$\displaystyle\delta_{Z}\mathcal{P}(\phi)=\sum_{|I|=0}^{\infty}\frac{\partial(\delta_{Z}\phi^{a})}{\partial
x^{I}}\cdot\frac{\delta
P_{\mu_{1}\cdots\mu_{p}}\big{(}\phi,\\{\partial_{J}\phi\\}_{|J|\leq
k}\big{)}}{\delta(\partial_{I}\phi^{a})}\cdot\mathrm{d}x^{\mu_{1}}\wedge\cdots\wedge\mathrm{d}x^{\mu_{p}}$

recovering the formulas implicitly used in the physics literature. The latter
will be fully justified in section 7 and Lem. 7.7.

By construction and Cor. 6.6, this is a Lie algebra action via derivations
with respect to the (graded) algebra structure of currents. In section 7, this
action will be identified with the ``Lie derivative along $\mathcal{Z}$'', in
an appropriate sense. Integrating the resulting currents along submanifolds,
we get an induced action on charges and hence on the algebra of local
functions on field space.

######  Definition 6.9 (Action of local vector fields on local functions).

The Lie algebra of local vector fields
$\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$ acts on charges, the generators of
smooth local functions $C^{\infty}_{\mathrm{loc}}(\mathcal{F})$ of Def. 3.20
via the action on the corresponding current, and is extended as a derivation
to all of $C^{\infty}_{\mathrm{loc}}(\mathcal{F})$.

(i) If $X\in\mathcal{X}_{\mathrm{ev}}(J^{\infty}_{M}F)$ is an evolutionary
vector field with induced local vector field $\mathcal{Z}=X\circ
j^{\infty}\in\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$ and
$P\in\Omega^{p,0}(J^{\infty}_{M}F)$ is a horizontal $(p,0)$-form with induced
charge $\mathcal{P}_{\Sigma^{p}}=\int_{\Sigma^{p}}P\circ j^{\infty}\in
C^{\infty}_{\mathrm{loc}}(\mathcal{F})$, then

$\displaystyle\mathcal{Z}(\mathcal{P}_{\Sigma^{p}}):=\int_{\Sigma^{p}}\mathcal{Z}(\mathcal{P})=\int_{\Sigma^{p}}\mathbb{L}_{\mathsf{pr}Z}(P)\circ
j^{\infty}\in C^{\infty}_{\mathrm{loc}}(\mathcal{F})\,.$ (154)

(ii) If $P^{\prime}\in\Omega^{p^{\prime},0}(J^{\infty}_{M})$ is an other
horizontal $(p^{\prime},0)$ form with induced charge
$\mathcal{P}^{\prime}_{\Sigma^{p^{\prime}}}$, then

$\mathcal{Z}(\mathcal{P}_{\Sigma^{p}}\cdot\mathcal{P}^{\prime}_{\Sigma^{p^{\prime}}}):=\mathcal{Z}(\mathcal{P}_{\Sigma^{p}})\cdot\mathcal{P}^{\prime}_{\Sigma^{p^{\prime}}}+\mathcal{P}_{\Sigma^{p}}\cdot\mathcal{Z}(\mathcal{P}^{\prime}_{\Sigma^{p^{\prime}}})\,.$

######  Remark 6.10 (Tangent vectors and derivations of local functionals).

Let $Z_{\phi}=\partial_{t}\phi_{t}|_{t=0}\in
T_{\phi}(\mathcal{F})=\Gamma_{M}(VF)$ be a tangent vector at
$\phi\in\mathcal{F}$, represented by an $\mathbbm{R}^{1}$-plot
$\phi_{t}\in\mathbold{\Gamma}_{M}(F)(\mathbbm{R}^{1})$.

(i) Recall, as in Rem. 2.18, there is an induced $\mathbbm{R}$-valued
derivation of local functionals

$\displaystyle C^{\infty}_{\mathrm{loc}}(\mathcal{F})$
$\displaystyle\longrightarrow\mathbbm{R}$
$\displaystyle\mathcal{P}_{\Sigma^{p}}$
$\displaystyle\longmapsto\partial_{t}(\mathcal{P}_{\Sigma^{p}}\circ\phi_{t})|_{t=0}\,.$

(ii) Contrary to the case of general smooth functions on $\mathcal{F}$ as in
Rem. 2.18, the derivation on local functionals is clearly independent of the
representative $\mathbbm{R}^{1}$-plot. To see this, notice that for each $x\in
M$

$\displaystyle\partial_{t}(P\circ j^{\infty}\phi_{t}(x))|_{t=0}$
$\displaystyle=\iota_{\partial_{t}j^{\infty}\phi_{t}(x)|_{t=0}}\mathrm{d}_{V}P|_{j^{\infty}\phi_{0}(x)}$

as in Eq. (114), where the latter `prolongated' tangent vector
$\partial_{t}j^{\infty}\phi_{t}|_{t=0}\in\Gamma_{M}(VJ^{\infty}F)$ depends
only on $\partial_{t}\phi_{t}|_{t=0}\in\Gamma_{M}(VF)$, as can be seen
explicitly in Ex. 4.7. Notice that varying pointwise in $x\in M$ this
contraction only partially defines a`$(p,0)$-form on $J^{\infty}_{M}F$', that
is only along the image of $j^{\infty}\phi$. However, it does define a top-
form on the base $M$, denoted by

$\displaystyle\partial_{t}(P\circ j^{\infty}\phi_{t})|_{t=0}$
$\displaystyle=(j^{\infty}\phi)^{*}\big{(}\iota_{\partial_{t}j^{\infty}\phi_{t}|_{t=0}}\mathrm{d}_{V}P)$

where the pullback form acts as in Lem. 5.20. Thus, on the induced local
function $\mathcal{P}_{\Sigma^{p}}$ we have

$\displaystyle\partial_{t}(\mathcal{P}_{\Sigma^{p}}\circ\phi_{t})\big{|}_{t=0}=\int_{\Sigma^{p}}\partial_{t}(P\circ
j^{\infty}\phi_{t})\big{|}_{t=0}=\int_{\Sigma^{p}}(j^{\infty}\phi)^{*}\iota_{\partial_{t}j^{\infty}\phi_{t}|_{t=0}}\mathrm{d}_{V}P\,,$

which manifestly depends only on $\phi\in\mathcal{F}$ and
$Z_{\phi}=\partial_{t}\phi_{t}|_{t=0}\in T_{\phi}\mathcal{F}$.

(iii) This observation may be used as an alternative definition of the action
of local vector fields. Indeed, for a local vector field $\mathcal{Z}$ we may
instead define $\mathcal{Z}(\mathcal{P}_{\Sigma^{p}})$ by its value on field
configurations as

$\displaystyle\mathcal{Z}(\mathcal{P}_{\Sigma^{p}})(\phi):=\partial_{t}\phi_{t}|_{t=0}(\mathcal{P}_{\Sigma^{p}})=\int_{\Sigma^{p}}\partial_{t}(P\circ
j^{\infty}\phi_{t})|_{t=0}\,,$ (155)

where $\mathcal{Z}_{\phi}=Z\circ j^{\infty}_{M}(\phi)\in
T_{\phi}(\mathcal{F})$, is be represented as

$\mathcal{Z}_{\phi}=\partial_{t}\phi_{t}|_{t=0}\,,$

for some smooth $\mathbbm{R}^{1}$-plot of fields $\phi_{t}$ (Lem. 2.17). This
recovers the expression of Def. 6.9 since

$\partial_{t}(P\circ
j^{\infty}\phi_{t})|_{t=0}=(j^{\infty}\phi)^{*}\iota_{\partial_{t}j^{\infty}\phi_{t}|_{t=0}}\mathrm{d}_{V}P=(\iota_{\mathsf{pr}Z}\mathrm{d}_{V})\circ
j^{\infty}\phi=L_{\mathsf{pr}Z}P\circ j^{\infty}\phi\;.$

(iv) The same statements apply for $\mathbbm{R}^{k}$-plots. Note that this is
in contrast with a general vector field, which might not necessarily define an
action on smooth (or local) functionals, as described in (33). From this
description, the derivation extension of Def. 6.9 is deduced as a property
instead.

(v) For a local vector field $\mathcal{Z}=\partial_{t}\mathcal{P}_{t}|_{t=0}$
arising from a 1-parameter family of local diffeomorphisms as in Ex. 6.3, the
Lie algebra action corresponds to the differentiation of the pullback map

$\mathcal{P}_{t}^{*}:C^{\infty}_{\mathrm{loc}}(\mathcal{F})\longrightarrow
C^{\infty}_{\mathrm{loc}}(\mathbbm{R}^{1}\times\mathcal{F})\,,$

and hence is the infinitesimal version of the
$\mathrm{Diff}_{\mathrm{loc}}(\mathcal{F})$ action on
$C^{\infty}(\mathcal{F})$.

At this point, we are well-equipped to define the notion of an infinitesimal
local symmetry of a local Lagrangian field theory. With this notion at hand,
the statement and proof of Noether's First and Second theorems follow easily,
as a result on the bicomplex of the infinite jet bundle $J^{\infty}_{M}F$,
which naturally pulls back to the field space as a statement about local
currents and their charges.

######  Definition 6.11 (Infinitesimal local symmetry of Lagrangian field
theory).

An evolutionary vector field $X\in\mathcal{X}_{\mathrm{ev}}(J^{\infty}_{M}F)$
is an infinitesimal symmetry of a Lagrangian density
$L\in\Omega^{d,0}(J^{\infty}_{M}F)$ if

$\displaystyle\mathbb{L}_{\mathsf{pr}Z}L=\mathrm{d}_{H}K_{Z}$ (156)

for some $K_{Z}\in\Omega^{d-1,0}(J^{\infty}_{M}F)$. Equivalently, by Lem. 5.20
and Def. 6.8, the corresponding local vector field $\mathcal{Z}=Z\circ
j^{\infty}_{M}\in\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$ is an infinitesimal
local symmetry of the corresponding Lagrangian field theory
$(\mathcal{F},\mathcal{L})$ if

$\mathcal{Z}(\mathcal{L})=\mathrm{d}_{M}(\mathcal{K}_{Z})\,,$

i.e., if it preserves the Lagrangian up to a trivial local Lagrangian.

It is easy to see that the Lie bracket $[\mathcal{Z}_{1},\mathcal{Z}_{2}]$
(Def. 6.7) of any two infinitesimal local symmetries
$\mathcal{Z}_{1},\mathcal{Z}_{2}\in\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$ of
a local field theory $(\mathcal{F},\mathcal{L})$ is also a symmetry. This
follows since

$\displaystyle\mathbb{L}_{[\mathsf{pr}Z_{1},\mathsf{pr}Z_{2}]}L$
$\displaystyle=\mathbb{L}_{\mathsf{pr}Z_{1}}\,(\mathbb{L}_{\mathsf{pr}Z_{2}}L)-\mathbb{L}_{\mathsf{pr}Z_{2}}(\mathbb{L}_{\mathsf{pr}Z_{1}}L)=\mathbb{L}_{\mathsf{pr}Z_{1}}(\mathrm{d}_{H}K_{Z_{2}})-\mathbb{L}_{\mathsf{pr}Z_{2}}(\mathrm{d}_{H}K_{Z_{1}})$
(157)
$\displaystyle=\mathrm{d}_{H}(\mathbb{L}_{\mathsf{pr}Z_{1}}K_{Z_{2}}-\mathbb{L}_{\mathsf{pr}Z_{2}}K_{Z_{1}})\,,$

where we used symmetry assumption of $\mathcal{Z}_{1},\mathcal{Z}_{2}$, and
then the Cartan calculus for evolutionary vector fields of Cor. 6.6. That is,
the subspace $\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$ of local
vector fields consisting of local symmetries of a field theory
$(\mathcal{F},\mathcal{L})$ is a Lie subalgebra

$\displaystyle\big{(}\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F}),\,[-,-]\big{)}\longhookrightarrow\big{(}\mathcal{X}_{\mathrm{loc}}(\mathcal{F}),\,[-,-]\big{)}\,,$
(158)

which is the infinitesimal version of the (smooth) subgroup inclusion
$\mathrm{Diff}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})\hookrightarrow\mathrm{Diff}_{\mathrm{loc}}$
from part (c) of Rem. 3.25.

There is a class of infinitesimal local symmetries that exist for every local
Lagrangian field theory. For reasons that will become apparent shortly, these
are called trivial infinitesimal symmetries.

######  Example 6.12 (Trivial infinitesimal symmetries).

Let $L\in\Omega^{d,0}(J^{\infty}_{M}F)$ be an arbitrary Lagrangian density.
For any bundle map
$T:J^{\infty}_{M}F\rightarrow\wedge^{2}VF\otimes\wedge^{d}TM$ over $F$, there
is an induced evolutionary vector field
$T\cdot\mathrm{EL}\in\mathcal{X}_{\mathrm{ev}}(J^{\infty}_{M}F)$ given by the
composition of bundle maps over $F$

$\displaystyle T\cdot\mathrm{EL}:J^{\infty}_{M}F\xrightarrow{\quad
T\otimes\mathrm{EL}\quad}\wedge^{2}VF\otimes\wedge^{d}TM\otimes
V^{*}F\otimes\wedge^{d}T^{*}M\xlongrightarrow{\quad\sim\quad}(\wedge^{2}VF\otimes
V^{*}F)\otimes(\wedge^{d}TM\otimes\wedge^{d}T^{*}M)\longrightarrow VF$

where the second map is the isomorphism swapping the order of the fibers,
while the last map combines the fiberwise contractions
$\iota_{(-)}(-):\wedge^{2}VF\otimes V^{*}F\rightarrow VF$ and
$\iota_{(-)}(-):\wedge^{d}TM\otimes\wedge^{d}T^{*}M\xrightarrow{\sim}M\times\mathbbm{R}$.
In local coordinates, we have $T=T^{[ab]}\cdot\frac{\partial}{\partial
u^{a}}\wedge\frac{\partial}{\partial u^{b}}\cdot\frac{\partial}{\partial
x^{1}}\wedge\cdots\wedge\frac{\partial}{\partial x^{d}}$ for some collection
of (local) functions $T^{[ab]}\in C^{\infty}(J^{\infty}_{M}F)$ antisymmetric
in the indices, and similarly
$\mathrm{EL}=\mathrm{EL}_{a}\cdot\mathrm{d}_{F}u^{a}\wedge\mathrm{d}x^{1}\cdots\mathrm{d}x^{d}$,
thus the induced `trivial' evolutionary vector field is locally

$\displaystyle
T\cdot\mathrm{EL}=T^{[ab]}\cdot\mathrm{EL}_{b}\cdot\frac{\partial}{\partial
u^{a}}\,,$

which is the form that often appears in the physics literature.575757As
customary, the global structure of $M$, $F\rightarrow M$ and induced bundles
is often ignored – hence defining a trivial symmetry simply by a collection of
functions $K^{[ab]}$ antisymmetric in the indices. This is not sufficient to
define a global evolutionary vector field on non-trivial field bundles. By
construction, this is `trivially' a symmetry of the density $L$

$\displaystyle\mathbb{L}_{\mathsf{pr}(T\cdot\mathrm{EL})}L=\iota_{\mathsf{pr}(T\cdot\mathrm{EL})}\mathrm{d}_{V}L=\iota_{\mathsf{pr}(T\cdot\mathrm{EL})}\mathrm{EL}+\iota_{\mathsf{pr}(T\cdot\mathrm{EL})}\mathrm{d}_{H}\theta_{L}=0+\mathrm{d}_{H}\big{(}-\iota_{\mathsf{pr}(T\cdot\mathrm{EL})}\theta_{L}\big{)}$

where the first term vanishes by its local coordinate expression (see also
proof of Prop. 6.14),
$\iota_{\mathsf{pr}(T\cdot\mathrm{EL})}\mathrm{EL}=T^{[ab]}\cdot EL_{a}\cdot
EL_{b}$ and using symmetry and antisymmetry of the indices, while the second
becomes horizontally exact by the commutation relation for evolutionary vector
fields. Lastly, notice that any such evolutionary vector field vanishes on the
shell $S_{L}\hookrightarrow J^{\infty}_{M}F$ of the Lagrangian (Def. 5.12).
Equivalently, the corresponding local vector field
$\mathcal{K}\cdot\mathcal{E}\mathcal{L}:=(K\cdot EL)\circ j^{\infty}=(K\circ
j^{\infty})\cdot(EL\circ j^{\infty})$ is an infinitesimal symmetry of the
field theory $(\mathcal{F},\mathcal{L})$

$\displaystyle\mathcal{T}\cdot\mathcal{E}\mathcal{L}=\mathcal{T}^{[ab]}\cdot\mathcal{E}\mathcal{L}_{b}\cdot\frac{\delta}{\delta\phi^{a}}\hskip
14.22636pt\in\hskip
14.22636pt\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})\,,$

which vanishes on the smooth subspace of on-shell fields
$\mathcal{F}_{\mathcal{E}\mathcal{L}}\hookrightarrow\mathcal{F}.$ It is easy
to see that the Lie bracket (Def. 6.7) of any two trivial infinitesimal
symmetries is again trivial. In other words, the set of all trivial
infinitesimal local symmetries is a further Lie subalgebra of (158), denoted
by

$\displaystyle(\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathrm{triv}}(\mathcal{F}),\,[-,-])\longhookrightarrow(\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F}),\,[-,-])\,.$

The integrated (finite) version of this fact further justifies the name
`trivial'.

######  Example 6.13 (Differentiating finite local symmetries).

Consider the case where $\mathcal{D}_{t}=D_{t}\circ j^{\infty}$ is a
1-parameter (finite) local symmetry of a Lagrangian field theory
$(\mathcal{F},\mathcal{L})$ as in Def. 3.23, starting at the identity, and so

$\mathcal{L}\circ\mathcal{D}_{t}=\mathcal{L}+\mathrm{d}_{M}\mathcal{K}_{t}$

for some $\mathcal{K}_{t}=K_{t}\circ j^{\infty}$, where
$K_{t}:\mathbbm{R}^{1}\times J^{\infty}_{M}F\rightarrow\wedge^{d-1}T^{*}M$ is
a 1-parameter family of smooth bundle maps over $M$, with
$K_{0}=0_{M}:J^{\infty}_{M}\rightarrow\wedge^{d-1}T^{*}M$ the zero bundle map.
Then the interested reader may verify, proceeding as in Ex. 6.3, that the
differentiated version of the relation recovers exactly that of Def. 6.11

$\mathcal{Z}(\mathcal{L})=\mathrm{d}_{M}(\mathcal{K}_{\mathcal{Z}})\,,$

with585858Strictly speaking, $\partial_{t}K_{t}|_{t=0}$ is a map
$J^{\infty}_{M}F\rightarrow
V(\wedge^{d-1}T^{*}M)\cong\wedge^{d-1}T^{*}M\times_{M}\wedge^{d-1}T^{*}M$
which covers $K_{0}=0_{M}$ via
$\mathsf{pr}_{1}:\wedge^{d-1}T^{*}M\times_{M}\wedge^{d-1}T^{*}M\rightarrow\wedge^{d-1}T^{*}M$.
Thus it is completely determined by the projection to the section factor,
which is the one we tacitly mean above.
$K_{Z}=\partial_{t}K_{t}|_{t=0}:J^{\infty}_{M}F\rightarrow\wedge^{d-1}(T^{*}M)$
and $\mathcal{Z}=Z\circ j^{\infty}$, for $Z=\partial_{t}D_{t}|_{t=0}$. In
other words, local infinitesimal symmetries of a Lagrangian field theory are
indeed the infinitesimal version of finite local symmetries. In particular, if
a trivial infinitesimal local symmetry
$\mathcal{T}\cdot\mathcal{E}\mathcal{L}$ of Ex. 6.12 is integrable, then the
corresponding symmetry $\mathcal{P}_{t}$ is necessarily the
identity595959Strictly, only for
$t\in(-\epsilon,\epsilon)\subset\mathbbm{R}^{1}$ for some
$\epsilon\in\mathbbm{R}$. on the subspace on-shell fields
$\mathcal{F}_{\mathcal{E}\mathcal{L}}$, since
$\mathcal{T}\cdot\mathcal{E}\mathcal{L}=\partial_{t}\mathcal{D}_{t}|_{t=0}$
vanishes on $\mathcal{F}_{\mathcal{E}\mathcal{L}}\hookrightarrow\mathcal{F}$.

The main and most famous application of infinitesimal local symmetries is via
Noether's First Theorem, by which each symmetry produces a conserved current.

######  Proposition 6.14 (Noether's First Theorem).

[No18] Let $Z\in\mathcal{X}_{\mathrm{ev}}(J^{\infty}_{M}F)$ be an
infinitesimal symmetry with $\mathbb{L}_{\mathsf{pr}Z}L=\mathrm{d}_{H}K_{Z}$.

(i) Then the $(d-1,0)$-form

$\displaystyle P_{Z}:=K_{Z}+\iota_{\mathsf{pr}Z}\theta_{L}$ (159)

satisfies

$\displaystyle\mathrm{d}_{H}P_{Z}=\iota_{\mathsf{pr}Z}EL=\langle
EL,Z\rangle\,,$

where $\mathrm{d}_{V}L=EL+\mathrm{d}_{H}\theta_{L}$ as in (5.1), and the
latter pairing is the duality bundle map
$\langle-,-\rangle:(\wedge^{d}T^{*}M\otimes V^{*}F)\otimes
VF\rightarrow\wedge^{d}T^{*}M$ over $M$. In particular, $P_{Z}$ is
horizontally closed on the shell of $\mathrm{S}_{L}\hookrightarrow
y(J^{\infty}_{M}F)$.

(ii) It follows the induced current

$\displaystyle\mathcal{P}_{Z}:=\mathcal{K}_{\mathcal{Z}}+\iota_{\mathcal{Z}}\theta_{\mathcal{L}}\equiv\mathcal{K}_{\mathcal{Z}}+(\iota_{\mathsf{pr}Z}\theta_{L})\circ
j^{\infty}$

satisfies

$\displaystyle\mathrm{d}_{M}\mathcal{P}_{Z}=\langle\mathcal{E}\mathcal{L},\mathcal{Z}\rangle$

and so it is conserved on the space of on-shell fields
$\mathcal{F}_{\mathrm{EL}=0}\hookrightarrow\mathcal{F}$.

###### Proof.

We have

$\displaystyle\mathrm{d}_{H}(P_{Z})$
$\displaystyle=\mathrm{d}_{H}K_{Z}+\mathrm{d}_{H}\iota_{\mathsf{pr}Z}\theta_{L}=\mathbb{L}_{\mathsf{pr}Z}L-\iota_{\mathsf{pr}Z}\mathrm{d}_{H}\theta_{L}$
$\displaystyle=\iota_{\mathsf{pr}Z}\mathrm{d}_{V}L-\iota_{\mathsf{pr}Z}\mathrm{d}_{H}\theta_{L}=\iota_{\mathsf{pr}Z}EL+\iota_{\mathsf{pr}Z}\mathrm{d}_{H}\theta_{L}-\iota_{\mathsf{pr}Z}\mathrm{d}_{H}\theta_{L}$
$\displaystyle=\iota_{\mathsf{pr}Z}EL\,,$

where the first line follows by the symmetry assumption and the second by the
commutation relation $[\mathrm{d}_{H},\iota_{\mathsf{pr}Z}\theta_{L}]=0$ for
evolutionary vector fields. The identification $\iota_{\mathsf{pr}Z}EL=\langle
EL,Z\rangle$ follows since $EL$ is a source form, and in particular
corresponds to a bundle map $J^{\infty}_{M}F\rightarrow
V^{*}F\otimes\wedge^{d}T^{*}M$ over M, according to Lem. 5.8. Explicitly,
$EL=EL^{a}\cdot\mathrm{d}_{V}u^{a}\wedge\mathrm{d}x^{1}\wedge\cdots\wedge\mathrm{d}x^{d}$
locally, and hence only the first component of
$\mathsf{pr}Z=Z^{a}\frac{\partial}{\partial u^{a}}+\sum_{|I|\geq
1}^{\infty}D_{I}(Z^{a})\frac{\partial}{\partial u^{a}_{I}}$ contributes to the
contraction, i.e., the component of its evolutionary vector field $Z$. The
induced current $\mathcal{P}_{Z}=P_{Z}\circ j^{\infty}$ is on-shell conserved
by Stokes' Theorem 5.21. In particular the formula for
$\mathrm{d}_{M}\mathcal{P}_{Z}$ follows immediately since $\langle
EL,Z\rangle\circ j^{\infty}_{M}=\langle EL\circ j^{\infty},Z\circ
j^{\infty}\rangle=\langle\mathcal{E}\mathcal{L},\mathcal{Z}\rangle.$ ∎

As with Eq. (140), we have used the notation for the pullback of the
contraction $\iota_{\mathsf{pr}Z}\theta_{L}$ to field space $\mathcal{F}$

$\displaystyle\iota_{\mathcal{Z}}\theta_{\mathcal{L}}:=(\iota_{\mathsf{pr}Z}\theta_{L})\circ
j^{\infty}\,.$ (160)

The underlying mathematical meaning of this notation will become apparent in
section 7, as an actual contraction of the local vector field
$\mathcal{Z}:=Z\circ j^{\infty}$ and the `local form $\theta_{\mathcal{L}}$'
on $\mathcal{F}\times M$ (see Eq. (7.1)). For the moment, we will only need it
and treat it as a useful notation.

The calculation may be read in reverse, implying the converse statement:
Suppose there exists a current $\mathcal{P}_{Z}$ which is on-shell preserved,
in such a way that
$\mathrm{d}_{M}\mathcal{P}_{Z}=\langle\mathcal{E}\mathcal{L},\mathcal{Z}\rangle$
for some local vector field $\mathcal{Z}=Z\circ j^{\infty}_{M}$. Then the
vector field is an infinitesimal symmetry of the local Lagrangian with
$\mathcal{Z}(\mathcal{L})=\mathrm{d}_{M}\big{(}\mathcal{P}_{Z}-(\iota_{\mathsf{pr}Z}\theta_{L}\circ
j^{\infty}_{M})\big{)}$. An immediate `trivial' application of Noether's First
Theorem is on the trivial symmetries from Ex. 6.12, whereby
$\mathcal{K}_{\mathcal{T}\cdot\mathcal{E}\mathcal{L}}=-(\iota_{\mathsf{pr}T\cdot
EL}\theta_{L})$ and hence
$\mathcal{P}_{\mathcal{T}\cdot\mathcal{E}\mathcal{L}}=0$ identically. Strictly
speaking, an infinitesimal symmetry
$\mathcal{Z}\in\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$ actually
defines a family of conserved currents: One may add an arbitrary horizontally
closed $(d-1,0)$-form $K_{Z}$ and similarly
$\iota_{\mathsf{pr}Z}\mathrm{d}_{M}T^{\prime}$ for an arbitrary horizontally
(closed and hence) exact $(d-1,1)$-form.

######  Example 6.15 (O($n$)-model conserved currents).

Consider the case of the O(n)-model Lagrangian $\mathcal{L}$ from Ex. 3.10,
and the vector fields
$\mathcal{Z}^{A}(\phi)=A^{a}_{\,b}\cdot\phi^{b}\cdot\frac{\delta}{\delta\phi^{a}}$
from Ex. 2.24 and
$\mathcal{Z}^{\nu}(\phi)=\mathbb{L}_{\nu}(\phi)^{a}\cdot\frac{\delta}{\delta\phi^{a}}=\nu^{\mu}\cdot\partial_{\mu}\phi^{a}\cdot\frac{\delta}{\delta\phi^{a}}$
from Ex. 2.25. These are local with corresponding the evolutionary vector
fields $Z^{A}=A^{a}_{\,b}\cdot u^{b}\cdot\frac{\partial}{\partial u^{a}}$ and
$Z^{\nu}(\phi)=\nu^{\mu}\cdot u^{a}_{\mu}\cdot\frac{\partial}{\partial
u^{a}}$, respectively.

(i) Choosing $A=(A^{a}_{\,b})\in\mathfrak{o}(n)$, it is immediate that
$\mathcal{Z}^{A}(\mathcal{L})(\phi):=\mathbb{L}_{\mathsf{pr}Z^{A}}(L)\big{(}j^{\infty}\phi)=0$
due to the $O(n)$-invariance of the Euclidean product $\langle-,-\rangle$ on
the target $W$. Recalling the variational decomposition $\delta
L=EL+\mathrm{d}_{H}\theta_{L}$ with
$\theta_{L}=-\langle\mathrm{d}_{V}u,\star\mathrm{d}_{H}u\rangle_{g}$ from Eq.
(132), the corresponding on-shell conserved current is

$\displaystyle\mathcal{P}_{\mathcal{Z}^{A}}(\phi)$
$\displaystyle=0+\iota_{\mathcal{Z}^{A}}\theta_{\mathcal{L}}(\phi):=(\iota_{\mathsf{pr}}\theta_{L})(j^{\infty}\phi)$
$\displaystyle=-\big{(}Z^{A,a}\wedge\star\mathrm{d}_{H}u_{a}\big{)}\circ
j^{\infty}\phi=-\mathcal{Z}^{A}(\phi)^{a}\wedge\star\mathrm{d}_{M}\phi$
$\displaystyle=-A^{a}_{\,b}\cdot\phi^{b}\wedge\star\mathrm{d}_{M}\phi_{a}\;.$

(ii) Choosing $\nu=\nu^{\mu}\cdot\frac{\partial}{\partial
x^{\mu}}\in\mathcal{X}(M)$ to be a Killing vector field of the metric $g$,
$\mathbb{L}_{\nu}(g)=0$, it follows that
$\mathcal{Z}^{\nu}(\mathcal{L})(\phi):=\mathbb{L}_{\mathsf{pr}Z^{\nu}}(L)\big{(}j^{\infty}\phi)=\mathbb{L}_{\nu}\big{(}\mathcal{L}(\phi)\big{)}=\mathrm{d}_{M}\iota_{\nu}\big{(}\mathcal{L}(\phi)\big{)}$.
The corresponding on-shell conserved current is

$\displaystyle\mathcal{P}_{\mathcal{Z}^{\nu}}(\phi)$
$\displaystyle=\iota_{\nu}\big{(}\mathcal{L}(\phi)\big{)}+\iota_{\mathcal{Z}^{\nu}}\theta_{\mathcal{L}}(\phi)$
$\displaystyle=\iota_{\nu}\big{(}\mathcal{L}(\phi)\big{)}-\mathbb{L}_{\nu}(\phi)^{a}\wedge\star\mathrm{d}_{M}\phi_{a}\,,$

which may be further expanded in local coordinates as in Ex. 3.10. In the case
of Minkowski spacetime with its Killing Lie algebra being the Poincaré Lie
algebra, these currents comprise the energy-momentum tensor.

######  Remark 6.16 (Integrating infinitesimal local symmetries).

Although every 1-parameter local diffeomorphism induces a local vector field
on field space (Ex. 6.3), it is not necessarily the case that every local
vector field integrates to a 1-parameter local diffeomorphism. Indeed, even in
the case where $\mathcal{Z}=Z\circ j^{\infty}$ for some $Z$ that globally
factors through a finite order jet $J^{k}_{M}F$, it generally only integrates
to a local flow on $J^{k}_{M}F$, which is not enough to define a
diffeomorphism on the full field space $\mathcal{F}=\mathbold{\Gamma}_{M}(F)$.
Moreover, it might be that a local vector field integrates to a spacetime
covariant diffeomorphism on $\mathcal{F}$ instead (see Lem. 6.17).
Nevertheless, we stress that Noether's First theorem (and Second, see Prop.
6.23) applies for any infinitesimal local symmetries, and the existence of
corresponding conserved charges is independent of the integrability
properties.

Let us close off this subsection by proving how the infinitesimal versions of
finite spacetime covariant symmetries (Def. 3.23) are in fact local
infinitesimal symmetries, justifying our focus – and generally of the physics
literature – on the latter.

######  Proposition 6.17 (Infinitesimal spacetime covariant symmetries are
local).

Let $\mathcal{D}_{t}$ be a smooth 1-parameter family of spacetime covariant
symmetries of a classical field theory $(\mathcal{F},\mathcal{L})$ starting at
the identity. That is,

$\mathcal{L}\circ\mathcal{D}_{t}=f_{t}^{*}\circ\mathcal{L}+f^{*}_{t}\circ\mathrm{d}_{M}\mathcal{K}_{t}$

where $\mathcal{D}_{t}=D_{t}\circ
j^{\infty}(-)\circ(\mathrm{id},f^{-1}_{t})\in[\mathcal{F},\mathcal{F}](\mathbbm{R}^{1})$
is induced by a smooth 1-parameter family of bundle maps

|
---|---
$\textstyle{\mathbbm{R}^{1}\times
J^{\infty}_{M}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{D_{t}}$$\textstyle{F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbbm{R}^{1}\times
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{t}}$$\textstyle{M\,.}$

covering a 1-parameter family of diffeomorphisms $f_{t}:\mathbbm{R}^{1}\times
M\rightarrow M$, for some $\mathcal{K}_{t}=K_{t}\circ j^{\infty}$ where
$K_{t}:\mathbbm{R}^{1}\times J^{\infty}_{M}F\rightarrow\wedge^{d-1}T^{*}M$ is
a 1-parameter family of smooth bundle maps over $M$. Then:

(i) The vector field
$\partial_{t}\mathcal{D}_{t}|_{t=0}:\mathcal{F}\rightarrow T\mathcal{F}$ is
local

$\displaystyle\partial_{t}\mathcal{D}_{t}|_{t=0}=\mathcal{Z}_{\mathrm{ev}}=Z_{\mathrm{ev}}\circ
j^{\infty}\hskip 14.22636pt\in\hskip
14.22636pt\mathcal{X}_{\mathrm{loc}}(\mathcal{F})\,,$

for an evolutionary vector field $Z_{\mathrm{ev}}:J^{\infty}_{M}F\rightarrow
VF$ over F.

(ii) The induced infinitesimal action is an infinitesimal local symmetry of
$(\mathcal{F},\mathcal{L})$

$\displaystyle\partial_{t}(\mathcal{L}\circ\mathcal{D}_{t})|_{t=0}\equiv\mathcal{Z}_{\mathrm{ev}}(\mathcal{L})=\mathrm{d}_{M}(\mathcal{K}_{Z_{\mathrm{ev}}})$

for some $K_{Z_{\mathrm{ev}}}:J^{\infty}_{M}F\rightarrow\wedge^{d-1}T^{*}M$
over M.

###### Proof.

We show the calculation at the level of $*$-plots of fields, with that of
higher $\mathbbm{R}^{k}$-plots being analogous. Running through the
differentiation of Ex. 2.23 (see also Ex. 6.3), it follows that

$\displaystyle\partial_{t}\mathcal{D}_{t}(\phi)$
$\displaystyle=\partial_{t}D_{t}|_{t=0}\circ
j^{\infty}(\phi)\,\circ\,\mathrm{id}_{M}+\partial_{t}\big{(}\pi^{\infty}_{0}\circ
j^{\infty}\phi\,\circ\,f_{t}^{-1}\big{)}|_{t=0}$
$\displaystyle=\partial_{t}\big{(}\pi^{\infty}_{0}\circ\mathsf{pr}D_{t})|_{t=0}\,\circ\,j^{\infty}\phi+\mathrm{d}(\pi^{\infty}_{0}\,\circ\,j^{\infty}\phi)\circ\big{(}\partial_{t}f^{-1}_{t}|_{t=0}\big{)}$
$\displaystyle=\mathrm{d}\pi^{\infty}_{0}\,\circ\,(\partial_{t}\mathsf{pr}D_{t}|_{t=0})\,\circ\,j^{\infty}\phi+\mathrm{d}\pi^{\infty}_{0}\,\circ\,\mathrm{d}(j^{\infty}\phi)\big{(}\partial_{t}f^{-1}_{t}|_{t=0}\big{)}$
$\displaystyle=:\mathrm{d}\pi^{\infty}_{0}\,\circ\,\mathsf{pr}Z\circ
j^{\infty}\phi-\mathrm{d}\pi^{\infty}_{0}\,\circ\,H(X)\circ j^{\infty}\phi\,,$

where the first line follows by the product rule, the second by the definition
(Def. 3.16) of the prolongation $\mathsf{pr}D_{t}:\mathbbm{R}^{1}\times
J^{\infty}_{M}F\rightarrow J^{\infty}_{M}F$ and the chain rule, while the
third again by the chain rule. The last line follows by defining the vector
field
$\mathsf{pr}Z:=\partial_{t}\mathsf{pr}D_{t}|_{t=0}:J^{\infty}_{M}F\rightarrow
TJ^{\infty}_{M}F$ and recalling606060We use
$H(X)_{j^{\infty}_{p}\phi}=H_{j^{\infty}_{p}\phi}\big{(}X(j^{\infty}_{p}\phi)\big{)}$
as a shorthand for
$H\big{(}(\pi^{\infty}_{M})^{*}X\big{)}(j^{\infty}_{p}\phi)=H_{j^{\infty}_{p}\phi}\big{(}(\pi^{\infty}_{M})^{*}X(j^{\infty}_{p}\phi)\big{)}$,
since the splitting is really a map $H:J^{\infty}_{M}F\times_{M}TM\rightarrow
TJ^{\infty}_{M}F$. that
$H_{j^{\infty}_{p}\phi}\big{(}X(j^{\infty}_{p}\phi)\big{)}:=\mathrm{d}(j^{\infty}\phi)_{p}X_{p}$
defines the horizontal lift of tangent vectors on $M$ to tangent vectors on
$J^{\infty}_{M}F$ (see Eq. (87), Prop. 4.8), where we define
$X:=\partial_{t}f_{t}|_{t=0}:M\rightarrow TM$, hence that
$\partial_{t}f^{-1}_{t}|_{t=0}=-X$.

We highlight that the vector field $\mathsf{pr}Z:J^{\infty}_{M}F\rightarrow
TJ^{\infty}_{M}F$ is not vertical and in particular not the prolongation of
some evolutionary vector field (unless $f_{t}=\mathrm{id}_{M}$). However, the
horizontal splitting of $TJ^{\infty}_{M}F$ decomposes the vector field as
$\mathsf{pr}Z=(\mathsf{pr}Z)_{V}+H(X)$, with the horizontal component being
necessarily the lift of $X$ since $Z$ covers $X$ – by construction. Working in
local coordinates (see (92)) and using the local characterization of
prolongated evolutionary vector fields (149), one sees that the vertical
component is in fact necessarily the prolongation
$(\mathsf{pr}Z)_{V}=\mathsf{pr}Z_{\mathrm{ev}}$ of an evolutionary vector
field $Z_{\mathrm{ev}}:J^{\infty}_{M}F\rightarrow VF$ over $F$ (see also
[An89, Prop. 1.20] for detailed coordinate formulas). Thus

$\displaystyle\partial_{t}\mathcal{D}_{t}|_{t=0}$
$\displaystyle=\mathrm{d}\pi^{\infty}_{0}\circ\mathsf{pr}Z_{\mathrm{ev}}\circ
j^{\infty}+\mathrm{d}\pi^{\infty}_{0}\circ H(X)\circ
j^{\infty}-\mathrm{d}\pi^{\infty}_{0}\circ H(X)\circ j^{\infty}$
$\displaystyle=Z_{\mathrm{ev}}\circ j^{\infty}=:\mathcal{Z}_{\mathrm{ev}}$

is in fact a local vector field $\mathcal{F}$. Completely analogously, it
follows that the infinitesimal action 616161A priori, the left-hand side is
defined only through Rem. 155. is given by

$\displaystyle\partial_{t}\big{(}\mathcal{L}\circ\mathcal{D}_{t}\big{)}|_{t=0}$
$\displaystyle=\mathsf{pr}Z_{\mathrm{ev}}(L)\circ
j^{\infty}+\mathbb{L}_{X}(\mathcal{L})-\mathbb{L}_{X}\circ(\mathcal{L})$
$\displaystyle=\mathcal{Z}_{\mathrm{ev}}(\mathcal{L})\,.$

with the latter action being that of Def. 6.8.

Assuming further that $\mathcal{D}_{t}$ is a 1-parameter covariant symmetry of
the Lagrangian, we furthermore get

$\displaystyle\partial_{t}\big{(}\mathcal{L}\circ\mathcal{D}_{t}\big{)}|_{t=0}$
$\displaystyle=\partial_{t}\big{(}f^{*}_{t}\circ\mathcal{L}+f^{*}_{t}\circ\mathrm{d}_{M}\mathcal{K}_{t})|_{t=0}$
$\displaystyle=\mathbb{L}_{X}(\mathcal{L})+\mathbb{L}_{X}(\mathrm{d}_{M}\mathcal{K}_{0})+\mathrm{d}_{M}(\partial_{t}\mathcal{K}_{t}|_{t=0})$
$\displaystyle=\mathrm{d}_{M}\big{(}\iota_{X}\mathcal{L}+\iota_{X}\mathrm{d}_{M}\mathcal{K}_{0}+\dot{\mathcal{K}}_{0}\big{)}\,,$

where in the first line we used the product rule, then the Cartan calculus on
$M$ and by defining $\dot{\mathcal{K}}_{0}=\dot{K}_{0}\circ j^{\infty}$ where
$\dot{K}_{0}:=\partial_{t}K_{t}|_{t=0}:J^{\infty}_{M}F\rightarrow\wedge^{d}T^{*}M$
(see Ex. 6.13). Lastly, it follows immediately in coordinates that
$\iota_{X}\mathcal{L}(\phi):=\iota_{X}(L\circ
j^{\infty}\phi)=(\iota_{H(X)}L)\circ j^{\infty}\phi$ since the Lagrangian
density $L$ is a horizontal form, and similarly for the second term. Combining
this with the compatibility of Lem. 5.20, we arrive at

$\partial_{t}\big{(}\mathcal{L}\circ\mathcal{D}_{t}\big{)}|_{t=0}=\mathrm{d}_{M}(K_{Z_{\mathrm{ev}}}\circ
j^{\infty})$

where
$K_{Z_{\mathrm{ev}}}=\iota_{H(X)}\big{(}L+\mathrm{d}_{H}K_{0}\big{)}+\dot{K}_{0}\,\in\,\Omega^{d-1,0}(J^{\infty}_{M}F)$,
which completes the proof. ∎

The above, somewhat abstract, statement applies in virtually all local field
theories with spacetime covariant symmetries (General relativity, Yang–Mills,
Chern–Simons). It is considerably easier to interpret the calculation in
explicit examples.

######  Example 6.18 (Differentiating spacetime symmetry of O($n$)-model).

Recall the O($n$)-model field theory of Ex. 3.10, with field space
$\mathcal{F}=[M,W]$ and local Lagrangian

$\mathcal{L}(\phi)=\tfrac{1}{2}\big{(}\langle\mathrm{d}_{M}\phi\,,\,\mathrm{d}_{M}\phi\rangle_{g}+c_{2}\cdot\langle\phi,\phi\rangle+\tfrac{1}{2}c_{4}\cdot(\langle\phi,\phi\rangle)^{2}\big{)}\cdot\mathrm{d}\mathrm{vol}_{g}\,,$

where $g$ is a background metric on $M$ and $\langle-,-\rangle$ an inner
product on $W$. By Ex. 3.24, any isometry $f:M\rightarrow M$ induces a
spacetime covariant symmetry via the pullback action
$\mathcal{D}=f^{*}:[M,W]\rightarrow[M,W]$

$\displaystyle\mathcal{L}\circ f^{*}(\phi)=f^{*}\circ\mathcal{L}(\phi)\,.$
(161)

Consider any 1-parameter family of isometries $f_{t}:\mathbbm{R}^{1}\times
M\rightarrow M$ starting at the identity, with induced Killing vector field
$v=\partial_{t}f_{t}|_{t=0}\in\mathcal{X}(M)$ on spacetime. Differentiating
the corresponding 1-parameter family action on $\mathcal{L}$, i.e., the left-
hand side of Eq. (161), it immediately follows that

$\displaystyle\partial_{t}\mathcal{L}(f^{*}_{t}\phi)|_{t=0}$
$\displaystyle=\big{(}\langle\mathrm{d}_{M}(\mathbb{L}_{v}\phi)\,,\,\mathrm{d}_{M}(\phi)\rangle_{g}+c_{2}\cdot\langle\mathbb{L}_{v}\phi,\phi\rangle+c_{4}\cdot(\langle\mathbb{L}_{v}\phi,\phi\rangle)^{2}\big{)}\cdot\mathrm{d}\mathrm{vol}_{g}\,.$

Next recall the vector field on field space
$\mathcal{Z}^{v}:\mathcal{F}\rightarrow T\mathcal{F}$ from Ex. 2.25
corresponding to
$f_{t}^{*}:\mathbbm{R}^{1}\times\mathcal{F}\rightarrow\mathcal{F}$. It is
immediately seen to be local as

$\mathcal{Z}^{\nu}(\phi)=\mathbb{L}_{v}(\phi^{a})\cdot\frac{\partial}{\partial
u^{a}}=Z^{\nu}\circ j^{\infty}(\phi)$

for $Z^{\nu}:J^{\infty}_{M}F\rightarrow VF$ defined locally by $\nu^{\mu}\cdot
u^{a}_{\mu}\cdot\frac{\partial}{\partial u^{a}}$. It follows by the explicit
formula above that the infinitesimal action the diffeomorphism
$f^{*}:\mathcal{F}\rightarrow\mathcal{F}$ on the Lagrangian is equivalently
given by the action of the corresponding local vector field (Def. 6.8)

$\partial_{t}\mathcal{L}(f^{*}_{t}\phi)|_{t=0}=\mathcal{Z}^{\nu}(\mathcal{L})(\phi)=\big{(}\langle\mathrm{d}_{M}(\mathbb{L}_{v}\phi)\,,\,\mathrm{d}_{M}(\phi)\rangle_{g}+c_{2}\cdot\langle\mathbb{L}_{v}\phi,\phi\rangle+c_{4}\cdot(\langle\mathbb{L}_{v}\phi,\phi\rangle)^{2}\big{)}\cdot\mathrm{d}\mathrm{vol}_{g}\,.$

At this point, either by using the fact that $\nu\in\mathcal{X}(M)$ is a
Killing vector field or by differentiating the right-hand side of Eq. (161),
it furthermore follows that the infinitesimal action satisfies

$\partial_{t}\mathcal{L}(f^{*}_{t}\phi)|_{t=0}=\mathcal{Z}^{\nu}(\mathcal{L})(\phi)=\mathrm{d}_{M}\big{(}\iota_{\nu}\mathcal{L}(\phi)\big{)}\,,$

i.e., an infinitesimal local symmetry as per Lem. 6.17.

###  6.2 Infinitesimal gauge symmetries and Noether's Second Theorem

There is a class of infinitesimal local symmetries that induce redundancies in
the physical interpretation of classical field theories, in the sense that
they obstruct (Prop. 6.33) the existence of a `Cauchy surface' (Def. 6.30),
i.e., a well-defined set of initial conditions which induce a unique evolution
of fields via the Euler–Lagrange equations. These are symmetries that can be
freely parametrized by `gauge' parameters which are functions, or more
generally sections of vector bundles, on the spacetime $M$. Such symmetries
induce currents that are conserved off-shell, which may equivalently be
interpreted as inducing interrelations between the components of the
Euler–Lagrange differential operator.

######  Lemma 6.19 (Motivating case of a gauge symmetry).

Let $\mathcal{Z}=Z\circ j^{\infty}\in\mathcal{X}(\mathcal{F})$ be a local
vector field. Suppose the local vector field $\mathcal{Z}_{f}=Z_{f}\circ
j^{\infty}:=(f\cdot Z)\circ j^{\infty}\in\mathcal{X}(\mathcal{F})$ is an
infinitesimal symmetry of the smooth Lagrangian
$\mathcal{L}:\mathcal{F}\rightarrow\Omega^{d}_{\mathrm{Vert}}(M)$, for any
$f\in C^{\infty}(M)$. That is,

$\mathcal{Z}_{f}(\mathcal{L})=\mathrm{d}_{M}\mathcal{K}_{Z_{f}}\,,$

for some $K_{Z_{f}}\in\Omega^{d-1,0}(J^{\infty}_{M}F)$, for every $f\in
C^{\infty}(M)$.

(i) Then the $(d-1)$-form current
$\mathcal{P}_{Z}:\mathcal{F}\rightarrow\Omega^{d-1}_{\mathrm{Vert}}(M)$
corresponding to $\mathcal{Z}\in\mathcal{X}(\mathcal{F})$ is conserved off-
shell

$\mathrm{d}_{M}\mathcal{P}_{Z}=0\,:\,\mathcal{F}\longrightarrow\Omega^{d}_{\mathrm{Vert}}(M)\,,$

and not only on the smooth space of on-shell fields
$\mathcal{F}_{\mathcal{E}\mathcal{L}}\hookrightarrow\mathcal{F}$.

(ii) Equivalently, there exist the off-shell relation between the Euler-
Lagrange differential operators of $\mathcal{L}$

$\langle\mathcal{E}\mathcal{L},\mathcal{Z}\rangle=\mathcal{E}\mathcal{L}_{a}\cdot\mathcal{Z}^{a}=0\,.$

###### Proof.

Since $(f\cdot Z)\circ j^{\infty}$ is a symmetry of $\mathcal{L}$, by
Noether's First Theorem

$\displaystyle\mathrm{d}_{M}\mathcal{P}_{f\cdot Z}$
$\displaystyle=\langle\mathcal{E}\mathcal{L},(f\cdot Z)\circ
j^{\infty}\rangle=\langle\mathcal{E}\mathcal{L},Z\circ j^{\infty}\rangle\cdot
f$ $\displaystyle=\mathrm{d}_{M}\mathcal{P}_{Z}\cdot f$

for any $f\in C^{\infty}(M)$. In other words, for any field configuration
$\phi\in\mathcal{F}$ (and any higher plot),
$\mathrm{d}_{M}\mathcal{P}_{Z}(\phi)\cdot f\in\Omega^{d}(M)$ is an exact top-
form for all $f\in C^{\infty}(M)$. Integrating over closed balls
$\bar{B}^{d}_{p}\subset M$ around any point $p\in M$ in the base manifold,

$\int_{\bar{B}^{d}_{p}}\mathrm{d}_{M}\mathcal{P}_{Z}(\phi)\cdot f=0$

for all $f\in C^{\infty}(M)$, which implies that
$\mathrm{d}_{M}\mathcal{P}_{Z}(\phi)|_{B^{d}_{p}}=0$ in the interior. Thus,
$\mathcal{P}_{Z}(\phi)$ is locally - and hence globally closed for any field
configuration $\phi\in\mathcal{F}$. 626262Strictly speaking, this only shows
that $\mathcal{P}_{Z}(\phi)$ is closed in the interior $\mathrm{int}(M)\subset
M$. It follows that, as a smooth current on field space,

$0=\mathrm{d}_{M}\mathcal{P}_{Z}:\mathcal{F}\longrightarrow\Omega^{d}_{\mathrm{Vert}}(M)\,,$

and so $\mathcal{P}_{Z}$ is conserved off-shell. By the formula
$\mathrm{d}_{M}\mathcal{P}_{Z}=\langle\mathcal{E}\mathcal{L},\mathcal{Z}\rangle$,
being conserved off-shell is equivalent to the identities

$\langle\mathcal{E}\mathcal{L},\mathcal{Z}\rangle=\mathcal{E}\mathcal{L}_{a}\cdot\
\mathcal{Z}^{a}=0$

between the components of the smooth Euler–Lagrange operator. ∎

The illustration above captures the main features of gauge symmetries. The
`gauge' parameters $f\in C^{\infty}(M)\cong\Gamma_{M}(M\times\mathbbm{R})$
parametrize an (infinite-dimensional) family of symmetries
$\\{\mathcal{Z}_{f}:=(f\cdot Z)\circ
j^{\infty}\\}\subset\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$, whose dependence
on $f$ is local. In the above example, $\mathcal{Z}_{f}$ depends on f via its
value at points, i.e., the zero-jet of $f$. These induce non-trivial
interrelations between the components of the Euler–Lagrange operator, and the
resulting current is conserved off-shell.

More generally, an infinitesimal local gauge symmetry is an infinite-
dimensional subspace $\\{\mathcal{R}_{e}:=R_{e}\circ
j^{\infty}\\}\subset\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$
parametrized by `gauge parameters', sections $e\in\Gamma_{M}(E)$ of a gauge
parameter vector bundle $E\rightarrow M$, such that the dependence of each
$\mathcal{R}_{e}$ on $e$ is linear and local, i.e., via the infinite jet
prolongation of $e$. The resulting currents satisfy similar off-shell
properties, which are equivalent to a set of identities relating the
components of the Euler–Lagrange operator. We now turn to make this intuition
precise.

######  Definition 6.20 (Infinitesimal gauge symmetry).

A (parametrized) collection of infinitesimal local gauge symmetries of a
Lagrangian density $L$ is an infinite-dimensional subspace of evolutionary
vector fields on $J^{\infty}_{M}F$ that may be identified with the image of a
$\mathbbm{R}$-linear map

$\displaystyle R_{(-)}:\Gamma_{M}(E)$
$\displaystyle\longrightarrow\mathcal{X}_{\mathrm{ev}}(J^{\infty}_{M}F)\,$
$\displaystyle e$ $\displaystyle\longmapsto
R\circ(j^{\infty}e,\mathrm{id}_{J^{\infty}_{M}F})$

where $E\rightarrow M$ is a ``gauge parameter'' vector bundle over M, and

$\textstyle{J^{\infty}_{M}E\times_{M}J^{\infty}_{M}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
22.76228ptR}$$\textstyle{VF\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F}$

is a smooth bundle map linear in the $J^{\infty}_{M}E$ fibers, 636363The fiber
product $J^{\infty}_{M}G\times_{M}J^{\infty}_{M}F$ is taken in
$\mathrm{LocProMan}\hookrightarrow\mathrm{FrMfd}$. Equivalently, it is the
infinite jet bundle $J^{\infty}_{M}(G\times_{M}F)$ on the fiber product of the
bundles $G$ and $F$, and so manifestly a locally pro-manifold. such that each
evolutionary vector field

$R_{e}:J^{\infty}_{M}F\cong
M\times_{M}J^{\infty}_{M}F\xlongrightarrow{\;\;\big{(}j^{\infty}e,\;\mathrm{id}_{J^{\infty}_{M}F}\big{)}\;\;}J^{\infty}_{M}E\times_{M}J^{\infty}_{M}F\xlongrightarrow{\;R\;}VF\,$

is a symmetry of the Lagrangian density

$\mathbb{L}_{\mathsf{pr}R_{e}}L=\mathrm{d}_{H}K_{R_{e}}\,,$

for some $K_{R_{e}}\in\Omega^{d-1,0}(J^{\infty}_{M}F)$ with local dependence
on $e$, i.e., given by a bundle map
$K_{R}:J^{\infty}_{M}E\times_{M}J^{\infty}_{M}F\rightarrow\wedge^{d-1}T^{*}M$
over $M$.

Composing with the jet prolongation along $F$, this corresponds to a subspace
of local vector fields of $\mathcal{F}$, given by the image of the (smooth)
$\mathbbm{R}$-linear map

$\mathcal{R}_{(-)}:=R_{(-)}\circ
j^{\infty}:\Gamma_{M}(E)\longrightarrow\mathcal{X}_{\mathrm{loc}}(\mathcal{F})\,,$

such that for each $e\in\Gamma_{M}(E)$ it defines a local symmetry of the
local Lagrangian $\mathcal{L}$

$\mathcal{R}_{e}(\mathcal{L})=\mathrm{d}_{M}\mathcal{K}_{R_{e}}\,.$

Let us expand the definition in local coordinates, where it appears in a more
familiar form. Let $\\{x^{\mu},c^{\beta}\\}$ be a compatible coordinate chart
for $E\rightarrow M$ with induced coordinates
$\big{\\{}x^{\mu},\\{c^{\beta}_{K}\\}_{0\leq|K|}\big{\\}}$ for
$J^{\infty}_{M}E\rightarrow M$. Then by the linearity assumption, the gauge
parameter bundle map $R:J^{\infty}_{M}E\times_{M}J^{\infty}_{M}F\rightarrow
VF$ is locally of the form

$\displaystyle R$ $\displaystyle=\sum_{|K|=0}^{\infty}c^{\beta}_{K}\cdot
R^{aK}_{\beta}\cdot\frac{\partial}{\partial u^{a}}$ (162)
$\displaystyle=\Big{(}c^{\beta}\cdot
R_{\beta}^{a}+c^{\beta}_{\mu}\cdot{R}_{\beta}^{a\mu}+c^{\beta}_{\mu_{1}\mu_{2}}\cdot
R_{\beta}^{a\mu_{1}\mu_{2}}+\cdots\Big{)}\cdot\frac{\partial}{\partial u^{a}}$

with the sum necessarily (locally) terminating, where each of the coefficients
$\\{R^{a}_{\beta},R^{a\mu}_{\beta},R^{a\mu_{1}\mu_{2}}_{\beta},\cdots\\}$ is a
smooth function on $J^{\infty}_{M}F$, and hence also of (locally) finite order
in the coordinates $\big{\\{}x^{\mu},\\{u^{a}_{I}\\}_{0\leq|I|}\big{\\}}$.
Thus, for a gauge parameter $e\in\Gamma_{M}(E)$ the corresponding evolutionary
vector field takes the form

$\displaystyle R_{e}$ $\displaystyle=\sum_{|K|=0}^{\infty}\frac{\partial
e^{\beta}}{\partial x^{K}}\cdot R^{aK}_{\beta}\cdot\frac{\partial}{\partial
u^{a}}$ $\displaystyle=\Big{(}e^{\beta}\cdot R_{\beta}^{a}+\frac{\partial
e^{\beta}}{\partial x^{\mu}}\cdot{R}_{\beta}^{a\mu}+\frac{\partial
e^{\beta}}{\partial x^{\mu_{1}}\partial x^{\mu_{2}}}\cdot
R_{\beta}^{a\mu_{1}\mu_{2}}+\cdots\Big{)}\cdot\frac{\partial}{\partial
u^{a}}\,.$

By abuse of notation, the corresponding local vector field on $\mathcal{F}$ is
often denoted by

$\displaystyle\mathcal{R}_{e}(\phi)$
$\displaystyle=\sum_{|K|=0}^{\infty}\frac{\partial e^{\beta}}{\partial
x^{K}}\cdot\mathcal{R}^{aK}_{\beta}(\phi)\cdot\frac{\delta}{\delta\phi^{a}}=\sum_{|K|=0}^{\infty}\frac{\partial
e^{\beta}}{\partial x^{K}}\cdot
R^{aK}_{\beta}(\phi,\\{\partial_{I}\phi\\}_{|I|\leq
k})\cdot\frac{\delta}{\delta\phi^{a}}$
$\displaystyle=\Big{(}e^{\beta}\cdot\mathcal{R}_{\beta}^{a}(\phi)+\frac{\partial
e^{\beta}}{\partial
x^{\mu}}\cdot\mathcal{R}_{\beta}^{a\mu}(\phi)+\frac{\partial
e^{\beta}}{\partial x^{\mu_{1}}\partial
x^{\mu_{2}}}\cdot\mathcal{R}_{\beta}^{a\mu_{1}\mu_{2}}(\phi)+\cdots\Big{)}\cdot\frac{\delta}{\delta\phi^{a}}\,,$

or even as an `infinitesimal transformation of the field'

$\delta_{R_{e}}\phi^{a}=e^{\beta}\cdot\mathcal{R}_{\beta}^{a}(\phi)+\frac{\partial
e^{\beta}}{\partial
x^{\mu}}\cdot\mathcal{R}_{\beta}^{a\mu}(\phi)+\frac{\partial
e^{\beta}}{\partial x^{\mu_{1}}\partial
x^{\mu_{2}}}\cdot\mathcal{R}_{\beta}^{a\mu_{1}\mu_{2}}(\phi)+\cdots\,,$

exactly matching the formulae appearing in the physics literature (see e.g.
[HT92]).

######  Example 6.21 (Electromagnetic gauge symmetry).

The archetypical example of a gauge symmetry is that of pure electromagnetism,
expressed in terms of gauge potentials. Consider spacetime $(M,g)$ as a
(pseudo)-Riemannian $d$-dimensional manifold, and field bundle as the
cotangent bundle $F=T^{*}M$ with induced coordinates $\\{x^{\mu},u_{\mu}\\}$.
The smooth set of electromagnetic fields is
$\mathcal{F}=\mathbold{\Gamma}_{M}(T^{*}M)\cong\Omega^{1}_{\mathrm{Vert}}(M)$,
and the local Lagrangian of pure electromagnetism is

$\displaystyle\mathcal{L}(A)$ $\displaystyle:=\tfrac{1}{2}\cdot
F_{A}\wedge\star F_{A}=\tfrac{1}{2}\langle
F_{A},F_{A}\rangle_{g}\cdot\mathrm{d}\mathrm{vol}_{g}\equiv\tfrac{1}{2}\langle\mathrm{d}_{M}A,\mathrm{d}_{M}A\rangle_{g}\cdot\mathrm{d}\mathrm{vol}_{g}$
$\displaystyle=\tfrac{1}{2}g^{\mu\nu}g^{\sigma\rho}\cdot\partial_{\sigma}A_{\mu}\cdot\partial_{\rho}A_{\nu}\cdot\sqrt{|g|}\cdot\mathrm{d}x^{1}\cdots\mathrm{d}x^{d}\,,$

where $F_{A}:=\mathrm{d}_{M}A$ is the field strength646464For a spacetime of
the form $\mathbbm{R}^{1}_{t}\times N$, the traditional electric and magnetic
field strengths are recovered via the decomposition
$F=E\wedge\mathrm{d}t+B=E_{it}\mathrm{d}x^{i}\wedge\mathrm{d}t+B_{ij}\mathrm{d}x^{i}\wedge\mathrm{d}x^{j}$.
(curvature) of the gauge potential (connection) $A\in\Omega^{1}(M)$. Since
$F=T^{*}M$ is a vector bundle it follows that $VF\cong
T^{*}M\times_{M}T^{*}M$, and so
$T\mathcal{F}\cong\mathcal{F}\times\mathcal{F}$ (see also Rem. 2.16). Hence,
for each function $e\in C^{\infty}(M)\cong\Gamma_{M}(M\times\mathbbm{R})$ on
spacetime there exists the constant vector field

$\displaystyle\mathcal{R}_{e}:\mathcal{F}$ $\displaystyle\longrightarrow
T\mathcal{F}\cong\mathcal{F}\times\mathcal{F}$ $\displaystyle A$
$\displaystyle\longmapsto(A,\mathrm{d}_{M}e)\,,$

which in local coordinates may be represented by

$\mathcal{R}_{e}(A):=\partial_{\mu}e\cdot\frac{\delta}{\delta
A_{\mu}}=\partial_{\mu}e\cdot\delta^{\mu}_{\nu}\cdot\frac{\delta}{\delta
A_{\nu}}\qquad\in\qquad\mathcal{X}_{\mathrm{loc}}(\mathcal{F})\,.$

Note that the dependence on the chosen function
$e\in\Gamma_{M}(M\times\mathbbm{R})$ is through
$J^{1}_{M}(M\times\mathbbm{R})$. It is infinitesimal symmetry of the
Lagrangian as can be checked in coordinates via Def. 6.8, or directly by the
nilpotency of the de Rham differential

$\displaystyle\mathcal{R}_{e}(\mathcal{L})(A)$
$\displaystyle=\tfrac{1}{2}\big{(}\mathrm{d}_{M}(\mathrm{d}_{M}e)\wedge\star\mathrm{d}_{M}A+\mathrm{d}_{M}A\wedge\star\mathrm{d}_{M}(\mathrm{d}_{M}e)\big{)}$
$\displaystyle=\mathrm{d}_{M}^{2}e\wedge\star\mathrm{d}_{M}A=0\,.$ (163)

Thus, it defines a parametrized gauge symmetry, with gauge parameter bundle
$E=M\times\mathbbm{R}$ and coordinates $\\{x^{\mu},c\\}$, and the
corresponding bundle map
$R:J^{\infty}_{M}(M\times\mathbbm{R})\times_{M}J^{\infty}_{M}(T^{*}M)\longrightarrow
T^{*}M\times_{M}T^{*}M\cong V(T^{*}M)$ over $T^{*}M$ given locally by

$c_{\mu}\cdot\delta^{\mu}_{\nu}\cdot\frac{\partial}{\partial u_{\mu}}\,.$

For completeness, let us note the electromagnetism Euler–Lagrange operator,
i.e., the (free) Maxwell equations take the form

$\displaystyle\mathcal{E}\mathcal{L}(A)=\mathrm{d}_{M}\star
F_{A}=\mathrm{d}_{M}\star\mathrm{d}_{M}A\qquad\in\qquad\Omega^{d-1}(M),$ (164)

where
$\Omega^{d-1}(M)\cong\Gamma_{M}(\wedge^{d}T^{*}M\times_{M}TM)\cong\Gamma_{M}(\wedge^{d}T^{*}M\otimes
V^{*}F)$, as expected. The current corresponding to each infinitesimal
symmetry $\mathcal{R}_{e}\in\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$ is given
by

$\displaystyle\mathcal{P}_{R_{e}}(A)=\mathrm{d}_{M}e\wedge\star\mathrm{d}_{M}A\,,$
(165)

which is obviously conserved on $\mathcal{F}_{\mathcal{E}\mathcal{L}}$.

######  Remark 6.22 (The Lie algebra all infinitesimal gauge symmetries).

For the sake of completeness, we comment on some abstract aspects of gauge
symmetries, defined via arbitrary parametrizations as above. Some of these
points are explicitly mentioned and further expanded in [HT92], while others
are implicitly necessary and slightly supplement the (local coordinate)
description of gauge symmetries therein.

(i) The actual gauge symmetries
$\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathcal{R}}(\mathcal{F})\subset\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$
are identified by the image of some local $\mathbbm{R}$-linear map
$\mathcal{R}_{(-)}:\Gamma_{M}(E)\rightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$
as in Def. 6.20. In particular, the gauge parameter $E\rightarrow M$ bundle
and corresponding parametrization bundle map $R$ are not unique.656565 For
instance, let $E^{\prime}\rightarrow M$ be any other vector bundle and
consider the map
$\tilde{\mathcal{R}}_{(-)}:\Gamma_{M}(E\times_{M}E^{\prime})\cong\Gamma_{M}(E)\oplus\Gamma_{M}(E^{\prime})\rightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$
acting via $\mathcal{R}_{(-)}$ on the first component and trivially on the
second. Furthermore, it might be that some gauge symmetries are in the image
of some parametrization
$\mathcal{R}_{(-)}:\Gamma_{M}(E)\rightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$,
but not in the image of another
$\hat{\mathcal{R}}_{(-)}:\Gamma_{M}(\hat{E})\rightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$.
666666For instance, take $\hat{E}\rightarrow M$ to be a proper subbundle of
$E\rightarrow M$ for an injective
$\mathcal{R}_{(-)}:\Gamma_{M}(E)\rightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$.
Then the restricted parametrization
$\hat{\mathcal{R}}_{(-)}:=\mathcal{R}_{(-)}|_{\hat{E}}:\Gamma_{M}(\hat{E})\rightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$
identifies strictly less gauge symmetries via its image.

(ii) By Eq. (158), it follows that for any two gauge parameters
$e_{1},e_{2}\in\Gamma_{M}(E)$ of a (parametrized) gauge symmetry $R$, the
commutator $[\mathcal{R}_{e_{1}},\mathcal{R}_{e_{2}}]$ is also a local
symmetry of $(\mathcal{F},\mathcal{L})$. Crucially, however, it is not
necessarily possible to express $[\mathcal{R}_{e_{1}},\mathcal{R}_{e_{2}}]$ as
$R_{e_{3}}$ for some other gauge parameter $e_{3}\in\Gamma_{M}(E)$.676767This
is possible for all gauge parameters if there exists a local ‘bracket map’
$[-,-]^{\mathcal{R}}_{(-)}:\Gamma_{M}(E)\times\Gamma_{M}(E)\times\mathcal{F}\rightarrow\Gamma_{M}(J^{\infty}E)$
such that
$\mathcal{R}_{[e_{1},e_{2}]^{\mathcal{R}}_{(\phi)}}(\phi)=[\mathcal{R}_{e_{1}}(\phi),\mathcal{R}_{e_{1}}(\phi)]_{\mathcal{X}_{\mathrm{loc}}(\mathcal{F})}$.
The bracket generally might depend on the dynamical fields (in a local
manner), and so the composition
$\Gamma_{M}(E)\times\Gamma_{M}(E)\times\mathcal{F}\rightarrow\Gamma_{M}(J^{\infty}E)\times\mathcal{F}\rightarrow
T\mathcal{F}$ defines a bundle map over $\mathcal{F}$, and so (generally) a
Lie algebroid structure, rather than a Lie algebra. Nevertheless, for many
physical examples (e.g. General Relativity, Yang-Mills, Chern-Simons theories)
this happens to be an actual Lie algebra map. Nevertheless, commutators of
(parametrized) gauge symmetries are also considered to be gauge symmetries.

(iii) We have defined an infinite-dimensional family of local symmetries, the
trivial symmetries
$\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathrm{triv}}(\mathcal{F})\hookrightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$
of Ex. 6.12. Strictly speaking, these are not parametrized by sections of some
vector bundle over $M$, but instead by vector bundle maps out
$K:J^{\infty}_{M}F\rightarrow\wedge^{2}VF\otimes\wedge^{d}TM$ over $F$.
Nevertheless, trivial symmetries are also considered to be gauge symmetries.

(iv) It follows that formally, the Lie algebra of all (implicit) infinitesimal
gauge symmetries is the minimal subalgebra (in fact ideal) of local
symmetries, containing all parametrized local symmetries
$\big{\\{}\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathcal{R}}(\mathcal{F})\hookrightarrow\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})\big{\\}}_{\mathcal{R}}$
and the trivial symmetries
$\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})$,

$\big{(}\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathrm{gauge}}\,,[-,-]\big{)}\longhookrightarrow\big{(}\mathcal{X}^{\mathcal{L}}_{\mathrm{loc}}(\mathcal{F}),\,[-,-]\big{)}\,.$

(v) A parametrized gauge symmetry
$\mathcal{R}_{(-)}:\Gamma_{M}(E)\rightarrow\mathcal{X}^{\mathcal{L}}_{\mathrm{loc}}(\mathcal{F})$
is called a generating set if any infinitesimal gauge symmetry
$\mathcal{Z}\in\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathrm{gauge}}$ may be
written as

$\mathcal{Z}=R\circ(C\times_{F}\mathrm{id}_{J^{\infty}_{M}F})\circ
j^{\infty}+\mathcal{K}\cdot\mathcal{E}\mathcal{L}$

for some bundle maps $C:J^{\infty}_{M}F\rightarrow J^{\infty}_{M}E$ and
$K:J^{\infty}_{M}F\rightarrow\wedge^{2}VF\otimes\wedge^{d}TM$. In simple words
the vector field $\mathcal{Z}(\phi)$ may be expressed via $\mathcal{R}$ using
a `field dependent gauge parameter' $\mathcal{C}(\phi)=C(j^{\infty}\phi)$, up
to a trivial gauge transformation
$\mathcal{K}\cdot\mathcal{E}\mathcal{L}(\phi)$, and so in local coordinates

$\delta_{Z}\phi^{a}=\Big{(}\mathcal{C}^{\beta}(\phi)\cdot\mathcal{R}_{\beta}^{a}(\phi)+\frac{\partial\mathcal{C}^{\beta}(\phi)}{\partial
x^{\mu}}\cdot\mathcal{R}_{\beta}^{a\mu}(\phi)+\cdots\Big{)}+\mathcal{K}^{[ab]}(\phi)\cdot\mathcal{E}\mathcal{L}_{b}(\phi)\,.$

There is a lot more to be said along these lines, which is however outside the
scope of this manuscript. For more details and explicit examples of the above
concepts, albeit in the topologically trivial cases, see [HT92].

The abstract characterization of the full Lie algebra of (implicit)
infinitesimal gauge symmetries (Rem. 6.22 (iv)) is great as an abstract
backbone, but is not directly amenable to explicit applications. In practice,
one always treats gauge symmetries via explicit parametrizations. Noether's
Second Theorem may be viewed as a way of detecting the existence of
(parametrized) gauge symmetries of a local field theory
$(\mathcal{F},\mathcal{L})$, and may be deduced immediately as an application
of Noether's First (Prop. 6.14), essentially via an integration by parts
argument.

######  Proposition 6.23 (Noether's Second Theorem).

Let

$\mathcal{R}_{(-)}:\Gamma_{M}(E)\longrightarrow\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$

be an $\mathbbm{R}$-linear map defined by
$\mathcal{R}_{e}:=R\circ\big{(}j^{\infty}e,j^{\infty}(-)\big{)}$ for some
bundle map

$\textstyle{J^{\infty}_{M}E\times_{M}J^{\infty}_{M}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\hskip
19.91684ptR}$$\textstyle{VF\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{F}$

linear in the $J^{\infty}_{M}E$ fibers.

(i) Then $\mathcal{R}_{(-)}$ parametrizes a collection of infinitesimal gauge
symmetries of a local field theory $(\mathcal{F},\mathcal{L})$ if and only if
the Euler–Lagrange differential operator satisfies the ``Noether identity"

$0=\mathcal{N}^{R}\circ(\mathcal{E}\mathcal{L}\times_{\mathcal{F}}\mathrm{id}_{\mathcal{F}})\quad:\quad\Gamma_{M}(F)\longrightarrow\Gamma_{M}(V^{*}F\otimes\wedge^{d}T^{*}M)\times\Gamma_{M}(F)\longrightarrow\Gamma_{M}(E^{*}\otimes\wedge^{d}T^{*}M)\,,$

where
$\mathcal{N}^{R}:\Gamma_{M}(V^{*}F\otimes\wedge^{d}T^{*}M)\times\Gamma_{M}(F)\rightarrow\Gamma_{M}(E^{*}\otimes\wedge^{d}T^{*}M)$
is the formal adjoint differential operator (see, e.g., [CH89, §V.1.3]) of
$\mathcal{R}=\Gamma_{M}(E)\times\Gamma_{M}(F)\rightarrow\Gamma_{M}(VF)$.

(ii) Explicitly, $\mathcal{N}^{R}$ is defined via (cohomological) integration
by parts686868In terms of jet bundles, this calculation corresponds to lifting
the Lagrangian density via the projection
$J^{\infty}_{M}E\times_{M}J^{\infty}_{M}F\rightarrow J^{\infty}_{M}F$, and
applying the interior Euler operator (Def. 5.9) on
$J^{\infty}_{M}(E\times_{M}F)$. This is, in particular, one way to justify the
global validity of the local formulas which follow. with respect to
$e\in\Gamma_{M}(E)$ so that for any
$\tilde{\phi}\in\Gamma_{M}(V^{*}F\otimes\wedge^{d}T^{*}M)$,
$\phi\in\Gamma_{M}(F)$ and $e\in\Gamma_{M}(E)$

$\langle\tilde{\phi},\,\mathcal{R}_{e}(\phi)\rangle_{VF}=\langle\mathcal{N}^{R}(\tilde{\phi},\phi),\,e\rangle_{E}+\mathrm{d}_{M}\mathcal{J}(\phi,\tilde{\phi},e)\,,$

for some differential operator
$\mathcal{J}:\Gamma_{M}(F)\times\Gamma_{M}(V^{*}F\otimes\wedge^{d}T^{*}M)\times\Gamma_{M}(E)\rightarrow\Omega^{d-1}(M)$
which is, in particular, $\mathbbm{R}$-linear in the second and third entries.

(iii) In local coordinates,
$\langle\mathcal{N}^{R}(\tilde{\phi},\phi),\,e\rangle_{E}=\sum_{|J|=0}^{\infty}(-1)^{|J|}\cdot\frac{\partial}{\partial
x^{J}}\big{(}\tilde{\phi}_{a}\cdot\mathcal{R}^{aJ}_{b}(\phi)\big{)}\cdot
e^{b}\cdot\mathrm{d}x^{1}\cdots\mathrm{d}x^{d}$ and so, for all $b$, the
condition reads

$0=\sum_{|J|=0}^{\infty}(-1)^{|J|}\cdot\frac{\partial}{\partial
x^{J}}\big{(}\mathcal{E}\mathcal{L}_{a}(\phi)\cdot\mathcal{R}^{aJ}_{b}(\phi)\big{)}\,.$

###### Proof.

If $\mathcal{R}_{e}$ defines a symmetry, then by Noether's First Theorem, the
corresponding charge satisfies

$\displaystyle\mathrm{d}_{M}\mathcal{P}_{R_{e}}(\phi)=\langle\mathcal{E}\mathcal{L}(\phi),\,\mathcal{R}_{e}(\phi)\rangle_{VF}=\sum_{|J|=0}^{\infty}\mathcal{E}\mathcal{L}_{a}(\phi)\cdot\frac{\partial
e^{b}}{\partial x^{J}}\cdot\mathcal{R}^{aJ}_{b}(\phi)\,.$

`Integrating by parts' (cohomologically) on the right-hand side with respect
to the gauge parameter $e^{b}$,

$\displaystyle\mathrm{d}_{M}\mathcal{P}_{R_{e}}(\phi)$
$\displaystyle=\langle\mathcal{N}^{R}\big{(}\mathcal{E}\mathcal{L}(\phi),\phi\big{)},\,e\rangle_{E}+\mathrm{d}_{M}\mathcal{J}\big{(}\phi,\mathcal{E}\mathcal{L}(\phi),e\big{)}$
$\displaystyle=e^{b}\cdot\sum_{|J|=0}^{\infty}(-1)^{|J|}\cdot\frac{\partial}{\partial
x^{J}}\big{(}\mathcal{E}\mathcal{L}_{a}(\phi)\cdot\mathcal{R}^{aJ}_{b}(\phi)\big{)}+\mathrm{d}_{M}\mathcal{J}\big{(}\phi,\mathcal{E}\mathcal{L}(\phi),e\big{)}\,,$

one sees that
$\langle\mathcal{N}^{R}\big{(}\mathcal{E}\mathcal{L}(\phi),\phi\big{)},\,e\rangle_{E}=\mathrm{d}_{M}\big{(}\mathcal{P}_{R_{e}}(\phi)+\mathcal{J}(\phi,\mathcal{E}\mathcal{L}(\phi),e)\big{)}$
is an exact form on $M$ for each field $\phi\in\mathcal{F}(*)$ and parameter
$e\in\Gamma_{M}(E)$. Proceeding as in the proof of Lem. 6.19, integrating
locally over balls $\bar{B}_{p}^{d}\subset M$ around each point $p\in M$
implies that

$\mathcal{N}^{R}\big{(}\mathcal{E}\mathcal{L}(\phi),\phi\big{)}=0$

on (the interior of) $M$, for all field configurations
$\phi\in\mathcal{F}(*)$. Conversely, reading the equations backward, the
vanishing identity implies that $\mathcal{R}_{e}$ is a symmetry for every
$e\in\Gamma_{M}(E)$. ∎

######  Corollary 6.24 (Trivial local currents from gauge symmetries).

(i) The conserved current $\mathcal{P}_{R_{e}}$ of a gauge symmetry
$\mathcal{R}_{e}\in\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathrm{gauge}}(\mathcal{F})$
satisfies

$\mathrm{d}_{M}\mathcal{P}_{R_{e}}=\mathrm{d}_{M}\mathcal{J}_{e}\,:\,\mathcal{F}\longrightarrow\Omega^{d}_{\mathrm{Vert}}(M)$

for some local
$\mathcal{J}_{e}:\mathcal{F}\rightarrow\Omega^{d-1}_{\mathrm{Vert}}(M)$ which
vanishes on the subspace
$\mathcal{F}_{\mathcal{E}\mathcal{L}}\hookrightarrow\mathcal{F}$ of on-shell
fields.

(ii) Furthermore, if the $(d-1)$-cohomology of $F$ vanishes, then
$\mathcal{P}_{R_{e}}$ is of the form

$\mathcal{P}_{R_{e}}=\mathcal{J}_{e}+\mathrm{d}_{M}\mathcal{T}_{e}$

for some local
$\mathcal{T}_{e}:\mathcal{F}\rightarrow\Omega^{d-2}_{\mathrm{Vert}}(M)$. In
particular, it defines a trivial on-shell local current
$\mathcal{P}_{R_{e}}|_{\mathcal{F}_{\mathcal{E}\mathcal{L}}}=\mathrm{d}_{M}\mathcal{T}_{e}$.

###### Proof.

By the proof of Prop. 6.23, we have the equality of $d$-form currents on the
field space $\mathrm{d}_{M}\mathcal{P}_{R_{e}}=\mathrm{d}_{M}\mathcal{J}_{e}$,
where

$\mathcal{J}_{e}:=\mathcal{J}\big{(}-,\mathcal{E}\mathcal{L}(-),e\big{)}\,:\,\mathcal{F}\longrightarrow\Omega^{d-1}_{\mathrm{Vert}}(M)$

is a $(d-1)$-current whose dependence is linear in the second entry, and hence
vanishes on-shell.

Now, both $\mathcal{P}_{R_{e}}$ and $\mathcal{J}_{e}$ arise from bi-
differential operators $\mathcal{P}_{R_{(-)}},\,\mathcal{J}_{(-)}\,:\,$
$\mathcal{F}\times\mathbold{\Gamma}_{M}(E)\rightarrow\Omega^{d-1}_{\mathrm{Vert}}(M)$
induced by horizontal $(d-1)$-forms on $J^{\infty}(E\times_{M}F)$, and so by
Prop. 5.18 if the $(d-1)$-cohomology of $E\times_{M}F$ vanishes, then
$\mathrm{d}_{M}(\mathcal{P}_{R_{e}}-\mathcal{J}_{e})=0$ implies

$\mathcal{P}_{R_{e}}=\mathcal{J}_{e}+\mathrm{d}_{M}\mathcal{T}_{e}$

for some
$\mathcal{T}_{e}:\mathcal{F}\rightarrow\Omega^{d-2}_{\mathrm{Vert}}(M)$. Note
that, since $E\times_{M}F$ is a (contractible) vector bundle over $F$, their
cohomologies coincide. Since $\mathcal{K}_{e}$ vanishes on the smooth subspace
of on-shell fields, it follows that

$\mathcal{P}_{R_{e}}|_{\mathcal{F}_{\mathcal{E}\mathcal{L}}}=0+\mathrm{d}_{M}\mathcal{T}_{e}|_{\mathcal{F}_{\mathcal{E}\mathcal{L}}}\,:\,\mathcal{F}_{\mathcal{E}\mathcal{L}}\longrightarrow\Omega^{d-1}_{\mathrm{Vert}}(M)\,.$

∎

The vanishing of the $(d-1)$-cohomology in the above corollary is usually
satisfied explicitly for physical theories (for instance, for field bundles
being vector bundles over Minkowski spacetime). If instead only the
$(d-1)$-cohomology of the spacetime vanishes, then the current is pointwise
(in the field space $\mathcal{F}_{\mathcal{E}\mathcal{L}}$)
$\mathrm{d}_{M}$-exact, but perhaps not with a local dependence in the field
(Rem. 5.23). In [Zu86] the statement is claimed without a proof and no mention
of the cohomology assumption. In [Blo23] the statement is claimed as in
[Zu86], but we are not able to follow the proof therein.

######  Example 6.25 (Noether identity for electromagnetism).

Consider the case of electromagnetism from Ex. 6.21. Cohomologically
integrating by parts with respect to the gauge parameter the gauge invariance
equation Eq. (6.21), we get

$\displaystyle 0$
$\displaystyle=\mathrm{d}_{M}^{2}e\wedge\star\mathrm{d}_{M}A$
$\displaystyle=\mathrm{d}_{M}\big{(}\mathrm{d}_{M}e\wedge\star\mathrm{d}_{M}A\big{)}+\mathrm{d}_{M}e\wedge\mathrm{d}_{M}\star\mathrm{d}_{M}A$
$\displaystyle=\mathrm{d}_{M}\mathcal{P}_{R_{e}}(A)+\mathrm{d}_{M}e\wedge\mathcal{E}\mathcal{L}(A)$
$\displaystyle=\mathrm{d}_{M}\mathcal{P}_{R_{e}}(A)+\mathrm{d}_{M}\big{(}e\wedge\mathcal{E}\mathcal{L}(A)\big{)}-e\wedge\mathrm{d}_{M}\big{(}\mathcal{E}\mathcal{L}(A)\big{)}\,,$

where we identified the conserved current (165) and the Euler–Lagrange
operator (164). It follows that the off-shell Noether identity corresponding
to the electromagnetic gauge symmetry is given by

$\mathcal{N}^{R}(\mathcal{E}\mathcal{L}(A)):=\mathrm{d}_{M}\circ\mathcal{E}\mathcal{L}(A)=\mathrm{d}_{M}(\mathrm{d}_{M}\star\mathrm{d}_{M}A)=0\,,$

and the corresponding current satisfies the off-shell identity

$\mathrm{d}_{M}\mathcal{P}_{R_{e}}=\mathrm{d}_{M}\mathcal{J}\big{(}A,\mathcal{E}\mathcal{L}(A),e\big{)}:=\mathrm{d}_{M}\big{(}-e\wedge\mathcal{E}\mathcal{L}(A)\big{)}\,.$

######  Remark 6.26 (Physical observables from gauge symmetries).

The potentiality of a trivial resulting current does not imply that gauge
symmetries do not produce meaningful observables via their charges over
compact submanifolds with boundary or non-compact submanifolds with suitable
asymptotics, for choices of gauge parameters with support on the boundary or
suitable asymptotic support, respectively (e.g. [BB01][ABS15]). Although we do
not pursue reviewing this matter here, this story should also naturally take
place in this generalized smooth setting.

######  Remark 6.27 (Nomenclature of local and global symmetries).

The adverbs ``local" and ``global" have several, but related meanings in the
physics literature. In this manuscript, we use local as in the mathematics
literature, to mean factoring through an infinite jet bundle, or to indicate
that a certain quantity is defined only on a subspace of an ambient space. In
the physics literature, gauge symmetries are often referred to simply as
`local symmetries' in the sense that they are freely parametrized by (gauge)
parameters that are not `constant' through space-time. In other words, their
value may be chosen to vary `locally from point to point in space-time. In
contrast, the term `global symmetry' (also called rigid) is reserved for
symmetries that are not part of a gauge symmetry, meaning they can be only
parametrized by a finite (or countable) dimensional parameter vector space.
More formally, the set of global or rigid infinitesimal symmetries of a
Lagrangian is defined as the quotient of all local infinitesimal symmetries by
the (ideal) of gauge symmetries (Rem. 6.22 (iv))

$\mathcal{X}_{\mathrm{loc}}^{\mathcal{L},\mathrm{rig}}(\mathcal{F}):=\mathcal{X}_{\mathrm{loc}}^{\mathcal{L}}(\mathcal{F})\,/\,\mathcal{X}^{\mathcal{L},\mathrm{gauge}}_{\mathrm{loc}}(\mathcal{F})\,.$

###  6.3 Cauchy surfaces and obstructions by gauge symmetries

The all-important feature of gauge symmetries, both physically and
mathematically, is that they `obstruct the unique evolution' of fields obeying
the Euler–Lagrange equations. Let us make this mathematically precise in our
current context of smooth sets.

######  Definition 6.28 (Sections on infinitesimal neighborhood of
submanifold).

Let $\iota_{\Sigma^{p}}:\Sigma^{p}\hookrightarrow M$ be a submanifold of
spacetime. The sections on the infinitesimal neighborhood of $\Sigma^{p}$ of
the bundle $F\rightarrow M$ is defined as the set of sections of the infinite
jet bundle $J^{\infty}_{M}F$ over $\Sigma^{p}$,

$\displaystyle\Gamma_{\Sigma^{p}}\big{(}J^{\infty}_{M}F|_{\Sigma^{p}}\big{)}\cong\big{\\{}\psi:\Sigma^{p}\rightarrow
J^{\infty}_{M}F\;\big{|}\;\pi_{M}\circ\psi=\iota_{\Sigma^{p}}\big{\\}}\,.$
(166)

The corresponding smooth set
$\mathbold{\Gamma}_{\Sigma^{p}}\big{(}J^{\infty}_{M}F|_{\Sigma^{p}}\big{)}$
follows as in Def. 2.12.

The nomenclature will become apparent in [GS24], whereby the `infinitesimal
neighborhood of $\Sigma^{p}$ in $M$' will be a bona-fide (thickened) smooth
subspace of $M$, over which sections of $F$ (and not $J^{\infty}_{M}F$)
correspond to the above set. For now, it is enough to note the interpretation
of the definition: Any section $\psi:\Sigma^{p}\rightarrow
J^{\infty}_{M}F|_{\Sigma^{p}}$ (smoothly) determines a particular instance of
the potential values for sections $\phi$ of $F\rightarrow M$ and their
derivatives to any order on the submanifold $\Sigma^{p}$,
$\psi(x)=[j^{\infty}_{x}\phi]$. In other words, one may ask whether there
exists a field $\phi\in\mathcal{F}(*)=\Gamma_{M}(F)$ whose jet restricts to
the chosen section $\psi$ on the infinitesimal neighborhood of $\Sigma^{p}$.
Along these lines, choosing the submanifold to be of codimension-1 and
demanding further compatibility with the Euler–Lagrange equations, we may
define the smooth set of initial conditions on a submanifold $\Sigma^{d-1}$
for a given Lagrangian field theory $(\mathcal{F},\mathcal{L})$ on the
spacetime $M$.

######  Definition 6.29 (Initial data).

Let $(\mathcal{F},\mathcal{L})$ be a local Lagrangian field theory on $M$ and
$\Sigma^{d-1}\hookrightarrow M$ a submanifold of codimension $1$. The (smooth)
set of initial data of $(\mathcal{F},\mathcal{L})$ on $\Sigma^{d-1}$ is
defined by the subset of (plots of) sections over the infinitesimal
neighborhood of $\Sigma^{d-1}$ that factor through the shell
$S_{L}\hookrightarrow J^{\infty}_{M}F$ (Def. 5.12) of the Lagrangian L,

$\displaystyle\mathrm{InData}_{\mathcal{L}}(\Sigma^{d-1}):=\mathbold{\Gamma}_{\Sigma^{d-1},EL=0}\big{(}J_{M}^{\infty}F|_{\Sigma^{d-1}}\big{)}\longhookrightarrow\mathbold{\Gamma}_{\Sigma^{d-1}}\big{(}J_{M}^{\infty}F|_{\Sigma^{d-1}}\big{)}\,.$
(167)

In particular, a point $\psi$ of $\mathrm{InData}_{\mathcal{L}}(\Sigma^{d-1})$
is a section of the form

$\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
10.40181pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern 28.40186pt\raise
0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{S_{L}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
62.92134pt\raise-14.68324pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
97.71304pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
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0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{J^{\infty}_{M}F\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\kern
79.713pt\raise-15.23259pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy<EMAIL_ADDRESS>0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\Sigma^{d-1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces{}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
10.2563pt\raise-3.57135pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{\psi}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
28.40186pt\raise-4.36377pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}\ignorespaces\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces{\hbox{\lx@xy@drawline@}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces{\hbox{\kern
62.92134pt\raise-19.3649pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
33.66158pt\raise-19.3649pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{}$}}}}}}}{\hbox{\kern
62.92134pt\raise-19.3649pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise 0.0pt\hbox{$\textstyle{M}$}}}}}}}{\hbox{\kern
107.67624pt\raise-19.3649pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{}$}}}}}}}\ignorespaces}}}}\ignorespaces\,,$

and similarly for higher plots.

There is a natural restriction map from the space of fields to sections over
the infinitesimal neighborhood of $\Sigma^{d-1}$, given by the compostion
$(-)|_{\Sigma^{d-1}}\circ
j^{\infty}:\mathcal{F}\rightarrow\mathbold{\Gamma}_{M}(J^{\infty}_{M}F)\rightarrow\mathbold{\Gamma}_{\Sigma^{d-1}}\big{(}J_{M}^{\infty}F|_{\Sigma^{d-1}}\big{)}$.
By definition of the space of initial data on $\Sigma^{d-1}$, this descends to
a map

$\displaystyle(-)|_{\Sigma^{d-1}}\circ
j^{\infty}:\mathcal{F}_{\mathcal{E}\mathcal{L}}\longrightarrow\mathrm{InData}_{\mathcal{L}}(\Sigma^{d-1})$
(168)

since any on-shell field $\phi:M\rightarrow F$ field factors through the shell
via the jet prolongation, and hence its restriction to any submanifold does as
well.

######  Definition 6.30 (Cauchy surface).

A (connected) codimension-1 submanifold $\Sigma^{d-1}\hookrightarrow M$ is a
Cauchy surface of the Lagrangian field theory $(\mathcal{F},\mathcal{L})$ if
the smooth map

$\displaystyle\mathrm{Cau}_{\Sigma^{d-1}}:=(-)|_{\Sigma^{d-1}}\circ
j^{\infty}:\mathcal{F}_{\mathcal{E}\mathcal{L}}\xlongrightarrow{\sim}\mathrm{InData}_{\mathcal{L}}(\Sigma^{d-1})$
(169)

is a diffeomorphism.

The interpretation is exactly as in the physics literature. At the level of
$*$-plots, this entails that there is a bijection between fields
$\phi:M\rightarrow F$ that satisfy the Euler–Lagrange equations
$\mathcal{E}\mathcal{L}(\phi)=0$ on $M$, and the set of initial conditions
$\psi:\Sigma^{d-1}\rightarrow J^{\infty}_{M}F$ consistent with the
Euler–Lagrange equations, $\mathrm{Im}(\psi)\subset S_{L}\hookrightarrow
J^{\infty}_{M}F$. In simple words, the derivatives of the on-shell fields
along a Cauchy surface serve as a parametrization of the full space of on-
shell fields. In our current setting of smooth sets, we see that this
parametrization, if it exists, is in fact automatically smooth.

######  Remark 6.31 (On the order of initial data).

In practice, for an Euler–Lagrange differential operator of order $k$, it is
sufficient to consider (compatible) initial data, in terms of jets along a
hypersurface, up to order $k-1$. We do not make this explicit, as we avoid
talking about a fixed order of the Lagrangian and the induced Euler–Lagrange
operator altogether.

######  Example 6.32 (Particle mechanics Cauchy surface).

As a check, we consider the particularly simple example of free (non-
relativistic) particle mechanics. Although everything we say generalizes
straightforwardly for a free particle moving in an arbitrary Riemannian
manifold $(M,g)$, for the sake of exposition we write formulas explicitly only
for the case of flat, Euclidean space $(\mathbbm{R}^{d},\delta)$. This is, in
fact, a special case of the O($n$)-model from Ex. 3.10 via the following
choices. Choosing the spacetime to be only `time' $M=\mathbbm{R}^{1}_{t}$ with
its canonical metric, then the Euclidean target
$W\cong(\mathbbm{R}^{d},\delta)$ is interpreted as the `space', and the field
space is the smooth path space
$\mathcal{F}=\mathbold{P}(\mathbbm{R}^{d})=[\mathbbm{R}^{1}_{t},\mathbbm{R}^{d}]$.
In the case where the coupling constants vanish, the Lagrangian reduces to the
free particle Lagrangian

$\displaystyle\mathcal{L}(\gamma)$
$\displaystyle=\langle\mathrm{d}_{t}\gamma,\star\mathrm{d}_{t}\gamma\rangle=\langle\partial_{t}\gamma,\partial_{t}\gamma\rangle\cdot\mathrm{d}t=\partial_{t}\gamma^{a}\cdot\partial_{t}\gamma_{a}\cdot\mathrm{d}_{t}\,.$

The corresponding Euler–Lagrange operator is simply Newton's Law

$\displaystyle\mathcal{E}\mathcal{L}(\gamma)$
$\displaystyle=\mathrm{d}_{t}\star\mathrm{d}_{t}\gamma=\partial_{t}^{2}\gamma\cdot\mathrm{d}_{t}\,,$

and the on-shell space of solutions consists of (plots of) straight lines in
$\mathbbm{R}^{d}$

$\displaystyle\mathcal{F}_{\mathcal{E}\mathcal{L}}=\mathrm{Lines}(\mathbbm{R}^{d})\longhookrightarrow[\mathbbm{R}_{t}^{1},\mathbbm{R}^{d}]\,,$

whose *-plots are simply straight lines $\gamma=v\cdot
t+c:\mathbbm{R}_{t}\rightarrow\mathbbm{R}^{d}$. Similarly,
$\mathbbm{R}^{k}$-plots are parametrized straight lines $\gamma^{k}=v^{k}\cdot
t+c^{k}:\mathbbm{R}^{k}\times\mathbbm{R}_{t}\rightarrow\mathbbm{R}^{d}$, for
arbitrary functions $v^{k}\in C^{\infty}(\mathbbm{R}^{k})$ and $c^{k}\in
C^{\infty}(\mathbbm{R}^{k})$.

The space of positions and velocities at any time instant
$t_{0}\in\mathbbm{R}^{1}_{t}$ should be a copy of the tangent bundle
$T\mathbbm{R}^{d}$, and as is well known (and obvious) the position and
velocity at a time instant fully determine the set of straight lines. Thus, it
is naturally expected that the full space of on-shell fields
$\mathrm{Lines}(\mathbbm{R}^{d})$ is diffeomorphic to the tangent
$T\mathbbm{R}^{d}$, in an appropriate sense. To see how this description
arises from our definition, notice that the Euler–Lagrange source form
$EL=u_{tt}^{a}\cdot\mathrm{d}_{V}u_{a}\cdot\mathrm{d}t\in\Omega^{d=1,1}\big{(}J^{\infty}_{\mathbbm{R}^{1}_{t}}(\mathbbm{R}^{1}_{t}\times\mathbbm{R}^{d})\big{)}$
imposes vanishing conditions on all jets of order $k\geq 2$, and no conditions
for $k\leq 1$. Hence its shell $S_{L}$ is canonically identified with the
first jet bundle under the canonical embedding

$\displaystyle
J^{1}_{\mathbbm{R}_{t}}(\mathbbm{R}^{1}_{t}\times\mathbbm{R}^{d})$
$\displaystyle\longhookrightarrow
J^{\infty}_{\mathbbm{R}^{1}_{t}}(\mathbbm{R}^{1}_{t}\times\mathbbm{R}^{d})$
$\displaystyle\big{(}\gamma(t_{0}),\dot{\gamma}(t_{0})\big{)}$
$\displaystyle\longmapsto\big{(}\gamma(t_{0}),\dot{\gamma}(t_{0}),0,\dots
0\big{)}\,,$

which makes sense globally in this case of trivial bundles. Now, connected
submanifolds of codimension-1 of the base $M=\mathbbm{R}^{1}_{t}$ are
necessarily single points, i.e., instants of time
$t_{0}\in\mathbbm{R}^{1}_{t}$. Hence the smooth set of initial data
$\mathrm{InData}_{\mathcal{L}}(\\{0\\})$ (Def. 6.29) along
$\\{t_{0}=0\\}\hookrightarrow\mathbbm{R}^{1}_{t}$, has $*$-plots $\psi$ that
factor through the shell

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and similarly for higher plots. Note that such sections are simply points (and
plots) of the first jet manifold
$J^{1}_{\mathbbm{R}_{t}}(\mathbbm{R}^{1}_{t}\times\mathbbm{R}^{d})$, and so
the smooth set of initial data is simply the Yoneda embedding of the first jet
space $\mathrm{InData}_{\mathcal{L}}(\\{0\\})\cong
y(J^{1}_{\mathbbm{R}_{t}}(\mathbbm{R}^{1}_{t}\times\mathbbm{R}^{d}))$. But the
manifold of first jets of curves is by definition the tangent bundle
$J^{1}_{\mathbbm{R}_{t}}(\mathbbm{R}^{1}_{t}\times\mathbbm{R}^{d})\cong_{\mathrm{SmthMfd}}T\mathbbm{R}^{d}$,
and so it follows that

$\displaystyle\mathrm{InData}_{\mathcal{L}}(\\{0\\})\cong_{\mathrm{SmthSet}}y(T\mathbbm{R}^{d})\,.$

In other words, the smooth set of initial conditions is simply (the Yoneda
embedding of) the tangent bundle $T\mathbbm{R}^{d}$ of the target
$\mathbbm{R}^{d}$, which shows immediately that the restriction

$\displaystyle\mathrm{Cau}_{\\{0\\}}:=(-)|_{\\{t=0\\}}\circ
j^{1}:\mathcal{F}_{\mathcal{E}\mathcal{L}}$
$\displaystyle\;\longrightarrow\;\mathrm{InData}_{\mathcal{L}}(\\{0\\})\cong
y(T\mathbbm{R}^{d})$ (170) $\displaystyle\gamma^{k}=v^{k}\cdot t+c^{k}$
$\displaystyle\;\longmapsto\;(c^{k},v^{k})\,,$

is a diffeomorphism. Here the choice of time instant
$\\{t_{0}\\}\hookrightarrow\mathbbm{R}^{1}_{t}$ is immaterial, and hence any
time instant is a Cauchy surface – but through a different Cauchy isomorphism!
We will revisit this setting in Ex. 7.42 below.

Generally, neither the surjectivity nor the injectivity of the underlying
point-set map (168) to the initial data is guaranteed, both of which depend
highly on the detailed form of the given Euler–Lagrange differential operator
$\mathcal{E}\mathcal{L}$. In non-technical terms, surjectivity says that for
any choice of initial data $\psi:\Sigma^{d-1}\rightarrow J^{\infty}_{M}F$,
there exists some field configuration $\phi:M\rightarrow F$ whose jet
prolongation restricts to $\psi$ on $\Sigma^{d-1}$, and so $\phi$ is thought
as the `evolution' of the initial condition $\psi$ on $M$ under the
Euler–Lagrange equations $\mathcal{E}\mathcal{L}=0$. Injectivity says that in
the cases of initial conditions $\psi$ for which some evolution $\phi$ exists,
then it is also unique.

The general study of such well-posedness questions is quite delicate and falls
within the realm of analysis of PDEs, which is outside of the scope of this
manuscript. However, an immediate general result that evades analytical
arguments is that finite gauge symmetries obstruct the existence of Cauchy
surfaces – by ruling out the injectivity of the underlying point-set map.

######  Proposition 6.33 (Gauge symmetry obstructs Cauchy surface).

Let
$\mathcal{R}_{(-)}:\Gamma_{M}(E)\rightarrow\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$
be an infinitesimal (local) gauge symmetry of a Lagrangian field theory
$(\mathcal{F},\mathcal{L})$ and $\Sigma^{d-1}\hookrightarrow M$ a
codimension-1 submanifold. Let $e\in\Gamma_{M}(E)$ be a (nonzero) gauge
parameter with support contained strictly within
$\mathrm{int}(M-\Sigma^{d-1})$. Then, assuming the infinitesimal symmetry
$\mathcal{R}_{e}$ is (a) integrable and (b) not zero on
$\mathcal{F}_{\mathcal{E}\mathcal{L}}$ (hence not trivial), the restriction to
initial data

$(-)|_{\Sigma^{d-1}}\circ
j^{\infty}:\mathcal{F}_{\mathcal{E}\mathcal{L}}\longrightarrow\mathrm{InData}_{\mathcal{L}}(\Sigma^{d-1})$

is not a diffeomorphism. In other words, the submanifold $\Sigma^{d-1}$ is not
a Cauchy surface.

###### Proof.

Denote by
$\mathcal{D}_{(e)}:=\mathcal{D}_{(e),t=1}:\mathcal{F}\rightarrow\mathcal{F}$
the diffeomorphism which is the integrated version of the local
$\mathcal{R}_{e}$ vector field, i.e.,
$\partial_{t}\mathcal{D}_{(e),t}|_{t=0}=\mathcal{R}_{e}$. This exists by
assumption and hence is in particular a local symmetry (Rem. 6.13). By Prop.
5.27, it preserves the on-shell space of fields
$\mathcal{D}_{(e)}:\mathcal{F}_{\mathcal{E}\mathcal{L}}\xrightarrow{\sim}\mathcal{F}_{\mathcal{E}\mathcal{L}}$,
and so if $\phi\in\mathcal{F}_{\mathcal{E}\mathcal{L}}(*)$ is an on-shell
field, then so is $\mathcal{D}_{(e)}(\phi)$. Furthermore, the infinitesimal
symmetry $\mathcal{R}_{e}$ is by assumption not zero on
$\mathcal{F}_{\mathcal{E}\mathcal{L}}$ (hence nontrivial, Ex. 6.12, Ex. 6.13).
Consequently, $\mathcal{D}_{(e)}$ is not the identity map
$\mathcal{D}_{(e)}\neq\mathrm{id}_{\mathcal{F}_{\mathcal{E}\mathcal{L}}}$ on
the space of on-shell fields, so that (without loss of generality)

$\mathcal{D}_{(e)}(\phi)\neq\phi\,.$

Next, since the support of $e$ is strictly within
$\mathrm{int}(M-\Sigma^{d-1})$, it follows that for any $p\in\Sigma^{d-1}$
there exists some open neighborhood $U_{p}\subset M$ on which $e$ vanishes,
$e|_{U_{p}}=0$. It follows that, by the $\mathbbm{R}$-linear dependence on $e$
and the petit sheaf nature of the field space $\mathcal{F}$ (Rem. 2.14), the
local vector field $\mathcal{R}_{e}$ vanishes on $U_{p}$, i.e.,
$\mathcal{R}_{e}|_{U_{p}}=0\in\mathcal{X}_{\mathrm{loc}}(\mathcal{F}|_{U_{p}})$.
Thus, the corresponding integrated diffeomorphism must be the identity when
restricted to $U_{p}\subset M$,

$\mathcal{D}_{(e)}|_{U_{p}}=\mathrm{id}_{\mathcal{F}|_{U_{p}}}\,:\,\mathcal{F}_{\mathcal{E}\mathcal{L}}|_{U_{p}}\longrightarrow\mathcal{F}_{\mathcal{E}\mathcal{L}}|_{U_{p}}\,,$

for some open neighborhood around $p\in\Sigma^{d-1}$, for every
$p\in\Sigma^{d-1}$.

Finally, to prove the claim of the proposition, it is enough to prove the map
is not bijective on $*$-plots. Let
$\phi\in\mathcal{F}_{\mathcal{E}\mathcal{L}}(*)$ be any on-shell field
configuration. Since local diffeomorphism on $\mathcal{F}$ are in particular
maps of (petit) sheaves on $M$, we have

$\mathcal{D}_{(e)}(\phi)|_{U_{p}}=\mathcal{D}_{(e)}|_{U_{p}}(\phi|_{U_{p}})=\mathrm{id}_{\mathcal{F}|_{U_{p}}}(\phi|_{U_{p}})=\phi|_{U_{p}}\in\mathcal{F}_{\mathcal{E}\mathcal{L}}|_{U_{p}}\,$

by which it follows that the infinite jets coincide

$j^{\infty}_{p}\big{(}\mathcal{D}_{(e)}(\phi)\big{)}=j^{\infty}_{p}\phi\,\in
J^{\infty}_{M}F\,,$

for every $p\in\Sigma^{d-1}$. That is, both on-shell fields
$\mathcal{D}_{(e)}(\phi)$ and $\phi$ define the same initial data on
$\Sigma^{d-1}$

$j^{\infty}\big{(}\mathcal{D}_{(e)}\phi)|_{\Sigma^{d-1}}=j^{\infty}\phi|_{\Sigma^{d-1}}\,:\,\Sigma^{d-1}\longrightarrow
S_{L}\hookrightarrow J^{\infty}_{M}F\,.$

Therefore, there are two different on-shell configurations
$\mathcal{D}_{(e)}(\phi),\,\phi$ with the same initial data
$j^{\infty}\phi|_{\Sigma^{d-1}}$, hence the map

$(-)|_{\Sigma^{d-1}}\circ
j^{\infty}:\mathcal{F}_{\mathcal{E}\mathcal{L}}(*)\longrightarrow\mathrm{InData}_{\mathcal{L}}(\Sigma^{d-1})(*)$

is not injective. ∎

######  Corollary 6.34 (Nonexistence of Cauchy surfaces).

Let
$\mathcal{R}_{(-)}:\Gamma_{M}(E)\rightarrow\mathcal{X}_{\mathrm{loc}}(\mathcal{F})$
be an infinitesimal (local) gauge symmetry of a Lagrangian field theory
$(\mathcal{F},\mathcal{L})$ such that $\mathcal{R}_{e}$ is (a) integrable and
(b) not zero on $\mathcal{F}_{\mathcal{E}\mathcal{L}}$, for all gauge
parameters $e\in\Gamma_{M}(F)$ of compact support. Then there does not exist
any Cauchy surface for $(\mathcal{F},\mathcal{L})$.

###### Proof.

Let $\Sigma^{d-1}$ be a codimension-$1$ submanifold and choose (nonzero) a
gauge parameter $e\in\Gamma_{M}(E)$ with compact support in
$\mathrm{int}(M-\Sigma^{d-1})$. Such sections exist since $E\rightarrow M$ is
a locally trivial bundle, by employing bump functions. By the assumptions on
$\mathcal{R}_{e}$, the result of Prop. 6.33 applies. The argument applies for
any candidate submanifold $\Sigma^{d-1}$, and so the result follows. ∎

######  Example 6.35 (Instances of nonexistence).

The assumptions of Cor 6.34 are satisfied in all examples of fundamental field
theories with gauge symmetries. We indicate the corresponding ingredients
without spelling out the details in two such cases:

(i) For the theory of General Relativity, the infinitesimal gauge symmetries
are parametrized by vector fields $\mathcal{X}(M)=\Gamma_{M}(TM)$ on
spacetime. Those of compact support $\mathcal{X}_{c}(M)\subset\mathcal{X}(M)$
(and their induced vector fields on $\mathcal{F}=\mathrm{Met}(M)$) arise by
differentiating diffeomorphisms $\mathrm{Diff}_{c}(M)\subset\mathrm{Diff}(M)$
(and the induced action on $\mathrm{Met}(M)$)696969Strictly speaking, the
(finite) gauge symmetries in the case of General Relativity are spacetime
covariant symmetries (Def. 3.23). Nevertheless, the on-shell space of fields
is still preserved (Prop. 5.41), and hence Prop. 6.33 and Cor. 6.34 apply with
minor modifications.  which are the identity outside a compact submanifold.

(ii) Similarly for Yang-Mills theory in the trivial topological sector, the
infinitesimal gauge symmetries are parametrized by
$C^{\infty}(M,\mathfrak{g})=\Gamma_{M}(M\times\mathfrak{g})$. The gauge
parameters of compact support $C^{\infty}_{c}(M,\mathfrak{g})\subset
C^{\infty}(M,\mathfrak{g})$ arise by differentiating maps
$C^{\infty}_{c}(M,G)\subset C^{\infty}(M,G)$ which map to the identity $e\in
G$ outside a compact submanifold (along with the induced vector fields on
$\mathcal{F}=\Omega^{1}(M,\mathfrak{g})$ by differentiating the induced finite
diffeomorphisms on $\Omega^{1}(M,\mathfrak{g})$, respectively). A special case
of this is (abelian) $G=U(1)$ Yang-Mills theory in the guise of
electromagnetism from Ex. 6.21. By the linearity of the field space
$\mathcal{F}=\Omega^{1}_{\mathrm{Vert}}(M)$ and the constancy of each vector
field $\mathcal{R}_{e}:\mathcal{F}\rightarrow
T\mathcal{F}\cong\mathcal{F}\times\mathcal{F}$, given by
$A^{k}\mapsto\mathrm{d}_{M}e$, the diffeomorphism corresponding to the
integrated infinitesimal transformation is given on $\mathbbm{R}^{k}$-plots of
fields by

$\displaystyle\mathcal{D}_{(e)}:\Omega_{\mathrm{Vert}}^{1}(M)$
$\displaystyle\longrightarrow\Omega_{\mathrm{Vert}}^{1}(M)$ $\displaystyle
A^{k}$ $\displaystyle\longmapsto A^{k}+\mathrm{d}_{M}e\,.$

######  Remark 6.36 (Gauge equivalence classes).

To preserve the assumption that classical physics is deterministic, i.e., with
a unique `time evolution', one is forced to consider any two field
configuration $\phi_{1},\phi_{2}\in\mathcal{F}(*)$ of a local field theory
$(\mathcal{F},\mathcal{L})$ which are related by a (finite) gauge
transformation as being equivalent. It follows that a physically viable
observable should assign the same value for any gauge equivalent field
configuration. One way to implement this rigorously is to consider the
quotient set of fields by gauge transformations, or in our setting, the
quotient smooth set – a colimit construction which exists in smooth sets

$\displaystyle\mathcal{F}\,/\,\mathrm{Diff}^{\mathcal{L},\mathrm{gauge}}(\mathcal{F})\hskip
14.22636pt\in\hskip 14.22636pt\mathrm{SmthSet}$ (171)

on which all (gauge invariant) local smooth functions out of $\mathcal{F}$
descend to. We note that for this quotient construction to make sense, it is
crucial that the Euler–Lagrange operator is covariant under local (and
spacetime covariant) symmetries, which is indeed the case (137). For the case
of electromagnetism, this is the smooth set corresponding to the quotient
$\Omega^{1}(M)/\mathrm{d}_{M}\Omega^{0}(M)$.

######  Remark 6.37 (On gauge equivalence and redundancy).

We briefly remark on two ways that the viewpoint of gauge symmetries as mere
redundancies can be insufficient. The concepts involved in the discussion
below will be rigorously fleshed out in [GSS24].

(i) This treatment is sufficient for most purposes of classical field theory
with fields being sections of fiber bundles
$\mathcal{F}=\mathbold{\Gamma}_{M}(F)$. However, for the purposes of `gauge
fixing'707070Roughly, this means finding an equivalent local field theory
$(\mathcal{F}^{\prime},\mathcal{L}^{\prime})$ with no gauge symmetries.
Equivalency, among other conditions, means that the two underlying on-shell
spaces should be diffeomorphic
$\mathcal{F}_{\mathcal{E}\mathcal{L}}/\mathrm{Diff}^{\mathcal{L},\mathrm{gauge}}(\mathcal{F})\cong\mathcal{F}^{\prime}_{\mathcal{E}\mathcal{L}^{\prime}=0}$.
and quantization it is extremely useful to consider this as a quotient in a
homotopical/higher geometrical sense, thus resulting into a smooth groupoid
instead (see section 8). The infinitesimal/perturbative (and algebraic)
incarnation of this construction appears in the physics literature under the
name BRST complex (see [HT92]).

(ii) There is another, related, sense in which the phrase `gauge equivalence
is redundancy' should be carefully interpreted. Throughout this text, we have
dealt with field spaces being sections of fiber bundles
$\mathcal{F}=\mathbold{\Gamma}_{M}(F)$ over the spacetime $M$, which are by
definition (petit) sheaves on $M$ (Rem. 2.14). This description includes, in
particular, globally defined $G$-gauge fields
$\Omega^{1}(M,\mathfrak{g})\cong\Gamma_{M}\big{(}T^{*}M\times_{M}(M\times\mathfrak{g})\big{)}$.
In other words, principal $G$-bundle connections over $M$ in the trivial
topological sector. Taking into account dynamics via a Lagrangian which is
invariant under the action of $C^{\infty}(M,G)$, one may (or may not) consider
the corresponding homotopy quotient groupoid as the actual physical field
space. However, gauge fields in the non-trivial topological sector, i.e.,
$G$-connections on $P\rightarrow M$ for an arbitrary $G$-bundle (e.g., Dirac
monopole in electromagnetism, instantons in Yang-Mills theory), do not follow
this description. In particular, the description of a single gauge field $A$
as an object living on spacetime717171As opposed to its equivalent description
of a global $1$-form $\tilde{A}\in\Omega^{1}(P,\mathfrak{g})$ on the
corresponding principal $G$-bundle $P\rightarrow M$. $M$ (with respect to a
good open cover
$\mathop{\text{\smash{\raisebox{-9.03763pt}{\scalebox{1.0}[-1.0]{$\prod$}}}\vphantom{$\prod$}}}_{i\in
I}U_{i}\rightarrow M$) is given by a pair of families of

(a) locally defined fields
$\big{\\{}A_{i}\in\Omega^{1}(U_{i},\mathfrak{g})\big{\\}}_{i\in I}$ and

(b) locally defined ``transition maps''727272Corresponding to the isomorphisms
between two different trivializations of $P\rightarrow M$ over $U_{i}$ and
$U_{j}$. $\big{\\{}g_{ij}\in C^{\infty}(U_{ij},G)\big{\\}}_{i,j\in I}$ on the
intersections $U_{ij}=U_{i}\cap U_{j}$ such that

$A_{j}=g_{ji}\cdot A_{i}\cdot
g_{ji}^{-1}+g_{ji}\cdot\mathrm{d}g_{ji}^{-1}\quad\mathrm{and}\quad
g_{ij}=g_{ik}\cdot g_{kj}$ (172)

on overlaps $U_{ij}$ and triple overlaps $U_{ijk}=U_{i}\cap U_{j}\cap U_{k}$,
respectively. In other words, the collection of all $G$-gauge fields on $M$
does not form a (petit) mapping/section sheaf of Rem. 2.14. Instead, they may
be identified with an appropriate mapping space construction in a higher
categorical setting, via maps into a classifying moduli stack
$M\rightarrow\mathbf{B}G_{\mathrm{conn}}$ (202), hence forming a (petit)
mapping stack on $M$!

One notices that the ``transformations'' relating the
$\big{\\{}A_{i}\in\Omega^{1}(U_{i},\mathfrak{g})\big{\\}}$ via the locally
defined transition maps $\big{\\{}g_{ij}\in
C^{\infty}(U_{ij},G)\big{\\}}_{i,j\in I}$ have the same form as the action of
(locally defined) gauge transformations in the case of the trivial topological
sector. However, these transition maps are data entering the definition of the
field configuration. They are not to be viewed as redundancies and one should
not quotient by them. Indeed, if one defines instead the field space to be
equivalence classes $[A_{i}]$ of locally defined forms by such
transformations, then the information about the non-trivial topological
sectors is lost. In particular, if these (locally defined) transition maps
were merely redundancies, then all gauge fields would be necessarily
equivalent to a globally defined $A\in\Omega^{1}(M,\mathfrak{g})$.

The actual gauge equivalences that may be partially viewed as redundancies in
the vein of (i) are instead parametrized by locally defined families737373In
contrast to transition the transition maps, these are defined locally on the
open subsets $U_{i}$, rather than the intersections $U_{ij}$.
$\big{\\{}\lambda_{i}\in C^{\infty}(U_{i},G)\big{\\}}_{i\in I}$ acting on any
gauge field747474Corresponding to pulling back the connection on the total
space $P\rightarrow M$ along a principal $G$-bundle automorphism
$\bar{P}\rightarrow P$ over $M$. $A=\big{(}\\{A_{i}\\}_{i\in
I},\,\\{g_{ij}\\}_{i,j\in I}\big{)}$ by

$\displaystyle\\{A_{i}\\}_{i\in I}$
$\displaystyle\longmapsto\\{\lambda_{i}^{-1}\cdot
A_{i}\cdot\lambda_{i}+\lambda_{i}^{-1}\cdot\mathrm{d}\lambda_{i}\\}_{i\in I}$
$\displaystyle\\{g_{ij}\\}_{i,j\in I}$
$\displaystyle\longmapsto\\{\lambda^{-1}_{i}\cdot
g_{ij}\cdot\lambda_{j}\\}_{i,j\in I}\,.$

In fact, these natural automorphisms of the full nonperturbative field space
consisting of $G$-gauge fields of arbitrary topological sectors are
automatically encoded in the (smooth) mapping stack/groupoid construction
$[M,\mathbf{B}G_{\mathrm{conn}}]$ as homotopies/higher morphisms between the
two gauge field configurations [FSS12][FSS13][FSS14], as we will make further
explicit in future installments of this series.

##  7 Presymplectic structure of local field theories

The setting of smooth sets is the natural arena to make fully rigorous the
observation of [Zu86] (considerably expanded in [DF99]), that the usual
manipulations of classical local field theory (for example as described in the
previous sections) may be concisely expressed via the bicomplex of local
differential forms on the product `smooth space' $\mathcal{F}\times M$ of off-
shell fields and the spacetime, which arises as the `pullback' of the
variational bicomplex on $J^{\infty}_{M}F$.

The point is that in our current context the spaces in question exist as
smooth sets. There is a natural notion of a smooth tangent bundle on
$\mathcal{F}\times M$, and hence differential forms on it, whereby one can
genuinely pullback forms on $J^{\infty}_{M}F$ via the smooth prolongated
evaluation map
$\mathrm{ev}\circ(j^{\infty}\times\mathrm{id}_{M}):\mathcal{F}\times
M\rightarrow J^{\infty}_{M}F$, hence identifying a subset of differential
forms with an induced bi-complex structure. This recovers the bicomplex of
local forms as implicitly used in [Zu86] and [DF99]. For any local field
theory $(\mathcal{F},\mathcal{L})$, the main additional insight of this
formalism is the study of the induced presymplectic $(d-1,2)$-form
$\omega_{\mathcal{L}}$ on $\mathcal{F}\times M$, its restriction to
$\mathcal{F}_{\mathcal{E}\mathcal{L}}\times M$ and its transgression to an on-
shell presymplectic $2$-form
$\int_{\Sigma^{d-1}}\omega_{\mathcal{L}}|_{\mathcal{E}\mathcal{L}}$ on
$\mathcal{F}_{\mathcal{E}\mathcal{L}}$ upon integration along an appropriate
submanifold, yielding the covariant phase space of a local field theory as a
presymplectic smooth set.

As previously, we rigorously define the relevant structures within the
category of smooth sets, while also describing the exact correspondence with
the formulas appearing in the physics literature.757575A similar formalization
is described in [Blo23], which however takes place in ‘pro-diffeological’
spaces. And hence in turn pro-smooth sets, rather than actual smooth sets.
This is not ideal and has its limitations – see Rem. 3.8. Further
discrepancies in definitions and results in comparison to our approach will be
noted throughout this section. However, a discussion of the presymplectic and
induced Poisson structures does not appear therein. Our presentation includes
a detailed discussion of the off-shell presymplectic current of field
theories, and the induced bracket structures of Noether and Hamiltonian local
currents, which in good situations restricts to the on-shell product
$\mathcal{F}_{\mathcal{E}\mathcal{L}}\times M$. Finally, by transgressing the
presymplectic current to an actual presymplectic $2$-form on $\mathcal{F}$, we
describe the analogous induced Poisson bracket on Hamiltonian functionals. In
good situations, this restricts to the covariant Poisson bracket of
Hamiltonian functionals on the space on-shell fields
$\mathcal{F}_{\mathcal{E}\mathcal{L}}$, i.e., the Poisson bracket
corresponding to the presymplectic structure of the covariant phase space.

In the mathematical physics literature, early accounts for the covariant phase
space approach include [So77][Ki73][KS76] [Ga72][GK74][KT79]. It was
rediscovered in [Wi86][Zu86], see also [CW87]. Detailed historical accounts,
with relevant references, of the development of the covariant phase space
formalism can be found in [Gi23] and [Ka14]. In the latter, the approach is
via the infinite jet bundle, and hence closer to ours, where furthermore the
relation with Peierls bracket (defined in terms of Green functions, when these
exist) is explained in detail.

###  7.1 The local bicomplex and its Cartan calculus

We start off by defining the tangent bundle on the product smooth set767676In
this final section, we omit reference to the fully faithful Yoneda embedding
symbols. We hope that, by now, the reader should be able to tell from the
ambient context which category the objects we refer to live in
($\mathrm{SmthMfd}$, $\mathrm{LocProMan}$ or $\mathrm{SmthSet}$).
$\mathcal{F}\times M\in\mathrm{SmthSet}$ to be

$T(\mathcal{F}\times M):=T\mathcal{F}\times TM\hskip 14.22636pt\in\hskip
14.22636pt\mathrm{SmthSet}\,,$

and so at the level of points $T_{(\phi,p)}(\mathcal{F}\times
M)=T_{\phi}\mathcal{F}\times T_{p}M=\Gamma_{M}(\phi^{*}VF)\times T_{p}M\subset
T(\mathcal{F}\times M)(*)$. The intuition is that both tangent vectors on M
and $\mathcal{F}$ are given by `infinitesimal curves' in $M$ and $\mathcal{F}$
respectively, and thus so should tangent vectors on $\mathcal{F}\times M$.
This intuition will, once again, become a constructive definition in the
synthetic sense – recovering the above (see [GS24]). Viewing
$\mathcal{F}\times M$ as a trivial bundle over spacetime

---
$\textstyle{\mathcal{F}\times
M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{pr}_{2}}$$\textstyle{M\,,}$

we may think of the (by definition) splitting of the tangent bundle
$T(\mathcal{F}\times M)$ as a horizontal splitting along the projection to
$M$. In other words,

$\displaystyle V(\mathcal{F}\times M)\times_{\mathcal{F}\times
M}H(\mathcal{F}\times M):=(T\mathcal{F}\times M)\times_{\mathcal{F}\times
M}(\mathcal{F}\times TM)\cong T\mathcal{F}\times TM\,,$ (173)

and so $T\mathcal{F}\times M\hookrightarrow T\mathcal{F}\times TM$ is the
vertical subbundle while $\mathcal{F}\times TM\hookrightarrow
T\mathcal{F}\times TM$ is the horizontal subbundle.

By definition, differential $m$-forms on both the spacetime manifold $M$ and
the infinite jet bundle $J^{\infty}_{M}F$ (Def. 4.13) are given by fiber-wise
linear bundle maps out of their tangent bundle (in $\mathrm{SmthSet}$). We
define differential forms on $\mathcal{F}\times M$ in the same vein and in
line with differential forms on $\mathcal{F}$ from Def. 2.27.

######  Definition 7.1 (Forms on $\mathcal{F}\times M$).

The set of differential m-forms on $\mathcal{F}\times M$ is defined as

$\displaystyle\Omega^{m}(\mathcal{F}\times
M):=\mathrm{Hom}^{\mathrm{fib.lin.an.}}_{\mathrm{SmthSet}}\big{(}T^{\times
m}(\mathcal{F}\times M)\,,\,\mathbbm{R}\big{)}\,,$ (174)

i.e., smooth real-valued, fiber-wise linear antisymmetric maps with respect to
the fiber-wise linear structure on the $m$-fold fiber product

$T^{\times m}(\mathcal{F}\times M):=T(\mathcal{F}\times
M)\times_{\mathcal{F}\times M}\cdots\times_{\mathcal{F}\times
M}T(\mathcal{F}\times M)$

of the tangent bundle over the $\mathcal{F}\times M$.

As usual, the collection of differential forms of all degrees forms a graded
$\mathbbm{R}$-vector space

$\Omega^{\bullet}(\mathcal{F}\times
M):=\bigoplus_{m\in\mathbbm{N}}\Omega^{m}(\mathcal{F}\times M)\,.$

Furthermore, it is straightforward to define what a horizontal and vertical
form on $\mathcal{F}\times M$ should be. For instance, as with the infinite
jet bundle (Def. 5.1), a horizontal $1$-form $\omega$ on $\mathcal{F}\times M$
is one that vanishes under the restriction

$\displaystyle\omega|_{V(\mathcal{F}\times M)}:T\mathcal{F}\times
M\longhookrightarrow T\mathcal{F}\times TM\longrightarrow\mathbbm{R}\;,$ (175)

and analogously for vertical $1$-forms, and even further for horizontal /
vertical $m$-forms on $\mathcal{F}\times M$. At this point, the development of
a bi-complex structure and, furthermore, a corresponding Cartan calculus on
$\mathcal{F}\times M$ reaches several stumbling blocks:

######  Remark 7.2 (Caveats with the `bicomplex of $\mathcal{F}\times M$').

Both [Zu86] and [DF99] seem to assume (albeit unnecessarily) the existence of
a bi-complex structure on the set of all differential forms777777Although they
do not explicitly define what this set consists of. We presume they mean a
version of our description. $\Omega^{\bullet}(\mathcal{F}\times M)$, along
with a Cartan Calculus with respect to vector fields on $\mathcal{F}\times M$.
However, the (potential) existence of this structure suffers from the
following ambiguities:

(a) Even though there is a notion of horizontal/vertical $1$-forms on
$\mathcal{F}\times M$, it is not obvious that every $1$-form $\omega$ on
$\mathcal{F}\times M$ splits as

$\omega\overset{?}{=}\omega_{V}+\omega_{H}\,.$

Recall, the reason this holds for finite-dimensional manifolds and the
infinite jet bundle is that $1$-forms, in these cases, are in 1-1
correspondence with maps of modules of vector fields (Rem. 4.12). This in turn
uses the paracompact structure of the underlying (Fréchet) manifolds to extend
tangent vectors on $J^{\infty}_{M}F$ to vector fields. This is not at all
obvious for the case of the field space $\mathcal{F}$, and in turn for
$\mathcal{F}\times M$. In other words, this seems to be a non-trivial obstacle
towards defining the bigrading on $\Omega^{\bullet}(\mathcal{F}\times M)$. A
potential proof that might overcome this is sketched in Rem. 7.38.
Nevertheless, there is a further crucial issue.

(b) There is no obvious definition for a `de Rham differential'

$\mathrm{d}_{\mathcal{F}}:\Omega^{m}(\mathcal{F})\longrightarrow\Omega^{m+1}(\mathcal{F})\,.$

Nevertheless, we note that for any vector field
$\mathcal{Z}:\mathcal{F}\rightarrow T\mathcal{F}$ there is a natural
contraction / interior product operation 787878This extends to higher degrees.
For $m=2$, the precomposition
$T\mathcal{F}\times_{\mathcal{F}}\mathcal{F}\xrightarrow{(\mathrm{id},\mathcal{Z})}T\mathcal{F}\times_{\mathcal{F}}T\mathcal{F}\rightarrow\mathbbm{R}$
defines a $1$-form by noting the canonical isomophism
$T\mathcal{F}\times_{\mathcal{F}}\mathcal{F}\cong T\mathcal{F}$ as a fiber
product over $\mathcal{F}$.

$\displaystyle\iota_{\mathcal{Z}}:\Omega^{1}(\mathcal{F})$
$\displaystyle\longrightarrow C^{\infty}(\mathcal{F})$ $\displaystyle\omega$
$\displaystyle\longmapsto\omega\circ\mathcal{Z}\,,$

which is however insufficient for a Cartan calculus in the absence of a
differential. Naturally, this is directly related to the observations of Rem.
2.18 and Eq. (33), since the would-be differential would necessarily define a
derivation / Lie derivative
$\mathbb{L}_{\mathcal{Z}}=\iota_{\mathcal{Z}}\mathrm{d}_{\mathcal{F}}$ on
$C^{\infty}(\mathcal{F})$. The same statements follow through for the case of
$\mathcal{F}\times M$, even without the further demand of an underlying bi-
grading structure.

(c) We note that the problems of (a), (b) are evaded if one instead takes as a
definition of forms on $\mathcal{F}$ that of the classifying space
$\mathbold{\Omega}^{m}_{\mathrm{dR}}$ (Def. 2.31). As explained therein, a de
Rham differential exists for this version of forms on $\mathcal{F}$, and
similarly for those on $\mathcal{F}\times M$. In fact, one can see that a
natural bicomplex structure exists797979For instance,
$\mathrm{Hom}_{\mathrm{SmthSet}}(\mathcal{F}\times
M,\,\mathbold{\Omega}^{1}_{\mathrm{dR}})\cong\mathrm{Hom}_{\mathrm{SmthSet}}\big{(}\mathcal{F},\,[M,\mathbold{\Omega}^{1}_{\mathrm{dR}}]\big{)}$
by the internal hom property. Evaluating on test probes, the mapping space on
the right is isomorphic to
$\big{(}y(\mathbbm{R})\hat{\otimes}\Omega^{1}(M)\big{)}\bigoplus\big{(}\mathbold{\Omega}^{1}_{\mathrm{dR}}\hat{\otimes}C^{\infty}(M)\big{)}$,
hence splitting a de Rham $1$-form
$\omega\in\Omega^{1}_{\mathrm{dR}}(\mathcal{F}\times M)$ as
$\omega=\omega^{1,0}+\omega^{0,1}$. This generalizes to de Rham m-forms, and
hence splitting the de Rham differential (Def. 2.32) as
$\mathrm{d}_{\mathrm{dR}}=\mathrm{d}_{\mathrm{dR}}^{\mathcal{F}}+\mathrm{d}_{\mathrm{dR}}^{M}$,
inducing a bi-complex structure. on
$\Omega^{\bullet}_{\mathrm{dR}}(\mathcal{F}\times
M):=\mathrm{Hom}_{\mathrm{SmthSet}}(\mathcal{F}\times
M,\,\mathbold{\Omega}^{\bullet}_{\mathrm{dR}})$. However, in the setting of
smooth sets the relation of the (classifying) de Rham forms with those of Def.
7.1 is unclear.808080As we have hinted before in Rem. 2.33, the situation is
different in the setting of infinitesimally thickened smooth sets where the
two notions will, indeed, be intricately related.  Furthermore, there is no
obvious interior product operation, which once again obstructs the
construction of a Cartan calculus.

The resolution, from our perspective, is that local classical field theory –
and hence all of the constructions and arguments of [Zu86][DF99] – only
requires the existence of a bi-complex structure (and a corresponding Cartan
Calculus) on the subset of local forms on $\mathcal{F}\times M$ (along with
the set of local vector fields), hence completely bypassing the above
conundrum. This local bicomplex does exist within our context of smooth sets,
and does indeed rigorously recover the statements and formulas of the above
references. To that end, recall the smooth evaluation map
$\mathrm{ev}:\mathbold{\Gamma}_{M}(F)\times M\longrightarrow F$ from Eq. (24).
There is an analogous smooth evaluation map, which we denote by the same
symbol,

$\displaystyle\mathrm{ev}:\mathbold{\Gamma}_{M}(J^{\infty}F)\times M$
$\displaystyle\longrightarrow J^{\infty}_{M}F$ $\displaystyle(\tilde{\phi},p)$
$\displaystyle\longmapsto\tilde{\phi}(p)\,,$

and similarly for higher plots. Precomposing along the smooth jet prolongation
$j^{\infty}:\mathcal{F}\rightarrow\mathbold{\Gamma}_{M}(J^{\infty}F)$, we may
define the smooth prolongated evaluation map with values in $J^{\infty}_{M}F$.

######  Definition 7.3 (Prolongated evaluation).

The prolongated evaluation map $\mathrm{ev}^{\infty}:\mathcal{F}\times
M\rightarrow J^{\infty}_{M}F$ is defined as the composition of maps of smooth
sets

$\displaystyle\mathrm{ev}^{\infty}:\mathcal{F}\times
M\xlongrightarrow{\quad(j^{\infty},\,\mathrm{id}_{M})\quad}\mathbold{\Gamma}_{M}(J^{\infty}F)\times
M\xlongrightarrow{\;\;\mathrm{ev}\;\;}J^{\infty}_{M}F\,.$

Explicitly, at the level of $*$-plots this is given by

$\mathrm{ev}^{\infty}(\phi,p)=j^{\infty}\phi(p)\hskip 14.22636pt\in\hskip
14.22636ptJ^{\infty}_{M}F\,,$

and similarly for higher plots.

At this point, we note that both the source $\mathcal{F}\times M$ and target
$J^{\infty}_{M}F$ of the prolongated evaluation consist of smooth spaces with
a well-defined tangent bundle, in terms of infinitesimal curves. 818181In
[GS24] all of the mentioned tangent bundles will be special instances of the
synthetic tangent bundle construction, making the intuition of infinitesimal
curves precise. The pushforward map motivated below will then follow naturally
as the synthetic pushforward of $\mathrm{ev}^{\infty}$. Moreover, by Lem.
2.17, we can represent any tangent vector $(\mathcal{Z}_{\phi},X_{p})\in
T_{\phi}\mathcal{F}\times T_{p}M\subset T\mathcal{F}\times TM$ by a line plot
$(\phi_{t},p_{t})\in\big{(}\mathcal{F}\times M\big{)}(\mathbbm{R}^{1}_{t})$
starting at $(\phi,p)$. Since tangent vectors on $J^{\infty}_{M}F$ are also
represented by line plots (Lem. 4.2), this suggests the existence of a
pushforward map along $\mathrm{ev}^{\infty}$. Indeed, we may calculate the
derivative of
$\mathrm{ev}^{\infty}\circ(\phi_{t},p_{t}):\mathbbm{R}^{1}\rightarrow
J^{\infty}_{M}F$ at $t=0$,

$\displaystyle\partial_{t}\big{(}\mathrm{ev}^{\infty}(\phi_{t},p_{t})\big{)}|_{t=0}$
$\displaystyle=\partial_{t}\big{(}j^{\infty}\phi_{t}\circ(\mathrm{id}_{\mathbbm{R}^{1}_{t}},p_{t})\big{)}|_{t=0}$
$\displaystyle=\partial_{t}(j^{\infty}\phi_{t})|_{t=0}(p_{t=0})+\partial_{t}\big{(}j^{\infty}\phi_{t=0}\circ(\mathrm{id}_{\mathbbm{R}^{1}_{t}},p_{t})\big{)}|_{t=0}$
$\displaystyle=\partial_{t}(j^{\infty}\phi_{t})|_{t=0}(p)+(\mathrm{d}j^{\infty}\phi)_{p}(\partial_{t}p_{t}|_{t=0})$
$\displaystyle=j^{\infty}\mathcal{Z}_{\phi}(p)+(\mathrm{d}j^{\infty}\phi)_{p}X_{p}\,,$

where the derivative is computed as in Lem. 4.2 and Ex. 4.7, hence via the
corresponding compatible family
$\\{j^{k}\phi_{t}\circ(\mathrm{id}_{\mathbbm{R}^{1}_{t}},p_{t}):\mathbbm{R}^{1}\rightarrow
J^{k}_{M}F\\}_{k\in\mathbbm{N}}$, whereby the chain rule applies – as done
implicitly in the second equality. The latter equalities show that the
derivative naturally combines the two previously introduced pushforward maps:
The `vertical pushforward' of Eq. (75) via the prolongation of the tangent
vector $\mathcal{Z}_{\phi}$ at $\phi\in\mathcal{F}$ evaluated at $p\in M$

$j^{\infty}\mathcal{Z}_{\phi}(p)\hskip 14.22636pt\in\hskip
14.22636ptV_{j^{\infty}_{p}\phi}J^{\infty}_{M}F\,,$

and the `horizontal pushforward' (horizontal lift) of Eq. (88) via the
differential of the prolongation of $\phi\in\mathcal{F}$ at $p\in M$

$(\mathrm{d}j^{\infty}\phi)_{p}X_{p}\hskip 14.22636pt\in\hskip
14.22636ptH_{j^{\infty}_{p}\phi}J^{\infty}_{M}F\,.$

The resulting tangent vector on $J^{\infty}_{M}F$ depends only on the tangent
vectors $(\mathcal{Z}_{\phi},X_{p})$, and so we may disregard the
representative line-plots and define the pushforward of $\mathrm{ev}^{\infty}$
directly as a map of tangent vectors.

######  Definition 7.4 (Pushforward of prolongated evaluation).

The pushforward of the prolongated evaluation
$\mathrm{ev}^{\infty}:\mathcal{F}\times M\rightarrow J^{\infty}_{M}F$ is
defined by

$\displaystyle\mathrm{d}\mathrm{ev}^{\infty}:T\mathcal{F}\times TM$
$\displaystyle\;\longrightarrow\;TJ^{\infty}_{M}F$
$\displaystyle(\mathcal{Z}_{\phi},X_{p})$
$\displaystyle\;\longmapsto\;j^{\infty}\mathcal{Z}_{\phi}(p)+(\mathrm{d}j^{\infty}\phi)_{p}X_{p}\,,$

and similarly for higher plots.

In a compatible coordinate chart for $J^{\infty}_{M}F$ around $p\in M$ where
$\mathcal{Z}_{\phi}=\mathcal{Z}_{\phi}^{a}\cdot\frac{\partial}{\partial
u^{a}}\in T_{\phi}\mathcal{F}$ and $X_{p}=X^{\mu}\cdot\frac{\partial}{\partial
x^{\mu}}|_{p}\in T_{p}M$, the pushforward is given by (see maps (75) and (88))

$\displaystyle\bigg{(}\mathcal{Z}_{\phi}^{a}\cdot\frac{\partial}{\partial
u^{a}},\,X^{\mu}\cdot\frac{\partial}{\partial
x^{\mu}}\Big{|}_{p}\bigg{)}\;\;\longmapsto\;\;\sum_{|I|=0}^{\infty}\frac{\partial\mathcal{Z}_{\phi}^{a}}{\partial
x^{I}}(p)\cdot\frac{\partial}{\partial
u^{a}_{I}}\Big{|}_{j^{\infty}_{p}\phi}+X^{\mu}\bigg{(}\frac{\partial}{\partial
x^{\mu}}\Big{|}_{j^{\infty}_{p}\phi}+\sum_{|I|=0}^{\infty}\frac{\partial\phi^{a}}{\partial
x^{I+\mu}}(p)\cdot\frac{\partial}{\partial
u^{a}_{I}}\Big{|}_{j^{\infty}_{p}\phi}\bigg{)}\,.$ (176)

We highlight that, by construction, the pushforward
$\mathrm{d}\mathrm{ev}^{\infty}$ respects the natural splitting of the tangent
bundles $T\mathcal{F}\times TM$ from Eq. (173), and that of $TJ^{\infty}_{M}F$
from Cor. 4.9, as bundles over $M$

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The idea now is to define the subset of local forms on $\mathcal{F}\times M$
by pulling back forms in the variational bi-complex on $J^{\infty}_{M}F$ via
$e_{\infty}:\mathcal{F}\times M\rightarrow J^{\infty}_{M}F$, by which we mean
precomposing differential forms with the pushforward map on tangent vectors.
In other words, the pullback local form
$(\mathrm{ev}^{\infty})^{*}\omega\in\Omega^{\bullet}(\mathcal{F}\times M)$ of
a differential form $\omega\in\Omega^{\bullet,\bullet}(J^{\infty}_{M}F)$ is
defined by the composition

$\displaystyle(\mathrm{ev}^{\infty})^{*}\omega:T\mathcal{F}\times
TM\xlongrightarrow{\;\;\mathrm{d}\mathrm{ev}^{\infty}\;\;}J^{\infty}_{M}F\xlongrightarrow{\;\;\omega\;\;}\mathbbm{R}\,.$
(177)

Since forms on $J^{\infty}_{M}F$ do have a well-defined bigrading, and the
pushforward map respects the respective splittings, it follows that the image
of the map of $\mathbbm{R}$-vector spaces

$\displaystyle(\mathrm{ev}^{\infty})^{*}:\Omega^{\bullet,\bullet}(J^{\infty}_{M}F)\longrightarrow\Omega^{\bullet}(\mathcal{F}\times
M)\,.$

induces a bi-grading – whose horizontal and vertical components coincide with
the corresponding notions defined (175) directly on $T\mathcal{F}\times TM$.
Thus, it follows that the vertical and horizontal differentials on
$J^{\infty}_{M}F$ induce two (anti-commuting) differentials on the image of
$(\mathrm{ev}^{\infty})^{*}$. In total, the pullback identifies a bi-complex
structure on the subspace of local forms on the smooth set $\mathcal{F}\times
M$.

######  Definition 7.5 (Bicomplex of local forms).

The bicomplex of local forms on $\mathcal{F}\times M$ is defined as the image
of the variational bicomplex on $J^{\infty}_{M}F$ under the pullback map
$(\mathrm{ev}^{\infty})^{*}$. That is,

$\displaystyle\Omega^{\bullet,\bullet}_{\mathrm{loc}}(\mathcal{F}\times
M):=\mathrm{Im}\big{(}(\mathrm{ev}^{\infty})^{*}\,:\,\Omega^{\bullet,\bullet}(J^{\infty}_{M}F)\longrightarrow\Omega^{\bullet}(\mathcal{F}\times
M)\big{)}\quad\subset\;\;\Omega^{\bullet}(\mathcal{F}\times M)\,,$

equipped with the induced horizontal and vertical differentials

$\mathrm{d}_{M}\big{(}(\mathrm{ev}^{\infty})^{*}\omega\big{)}:=(\mathrm{ev}^{\infty})^{*}(\mathrm{d}_{H}\omega)\,,\hskip
56.9055pt\delta\big{(}(\mathrm{ev}^{\infty})^{*}\omega\big{)}\equiv\mathrm{d}_{\mathcal{F}}\big{(}(\mathrm{ev}^{\infty})^{*}\omega\big{)}:=(\mathrm{ev}^{\infty})^{*}(\mathrm{d}_{V}\omega)\,,$

respectively.

By construction, the local bi-complex is a module over local functions, or
equivalently local $0$-forms on $\mathcal{F}\times M$, i.e.,
$C^{\infty}_{\mathrm{loc}}(\mathcal{F}\times
M)\cong\Omega^{0,0}_{\mathrm{loc}}(\mathcal{F}\times M)$. It is instructive to
spell out the form of the local bicomplex in local coordinates, which will
make contact with the formulas of [DF99]. To that end, for any 1-form
$\omega=\omega_{H}+\omega_{V}=(\omega_{H})_{\mu}\cdot\mathrm{d}x^{\mu}+\sum_{|I|=0}^{\infty}(\omega_{V})_{a}^{I}\cdot\mathrm{d}_{V}u^{a}_{I}$
on $J^{\infty}_{M}F$ and any tangent vector $(\mathcal{Z}_{\phi},X_{p})\in
T_{\phi}\mathcal{F}\times T_{p}M$, we may compute

$\displaystyle\big{(}(\mathrm{ev}^{\infty})^{*}\omega\big{)}|_{(\phi,p)}(\mathcal{Z}_{\phi},X_{p})$
$\displaystyle=\omega|_{j^{\infty}_{p}\phi}\Big{(}(\mathrm{d}\mathrm{ev}^{\infty})_{(\phi,p)}\big{(}\mathcal{Z}_{\phi},p\big{)}\Big{)}$
$\displaystyle=(\omega_{H}+\omega_{V})|_{j^{\infty}_{p}\phi}\big{(}j^{\infty}\mathcal{Z}_{\phi}(p)+(dj^{\infty}\phi)_{p}X_{p}\big{)}$
(178)
$\displaystyle=\omega_{V}|_{j^{\infty}_{p}\phi}\big{(}j^{\infty}\mathcal{Z}_{\phi}(p)\big{)}+\omega_{H}|_{j^{\infty}_{p}\phi}\big{(}(\mathrm{d}j^{\infty}\phi)_{p}X_{p}\big{)}$
$\displaystyle=\sum_{|I|=0}^{\infty}\big{(}\omega_{V}\big{)}_{a}^{I}(j^{\infty}_{p}\phi)\cdot\frac{\partial\mathcal{Z}_{\phi}^{a}}{\partial
x^{I}}(p)+\big{(}\omega_{H}\big{)}_{\mu}(j^{\infty}_{p}\phi)\cdot X^{\mu}\,,$

where in the third equality we used the vertical/horizontal properties of the
two tangent vectors, respectively, and finally used their coordinate form
(176). Note that, even though $e^{\infty}:\mathcal{F}\times M\rightarrow
J^{\infty}_{M}F$ might not be surjective 828282In more detail, as per Rem.
2.14, this is really a map of petit sheaves (and hence the pullback too) –
i.e., over each $U\subset M$. Regardless of the (non)-existence of global
sections, the restricted map $ev^{\infty}|_{U}:\Gamma_{U}(F)\times
U\rightarrow J^{\infty}_{U}F$ might still not be surjective for all $U\subset
M$, and the comment applies for each open neighborhood. It is, however, an
epimorphism as a map of (petit) sheaves – i.e., surjective on stalks. the
differentials of a local form are well-defined, i.e., independent of the
chosen representative. This follows since pullback
$(\mathrm{ev}^{\infty})^{*}\omega$ local form depends manifestly only the
values of $\omega\in J^{\infty}_{M}F$ along the image of
$\mathrm{ev}^{\infty}$ (see (7.1)).

#### Standard notation for the local bicomplex.

Recalling the abuse of notation from (148), we may denote the vertical tangent
vector $\mathcal{Z}_{\phi}=\mathcal{Z}_{\phi}^{a}\cdot\frac{\partial}{\partial
u^{a}}\,\in\,T_{\phi}\mathcal{F}=\Gamma_{M}(\phi^{*}VF)$ as

$\mathcal{Z}_{\phi}=\mathcal{Z}^{a}\cdot\frac{\delta}{\delta\phi^{a}}\Big{|}_{(\phi,p)}\hskip
14.22636pt\in\hskip 14.22636ptT_{\phi}\mathcal{F}\longhookrightarrow
T_{\phi}\mathcal{F}\times T_{p}M\,.$

Following this trend, the pullback of a vertical $1$-form
$\omega_{V}=\sum_{|I|=0}^{\infty}(\omega_{V})_{a}^{I}\cdot\mathrm{d}_{V}u^{a}_{I}=\sum_{|I|=0}^{\infty}(\omega_{V})_{a}^{I}\big{(}x^{\mu},\\{u^{b}_{J}\\}_{|J|\leq
k}\big{)}\cdot\mathrm{d}_{V}u^{a}_{I}$ may be denoted as

$\displaystyle\big{(}(\mathrm{ev}^{\infty})^{*}\omega_{V}\big{)}:$
$\displaystyle=\sum_{|I|=0}^{\infty}(\omega_{V})_{a}^{I}\circ\mathrm{ev}^{\infty}\cdot(\mathrm{ev}^{\infty})^{*}\mathrm{d}_{V}u^{a}_{I}$
(179)
$\displaystyle=\sum_{|I|=0}^{\infty}(\omega_{V})_{a}^{I}\big{(}x^{\mu},\\{\partial_{J}\phi^{b}\\}_{|J|\leq
k}\big{)}\cdot\delta(\partial_{I}\phi^{a})\hskip 14.22636pt\in\hskip
14.22636pt\Omega^{0,1}_{\mathrm{loc}}(\mathcal{F}\times M)$

with the implicit understanding that the notation here838383The abuse of
notation would be slightly less confusing if we used the notation
$\hat{\phi}^{a}_{I}$ from Ex. 3.22. We comply with the standard physics
literature notation as in [DF99], to make direct contact with the formulas
therein. means
$\partial_{I}\phi^{a}:=(\mathrm{ev}^{\infty})^{*}u^{a}_{I}=u^{a}_{I}\circ\mathrm{ev}^{\infty}\in
C^{\infty}_{\mathrm{loc}}(\mathcal{F}\times M)$, and so

$\delta(\partial_{I}\phi^{a})\equiv\delta(u^{a}_{I}\circ\mathrm{ev}^{\infty}):=(\mathrm{ev}^{\infty})^{*}\mathrm{d}_{V}u^{a}_{I}\,.$

Under this abuse of notation, the rigorous calculation of (7.1) justifies the
following useful (formal) manipulation from [DF99]

$\displaystyle\delta(\partial_{I}\phi^{a})\big{|}_{\phi,p}\bigg{(}\mathcal{Z}^{b}_{\phi}\cdot\frac{\delta}{\delta\phi^{b}}\Big{|}_{(\phi,p)}\bigg{)}=\partial_{I}\bigg{(}\delta\phi^{a}\Big{(}\mathcal{Z}^{b}_{\phi}\cdot\frac{\delta}{\delta\phi^{b}}\Big{)}\\!\bigg{)}(p)=\partial_{I}\mathcal{Z}^{a}_{\phi}(p)=\frac{\partial\mathcal{Z}^{a}_{\phi}}{\partial
x^{I}}(p)\,,$ (180)

where, as explained, the partial derivative symbol on the left-hand side is a
formal symbol, while on the right-hand side it is an actual derivative. In
other words, with the caveats and abuse of notation mentioned, the `vertical
differential commutes with partial derivatives'

$\displaystyle``\,\delta(\partial_{I}\phi^{a})=\partial_{I}(\delta\phi^{a})\,"\,,$

which is precisely how these symbols are usefully manipulated in explicit
calculations appearing in the literature.

Along the same lines, given a smooth function
$f=f\big{(}x,\\{u^{b}_{J}\\}_{|J|\leq k}\big{)}\in
C^{\infty}(J^{\infty}_{M}F)$ with induced local function
$(\mathrm{ev}^{\infty})^{*}f\in\Omega^{0,0}_{\mathrm{loc}}(\mathcal{F}\times
M)$, again by (7.1), the corresponding local vertical 1-form may be computed
and denoted as

$\displaystyle\delta\big{(}(\mathrm{ev}^{\infty})^{*}f):$
$\displaystyle=(\mathrm{ev}^{\infty})^{*}\mathrm{d}_{V}f=\sum_{|I|=0}^{\infty}\frac{\partial
f}{\partial
u^{a}_{I}}\circ\mathrm{ev}^{\infty}\cdot(\mathrm{ev}^{\infty})^{*}\mathrm{d}_{V}u^{a}_{I}$
$\displaystyle=\sum_{|I|=0}^{\infty}\frac{\delta
f\big{(}x,\\{\partial_{J}\phi^{b}\\}_{|J|\leq
k}\big{)}}{\delta(\partial_{I}\phi^{a})}\cdot\delta(\partial_{I}\phi^{a})$
(181)

On the other hand, the corresponding local horizontal 1-form is given by

$\displaystyle\mathrm{d}_{M}\big{(}(\mathrm{ev}^{\infty})^{*}f):$
$\displaystyle=(\mathrm{ev}^{\infty})^{*}\mathrm{d}_{H}f=(\mathrm{ev}^{\infty})^{*}(D_{\mu}f\cdot\mathrm{d}_{H}x^{\mu})=D_{\mu}f\circ\mathrm{ev}^{\infty}\cdot(\mathrm{ev}^{\infty})^{*}\mathrm{d}_{H}x^{\mu}\,,$

which at any point $(\phi,p)\in\mathcal{F}\times M$, again by (7.1), is
further given by

$\displaystyle\mathrm{d}_{M}\big{(}(\mathrm{ev}^{\infty})^{*}f)|_{(\phi,p)}$
$\displaystyle=D_{\mu}f(j^{\infty}_{p}\phi)\cdot\mathrm{d}x^{\mu}\big{|}_{p}=\Big{(}\frac{\partial
f}{\partial x^{\mu}}\circ
j^{\infty}\phi\Big{)}(p)\cdot\mathrm{d}x^{\mu}\Big{|}_{p}$
$\displaystyle=\mathrm{d}_{M}(f\circ j^{\infty}\phi)|_{p}$

as a map $T_{\phi}\mathcal{F}\times
T_{p}M\xrightarrow{\mathsf{pr}_{2}}T_{p}M\rightarrow\mathbbm{R}$, where in the
second equality we used the chain rule (4.10). This justifies the notation
$\mathrm{d}_{M}$ for the horizontal differential on
$\Omega^{\bullet,\bullet}_{\mathrm{loc}}(\mathcal{F}\times M)$ – it is indeed
simply the de Rham differential along the spacetime $M$, computed at each
fixed field configuration $\phi\in\mathcal{F}$

$\displaystyle\mathrm{d}_{M}\big{(}(\mathrm{ev}^{\infty})^{*}f\big{)}=\frac{\partial
f\big{(}x,\\{\partial_{J}\phi^{b}\\}_{|J|\leq k}\big{)}}{\partial
x^{\mu}}\cdot\mathrm{d}x^{\mu}$ (182)

In these terms, the compatibility of the differentials from Prop. 5.20 is then
simply the further pullback, 848484The pullback of the inclusion
$\iota_{\phi}:M\hookrightarrow\mathcal{F}\times M$ is again defined via its
pushforward $\mathrm{d}\iota_{\phi}:T_{p}M\hookrightarrow
T_{\phi}\mathcal{F}\times T_{p}M$. It follows the pullback is given by
$\iota_{\phi}^{*}=\mathrm{ev}_{\phi}\circ\mathsf{pr}_{H}:\Omega^{\bullet,\bullet}_{\mathrm{loc}}(\mathcal{F}\times
M)\rightarrow\Omega^{\bullet,0}_{\mathrm{loc}}(\mathcal{F}\times
M)\rightarrow\Omega^{\bullet}(M)$. for any $\phi\in\mathcal{F}$, along