega(\phi(w),\delta w)=\omega^{\sharp}(\phi(w))(\delta w),$

so

$\delta\psi=\omega^{\sharp}\circ\phi\colon W\to W^{*}$

as a one-form on $W$. Since $\omega^{\sharp}$ is injective, it is possible to
recover $\phi$ out of $\delta\psi$.

Vice versa, suppose we have a splitting $V=W\oplus W^{\prime}$, already
realized as a subspace of $W\oplus W^{*}$ with canonical symplectic structure,
and a quadratic form $\psi$ on $W$. If $\delta\psi(W)\subseteq W^{\prime}$,
then the graph $L_{\psi}$ of $\delta\psi$ is a split lagrangian subspace of
$V$ and $\psi$ is called a generating function for it.

######  Example A.8.

Let $I$ be the closed interval $[-1/2,1/2]$ and $V\mathrel{\mathop{\mathchar
58\relax}}=C^{0}(I)\oplus C^{0}(I)$ as a real vector space. Define

$\omega(f\oplus g,h\oplus k)=\int_{I}(fk-gh)\,\mathrm{d}x.$

Then $(V,\omega)$ is split symplectic with $W=W^{\prime}=C^{0}(I)$ (more
properly $W=C^{0}(I)\oplus\\{0\\}$ and $W^{\prime}=\\{0\\}\oplus C^{0}(I)$).
The quadratic form $\psi(f)=\frac{1}{2}\int_{I}f^{2}\,\mathrm{d}x$ has
differential $\delta_{f}\psi=f$, so it is a generating function for the
diagonal $L_{\psi}=\\{f\oplus f,\ f\in C^{0}(I)\\}$. On the other hand, the
quadratic form $\tilde{\psi}(f)=\frac{1}{2}f(0)^{2}$, $f\in W$, has
differential $\delta_{f}\tilde{\psi}=f(0)\delta$, with $\delta$ the delta
function at $0$, which is an element of $W^{*}$ but not of $W^{\prime}$, so it
is not an allowed generating function.

####  A.1.3. Symplectic reduction

We will only consider the particular case of coisotropic reduction, which is
of interest for this paper.

Let $C$ be a coisotropic subspace of $(V,\omega)$. Then the kernel of the
restriction of $\omega$ to $C$ is precisely the orthogonal subspace
$C^{\perp}$. If follows that $\underline{C}\mathrel{\mathop{\mathchar
58\relax}}=C/C^{\perp}$, the symplectic reduction of $C$, is endowed with the
symplectic form

(123) $\underline{\omega}([v],[w])=\omega(v,w),\quad v\in[v],\ w\in[w].$

Denote by $\pi$ the canonical projection $C\to\underline{C}$. If $L$ is a
subspace of $V$, we set $\underline{L}\mathrel{\mathop{\mathchar
58\relax}}=\pi(L\cap C)\subseteq\underline{C}$.

######  Lemma A.9 (Relative linear reduction).

If $L$ is isotropic in $V$, then $\underline{L}$ is isotropic in
$\underline{C}$. In the finite-dimensional case, if $L$ is lagrangian in $V$,
then $\underline{L}$ is lagrangian in $\underline{C}$.

###### Proof.

The first statement may be easily proved observing that, if $[v]$ and $[w]$
are in $\underline{L}$, then every $v\in[v]$ and every $w\in[w]$ lie in $L\cap
C$ and hence in $L$. Therefore, $\omega(v,w)=0$ and, by (123), we get
$\underline{\omega}([v],[w])=0$, so $\underline{L}$ is isotropic.

A different proof, which also leads to the second statement, is based on the
following:

$\pi^{-1}(\underline{L}^{\perp})=(\pi^{-1}(\underline{L}))^{\perp}\cap
C=(L\cap C+C^{\perp})^{\perp}\cap C=\\\ =(L\cap C)^{\perp}\cap
C\supset(L^{\perp}+C^{\perp})\cap C=L^{\perp}\cap C+C^{\perp}.$

In the last equality we have used the obvious fact that $(A+B)\cap C=A\cap
C+B$ for any three subspaces $A$, $B$, and $C$ with $B\subset C$. Thus,
$\underline{L}^{\perp}\supset\pi(L^{\perp}\cap C)\supset\pi(L\cap
C)=\underline{L}$. All inclusions are equalities under the assumptions that
$V$ is finite-dimensional and $L$ is lagrangian. ∎

In infinite dimensions, it may happen that $L$ is lagrangian but
$\underline{L}$ is not:

######  Example A.10.

Consider the symplectic space of Example A.8 and

$C\mathrel{\mathop{\mathchar 58\relax}}=\\{f\oplus g\in
V\mathrel{\mathop{\mathchar 58\relax}}g(0)=0\\}.$

One has that $C^{\perp}=\\{0\\}$, so $C$ is coisotropic and $\underline{C}=C$.
Let

$L\mathrel{\mathop{\mathchar 58\relax}}=\left\\{f\oplus g\in
V\mathrel{\mathop{\mathchar 58\relax}}\int_{I}f\;\mathrm{d}x=0,\ g\text{\
constant}\right\\}.$

One can easily show that $L$ is lagrangian, actually split lagrangian, in $V$,
but $\underline{L}$ is not lagrangian in $\underline{C}$.

######  Remark A.11 (Generalized generating functions in the linear case).

Suppose we have a sum $V\oplus Z$ of symplectic spaces with $Z=B\oplus
B^{\prime}$ split. First observe that $C\mathrel{\mathop{\mathchar
58\relax}}=V\oplus B\oplus\\{0\\}$ is coisotropic in $V\oplus Z$ with
$C^{\perp}=\\{0\\}\oplus B\oplus\\{0\\}$, so $\underline{C}=V$. Now assume
that also $V=W\oplus W^{\prime}$ is split and rearrange $V\oplus Z$ as the
split symplectic space $(W\oplus B)\oplus(W^{\prime}\oplus B^{\prime})$. Let
$L_{\psi}$ be the split lagrangian subspace of $V\oplus Z$ with generating
function a quadratic function $\psi$ on $W\oplus B$. It then follows that
$\underline{L}_{\psi}$ is isotropic in $V$. If it is split lagrangian (as it
automatically happens in the finite-dimensional case), then one says that
$\psi$ is a generalized generating function for $\underline{L}_{\psi}$,
depending on extra parameters in $B$.

######  Example A.12.

Consider $V=W\oplus W^{*}$ and $Z=B\oplus B^{*}$ with $W$ and $B$ finite-
dimensional. Let $F\colon W\to B^{*}$ be a linear map and let $F^{*}\colon
B\to W^{*}$ denote its dual. Then the quadratic form $\psi(w\oplus b)=(Fw,b)$,
with $(\ ,\ )$ denoting the pairing of a space with its dual, is a generalized
generating function for $\underline{L}_{\psi}=\ker
F\oplus\text{im}\,F^{*}=\ker F\oplus(\ker F)^{0}$, with 0 denoting the
annihilator space. Unless $F$ is the zero map, $\underline{L}_{\psi}$ cannot
be generated by a function on $W$ only, so extra parameters are indeed
necessary.

######  Example A.13.

The above example may be extended to the infinite-dimensional case as follows.
Let $F\colon W\to B^{\prime}\subseteq B^{*}$ and $F^{\prime}\colon B\to
W^{\prime}\subseteq W^{*}$ be linear maps such that
$\iota_{W^{\prime}}F^{\prime}=(\iota_{B^{\prime}}F)^{*}\iota$, where
$\iota_{B^{\prime}}\colon B^{\prime}\to B^{*}$ and $\iota_{W^{\prime}}\colon
W^{\prime}\to W^{*}$ are the given inclusions and $\iota\colon B\to B^{**}$ is
the canonical inclusion. Then the quadratic form $\psi(w\oplus
b)=(\iota_{B^{\prime}}Fw,b)=(\iota_{W^{\prime}}F^{\prime}b,w)$ is a generating
function for a lagrangian subspace $L_{\psi}$ of $V\oplus Z$. The
corresponding isotropic subspace $\underline{L}_{\psi}$ of $V$ is $\ker
F\oplus\text{im}\,F^{\prime}$.

######  Example A.14.

A simple illustration of the last example is the following. Let $\Sigma$ be a
closed, oriented two-manifold and let $W=W^{\prime}=\Omega^{1}(\Sigma)$,
$B=\Omega^{0}(\Sigma)$, $B^{\prime}=\Omega^{2}(\Sigma)$, with symplectic forms
induced by integration over $\Sigma$:

$\omega(\alpha\oplus\alpha^{\prime},\beta\oplus\beta^{\prime})=\int_{\Sigma}(\alpha\wedge\beta^{\prime}-\beta\wedge\alpha^{\prime}).$

The linear maps $F=\mathrm{d}\colon\Omega^{1}(\Sigma)\to\Omega^{2}(\Sigma)$
and $F^{\prime}=\mathrm{d}\colon\Omega^{0}(\Sigma)\to\Omega^{1}(\Sigma)$
satisfy the assumptions, so we have the generalized generating function
$\psi(w\oplus b)=\int_{\Sigma}\mathrm{d}w\wedge b$, with
$w\in\Omega^{1}(\Sigma)$ and $b\in\Omega^{0}(\Sigma)$. We get
$\underline{L}_{\psi}=\Omega^{1}_{\text{cl}}(\Sigma)\oplus\Omega^{1}_{\text{ex}}(\Sigma)$,
where the subscripts denote closed and exact forms. In this example
$\underline{L}_{\psi}$ is actually a split lagrangian. This follows from the
Hodge decomposition theorem. In fact, putting a metric on $\Sigma$ one can
define the Hodge $*$ operator and the codifferential $\mathrm{d}^{*}$. As an
isotropic complement of $\underline{L}_{\psi}$ one can then take
$\Omega^{1}_{\text{coex}}(\Sigma)\oplus\Omega^{1}_{\text{cocl}}(\Sigma)$.

####  A.1.4. Canonical relations

Canonical relations are a generalization of symplectomorphisms.

Recall that a linear map $\phi$ between $(V,\omega_{V})$ and $(W,\omega_{W})$
is called a symplectomorphism if it is an isomorphism and respects the
symplectic forms as in (120). It follows that the graph of $\phi$,

$L_{\phi}=\\{v\oplus\phi(v),\ v\in V\\},$

is a split lagrangian subspace of $\bar{V}\oplus W$, where $\bar{V}$ denotes
$V$ with symplectic form $-\omega_{V}$. In fact, we have

$\omega_{\bar{V}\oplus
W}(v_{1}\oplus\phi(v_{1}),v_{2}\oplus\phi(v_{2}))=-\omega_{V}(v_{1},v_{2})+\omega_{W}(\phi(v_{1}),\phi(v_{2}))=0,$

so $L_{\phi}$ is isotropic. Note that $L_{-\phi}$ is also isotropic and, since
$\phi$ is an isomorphism, it is a complement to $L_{\phi}$.

More generally, one calls any (split) lagrangian subspace $L$ of
$\bar{V}\oplus W$ a (split) canonical relation from $V$ to $W$. One also
writes

(124) $L\colon V\nrightarrow W.$

If $V=V_{1}\oplus V_{2}$ and $W=W_{1}\oplus W_{2}$ are split symplectic
spaces, then a lagrangian subspace $L$ of $\bar{V}\oplus W$ might have a
(generalized) generating function $\psi$, a quadratic form on $V_{1}\oplus
W_{1}(\oplus B)$, which is then called a (generalized) generating function for
the canonical relation $L$.

Here is an example of a canonical relation that is important for our
applications:

######  Lemma A.15.

Let $C$ be a coisotropic subspace of $V$. Then the subspace
$L_{C}\mathrel{\mathop{\mathchar 58\relax}}=\\{(v,v^{\prime})\in C\oplus C\ |\
v^{\prime}-v\in C^{\perp}\\}$ is isotropic in $\bar{V}\oplus V$. If $V$ is
finite-dimensional, then $L$ is lagrangian, hence a canonical relation from
$V$ to itself.

###### Proof.

For $v_{1},v_{2}\in C$ and $w_{1},w_{2}\in C^{\perp}$ we have
$\omega(v_{1}+w_{1},v_{2}+w_{2})-\omega(v_{1},v_{2})=0$, so $L_{C}$ is
isotropic. If $\dim V=2n$ and $\dim C=k$, then $\dim L_{C}=\dim C+\dim
C^{\perp}=k+(2n-k)=2n$, so $2\dim L_{C}=\dim(\bar{V}\oplus V)$. ∎

For an example where $L_{C}$ is not lagrangian, consider $C$ as in Example
A.10. Since $C^{\perp}=\\{0\\}$, we have that $L_{C}=\Delta_{C}$, the diagonal
in $C\oplus C$. On the other hand, $L_{C}^{\perp}=\Delta_{V}$, the diagonal in
$V\oplus V$, which properly contains $\Delta_{C}$.

Another relation between canonical relations and coisotropic submanifolds is
the following result (observed in [18] in the context of field theory).

######  Lemma A.16.

Let $L$ be a lagrangian subspace of $\bar{V}\oplus V$. Define
$C\mathrel{\mathop{\mathchar 58\relax}}=\pi_{1}(L)\subseteq V$, with $\pi_{1}$
the projection to the first summand. Then $C$ is coisotropic.

Note that this result holds also in the infinite-dimensional case.

###### Proof.

By definition we have that $v$ belongs to $C$ iff there is a $w\in V$ such
that $(v,w)$ belongs to $L$. Now let $\tilde{v}$ be in $C^{\perp}$, i.e.,
$\omega(\tilde{v},v)=0$ $\forall v\in C$. This implies

$\omega(\tilde{v},v)-\omega(0,w)=0\ \forall(v,w)\in L.$

Therefore, $(\tilde{v},0)\in L^{\perp}=L$, which implies $\tilde{v}\in C$. ∎

###  A.2. Symplectic manifolds

We now extend the above results to manifolds.

######  Definition A.17 (Symplectic manifolds).

A symplectic form $\omega$ on a smooth manifold $M$ is a nondegenerate closed
two-form on it. By nondegenerate we mean that for each $x\in M$ the bilinear
form $\omega_{x}$ on $T_{x}M$ satisfies (119). Note that, for each $x\in M$,
the pair $(T_{x}M,\omega_{x})$ is then a symplectic space. The pair
$(M,\omega)$ is called a symplectic manifold.

We allow $M$ to be an infinite-dimensional manifold (for the examples in this
paper a Fréchet manifold). In almost all the examples of this paper the
symplectic manifold $M$ is actually a vector space, but we also consider
nonlinear maps and nonlinear lagrangian submanifolds.

######  Definition A.18 (Symplectomorphisms).

A symplectomorphism between symplectic manifolds $(M,\omega_{M})$ and
$(N,\omega_{N})$ is a diffeomorphism $\phi\colon M\to N$ such that
$\phi^{*}\omega_{N}=\omega_{M}$.

######  Definition A.19 (Special subspaces).

A submanifold $S$ of a symplectic manifold $M$ is called isotropic /
coisotropic / lagrangian / split lagrangian if $T_{x}S$ is a subspace of this
class in $T_{x}M$ for every $x\in S$. Note that, as these concepts are defined
in terms of tangent spaces, they make sense both for embedded and for immersed
submanifolds. (Unless we explicitly say otherwise, by submanifold we then mean
an immersed submanifold.)

######  Definition A.20 (The characteristic distribution).

The characteristic distribution of a coisotropic submanifold $C$ of $M$ is the
subbundle $C^{\perp}\mathrel{\mathop{\mathchar 58\relax}}=\cup_{x\in
C}(T_{x}C)^{\perp}$ of $TC$. It is a subbundle, hence a regular distribution,
because it is the kernel of $\omega_{C}^{\sharp}\colon TC\to T^{*}C$, where
$\omega_{C}$ is the restriction of the symplectic form to $C$.

######  Lemma A.21.

Let $C$ be a coisotropic submanifold of $M$. Then

  1. (1)

For every $X\in\Gamma(C^{\perp})$, we have $\operatorname{L}_{X}\omega_{C}=0$,
where $\operatorname{L}$ denotes the Lie derivative;

  2. (2)

The characteristic distribution $C^{\perp}$ is involutive.

###### Proof.

By definition $\iota_{X}\omega_{C}=0$. Since $\mathrm{d}\omega_{C}=0$, we get
$\operatorname{L}_{X}\omega_{C}=0$ by Cartan’s magic formula.

Next suppose that $Y$ is also a section of $C^{\perp}$. By Cartan’s calculus
we have

$\iota_{[Y,X]}\omega_{C}=[\iota_{Y},\operatorname{L}_{X}]\omega_{C},$

where we have also used the above statement. Therefore,
$[Y,X]\in\Gamma(C^{\perp})$. ∎

######  Remark A.22 (Split symplectic manifolds).

We are now interested in the generalization of the concept of split symplectic
space of Remark A.6: a split symplectic manifold. In the finite-dimensional
case, this is the notion of a cotangent bundle $T^{*}M$ with its canonical
symplectic form (this is the two-form which in local coordinates $q^{i}$ for
the base and $p_{i}$ for the fiber reads
$\mathrm{d}p_{i}\mathrm{d}q^{i}$).545454More intrinsically, the canonical
symplectic form is the differential of the canonical one-form $\theta$ with
$\theta_{(p,q)}(v)\mathrel{\mathop{\mathchar
58\relax}}=(p,\mathrm{d}_{(p,q)}\pi(v))$, with $q\in M$ and $p\in T^{*}_{q}M$,
where $\pi$ is the projection $T^{*}M\to M$. In the infinite-dimensional case,
more generally, we consider subbundles $\tilde{T}^{*}M$ of $T^{*}M$ with
symplectic form given by the restriction of the canonical symplectic form.

######  Remark A.23 (Polarizations).

A more flexible notion of splitting than that of Remark A.22, which is used in
the classical setting, is that of polarization, which is used for geometric
quantization. Namely, a (real) polarization on a symplectic manifold $M$ is an
integrable smooth distribution $D$ (i.e., a subbundle of $TM$) such that
$D_{x}$ is split lagrangian in $T_{x}M$ for every $x\in M$. One also uses the
notion of complex polarization, which is an integrable subbundle $D$ of the
complexification $TM\otimes\mathbb{C}$ of the tangent bundle such that $D_{x}$
is split lagrangian in $T_{x}M\otimes\mathbb{C}$ for every $x\in M$.

######  Remark A.24 (Generating functions).

Let $\tilde{T}^{*}M$ be a split symplectic manifold and $\phi$ a section (in
particular $\phi$ is a one-form on $M$). Its graph $L$ is then lagrangian,
actually embedded split lagrangian, if and only if $\phi$ is closed. If
$\phi=\mathrm{d}\psi$, then $\psi$ is called a generating function for $L$,
which we also denote as $L_{\psi}$. Note that a function $\psi$ on $M$ defines
an embedded split lagrangian submanifold $L_{\psi}$ of $\tilde{T}^{*}M$ under
the condition that $\mathrm{d}\psi$ is a section of it.

######  Remark A.25 (Symplectic reduction).

Next we consider symplectic reduction of coisotropic submanifolds. If $C$ is
coisotropic in $M$, then its characteristic distribution
$C^{\perp}\mathrel{\mathop{\mathchar 58\relax}}=\cup_{x\in C}(T_{x}C)^{\perp}$
is a regular involutive distribution on $C$ by Lemma A.21 If $C$ is defined by
constraints, one can verify that the characteristic distribution is spanned by
the hamiltonian vector fields of the constraints (which therefore must be in
involution under the Poisson bracket).555555The hamiltonian vector field of a
function $H$ is the unique vector field $X_{H}$ such that
$\iota_{X}\omega+\mathrm{d}H=0$. (Note that in the infinite-dimensional case
the existence of a hamiltonian vector field is an assumption, but its
uniqueness is guaranteed.) The Poisson bracket of two functions $H$ and $G$,
with hamiltonian vector fields $X_{H}$ and $X_{G}$, is the function
$\\{H,G\\}\mathrel{\mathop{\mathchar
58\relax}}=X_{G}(H)=\iota_{X_{H}}\iota_{X_{G}}\omega$. Its hamiltonian vector
field is $[X_{H},X_{G}]$.  If $M$ is a Banach manifold (e.g., it is finite-
dimensional), then the characteristic distribution is also integrable. In the
Fréchet case, integrability is not guaranteed. Assuming that the
characteristic distribution is integrable, we denote by $\underline{C}$ its
leaf space and by $\pi\colon C\to\underline{C}$ the canonical projection. If
$\underline{C}$ is smooth, than it has a unique symplectic structure
$\underline{\omega}$ such that
$\pi^{*}\underline{\omega}=\iota_{C}^{*}\omega$, with $\iota_{C}\colon C\to M$
the inclusion map. In particular, for each $\underline{x}\in\underline{C}$,
the symplectic form $\underline{\omega}_{\underline{x}}$ is defined as in
Section A.1.3.

######  Remark A.26 (Relative reduction).

Assume $\underline{C}$ to be smooth. If $L$ is an isotropic submanifold of $M$
and $\underline{L}\mathrel{\mathop{\mathchar 58\relax}}=\pi(L\cap C)$ is a
submanifold of $\underline{C}$, then, by Lemma A.9, $\underline{L}$ is an
isotropic submanifold of $\underline{C}$. If in addition $M$ is finite-
dimensional and $L$ is lagrangian, then $\underline{L}$ is also lagrangian.

######  Remark A.27 (Generalized generating functions).

Our main example, which extends Remark A.11 to the nonlinear case, will be the
product of two split symplectic manifolds,
$N=\tilde{T}^{*}M\times\tilde{T}^{*}B=\tilde{T}^{*}(M\times B)$, with
$C=\tilde{T}^{*}M\times B$ and hence $\underline{C}=\tilde{T}^{*}M$. Let
$\psi$ be a function on $M\times B$ such that the one-form $\mathrm{d}\psi$ is
a section of $\tilde{T}^{*}(M\times B)$ and let $L_{\psi}$ the split
lagrangian submanifold in $N$ it generates. If $\underline{L}_{\psi}$ is a
split lagrangian submanifold of $\underline{C}=\tilde{T}^{*}M$, then we say
that $\psi$ is a generalized generating function for $\underline{L}_{\psi}$.
In the finite-dimensional case, one only has to check that that
$\underline{L}_{\psi}$ is a submanifold because it is then automatically split
lagrangian. Some conditions for this to happen are presented in [9, Section
4.3]. There, more generally, $B$ is just assumed to be the typical fiber of a
submersion $\pi\colon E\to M$ and the triple $(E,\pi,\psi)$, such that
$\underline{L}_{\psi}$ is a submanifold, is called a Morse family for
$\underline{L}_{\psi}$. Explicitly, if we denote by $q^{i}$ and $T^{\alpha}$
the coordinates on $M$ and $B$, respectively, and by $p_{i}$ and $E_{\alpha}$
their momenta, the lagrangian submanifold $L_{\psi}$ is given by the equations
$p_{i}=\frac{\partial\psi}{\partial q^{i}}$ and
$E_{\alpha}=\frac{\partial\psi}{\partial T^{\alpha}}$. Intersecting with
$C=\tilde{T}^{*}M\times B$ corresponds to imposing
$\frac{\partial\psi}{\partial T^{\alpha}}=0$. Therefore,
$\underline{L}_{\psi}$ is given by the equations

$p_{i}=\frac{\partial\psi}{\partial q^{i}},\qquad\frac{\partial\psi}{\partial
T^{\alpha}}=0.$

######  Definition A.28 (Canonical relations).

A (split) canonical relation $L$ between two symplectic manifolds
$(M_{1},\omega_{1})$ and $(M_{2},\omega_{2})$ is a (split) lagrangian
submanifold of $\bar{M}_{1}\times M_{2}$ (i.e., $M_{1}\times M_{2}$ with
symplectic form $\pi_{2}^{*}\omega_{2}-\pi_{1}^{*}\omega_{1}$ with
$\pi_{i}\colon M_{1}\times M_{2}\to M_{i}$ the canonical projections). In
particular, the graph of a symplectomorphism from $M_{1}$ to $M_{2}$ is an
embedded split canonical relation.

Here is an example (generalizing Lemma A.15):

######  Lemma A.29.

Let $C$ be a coisotropic submanifold of $M$ and assume that its involutive
characteristic distribution is actually integrable. We say that two points
$(x,x^{\prime})\in C\times C$ lie on the same leaf of the characteristic
distribution if there is a path in $C$ with endpoints $x$ and $x^{\prime}$ and
tangent vector at each point lying in the characteristic distribution. Then,
if

$L_{C}\mathrel{\mathop{\mathchar 58\relax}}=\\{(x,x^{\prime})\in C\times C\ |\
x,x^{\prime}\text{ lie on the same leaf of the characteristic
distribution}\\}$

is a submanifold, it is isotropic in $\bar{M}\times M$. If $M$ is finite-
dimensional, then $L$ is lagrangian, hence a canonical relation from $M$ to
itself.

###### Proof.

Let $(x,x^{\prime})$ be a point in $L_{C}$. By integrability, there is a
neighborhood $U$ of $x$ that is mapped to a convex open subspace of a vector
space in which the distribution corresponds to a family of linear subspaces.
We first consider the case where $x^{\prime}\in U$. The images $y$ and
$y^{\prime}$ of $x$ and $x^{\prime}$ can then be actually connected by a
straight line. We consider the constant vector field corresponding to this
direction and its flow $\Phi$ at time $1$, a translation mapping a
neighborhood of $y$ to a neighborhood of $y^{\prime}$ satisfying
$\Phi(y)=y^{\prime}$. Since $\Phi$ is a flow along a vector field in the
kernel of the image of the restriction of the symplectic form, it leaves the
latter invariant by Lemma A.21. Let $A$ be the differential of $\Phi$ at $y$
pulled back to $x$. Therefore, $A$ is a linear isomorphism $T_{x}C\to
T_{x^{\prime}}C$ satisfying

$\omega_{x}(v,w)=\omega_{x^{\prime}}(Av,Aw)\ \forall v,w\in T_{x}C.$

By construction $A(T_{x}C^{\perp})=T_{x^{\prime}}C^{\perp}$. Therefore, $A$
induces a symplectomorphism $\underline{A}$ between
$\underline{T_{x}C}\mathrel{\mathop{\mathchar
58\relax}}=T_{x}C/T_{x}C^{\perp}$ and
$\underline{T_{x^{\prime}}C}\mathrel{\mathop{\mathchar
58\relax}}=T_{x^{\prime}}C/T_{x^{\prime}}C^{\perp}$. The first consequence of
this is that

$(v,v^{\prime})\in T_{(x,x^{\prime})}L_{C}\iff v^{\prime}-Av\in
T_{x^{\prime}}C^{\perp}\iff[v^{\prime}]=\underline{A}[v].$

The second consequence is that $T_{(x,x^{\prime})}L_{C}$ is isotropic. In
fact, for $(v,v^{\prime}),(w,w^{\prime})\in T_{(x,x^{\prime})}L$ we get

$\omega_{x^{\prime}}(v^{\prime},w^{\prime})=\omega_{x^{\prime}}(Av,Aw)=\omega_{x}(v,w).$

For a generic point $(x,x^{\prime})$ in $L_{C}$, we have by definition a path
$\gamma\colon[0,1]\to C$ such that $\gamma(0)=x$, $\gamma(1)=x^{\prime}$ and
$\mathrm{d}_{t}\gamma\in T_{\gamma(t)}C^{\perp}$ $\forall t\in[0,1]$. By
compactness of $[0,1]$, we may find a finite partition
$0=t_{0}<t_{1}<t_{2}<\dots<t_{n}=1$ such that, for each $i=0,\dots,n-1$, the
points $\gamma(t_{i})$ and $\gamma(t_{i+1})$ lie in a neighborhood $U_{i}$ as
in the previous paragraph. Set $x_{i}\mathrel{\mathop{\mathchar
58\relax}}=\gamma(t_{i})$ and proceed as above. We then have that
$(v_{i},v_{i+1})\in T_{(x_{i},x_{i+1})}L_{C}$ iff
$[v_{i+1}]=\underline{A}_{i}[v_{i}]$. Therefore,

$(v,v^{\prime})\in
T_{(x,x^{\prime})}L_{C}\iff[v^{\prime}]=\underline{A}_{n-1}\dots\underline{A}_{1}\underline{A}_{0}[v]\iff
v^{\prime}-A_{n-1}\dots A_{1}A_{0}v\in T_{x^{\prime}}C^{\perp}.$

Given $v\in T_{x}C$, we set $[v_{i+1}]\mathrel{\mathop{\mathchar
58\relax}}=\underline{A}_{i}\dots\underline{A}_{1}\underline{A}_{0}[v]$ and
choose representatives $v_{i+1}\in[v_{i+1}]$. For
$(v,v^{\prime}),(w,w^{\prime})\in T_{(x,x^{\prime})}L$, we then have

$\omega_{x_{i+1}}(v_{i+1},w_{i+1})-\omega_{x_{i}}(v_{i},w_{i})=0\ \forall
i=0,\dots,n-1.$

Summing over $i$, we get

$\omega_{x^{\prime}}(v^{\prime},w^{\prime})-\omega_{x}(v,w)=0,$

which shows that $T_{(x,x^{\prime})}L_{C}$ is isotropic.

In the finite-dimensional case, we easily check, as in the proof of Lemma
A.15, that $\dim L=\frac{1}{2}\dim(\bar{M}\times M)$. ∎

We also have the following straightforward generalization of Lemma A.16.

######  Lemma A.30.

Let $L$ be a lagrangian submanifold of $\bar{M}\times M$. If
$C\mathrel{\mathop{\mathchar 58\relax}}=\pi_{1}(L)$, where $\pi_{1}$ is the
projection to the first factor, is a submanifold of $M$, then $C$ is
coisotropic.

###### Proof.

By definition we have that $x$ belongs to $C$ iff there is a $y\in M$ such
that $(x,y)$ belongs to $L$. Moreover, $v$ belongs to $T_{x}C$ iff there is a
$w\in T_{y}M$ such that $(v,w)$ belongs to $T_{(x,y)}L$. Now let $\tilde{v}$
be in $T_{x}C^{\perp}$, i.e., $\omega_{x}(\tilde{v},v)=0$ $\forall v\in
T_{x}C$. This implies

$\omega_{x}(\tilde{v},v)-\omega_{y}(0,w)=0\ \forall(v,w)\in T_{(x,y)}L.$

Therefore, $(\tilde{v},0)\in T_{(x,y)}L^{\perp}=T_{(x,y)}L$, which implies
$\tilde{v}\in T_{x}C$. ∎

######  Remark A.31 (Applications in field theory).

The above two lemmata, A.29 and A.30, are important for the applications in
field theory. In the finite-dimensional case, we have assumed in Section 4,
see Remark 4.1, that the constraints define a coisotropic submanifold $C$.
Then the evolution relation $L$ coincides with $L_{C}$ and is therefore
lagrangian by Lemma A.29. In the infinite-dimensional case, it is instead
better to start from the evolution relation $L$, which is automatically
isotropic. If we can prove that it is lagrangian (e.g., by the Hodge
decomposition theorem), as we can do in all the examples in Section 7, we then
get by Lemma A.30 that $C$ is coisotropic. Finally, we just have to check that
$L=L_{C}$.

If $M_{i}=\tilde{T}^{*}N_{i}$, $i=1,2$, then we may identify
$\bar{M}_{1}\times M_{2}$ with $\tilde{T}^{*}(N_{1}\times N_{2})$ (changing
the sign of the $\tilde{T}^{*}N_{1}$ components of the canonical symplectic
form). A (split) lagrangian submanifold $L$ of $\bar{M}_{1}\times M_{2}$ might
have a (generalized) generating function $\psi$ on $N_{1}\times N_{2}$, which
is then called a (generalized) generating function for the (split) canonical
relation $L$ from $M_{1}$ to $M_{2}$.

######  Remark A.32 (Generating function for a canonical relation).

Explicitly, if we denote by $q^{i}$ and $Q^{i}$ the coordinates on $N_{1}$ and
$N_{2}$, respectively, and by $p_{i}$ and $P_{i}$ their momenta, the canonical
relation defined by a generating function $\psi$ is given by the equations

$P_{i}=\frac{\partial\psi}{\partial Q^{i}},\qquad
p_{i}=-\frac{\partial\psi}{\partial q^{i}}.$

######  Example A.33 (Generating function for a change of polarization).

A particular case is when $M_{1}=M_{2}$ and $L$ is the graph of the identity
map, which is obviously a symplectomorphism, but we choose $N_{1}$ and $N_{2}$
to be different. If there is a generating function $\psi$ for $L$, with
respect to these splittings, then the coordinates on $\tilde{T}^{*}N_{1}$ and
$\tilde{T}^{*}N_{2}$ are related by the last equation.

######  Remark A.34 (Generalized generating function for a canonical
relation).

If we have instead a generalized generating function $\psi$ on $N_{1}\times
N_{2}\times B$, with coordinates $q^{i},Q^{i},T^{\alpha}$, then the
corresponding canonical relation is defined by the equations

$P_{i}=\frac{\partial\psi}{\partial Q^{i}},\qquad
p_{i}=-\frac{\partial\psi}{\partial q^{i}},\qquad\frac{\partial\psi}{\partial
T^{\alpha}}=0.$

######  Remark A.35 (Composition of canonical relations).

If we have a relation $L_{1}$ from $M_{1}$ to $M_{2}$ and a relation $L_{2}$
from $M_{2}$ to $M_{3}$ (i.e., $L_{1}\subseteq M_{1}\times M_{2}$ and
$L_{2}\subseteq M_{2}\times M_{3}$), we may set-theoretically compose them to
a relation $L_{2}\circ L_{1}$ from $M_{1}$ to $M_{3}$:

(125) $L_{2}\circ L_{1}\mathrel{\mathop{\mathchar
58\relax}}=\\{(x_{1},x_{3})\in M_{1}\times M_{3}\ |\ \exists x_{2}\in
M_{2}\mathrel{\mathop{\mathchar 58\relax}}(x_{1},x_{2})\in
L_{1}\,\mathrm{and}\,(x_{2},x_{3})\in L_{2}\\}.$

If $M_{1}$, $M_{2}$ and $M_{3}$ are manifolds and $L_{1}$ and $L_{2}$ are
submanifolds, there is no guarantee that $L_{2}\circ L_{1}$ is also a
submanifold. Let us assume this to be the case and also that $L_{1}$ and
$L_{2}$ are (split) canonical relations—i.e., $M_{1}$, $M_{2}$ and $M_{3}$ are
symplectic manifolds and $L_{1}$ and $L_{2}$ are (split) lagrangian
submanifolds of $\bar{M}_{1}\times M_{2}$ and $\bar{M}_{2}\times M_{3}$,
respectively. Then the composition (125) may be realized as a particular case
of Remark A.26. Namely, we consider the (split) lagrangian submanifold
$L\mathrel{\mathop{\mathchar 58\relax}}=L_{1}\times L_{2}$ and the coisotropic
submanifold $C\mathrel{\mathop{\mathchar
58\relax}}=M_{1}\times\Delta_{M_{2}}\times M_{3}$ in $\bar{M}_{1}\times
M_{2}\times\bar{M}_{2}\times M_{3}$, where
$\Delta_{M_{2}}\mathrel{\mathop{\mathchar 58\relax}}=\\{(x_{2},x_{2})\in
M_{2}\times M_{2}\\}$ denotes the diagonal submanifold. We clearly have
$\underline{C}=\bar{M}_{1}\times M_{3}$ and $\underline{L}=L_{2}\circ L_{1}$.
Therefore, in the finite-dimensional case $L$ is lagrangian, whereas in the
infinite-dimensional case, at least it is isotropic and one has to check
separately if it is also (split) lagrangian.

######  Remark A.36 (Composition of generating functions I: mixed
polarizations).

Let us assume that the composition of the (split) canonical relations $L_{1}$
and $L_{2}$ of Remark A.35 is a (split) canonical relation. Let us also assume
that $L_{1}$ and $L_{2}$ have (generalized) generating functions $\psi_{1}$
and $\psi_{2}$ in the following way. We assume $M_{i}=\tilde{T}^{*}N_{i}$,
$i=1,3$, $M_{2}=U\times U^{\prime}$ with $U$ and $U^{\prime}$ open subsets of
$V$ and $V^{\prime}\subseteq V^{*}$, respectively, and that $\psi_{1}\in
C^{\infty}(N_{1}\times U^{\prime}\times B)$ and $\psi_{2}\in
C^{\infty}(U\times N_{3}\times\tilde{B})$, where $B$ and $\tilde{B}$ are
parameter spaces. For notational simplicity, we work as in the finite-
dimensional case and avoid writing component indices. We denote the
coordinates as follows: $(p,q)$ for $\tilde{T}^{*}N_{1}$,
$(\tilde{p},\tilde{q})$ for $\tilde{T}^{*}N_{3}$, $T$ for $B$, and $\tilde{T}$
for $\tilde{B}$. For the first copy of $M_{2}$ we denote the coordinates in
$U\times U^{\prime}$ by $(Q,P)$ and for the second copy by
$(\hat{Q},\hat{P})$. Then $\psi_{1}$ is a function of $q,P,T$ and $L_{1}$ is
given by the equations

$Q=-\frac{\partial\psi_{1}}{\partial P},\quad
p=-\frac{\partial\psi_{1}}{\partial q},\quad\frac{\partial\psi_{1}}{\partial
T}=0,$

whereas $\psi_{2}$ is a function of $\hat{Q},\tilde{q},\tilde{T}$ and $L_{2}$
is given by the equations

$\tilde{p}=\frac{\partial\psi_{2}}{\partial\tilde{q}},\quad\hat{P}=-\frac{\partial\psi_{2}}{\partial\hat{Q}},\quad\frac{\partial\psi_{2}}{\partial\tilde{T}}=0.$

Since $C$ is defined by the equations $\hat{P}=P$ and $\hat{Q}=Q$, we see that
$L$ has generalized generating function, on $N_{1}\times N_{3}\times
B\times\tilde{B}\times U^{\prime}\times U$,

$\psi_{3}(q,\tilde{q};T,\tilde{T},P,Q)=\psi_{1}(q,P;T)+\psi_{2}(Q,\tilde{q};\tilde{T})+PQ,$

where the last term is defined via the pairing. Note that this is a
generalized generating function with parameter space $B\times\tilde{B}\times
M_{2}$. The conditions with respect to $M_{2}$, given by

$\frac{\partial\psi_{1}}{\partial
P}+Q=0\quad\text{and}\quad\frac{\partial\psi_{2}}{\partial Q}+P=0,$

are precisely the first condition for $L_{1}$ and the second condition for
$L_{2}$. If we can solve these equations with respect to $(P,Q)$, we also get
a (generalized) generating function $\underline{\psi}_{3}\in
C^{\infty}(N_{1}\times N_{3}\times B\times\tilde{B})$, which is the critical
value of $\psi_{3}$ at this point. Note that if $\psi_{1}$ and $\psi_{2}$ are
not generalized (i.e., $B$ and $\tilde{B}$ are one-point sets), then
$\underline{\psi}_{3}$ is also not generalized.

######  Remark A.37 (Composition of generating functions II: same
polarization).

In the setting of the preceding Remark A.36, we now assume that $\psi_{1}\in
C^{\infty}(N_{1}\times U\times B)$ and $\psi_{2}\in C^{\infty}(U\times
N_{3}\times\tilde{B})$. Now $\psi_{1}$ is a function of $q,Q,T$ and $L_{1}$ is
given by the equations

$P=\frac{\partial\psi_{1}}{\partial Q},\quad
p=-\frac{\partial\psi_{1}}{\partial q},\quad\frac{\partial\psi_{1}}{\partial
T}=0,$

whereas, as before, $\psi_{2}$ is a function of $\hat{Q},\tilde{q},\tilde{T}$
and $L_{2}$ is given by the equations

$\tilde{p}=\frac{\partial\psi_{2}}{\partial\tilde{q}},\quad\hat{P}=-\frac{\partial\psi_{2}}{\partial\hat{Q}},\quad\frac{\partial\psi_{2}}{\partial\tilde{T}}=0.$

Setting $\hat{P}=P$ and $\hat{Q}=Q$, yields

$\frac{\partial\psi_{1}}{\partial Q}+\frac{\partial\psi_{2}}{\partial Q}=0.$

Therefore, the composition $L$ has generalized generating function, on
$N_{1}\times N_{3}\times B\times\tilde{B}\times U$,

$\psi_{3}(q,\tilde{q};T,\tilde{T},Q)=\psi_{1}(q,Q;T)+\psi_{2}(Q,\tilde{q};\tilde{T}).$

Note that now the parameter space is $B\times\tilde{B}\times U$. Also in this
case, it might be possible to solve the critical equation in $Q$ and get a
(generalized) generating function $\underline{\psi}_{3}\in
C^{\infty}(N_{1}\times N_{3}\times B\times\tilde{B})$, which is not
generalized if $\psi_{1}$ and $\psi_{2}$ are not.

######  Remark A.38 (Evolution relations and generalized generating functions
in field theory).

Let us conclude with the application to field theory [38, 18]. Under some
regularity assumptions, a lagrangian field theory in $d+1$ dimensions produces
a (usually infinite-dimensional) symplectic manifold
$\mathcal{F}^{\partial}_{\Sigma}$ of boundary fields on every closed
$d$-manifold $\Sigma$ (roughly speaking, this is the space of fields and some
of their normal jets on which the boundary one-form obtained after integration
by parts from the variation of the action functional is the potential for a
symplectic form). The field theory on $I\times\Sigma$, where $I$ is an
interval, produces a subset $L$, which we call the evolution relation, of
$\mathcal{F}^{\partial}_{\Sigma}\times\mathcal{F}^{\partial}_{\Sigma}$ given
by the boundary data corresponding to possible solutions to the EL equations.
If $L$ is a submanifold, it is then automatically isotropic in
$\bar{\mathcal{F}^{\partial}_{\Sigma}}\times\mathcal{F}^{\partial}_{\Sigma}$.
In a good field theory, we require $L$ to be split lagrangian, so a canonical
relation from $\mathcal{F}^{\partial}_{\Sigma}$ to itself. (In the case of a
field theory with nondegenerate lagrangian, $L$ is the graph of the associated
hamiltonian flow, so it is an embedded split canonical relation.) In this
case, there might be a (generalized) generating function for $L$ with respect
to a split symplectic structure on $\mathcal{F}^{\partial}_{\Sigma}$.
Throughout this paper we have constructed (generalized) generating functions
for several examples from field theory. The discussion of the first part of
this paper shows how to construct a generalized generating function for the
evolution relation in terms of the HJ action. Note that in the infinite-
dimensional case one has to show separately (e.g., by Hodge decomposition
theorem) that it is actually split lagrangian.

######  Remark A.39 (Gauge-fixing fermion).

Another application of the formalism of (generalized) generating functions
occurs in the gauge-fixing procedure in the BRST or BV formalism, where the
symplectic manifold is actually a supermanifold and the symplectic form is odd
with respect to the internal grading. In this case a generating function
$\psi$ for a gauge-fixing lagrangian $L_{\psi}$ must be odd, and for this
reason it is usually called a gauge-fixing fermion. Usually, there is no
generating function on the base $\mathcal{F}$ of the BV space of fields
$\Pi\tilde{T}^{*}\mathcal{F}$, so one really has to resort to a generalized
generating function on $\mathcal{F}\times B$, where $B$ is a space of
parameters. For example, in abelian gauge theories (on a trivial principal
bundle) on a manifold $M$, one has
$\mathcal{F}=\Omega^{1}(M)\times\Pi\Omega^{0}(M)\ni(A,c)$, where $A$ is the
connection and $c$ is the ghost. We denote the corresponding momenta, with
opposite parity, by $A^{+}$ and $c^{+}$. To impose the Lorenz gauge fixing
$\mathrm{d}^{*}A=0$ (working in the complement of cohomology) actually means
defining $L$ as the conormal bundle of the submanifold of $\mathcal{F}$
defined by this equation, which clearly does not have a generating function.
One therefore introduces extra parameters $\bar{c}\in
B=\Pi\Omega^{\text{top}}(M)$ and the generalized generating function (the
gauge-fixing fermion) $\psi=\int_{M}\bar{c}\,\mathrm{d}^{*}A$. The
corresponding gauge-fixing lagrangian $L=\underline{L}_{\psi}$ is then
correctly given by the equations

$A^{+}=\mathrm{d}^{*}\bar{c},\quad c^{+}=0,\quad\mathrm{d}^{*}A=0.$

In field theory, one prefers to introduce the last condition (namely,
$\bar{c}^{+}=\frac{\partial\psi}{\partial\bar{c}}=0$) in terms of a Lagrange
multiplier $\lambda\in B^{\prime}=\Omega^{0}(M)$,

$\delta(\bar{c}^{+})=\int
D\lambda\mathrm{e}^{\frac{\mathrm{i}}{\hbar}\int_{M}\lambda\bar{c}^{+}},$

and then to regard $\int_{M}\lambda\bar{c}^{+}$ as a new term to be added to
the action. The extended BV space of fields eventually is
$\Pi\tilde{T}^{*}(\mathcal{F}\times B\times B^{\prime})$. (For a slightly more
general presentation of this, see [12, Sect. 3.2.1].)

###  A.3. Generalized Hamilton–Jacobi actions with infinite-dimensional
targets

We now give more details on the constructions of Section 7, in the case of an
infinite-dimensional target. This is based on results of [17] and [23].

The first thing we have to generalize is the target $T^{*}\mathbb{R}^{n}$ and
the term $p_{i}\mathrm{d}q^{i}$ in the action. We focus on the case when the
target is linear (as this is the case for all the examples in this paper) and
we take it to be a split symplectic space $V=W\oplus W^{\prime}$ as in Remark
A.6. More concretely, we assume we have it realized with $W^{\prime}\subseteq
W^{*}$ and canonical symplectic form $\omega$ induced from the pairing as in
(122). We may also write $\omega$ as the differential of the $1$-form $\alpha$
on $V$ defined by

$(\alpha(v\oplus a),w\oplus b)\mathrel{\mathop{\mathchar 58\relax}}=a(w).$

The map $(p,q)\colon[t_{a},t_{b}]\to T^{*}\mathbb{R}^{n}$ is now generalized
to a map $\phi\colon[t_{a},t_{b}]\to V$ and the term in the action
$\int_{t_{a}}^{t_{b}}p_{i}\mathrm{d}q^{i}$ is generalized to
$\int_{t_{a}}^{t_{b}}\phi^{*}\alpha$.

Next we have to generalize the Lagrange multipliers $e^{\alpha}$, the
contraints $H_{\alpha}$ and the second term in the action:
$\int_{t_{a}}^{t_{b}}e^{\alpha}H_{\alpha}$. For this we take a (possibly
infinite-dimensional) vector space $U$ and a subspace $U^{\prime}$ of its dual
$U^{*}$ such that the canonical pairing $\langle\ ,\ \rangle$ of $U^{\prime}$
with $U$ is still weakly nondegenerate. We assume we are given a (possibly
nonlinear) map $H\colon V\to U^{\prime}$ and introduce $e$ as a $1$-form on
$[t_{a},t_{b}]$ taking values in $U$. We can then write the action as

$S[\phi,e]=\int_{t_{a}}^{t_{b}}(\phi^{*}\alpha-\langle e,H\circ\phi\rangle).$

We denote by $\mathcal{F}$ the space of fields $(\phi,e)$ as an infinite-
dimensional Fréchet manifold and assume $S$ to be a smooth function on
$\mathcal{F}$. Integrating by parts, we may express its differential as

(126) $\delta S=\text{\sl EL}+\pi_{b}^{*}\alpha-\pi_{a}^{*}\alpha,$

where EL is again expressed as an integral over $[t_{a},t_{b}]$ and
$\pi_{b},\pi_{a}\colon\mathcal{F}\to V$ are the surjective submersions
$\pi_{b}\colon(\phi,e)\mapsto\phi(t_{b})$ and
$\pi_{a}\colon(\phi,e)\mapsto\phi(t_{a})$.

To define the EL equations we have to assume that, for every $x\in V$,
$\mathrm{d}_{x}H\in V^{*}\otimes U^{\prime}$ lies in the image of
$\omega^{\sharp}\otimes\operatorname{Id}\colon V\otimes U^{\prime}\to
V^{*}\otimes U^{\prime}$. We then have a unique $X_{x}\in V\otimes U^{\prime}$
such that $\omega^{\sharp}(X_{x})=\mathrm{d}_{x}H$. The EL equations then read

$\mathrm{d}\phi=\langle e,X\rangle,\quad H\circ\phi=0.$

Again we call the first equation the evolution equation and the second the
constraint.

######  Remark A.40.

In the case of field theory, the existence of the family $X$ of hamiltonian
vector fields is ensured by locality. Namely, $V$ and $U$ are (modeled on)
spaces of sections of fiber bundles over a closed manifold $\Sigma$. The
fields $(\phi,e)$ are then sections of fiber bundles over
$[t_{a},t_{b}]\times\Sigma$. Finally, $H$ and $\alpha$ are assumed to be local
(over $\Sigma$).

We can now define the evolution relation $L$ as the set of pairs of points in
$V$ that can be connected by solutions of the EL equations. We will assume $L$
to be a (possibly immersed) lagrangian submanifold.565656For the boundary
value problem to be well-defined, one should actually ask for $L$ to be split
lagrangian, but this is not needed in the present section. Note that $L$ does
not depend on the choice of splitting of $V$, since the EL equations are
defined only in terms of $H$ and the symplectic form.

######  Remark A.41.

That $L$ is isotropic is guaranteed, under some regularity assumptions.
Indeed, we have $L=\pi_{a}\times\pi_{b}(\mathcal{EL})$, where
$\mathcal{EL}\subset\mathcal{F}$ is the set of solutions to the EL equations.
If we assume that $\mathcal{EL}$ is a (possibly immersed) submanifold, then
(126) implies that $L$ is isotropic. In fact, with
$\pi\mathrel{\mathop{\mathchar 58\relax}}=\pi_{a}\times\pi_{b}$ and
$\alpha_{\bar{V}\oplus V}\mathrel{\mathop{\mathchar
58\relax}}=(-\alpha,\alpha)$, we have

$\iota_{\mathcal{EL}}^{*}\delta S=\pi^{*}\iota_{L}^{*}(\alpha_{\bar{V}\oplus
V}),$

with $\iota$s denoting the inclusion maps. This implies
$\pi^{*}\iota_{L}^{*}(\delta\alpha_{\bar{V}\oplus V})=0$ and, therefore,
$\iota_{L}^{*}(\delta\alpha_{\bar{V}\oplus V})=0$ because $\pi$ is a
submersion.

Let $\pi_{i}\colon V\oplus V\to V$, $i=1,2$, denote the two projections and
set $C_{i}=\pi_{i}(L)$. It then follows that
$C_{1}=C_{2}=C\mathrel{\mathop{\mathchar 58\relax}}=H^{-1}(0)$. In fact, since
the constraints have to be satisfied everywhere, they are satisfied in
particular at the endpoints, so $C_{i}\subseteq C$. On the other hand, for
$e=0$, we have the constant solutions $(c,c)$ with $c\in C$, so $C\subseteq
C_{i}$. Since, by definition, $L\subset C_{1}\times C_{2}$, we then have
$L\subset C\times C$.575757Note that, as a relation on $V$, $L$ is symmetric
and transitive, but in general not reflexive, and that $C$ is the largest
subset of $V$ on which $L$ restricts as an equivalence relation. Next we
assume that $C$ is a (possibly immersed) submanifold of $V$.

######  Lemma A.42.

$C$ is coisotropic in $V$.

###### Proof.

Fix $u\in C$ and $\hat{u}\in\pi_{1}^{-1}(u)\subset L$. We have assumed that
$T_{\hat{u}}L$ is lagrangian, and we want to prove that
$(T_{u}C)^{\perp}\subseteq T_{u}C$. We have

$T_{u}C=\\{v\in V\ |\ \exists w\in V\mathrel{\mathop{\mathchar
58\relax}}(v,w)\in T_{\hat{u}}L\\}.$

By definition, if $\tilde{v}\in V$ belongs to $(T_{u}C)^{\perp}$, then
$\omega(\tilde{v},v)=0$ for every $v\in T_{u}C$, i.e., $\omega(\tilde{v},v)=0$
for every $(v,w)\in T_{\hat{u}}L$. This implies
$(\tilde{v},0)\in(T_{\hat{u}}L)^{\perp}$. Since $L$ is lagrangian, we get
$(\tilde{v},0)\in T_{\hat{u}}L$, so $\tilde{v}\in T_{u}C$. ∎

Observe that, for every $t\in[t_{a},t_{b}]$, the right hand side of the
evolution equation lies in $(T_{\phi(t)}C)^{\perp}$. Since $C$ is coisotropic,
this is contained in $T_{\phi(t)}C$. As a consequence, a solution of the
evolution equation starting at some point of $C$ will never leave $C$, i.e.,
if the constraints are imposed at the initial (or final) endpoint, they will
be satisfied at every time. (In the finite-dimensional case, this statement
followed from (20).)

######  Remark A.43.

The above assumptions that $V$ is split and $L$ is lagrangian are satisfied in
the case of Chern–Simons theory thanks to the Hodge decomposition of
differential forms (actually one even proves that $L$ is split lagrangian).
One can also easily check that $L$ and $C$ are submanifolds.

We now move to the HJ action. For this we might want to choose a different
splitting $V=Z\oplus Z^{\prime}$ at the final endpoint. The HJ action as a
generalized generating function for $L$ will depend on the choices of
splitting, even though $L$ does not. We denote by $\beta$ the $1$-form
corresponding to this splitting. We assume that we have a generating function
$f$ on $W\oplus Z$ for the change of polarization, as in Example A.33. In
particular, $\alpha=\beta+\delta f$. If we define

$S^{f}\mathrel{\mathop{\mathchar 58\relax}}=S-\pi_{b}^{*}f,$

we get

$\delta S^{f}=\text{\sl EL}+\pi_{b}^{*}\beta-\pi_{a}^{*}\alpha.$

Finally we define $S^{f}_{\text{HJ}}$, as a function on $W\times
Z\times\Omega^{1}([t_{a},t_{b}],U)$, as $S^{f}$ evaluated on the assumedly
unique solution $\bar{\phi}$ of the evolution equation with given endpoint
conditions for $\phi$ in $W$ and $Z$. We then get

$\delta S^{f}_{\text{HJ}}=-\int_{t_{a}}^{t_{b}}\langle\delta
e,H\circ\bar{\phi}\rangle+\pi_{b}^{*}\beta-\pi_{a}^{*}\alpha,$

which shows that $S^{f}_{\text{HJ}}$ is a generalized generating function for
$L$. In general, it may be difficult to characterize the variations of $e$
that leave $S^{f}_{\text{HJ}}$ invariant, since we cannot use (20). This may
be however done explicitly, e.g., when $U$ is a Lie algebra and $H\colon V\to
U^{\prime}$ is an equivariant momentum map. In this case, which is enough to
treat Chern–Simons theory, one easily sees that $\delta
e=\mathrm{d}\gamma+[e,\gamma]$, with $\gamma$ a map $[t_{a},t_{b}]\to U$ that
vanishes at the endpoints, leaves $S^{f}_{\text{HJ}}$ invariant.

##  Appendix B The modified differential quantum master equation

Following [15] and [19, Section 4.2], we explain how the mQME and the
(appropriate notion of) invariance under deformations of gauge fixing arise
from the perturbative treatment of the theory. We will be rather sketchy and
rely on some knowledge of such references.

###  B.1. Assumptions

We focus on the case where the constraints are in involution with structure
constants; i.e., the constraints $H_{\alpha}$ are the components of an
equivariant momentum map as in (19):

(127)
$\\{H_{\alpha},H_{\beta}\\}=f_{\alpha\beta}^{\gamma}H_{\gamma},\quad\forall\alpha,\beta.$

Moreover, if we change the polarization via a generating function $f(q,Q)$, we
define the transformed constraints $\tilde{H}_{\alpha}$ as in (112), so we
have the relations

(128) $H_{\alpha}\left(\frac{\partial f}{\partial
q},q\right)=\tilde{H}_{\alpha}\left(-\frac{\partial f}{\partial
Q},Q\right),\quad\forall\alpha.$

For example, if we just interchange the $p$ and $q$ variables—$Q=p$, $P=-q$,
$f=qQ$—then we have the definition
$\tilde{H}_{\alpha}(P,Q)\mathrel{\mathop{\mathchar
58\relax}}=H_{\alpha}(Q,-P)$ and the relation
$H_{\alpha}(Q,q)=\tilde{H}_{\alpha}(-q,Q)$.

As already mentioned several times in the paper, our result is based on the
following

######  Assumption B.1.

We assume that the constraints $H_{\alpha}$ are quantized to the components of
a quantum equivariant momentum map as in (88), i.e,

(129) $\left[\widehat{H}_{\alpha},\widehat{H}_{\beta}\right]=\mathrm{i}\hbar
f^{\gamma}_{\alpha\beta}\widehat{H}_{\gamma},$

with

$\widehat{H}_{\alpha}=H_{\alpha}\left(\mathrm{i}\hbar\frac{\partial}{\partial
q},q\right),$

and that they are compatible, at the quantum level, with the generating
function $f(q,Q)$ of the change of polarization as in (113), i.e,

(130) $H_{\alpha}\left(\mathrm{i}\hbar\frac{\partial}{\partial
q},q\right)\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f}=\tilde{H}_{\alpha}\left(-\mathrm{i}\hbar\frac{\partial}{\partial
Q},Q\right)\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f},\quad\forall\alpha.$

######  Remark B.2.

Observe the following:

  1. (1)

In the above formulae, the standard ordering is assumed, i.e., all the
derivatives are placed to the right.

  2. (2)

The assumptions (129) and (130) can equivalently be written using star
products.

  3. (3)

The classical relations (127) automatically imply the quantum relations (129)
if the constraints $H_{\alpha}$ are linear or biaffine.

  4. (4)

The classical relations (128) automatically imply the quantum relations (130)
if

    1. (a)

the endpoint polarizations are linear (in particular, $f$ is bilinear in
$(q,Q)$), or

    2. (b)

the constraints $H_{\alpha}(p,q)$ are linear in $p$ and the transformed
constraints $\tilde{H}_{\alpha}(P,Q)$ are linear in $P$.

  5. (5)

If the assumptions (129) and (130) are not satisfied on the nose, one may try
to deform the $H_{\alpha}$s and the $\tilde{H}_{\alpha}$s with
$\hbar$-corrections in order to impose the relations, but there is no
guarantee that this is possible.

  6. (6)

If the change of polarization is not linear,
$\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f}$ might not be the integral kernel of
a unitary operator. One might try to deform $f$, if possible, with
$\hbar$-corrections in order to make the operator unitary. One should then try
to find deformations of the $H_{\alpha}$s and the $\tilde{H}_{\alpha}$s so as
to achieve the relations (129) and (130). Note that the classical condition of
point (4b) that the nondeformed $H_{\alpha}$s and $\tilde{H}_{\alpha}$s are
linear in the respective momenta is no longer enough for the relation (130) to
be satisfied on the nose, since the $\tilde{H}_{\alpha}$ are defined using the
nondeformed $f$.

###  B.2. The propagator and the mdQME

The starting point for our discussion is the propagator $\eta$, which is an
inverse of (more precisely, a parametrix for) the de Rham differential
$\mathrm{d}$, compatible with the endpoint conditions.

For example, in the quantization in Case II, the propagator is
$\eta(s,t)=\theta(s-t)$—the Heaviside step function—for all fields, see (93).
In Case I, because of the different endpoint conditions (and the choice of
$e$), the propagator in the ghost sector is $\eta(s,t)=\theta(s-t)-\phi(s)$,
where $\phi$ is a chosen function with the property
$\phi(t_{b})-\phi(t_{a})=1$.

In general, the propagator $\eta$ is a smooth function on the compactified
configuration space, see [31, 4], of two points on the interval
$I=[t_{a},t_{b}]$, i.e.,

$C_{2}(I)=\\{(s,t)\in I\times I\ |\ s\leq t\\}\sqcup\\{(s,t)\in I\times I\ |\
s\geq t\\},$

that satisfies the endpoint conditions and the limit $\eta|_{s\to
t^{+}}-\eta|_{s\to t^{-}}=1$ and such that $\mathrm{d}\eta=\chi$, where $\chi$
is a smooth $1$-form on $I\times I$ containing information on the cohomology
and the endpoint conditions.

The endpoint conditions are fixed throughout. However, the choices of $\chi$
and $\eta$ are not unique and correspond to different gauge fixings and
choices of residual fields. We can actually change $\eta$ by adding to it a
smooth function $g$ on $I\times I$ that satisfies the endpoint conditions and
simultaneously change $\chi$ by adding $\mathrm{d}g$ to it. It is convenient
to interpolate between the two propagators using a parameter $u\in
J\mathrel{\mathop{\mathchar 58\relax}}=[0,1]$ defining
$\eta_{u}\mathrel{\mathop{\mathchar 58\relax}}=\eta+ug$ and
$\chi_{u}=\chi+u\mathrm{d}g$. It is even better to define $\check{\eta}\in
C^{\infty}(C_{2}(I)\times J)$ as $\check{\eta}(s,t,u)=\eta_{u}(s,t)$ and
$\check{\chi}\in\Omega^{1}(C_{2}(I)\times J)$ via585858In our theory we use
two propagators—the propagator in the physical sector $(p,q,p^{+},q^{+})$ and
the propagator in the ghost sector $(e^{+},c,c^{+},e)$—and we may want to take
different choices for them. This means that we will have a pair
$(\check{\eta},\check{\chi})$ for the physical sector and a possibly different
one for the ghost sector.

(131) $\mathrm{d}\check{\eta}=\check{\chi},$

where now $\mathrm{d}$ denotes the total differential on
$\Omega^{\bullet}(C_{2}(I)\times J)$.

As the choice of $\chi_{u}$ is related to the choice of the residual fields,
we have to deform them accordingly. We next compute the partition “form” for
these choices and we denote it as $\check{Z}\in\Omega^{\bullet}(J)$. Note that
the zero-form component is the usual partition function, depending on the
parameter $u$.

We claim that (131) and Assumption B.1 imply that the partition form satisfies
the differential mQME (mdQME)

$(\Omega+\mathrm{i}\hbar\mathrm{d}_{J}+\hbar^{2}\Delta)\check{Z}=0,$

where $\mathrm{d}_{J}=\mathrm{d}u\frac{\partial}{\partial u}$ is the de Rham
differential on $J$. The zero-form part of the mdQME is the mQME for the
partition function (for every value of $u$), whereas the one-form part implies
that the partition function changes by an $(\Omega+\hbar^{2}\Delta)$-exact
term under a deformation of the data. These are the two statements we wanted
to prove. Therefore, we are just left with showing that the mdQME holds.

###  B.3. Proof of the mdQME

To prove this we simply apply $\mathrm{i}\hbar\mathrm{d}_{J}$ to $\check{Z}$
and observe that, for every integral in the Feynman diagram expansion, we can
use the generalized Stokes’s formula

$\mathrm{d}_{J}\int\ =\pm\int\mathrm{d}\ \ \pm\int_{\partial}\ ,$

where the signs depend on dimensions (number of vertices) and on the choices
of orientation.

The first contribution on the right hand side replaces the propagators, one by
one, by $\hat{\chi}$. The sum of all these diagrams corresponds to the
application of $\mathrm{i}\hbar\Delta$.

The second contribution is related to $\Omega$, as we are going to see. First
observe that we are considering integrals over boundaries of compactified
configuration spaces $C_{n}(I)$ of $n$ points on the interval
$I=[t_{a},t_{b}]$. Such boundaries are of two types.

Internal boundaries:

In this case $k>1$ points collapse together in the interior of $I$. Such a
boundary component fibers over $C_{n-k+1}(I)$ with a $(k-2)$-dimensional
fiber.

Endpoint boundaries:

In this case $k\geq 1$ points collapse together at one endpoint of $I$. Such a
boundary component fibers over $C_{n-k}(I)$ with a $(k-1)$-dimensional fiber.

The crucial point is that the residual fields and the forms $\hat{\chi}$,
which have no discontinuities, are basic in the above fibrations. Along the
fibers we only have to integrate the product of the propagators $\hat{\eta}$
whose both vertices are in the fiber. As the propagators are zero-forms, we
may get a nonzero integral only if

  1. (1)

we are at an internal boundary component with $k=2$, or

  2. (2)

we are at an endpoint boundary component with $k=1$.

In the first case, we have three kinds of collapsing diagrams (we call
physical vertices those containing the constraint hamiltonians—and their
derivatives—and ghost vertices those containing the structure constants):

  1. i.

Two physical vertices collapse together: these diagrams—which may have any
positive number of physical propagators—produce a contribution of the form
$[H_{\alpha},\stackrel{{\scriptstyle\star}}{{,}}H_{\beta}]$.

  2. ii.

One physical vertex collapses with a ghost vertex: these diagrams produce a
contribution of the form $f_{\alpha\beta}^{\gamma}H_{\gamma}$.

  3. iii.

Two ghost vertices collapse together: these diagrams sum up to zero thanks to
the Jacobi identity.

The contributions in i. and ii. cancel each other thanks to (129) in
Assumption B.1.

Therefore, we are only left with the case of a single bulk vertex collapsing
at one endpoint. We consider the following cases:

  1. (a)

Consider first the case of a ghost vertex collapsing, say, at the initial
endpoint. It is a very simple situatation, as the fiber contains one or two
ghost propagators, connecting the vertex to initial endpoint insertions. This
corresponds to the application of
$f_{\alpha\beta}^{\gamma}\,c^{\alpha}_{a}c^{\beta}_{a}\frac{\partial}{\partial
c^{\gamma}_{a}}$ or
$f_{\alpha\beta}^{\gamma}\,b_{\gamma}^{a}\frac{\partial^{2}}{\partial
b_{\alpha}^{a}\partial b_{\beta}^{a}}$, depending on the choice of endpoint
condition. The case of the final endpoint is similar.

  2. (b)

Next consider the case of a physical vertex collapsing at the initial
endpoint, where, by convention, we always choose a linear polarization.
Depending on the ghost initial endpoint conditions, the ghost propagator
stemming from the vertex is either connected to an initial endpoint
insertion—corresponding to the insertion of a $c_{a}$—or leaves the collapsing
diagram—corresponding to the application of a derivative with respect to a
$b^{a}$. Besides, there is an arbitrary number of physical propagators
stemming from the collapsing vertex, some connecting it to an initial endpoint
insertion and some leaving the collapsing diagram. Depending on the physical
initial endpoint conditions, summing over all such diagrams corresponds to the
application of $H_{\alpha}\left(\mathrm{i}\hbar\frac{\partial}{\partial
q_{a}},q_{a}\right)$ or
$H_{\alpha}\left(p^{a},-\mathrm{i}\hbar\frac{\partial}{\partial
p^{a}}\right)$. This together with the result of (a) for the initial endpoint
corresponds to the application of $\Omega_{a}$.

  3. (c)

Finally consider the case of a physical vertex collapsing at the final
endpoint. If the polarization is linear, this is very similar to case (b) and,
together with the result of (a) for the final endpoint, corresponds to the
application of $\Omega_{b}$. In general, we have instead several propagators
connecting the collapsing vertex to several $f$ insertions. This sums up to a
final endpoint insertion of
$H_{\alpha}\left(\mathrm{i}\hbar\frac{\partial}{\partial
q},q\right)\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f}|_{q=\hat{q}(t_{b}),Q=Q_{b}}$.595959If
the constraint hamiltonians $H_{\alpha}(p,q)$ are linear in $p$, we have at
most one physical propagator connecting the collapsing vertex to a final
endpoint insertion of $f$. This then simply sums up to a final endpoint
insertion of $H_{\alpha}(p^{0},\hat{q})$, with $p^{0}=\frac{\partial
f}{\partial q}(\hat{q}(t_{b}),Q_{b})$ as in (116), times the insertion of
$\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(\hat{q}(t_{b}),Q_{b})}$. By the
assumed relation (130), we have606060If the constraint hamiltonians are linear
in the respetive momenta, this is simply
$\begin{split}\tilde{H}_{\alpha}\left(-\mathrm{i}\hbar\frac{\partial}{\partial
Q_{b}},Q_{b}\right)\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(\hat{q}(t_{b}),Q_{b})}&=\tilde{H}_{\alpha}\left(P(p^{0},\hat{q}(t_{b})),Q_{b}\right)\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(\hat{q}(t_{b}),Q_{b})}\\\
&=H_{\alpha}(p^{0},\hat{q}(t_{b}))\,\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(\hat{q}(t_{b}),Q_{b})},\end{split}$
where we have used the classical relation (128) in the second equality.

$\tilde{H}_{\alpha}\left(-\mathrm{i}\hbar\frac{\partial}{\partial
Q_{b}},Q_{b}\right)\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f(\hat{q}(t_{b}),Q_{b})}=H_{\alpha}\left(\mathrm{i}\hbar\frac{\partial}{\partial
q},q\right)\mathrm{e}^{-\frac{\mathrm{i}}{\hbar}f}|_{q=\hat{q}(t_{b}),Q=Q_{b}},$

so this contribution, together with the result of (a) for the final endpoint,
corresponds to the application of $\Omega_{b}$ as in (115b).

###  B.4. Historical remarks

The first instance of the method described in this appendix for tracking the
changes under deformation of the gauge fixing goes back to [4] in the case of
Chern–Simons theory without boundary, without residual fields and with gauge
fixings parametrized by a Riemannian metric. The method was extended in [11]
to more general propagators. In [15] the case with residual fields was first
considered, and in [19, Section 4.2] the general method delineated here—for
general AKSZ theories, in presence of boundary and with residual fields—was
introduced. By this method the boundary-related operator $\Omega$ is
constructed in terms of boundary terms as explained above. The main difference
with the present study is that there the case of dimension larger than one was
considered, where some vanishing theorems simplify the computations. A
differential, compatible with the BV structure, was explicitly introduced in
[10] to keep track of changes in the choice of background, leading to the
differential QME (dQME). The incorporation of the boundary, along the lines of
[19], leading to the mdQME was finally presented in [14].

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