A first idea of quantum field theory -- Symmetries

## Symmetries
 {#Symmetries}

In this chapter we discuss these topics:

* _[Infinitesimal symmetries of the Lagrangian density](#InfinitesimalSymmetriesOfTheLagrangianDensity)_

* _[Infinitesimal symmetries of the presymplectic potential current](#InfinitesimalSymmetriesOfThePresymplecticPotentialCurrent)_

$\,$

We have introduced the concept of _[[Lagrangian field theories]]_ $(E,\mathbf{L})$ in terms of a [[field bundle]] $E$ equipped with a
[[Lagrangian density]] $\mathbf{L}$ on its [[jet bundle]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).
Generally, given any [[object]] equipped with some [[structure]], it is of paramount interest to determine
the [[symmetries]], hence the [[isomorphisms]]/[[equivalences]] of the object that preserve the given [[structure]]
(this is the "[[Erlanger program]]", [Klein 1872](Erlangen+program#Klein1872)).

The [[infinitesimal symmetries of the Lagrangian density]] (def. \ref{SymmetriesAndConservedCurrents} below)
send one [[field history]] to an [[infinitesimal|infinitesimally]] nearby one which is
"[[equivalence|equivalent]]" for all purposes of [[field theory]]. Among these are the _[[infinitesimal gauge symmetries]]_
which will be of concern [below](#GaugeSymmetries).
A central theorem of [[variational calculus]] says that [[infinitesimal symmetries of the Lagrangian]]
correspond to _[[conserved currents]]_, this is [[Noether theorem|Noether's theorem I]], prop. \ref{NoethersFirstTheorem} below.
These conserved currents constitute an [[Lie algebra extension|extension]] of the [[Lie algebra]]
of symmetries, called the _[[Dickey bracket]]_.

But in (eq:DerivativeOfLepageForm) we have seen that the [[Lagrangian density]]
of a [[Lagrangian field theory]] is just one component, in [[codimension]] 0, of an inhomogeneous
"[[Lepage form]]" which in [[codimension]] 1 is given by the [[presymplectic potential current]] $\Theta_{BFV}$ (eq:PresymplecticPotential).
(This will be conceptually elucidated, after we have introduced the [[local BV-complex]], in example \ref{DerivedPresymplecticCurrentOfRealScalarField} below.) This means that in [[codimension]] 1 we are to
consider infinitesimal [[on-shell]] symmetries of the [[Lepage form]] $\mathbf{L} + \Theta_{BFV}$. These are
known as _[[Hamiltonian vector fields]]_ (def. \ref{HamiltonianForms} below) and the analog of [[Noether's theorem|Noether's theorem I]]
now says that these correspond to _[[Hamiltonian differential forms]]_. The [[Lie algebra]] of these
infinitesimal symmetries is called the _[[Poisson bracket Lie n-algebra|local Poisson bracket]]_ (prop. \ref{LocalPoissonBracket} below).

**[[Noether theorem]] and [[Hamiltonian Noether theorem]]**

| $\,$ [[variational differential forms|variational form]] $\,$  |   $\,$ [[symmetry]] $\,$ | $\,$ [[Cartan's homotopy formula|homotopy formula]] $\,$ | $\,$ physical quantity $\,\,\,$ | $\,$ [[Lie n-algebra|local symmetry algebra]] $\,$ |
|----|----------|----------------------------|---------------------------------|------|
| [[Lagrangian density]] $\mathbf{L}$
 (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) | $\mathcal{L}_v \mathbf{L} = d \tilde J$  | $ d(\underset{= J_v}{\underbrace{\tilde J - \iota_v \Theta_{BFV}}}) = \iota_v \, \delta_{EL}\mathbf{L}$ | [[conserved current]] $J_v$
 (def. \ref{SymmetriesAndConservedCurrents}) | [[Dickey bracket]] |
| [[presymplectic current]] $\Omega_{BFV}$
 (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})  | $\mathcal{L}^{var}_v \Theta_{BFV} = \delta \tilde H$ | $\delta(\underset{= H_v}{\underbrace{\tilde H_v - \iota_v \Theta_{BFV}}}) = \iota_v \Omega_{BFV}$ | [[Hamiltonian differential form|Hamiltonian form]] $H_v$
 (def. \ref{HamiltonianForms}) | [[Poisson bracket Lie n-algebra|local Poisson bracket]]
 (prop. \ref{LocalPoissonBracket}) |

$\,$

In the chapter _[Phase space](#PhaseSpace)_ below we [[transgression|transgress]] this [[Poisson bracket Lie n-algebra|local Poisson bracket]]
of [[infinitesimal symmetries]] of the [[presymplectic potential current]] to the "global" [[Poisson bracket]] on the _[[covariant phase space]]_ (def. \ref{PoissonBracketOnHamiltonianLocalObservables} below). This is the structure which then [further below](#Quantization) leads over to the
[[quantization]] ([[deformation quantization]]) of the [[prequantum field theory]] to a genuine [[perturbative quantum field theory]].
However, it will turn out that there may be an [[obstruction]] to this construction, namely the existence of
special infinitesimal symmetries of the Lagrangian densities, called _implicit [[gauge symmetries]]_ (discussed [further below](#GaugeSymmetries)).

$\,$

**[[infinitesimal symmetries]] of the [[Lagrangian density]]**
 {#InfinitesimalSymmetriesOfTheLagrangianDensity}

+-- {: .num_defn #Variation}
###### Definition
**(variation)**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}).

A _variation_ is a [[vertical vector field]] $v$ on the [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) hence a vector field which
vanishes when evaluated in the [[horizontal differential forms]].

In the special case that the [[field bundle]] is [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle},
a variation is of the form

$$
  v = v^a \partial_{\phi^a} + v^a_{,\mu} \partial_{\phi^a_{,\mu}} + v^a_{\mu_1 \mu_2} \partial_{\phi^a_{\mu_1 \mu_2}} + \cdots
$$

=--

The concept of variation in def. \ref{Variation} is very general, in that it allows to vary the
field coordinates independently from the corresponding jets. This generality is necessary for
discussion of symmetries of [[presymplectic currents]] in def. \ref{HamiltonianForms} below.
But for discussion of symmetries of [[Lagrangian densities]] we are interested in
explicitly varying just the [[field (physics)|field]] coordinates (def. \ref{EvolutionaryVectorField} below)
and inducing from this the corresponding variations of the field derivatives (prop. \ref{EvolutionaryVectorFieldProlongation}) below.

In order to motivate the following definition \ref{EvolutionaryVectorField} of _[[evolutionary vector fields]]_ we follow remark \ref{ReplacingBundleMorphismsByDifferentialOperators} saying that concepts in [[variational calculus]] are obtained from their analogous concepts in plain [[differential calculus]] by replacing plain [[bundle morphisms]] by morphisms out of the [[jet bundle]]:

Given a [[fiber bundle]] $E \overset{fb}{\to} \Sigma$, then a _[[vertical vector field]]_ on $E$ is
a [[section]] of its [[vertical tangent bundle]] $T_\Sigma E$ (def. \ref{VerticalTangentBundle}), hence is a [[bundle morphism]]
of this form

$$
  \array{
    E && \overset{\text{vertical vector field}}{\longrightarrow} && T_\Sigma E
    \\
    & {}_{\mathllap{id}}\searrow && \swarrow
    \\
    && E
  }
$$

The variational version replaces the vector bundle on the left with its jet bundle:

+-- {: .num_defn #EvolutionaryVectorField}
###### Definition
**([[evolutionary vector fields]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles}). Then an _[[evolutionary vector field]]_ $v$ on $E$ is  "variational vertical vector field" on $E$, hence a smooth [[bundle]] [[homomorphism]]
out of the [[jet bundle]] (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime})

$$
  \array{
    J^\infty_\Sigma E
    && \overset{v}{\longrightarrow} &&
    T_\Sigma E
    \\
    & {}_{\mathllap{jb_{\infty,0}}}\searrow && \swarrow_{\mathllap{}}
    \\
    && E
  }
$$

to the [[vertical tangent bundle]] $T_\Sigma E \overset{}{\to} \Sigma$ (def. \ref{VerticalTangentBundle}) of $E \overset{fb}{\to} \Sigma$.

In the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] as in example \ref{TrivialVectorBundleAsAFieldBundle}, this means that an evolutionary vector field is
a [[tangent vector field]] (example \ref{TangentVectorFields}) on $J^\infty_\Sigma(E)$ of the special form

$$
  \begin{aligned}
    v
     & =
    v^a \partial_{\phi^a}
    \\
    & =
    v^a\left(
      (x^\mu), (\phi^a), (\phi^a_{,\mu}), \cdots
    \right) \partial_{\phi^a}
  \end{aligned}
  \,,
$$

where the [[coefficients]] $v^a \in C^\infty(J^\infty_\Sigma(E))$ are general [[smooth functions]]
on the [[jet bundle]] (while the cmponents are [[tangent vectors]] along the field coordinates $(\phi^a)$,
but not along the spacetime coordinates $(x^\mu)$ and not along the jet coordinates $\phi^a_{,\mu_1 \cdots \mu_k}$).

We write

$$
  \Gamma_E^{ev}\left( T_\Sigma E \right)
  \;\in\;
  \Omega^{0,0}_\Sigma(E) Mod
$$

for the space of evolutionary vector fields,
regarded as a [[module]] over the $\mathbb{R}$-[[associative algebra|algebra]]

$$
  \Omega^{0,0}_\Sigma(E)
  \;=\;
  C^\infty\left( J^\infty_\Sigma(E) \right)
$$

of [[smooth functions]] on the [[jet bundle]].

=--

An [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField}) describes
an infinitesimal change of field values _depending_ on, possibly, the point in spacetime and the values of the field
and all its derivatives (locally to finite order, by prop. \ref{JetBundleIsLocallyProManifold}).

This induces a corresponding infinitesimal change of the derivatives of the fields, called the _prolongation_
of the evolutionary vector field:

+-- {: .num_prop #EvolutionaryVectorFieldProlongation}
###### Proposition
**(prolongation of [[evolutionary vector field]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]].

Given an [[evolutionary vector field]] $v$ on $E$ (def. \ref{EvolutionaryVectorField})
there is a unique [[tangent vector field]] $\hat v$ (example \ref{TangentVectorFields}) on the [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) such that

1. $\hat v$ agrees on field coordinates (as opposed to jet coordinates) with $v$:

   $$
     (jb_{\infty,0})_\ast(\hat v) = v
     \,,
   $$

   which means in the special case that $E \overset{fb}{\to} \Sigma$ is a [[trivial vector bundle]] over [[Minkowski spacetime]]
   (example \ref{TrivialVectorBundleAsAFieldBundle})
   that $\hat v$ is of the form

   $$
     \label{GenericComponentsForProlongationOfEvolutionaryVectorField}
     \hat v
     \;=\;
     \underset{
       = v
     }{
     \underbrace{
     v^a \partial_{\phi^a}
     }}
     \,+\,
     \hat v^a_{\mu} \partial_{\phi^a_{,\mu}}
     +
     \hat v^a_{\mu_1 \mu_2} \partial_{\phi^a_{,\mu_1 \mu_2}}
     +
     \cdots
   $$

1. contraction with $\hat v$ (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[total derivative|total spacetime derivative]] (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}):

   $$
     \label{ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative}
     \iota_{\hat v} \circ d + d \circ \iota_{\hat v} = 0
     \,.
   $$

In particular [[Cartan's homotopy formula]] (prop. \ref{CartanHomotopyFormula})
for the [[Lie derivative]] $\mathcal{L}_{\hat v}$ holds with respect to the [[variational derivative]] $\delta$:

$$
 \label{HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField}
  \mathcal{L}_{\hat v} = \delta \circ \iota_{\hat v} + \iota_{\hat v} \circ \delta
$$

Explicitly, in the special case that the [[field bundle]] is a [[trivial vector bundle]] over [[Minkowski spacetime]] (example \ref{TrivialVectorBundleAsAFieldBundle}) $\hat v$ is given by

$$
  \label{ProlongationOfEvolutionaryVectorFieldExplicit}
  \hat v = \underoverset{n = 0}{\infty}{\sum} \frac{d^n v^a}{ d x^{\mu_1} \cdots d x^{\mu_n} } \partial_{\phi^a_{\mu_1 \cdots \mu_n}}
  \,.
$$

=--

+-- {: .proof}
###### Proof

It is sufficient to prove the coordinate version of the statement.
We prove this by [[induction]] over the maximal jet order $k$. Notice that the coefficient of $\partial_{\phi^a_{\mu_1 \cdots \mu_k}}$
in $\hat v$ is given by the contraction $\iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}$ (def. \ref{ContractionOfFormsWithVectorFields}).

Similarly (at "$k = -1$") the component of $\partial_{\mu_1}$ is given by $\iota_{\hat v} d x^{\mu}$. But by the second condition
above this vanishes:

$$
  \begin{aligned}
    \iota_{\hat v} d x^\mu
    & =
    d \iota_{\hat v} x^\mu
    \\
    & =
    0
  \end{aligned}
$$

Moreover, the coefficient of $\partial_{\phi^a}$ in $\hat v$ is fixed by the first condition above to be

$$
  \iota_{\hat v} \delta \phi^a
  =
  v^a
  \,.
$$

This shows the statement for $k = 0$. Now assume that the statement is true up to some $k \in \mathbb{N}$.
Observe that the coefficients of all $\partial_{\phi^a_{\mu_1 \cdots \mu_{k+1}}}$ are fixed by the
contractions with $\delta \phi^a_{\mu_1 \cdots \mu_{k} \mu_{k+1}} \wedge d x^{\mu_{k+1}}$. For this we find again from the second
condition and using $\delta \circ d + d \circ \delta = 0$ as well as the induction assumption that

$$
  \begin{aligned}
    \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_{k+1}} \wedge d x^{\mu_{k+1}}
    & =
    \iota_{\hat v}  \delta d \phi^a_{\mu_1 \cdots \mu_k}
    \\
    & =
    d \iota_{\hat v} \delta \phi^a_{\mu_1 \cdots \mu_k}
    \\
    &
    = d \frac{d^k v^a}{d x^{\mu_1} \cdots d x^{\mu_k}}
    \\
    & =
    \frac{d^{k+1}v^a }{d x^{\mu_1} \cdots d x^{\mu_{k+1}}} d x^{\mu_{k+1}}
    \,.
  \end{aligned}
$$

This shows that $\hat v$ satisfying the two conditions given exists uniquely.

Finally formula (eq:HomotopyFormulaForLieDerivativeAlongProlongationOfEvolutionaryVectorField) for the [[Lie derivative]] follows from the second of the two conditions with [[Cartan's homotopy formula]]
$\mathcal{L}_{\hat v} = \mathbf{d} \circ \iota_{\hat v} + \iota_{\hat v} \circ \mathbf{d}$ (prop. \ref{CartanHomotopyFormula})
together with $\mathbf{d} = \delta + d$ (eq:VariationalDerivative).

=--

+-- {: .num_prop #EvolutionaryVectorFieldLieAlgebra}
###### Proposition
**([[evolutionary vector fields]] form a [[Lie algebra]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[fiber bundle]]. For any two [[evolutionary vector fields]]
$v_1$, $v_2$ on $E$ (def. \ref{EvolutionaryVectorField}) the [[Lie bracket]] of [[tangent vector fields]]
of their prolongations $\hat v_1$, $\hat v_2$ (def. \ref{EvolutionaryVectorFieldProlongation})
is itself the prolongation $\widehat{[v_1, v_2]}$ of a unique evolutionary vector field $[v_1,v_2]$.

This defines the structure of a [[Lie algebra]] on evolutionary vector fields.

=--

+-- {: .proof}
###### Proof

It is clear that $[\hat v_1, \hat v_2]$ is still [[vertical vector field|vertical]], therefore, by
prop. \ref{EvolutionaryVectorFieldProlongation}, it is sufficient to show that contraction $\iota_{[v_1, v_2]}$ with this
vector field (def. \ref{ContractionOfFormsWithVectorFields}) anti-commutes with the [[horizontal derivative]] $d$,
hence that $[d, \iota_{[\hat v_1, \hat v_2]}] = 0$.

Now $[d, \iota_{[\hat v_1, \hat v_2]}]$ is an
operator that sends vertical 1-forms to horizontal 1-forms and vanishes on
horizontal 1-forms. Therefore it is sufficient to see that this operator
in fact also vanishes on all vertical 1-forms. But for this it is
sufficient that it commutes with the vertical derivative.
This we check by [[Cartan calculus]], using $[d,\delta] = 0$ and $[d, \iota_{\hat v_i}]=0$, by assumption:

$$
  \begin{aligned}
    {[ \delta, [ d,\iota_{[\hat v_1, \hat v_2]}] ]}
    & =
    - [d, [\delta, \iota_{[\hat v_1, \hat v_2]}]]
    \\
    & =
    - [d,  \mathcal{L}_{[\hat v_1, \hat v_2]}]
    \\
    & =
    -[d, [\mathcal{L}_{\hat v_1}, \iota_{\hat v_2}]  ]
    \\
    & =
    - [d, [ [\delta, \iota_{\hat v_1}], \iota_{\hat v_2}  ]]
    \\
    & =
    0
    \,.
  \end{aligned}
$$

=--

Now given an evolutionary vector field, we want to consider the [[flow]] that it
induces on the [[space of field histories]]:

+-- {: .num_defn #FlowOfFieldHistoriesAlongEvolutionaryVectorField}
###### Definition
**([[flow]] of [[field histories]] along [[evolutionary vector field]])**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles})
and let $v$ be an [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField})
such that the ordinary [[flow]] of its prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation})

$$
  \exp(t \hat v)
  \;\colon\; J^\infty_\Sigma(E) \longrightarrow J^\infty_\Sigma(E)
$$

exists on the [[jet bundle]] (e.g. if the order of derivatives of field coordinates that it depends
on is bounded).

For $\Phi_{(-)} \colon U_1 \to  \Gamma_\Sigma(E)$ a collection of  [[field histories]]
(hence a plot of the [[space of field histories]] (def. \ref{SupergeometricSpaceOfFieldHistories}) ) the _[[flow]]_ of $v$
through $\Phi_{(-)}$ is the [[smooth function]]

$$
  U_1 \times \mathbb{R}^1
   \overset{\exp(v)(\Phi_{(-)})}{\longrightarrow}
  \Gamma_\Sigma(E)
$$

whose unique factorization $\widehat{\exp(v)}(\Phi_{(-)})$ through the space of jets of field histories
(i.e. the [[image]] $im(j^\infty_\Sigma)$ of [[jet prolongation]], def. \ref{JetProlongation})

$$
  \array{
    && im(j^\infty_\Sigma) &\hookrightarrow& \Gamma_\Sigma(J^\infty_\Sigma(E))
    \\
    & {}^{\mathllap{\widehat{\exp(v)}(\Phi_{(-)})}} \nearrow& \downarrow^{\mathrlap{\simeq}}
    \\
    U_1 \times \mathbb{R}^1
      &\underset{ \exp(v)(\Phi) }{\longrightarrow}&
    \Gamma_{\Sigma}(E)_{}
  }
$$

takes a plot $t_{(-)} \;\colon\; U_2 \to \mathbb{R}^1$ of the [[real line]]
(regarded as a [[super formal smooth set|super smooth set]] via example \ref{SuperSmoothSetSuperCartesianSpaces}), to the plot

$$
  \label{LocalDataForFlowOfImplicitInfinitesimalGaugeSymmetry}
  (\exp(t(-) \hat v) \circ j^\infty_\Sigma(\Phi_{(-)})
    \;\colon\:
  U_1 \times U_2   \longrightarrow \Gamma_\Sigma\left( J^\infty_\Sigma(E) \right)
$$

of the [[smooth space|smooth]] [[space of sections]] of the [[jet bundle]].

(That $\exp(t(-) \hat v)$ indeed flows jet prolongations $j^\infty_\Sigma(\Phi(-))$ again to jet prolongations
is due to its defining relation to the [[evolutionary vector field]] $v$ from prop. \ref{EvolutionaryVectorFieldProlongation}.)

=--

+-- {: .num_defn #SymmetriesAndConservedCurrents}
###### Definition
**([[infinitesimal symmetries of the Lagrangian]] and [[conserved currents]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

Then

1. an _[[infinitesimal symmetry of the Lagrangian]]_ is an [[evolutionary vector field]] $v$ (def. \ref{EvolutionaryVectorField})
   such that the [[Lie derivative]] of the [[Lagrangian density]] along its
   prolongation $\hat v$ (prop. \ref{EvolutionaryVectorFieldProlongation}) is a [[total spacetime derivative]]:

   $$
     \mathcal{L}_{\hat v} \mathbf{L} \;=\; d \tilde J_{\hat v}
   $$

1. an _[[on-shell]] [[conserved current]]_ is a horizontal $p$-form $J \in \Omega^{p,0}_\Sigma(E)$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime})
   whose [[total derivative|total spacetime derivative]] vanishes on the [[prolonged shell]] (eq:ShellInJetBundle)

   $$
     d J\vert_{\mathcal{E}^\infty} \;=\; 0
     \,.
   $$

=--

+-- {: .num_prop #NoethersFirstTheorem}
###### Proposition
**([[Noether's theorem]] I)**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

If $v$ is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents})
with $\mathcal{L}_{\hat v} \mathbf{L} = d \tilde J_{\hat v}$, then

$$
  \label{NoetherCurrent}
  J_{\hat v} \coloneqq \tilde J_{\hat v} - \iota_{\hat v} \Theta_{BFV}
$$

is an [[on-shell]] [[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}), for $\Theta_{BFV}$
a presymplectic potential (eq:PresymplecticPotential) from def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}.

=--

([[Noether's theorem|Noether's theorem II]] is prop. \ref{NoetherIdentities} below.)

+-- {: .proof}
###### Proof

By [[Cartan's homotopy formula]] for the [[Lie derivative]] (prop. \ref{CartanHomotopyFormula})
and the decomposition of the variational derivative $\delta \mathbf{L}$ (eq:dLDecomposition)
and the fact that contraction $\iota_{\hat v}$ with the prolongtion of an evolutionary vector field vanishes on horizontal differential forms
(eq:GenericComponentsForProlongationOfEvolutionaryVectorField) and anti-commutes with the horizontal differential (eq:ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative), by def. \ref{EvolutionaryVectorField},
we may re-express the defining equation for the symmetry as follows:

$$
  \begin{aligned}
    d \tilde J_{\hat v}
    & =
    \mathcal{L}_{\hat v} \mathbf{L}
    \\
    & =
    \iota_{\hat v} \underset{= \delta_{EL}\mathbf{L} - d \Theta_{BFV}}{\underbrace{\mathbf{d} \mathbf{L}}}
    +
    \mathbf{d} \underset{= 0}{\underbrace{\iota_v \mathbf{L}}}
    \\
    & =
    \iota_{\hat v} \delta_{EL} \mathbf{L}
    + d \iota_{\hat v} \Theta_{BFV}
  \end{aligned}
$$

which is equivalent to

$$
  \label{CurrentNoetherConservation}
  d(\underset{= J_{\hat v}}{\underbrace{\tilde J_{\hat v} - \iota_{\hat v} \Theta_{BFV}}})
   \;=\;
  \iota_{\hat v} \delta_{EL}\mathbf{L}
$$

Since, by definition of the [[shell]] $\mathcal{E}$, the differential form on the right
vanishes on $\mathcal{E}$ this yields the claim.

=--

+-- {: .num_example #ScalarFieldEnergyMomentum}
###### Example
**([[energy-momentum]] of the [[scalar field]])**

Consider the [[Lagrangian field theory]] of the [[free field|free]] [[scalar field]] from def. \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}:

$$
  \mathbf{L}
  \;=\;
  \tfrac{1}{2}
  \left(
    \eta^{\mu \nu}\phi_{,\mu} \phi_{,\nu}
    -
    m^2 \phi^2
  \right)
  dvol_\Sigma
  \,.
$$

For $\nu \in \{0, 1, \cdots, p\}$ consider the vector field on the jet bundle given by

$$
  v_\nu
  \;\coloneqq\;
  \phi_{,\nu} \partial_{\phi}
  +
  \phi_{,\mu \nu} \partial_{\phi_{,\mu}}
  +
  \cdots
  \,.
$$

This describes infinitesimal translations of the fields in the direction of $\partial_\nu$.

And this is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}), since

$$
  \iota_{v_\nu} \mathbf{d}\mathbf{L}
  =
  d L \wedge \iota_{\partial_\nu} dvol_\Sigma
  \,.
$$

With the formula (eq:PresymplecticPotentialOfFreeScalarField) for the presymplectic potential

$$
  \Theta_{BFV}
    =
  \eta^{\mu \nu} \phi_{,\mu} \delta \phi \iota_{\partial_{\nu}} dvol_\Sigma
$$

it hence follows from [[Noether's theorem]] (prop. \ref{NoethersFirstTheorem}) that the corresponding
[[conserved current]] (def. \ref{SymmetriesAndConservedCurrents}) is

$$
  \begin{aligned}
    T_\nu
     & =
     L \, \iota_{\partial_\nu} dvol_\Sigma
     -
     \iota_{v_\nu}\Theta_{BFV}
     \\
     &
     =
     L \, \iota_{\partial_\nu} dvol_\Sigma
     -
     \eta^{\rho \mu} \phi_{,\rho} \phi_{,\nu}
     \,
     \iota_{\partial_\mu} dvol_\Sigma
     \\
     &  =
     (
     \underset{=: T^\mu_\nu}{
     \underbrace{
       \delta^\mu_\nu  L
       -
       \eta^{\rho \mu} \phi_{,\rho} \phi_{,\nu}
     }
     }
     )
     \,
     \iota_{\partial_\mu} dvol_\Sigma
   \end{aligned}
   \,.
$$

This [[conserved current]] is called the _[[energy-momentum tensor]]_.

=--

+-- {: .num_example #DiracCurrent}
###### Example
**([[Dirac current]])**

Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] in spacetime dimension $p + 1 = 3+1$ (example \ref{LagrangianDensityForDiracField})

$$
  \mathbf{L} = i \overline{\psi} \gamma^\mu \psi_{,\mu} \, dvol_\Sigma
  \,.
$$

Then the prolongation (prop. \ref{EvolutionaryVectorFieldProlongation}) of the [[evolutionary vector field]] (def. \ref{EvolutionaryVectorField})

$$
  v \;\coloneqq\; i \psi_\alpha \partial_{\psi_\alpha}
$$

is an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}). The [[conserved current]] that corresponds to this under [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}) is

$$
  i
  \overline{\psi} \gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma
  \;\in\;
  \Omega^{p,0}_{\Sigma}(E)
  \,.
$$

This is called the _[[Dirac current]]_.

=--

+-- {: .proof}
###### Proof

By equation (eq:ProlongationOfEvolutionaryVectorFieldExplicit) the prolongation of $v$ is

$$
  \hat v
    =
  i \psi_\alpha \partial_{\psi_\alpha}
    +
  i \psi_{\alpha,\mu} \partial_{\psi_{\alpha,\mu}}
    +
   \cdots
  \,.
$$

Therefore the [[Lagrangian density]] is strictly invariant under the [[Lie derivative]] along $\hat v$

$$
  \begin{aligned}
    \mathcal{L}_{\hat v} \left(
      i \overline{\psi} \gamma^\mu \psi_{,\mu}
    \right)
    dvol_\Sigma
    & =
    \underset{
      = i \cdot (-i) \overline{\psi} \gamma^\mu \psi_{,\mu}
    }{
    \underbrace{
      i \overline{i \psi} \gamma^\mu \psi_{,\mu}
    }
    }
    dvol_\Sigma
    +
    \underset{
      i \cdot i \overline{\psi} \gamma^\mu \psi_{,\mu}
    }{
    \underbrace{
      i \overline{\psi} \gamma^\mu (i \psi_{,\mu})
    }
    }
    dvol_\Sigma
    \\ & =
    0
    \,.
  \end{aligned}
$$

and so the formula for the corresponding conserved current (eq:NoetherCurrent) is

$$
  \begin{aligned}
    J_v
    & =
    - \iota_{\hat v}
    \left(
    \underset{
      -
      \overline{\psi}
      \gamma^\mu
      \delta \psi
      \,
      \iota_{\partial_\mu} dvol_\Sigma    }{
    \underbrace{
      \Theta_{BFV}
    }
    }
    \right)
    \\
    &
    = + i \overline{\psi}\gamma^\mu \psi \, \iota_{\partial_\mu} dvol_\Sigma
  \end{aligned}
  \,,
$$

where under the brace we used example \ref{PresymplecticCurrentDiracField} to identify the [[presymplectic potential]] for the [[free field theory|free]] [[Dirac field]].

=--

$\,$

Since an [[infinitesimal symmetry of a Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents}) by definition changes the Lagrangian only up to a [[total spacetime derivative]], and since the [[Euler-Lagrange equation|Euler-Lagrange]] [[equations of motion]] by construction depend on the [[Lagrangian density]] only up to a [[total spacetime derivative]] (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}), it is plausible that and [[infinitesimal symmetry of the Lagrangian]] preserves the [[equations of motion]] (eq:EulerLagrangeEquationGeneral), hence the [[shell]] (eq:ProlongedShellInJetBundle). That this is indeed the case is the statement of prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion} below.

To make the proof transparent, we now first introduce the concept of the _[[evolutionary derivative]]_ (def. \ref{FieldDependentDifferentialOperatorDerivative}) below and then observe that in terms of these the [[Euler-Lagrange derivative]] is in fact a [[derivation]] (prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}).

+-- {: .num_defn #FieldDependentSections}
###### Definition
**([[field (physics)|field]]-dependent [[sections]])**

For

$$
  E \overset{fb}{\longrightarrow} \Sigma
$$

a [[fiber bundle]] (def. \ref{FiberBundle}), regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}), and for

$$
  E' \overset{fb'}{\longrightarrow} \Sigma
$$

any other [[fiber bundle]] over the same base space ([[spacetime]]), we write

$$
  \Gamma_{J^\infty_\Sigma(E)}(E')
  \;\coloneqq\;
  \Gamma_{J^\infty_\Sigma(E)}( jb^\ast E' )
  \;=\;
  Hom_\Sigma(J^\infty_\Sigma(E), E')
  \;\simeq\;
  DiffOp(E,E')
$$

for the [[space of sections]] of the [[pullback of bundles]] of $E'$ to the [[jet bundle]] $J^\infty_\Sigma(E) \overset{jb}{\longrightarrow} \Sigma$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) along $jb$.

$$
  \Gamma_{J^\infty_\Sigma(E)}(E')
  \;=\;
  \left\{
    \array{
       && E'
       \\
       & {}^{\mathllap{}}\nearrow & \downarrow \mathrlap{fb'}
       \\
       J^\infty_\Sigma(E)
       &\underset{jb}{\longrightarrow}&
       \Sigma
    }
    \phantom{A}\,\,
  \right\}
  \,.
$$

(Equivalently this is the space of [[differential operators]] from sections of $E$ to sections of $E'$, according to prop. \ref{DifferentialOperator}. )

=--

In ([Olver 93, section 5.1, p. 288](evolutionary+derivative#Olver93)) the field dependent sections of def. \ref{FieldDependentSections}, considered in [[local coordinates]], are referred to as [[tuples]] of _differential functions_.

+-- {: .num_example #EvolutionaryVectorFieldsAsFieldDependentSections}
###### Example
**([[source forms]] and [[evolutionary vector fields]] are field-dependent sections)**

For $E \overset{fb}{\to} \Sigma$ a [[field bundle]], write $T_\Sigma E$ for its [[vertical tangent bundle]] (example \ref{VerticalTangentBundle}) and $T_\Sigma^\ast E$ for its [[dual vector bundle]] (def. \ref{DualVectorBundle}), the [[vertical cotangent bundle]].

Then the field-dependent sections of these bundles according to def. \ref{FieldDependentSections} are identified as follows:

* the space $\Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)$ contains the space of [[evolutionary vector fields]] $v$ (def. \ref{EvolutionaryVectorField}) as those bundle morphism which respect not just the projection to $\Sigma$ but also its factorization through $E$:

  $$
    \left(
      \array{
        && T_\Sigma E
        \\
        & {}^{\mathllap{v}}\nearrow & \downarrow^{\mathrlap{tb_\Sigma}}
        \\
        J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \underset{fb}{\longrightarrow}& \Sigma
      }
    \right)
    \;\in\;
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
  $$

* $\Gamma_{J^\infty_\Sigma(E)}( T^\ast_\Sigma E) \otimes \wedge^{p+1}_\Sigma(T^\ast \Sigma)$ contains the space of [[source forms]] $E$ (prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) as those bundle morphisms which respect not just the projection to $\Sigma$ but also its factorization through $E$:

  $$
    \left(
      \array{
        && T^\ast_\Sigma E
        \\
        & {}^{E}\nearrow & \downarrow^{\mathrlap{ctb_\Sigma}}
        \\
        J^\infty_\Sigma(E) &\underset{jb_{\infty,0}}{\longrightarrow}& E & \underset{fb}{\longrightarrow}& \Sigma
      }
    \right)
    \;\in\;
    \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
  $$

This makes manifest the duality pairing between [[source forms]] and [[evolutionary vector fields]]

$$
  \array{
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
    \otimes
    \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
    &\longrightarrow&
    C^\infty(J^\infty_\Sigma(E))
  }
$$

which in local coordinates is given by

$$
  (v^a \partial_{\phi^a} \,,\, \omega_a \delta \phi^a)
  \mapsto
  v^a \omega_a
$$

for $v^a, \omega_a \in C^\infty(J^\infty_\Sigma(E))$ [[smooth functions]] on the [[jet bundle]] (as in prop. \ref{JetBundleIsLocallyProManifold}).

=--

+-- {: .num_defn #FieldDependentDifferentialOperatorDerivative}
###### Definition
**([[evolutionary derivative of field-dependent section]])**

Let

$$
  E \overset{fb}{\to} \Sigma
$$

be a [[fiber bundle]] regarded as a [[field bundle]] (def. \ref{FieldsAndFieldBundles}) and let

$$
  V \overset{vb}{\to} \Sigma
$$

be a [[vector bundle]] (def. \ref{VectorBundle}). Then for

$$
  P \in \Gamma_{J^\infty_\Sigma(E)}(V)
$$

a field-dependent section of $E$ according to def. \ref{FieldDependentSections}, its _evolutionary derivative_ is the morphism

$$
  \array{
    \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma E)
      & \overset{  \mathrm{D}P }{\longrightarrow} &
    \Gamma_{J^\infty_\Sigma(E)}(V)
    \\
    v &\mapsto& \hat v(P)
  }
$$

which, under the identification of example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}, sense an [[evolutionary vector field]] $v$ to the [[derivative]] of $P$ (example \ref{TangentVectorFields}) along the prolongation [[tangent vector field]] $\hat v $ of $v$ (prop. \ref{EvolutionaryVectorFieldProlongation}).

In the case that $E$ and $V$ are [[trivial vector bundles]] over [[Minkowski spacetime]] with coordinates $((x^\mu), (\phi^a))$ and $((x^\mu),  (\rho^b))$, respectively (example \ref{TrivialVectorBundleAsAFieldBundle}), then by (eq:ProlongationOfEvolutionaryVectorFieldExplicit) this is given by

$$
  ((\mathrm{D}P)(v))^b
   \;=\;
  \left(
     v^a \frac{\partial P^b}{\partial \phi^a}
     +
     \frac{d v^a}{d x^\mu}
     \frac{\partial P^b}{\partial \phi^a_{,\mu}}
     +
      \frac{d^2 v^a}{d x^\mu d x^\nu}
      \frac{\partial  P^b}{\partial \phi^a_{,\mu \nu}}
      +
     \cdots
  \right)
$$

This makes manifest that $\mathrm{D}P$ may equivalently be regarded as a $J^\infty_\Sigma(E)$-dependent [[differential operator]] (def. \ref{DifferentialOperator}) from the [[vertical tangent bundle]] $T_\Sigma E$ (def. \ref{VerticalTangentBundle}) to $V$, namely a [[bundle homomorphism]] over $\Sigma$ of the form

$$
  \mathrm{D}_P
  \;\colon\;
  J^\infty_\Sigma(E) \times_\Sigma J^\infty_\Sigma T_\Sigma E
  \longrightarrow
  V
$$

in that

$$
  \label{FrechetDerivativeAsDifferentialOperatorEquality}
  \mathrm{D}_P(-,v)
  =
  \mathrm{D}P(v)
  =
  \hat v (P)
  \,.
$$

=--

([Olver 93, def. 5.24](evolutionary+derivative#Olver93))

+-- {: .num_example #DifferentialOperatorDerivativeOfLagrangianFunction}
###### Example
**([[evolutionary derivative]] of [[Lagrangian function]])**

Over [[Minkowski spacetime]] $\Sigma$ (def. \ref{MinkowskiSpacetime}), let $\mathbf{L} = L dvol \in \Omega^{p+1,0}_\Sigma(E)$ be a [[Lagrangian density]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}), with coefficient function regarded as a field-dependent section (def. \ref{FieldDependentSections}) of the [[trivial bundle|trivial]] [[real line bundle]]:

$$
  L \;\in \; \Gamma_{J^\infty_\Sigma}(\Sigma \times \mathbb{R})
  \,,
$$

Then the [[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators})

$$
  (\mathrm{D}_L)^\ast
    \;\colon\;
  J^\infty_\Sigma(E)\times_\Sigma (\Sigma \times \mathbb{R})^\ast
  \longrightarrow
  T_\Sigma^\ast E
$$

of its [[evolutionary derivative]], def. \ref{FieldDependentDifferentialOperatorDerivative}, regarded as a $J^\infty_\Sigma(E)$-dependent differential operator $\mathrm{D}_P$ from $T_\Sigma$ to $V$ and applied to the constant section

$$
  1 \in \Gamma_\Sigma(\Sigma \times \mathbb{R}^\ast)
$$

is the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral)

$$
  \delta_{EL}\mathbf{L}
  \;=\;
  \left(\mathrm{D}_{L}\right)^\ast(1)
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T_\Sigma^\ast)
   \simeq
   \Omega^{p+1,1}_\Sigma(E)_{source}
$$

via the identification from example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}.

=--

+-- {: .num_prop #EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}
###### Proposition
**([[Euler-Lagrange derivative]] is [[derivation]] via [[evolutionary derivatives]])**

Let $V \overset{vb}{\to} \Sigma$ be a [[vector bundle]] (def. \ref{VectorBundle}) and write $V^\ast \overset{}{\to} \Sigma$ for its [[dual vector bundle]] (def. \ref{DualVectorBundle}).

For field-dependent sections (def. \ref{FieldDependentSections})

$$
  \alpha \in \Gamma_{J^\infty_\Sigma(E)}(V)
$$

and

$$
  \beta^\ast \in \Gamma_{J^\infty_\Sigma(E)}(V^\ast)
$$

we have that the [[Euler-Lagrange derivative]] (eq:EulerLagrangeEquationGeneral) of their canonical pairing to a [[smooth function]] on the [[jet bundle]] (as in prop. \ref{JetBundleIsLocallyProManifold}) is the sum of the derivative of either one via the [[formally adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}) of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}) of the other:

$$
  \delta_{EL}( \alpha \cdot \beta^\ast )
  \;=\;
  (\mathrm{D}_\alpha)^\ast(\beta^\ast)
  +
  (\mathrm{D}_{\beta^\ast})^\ast(\alpha)
$$

=--

+-- {: .proof}
###### Proof

It is sufficient to check this in [[local coordinates]]. By the [[product law]] for [[differentiation]] we have

$$
  \begin{aligned}
    \frac{
      \delta_{EL} \left(\alpha \cdot \beta^\ast \right)
    }
    {
      \delta \phi^a
    }
    & =
    \frac{\partial \left(\alpha \cdot \beta^\ast \right)}{\partial \phi^a}
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \left( \alpha \cdot \beta^\ast \right)}{\partial \phi^a_{,\mu}}
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \left( \alpha \cdot \beta^\ast \right) }{\partial \phi^a_{,\mu \nu}}
    \right)
    -
    \cdots
    \\
    & =
    \phantom{+}
    \frac{\partial \alpha }{\partial \phi^a}
    \cdot \beta^\ast
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \alpha }{\partial \phi^a_{,\mu}}
       \cdot
       \beta^\ast
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \alpha  }{\partial \phi^a_{,\mu \nu}}
       \cdot
       \beta^\ast
    \right)
    -
    \cdots
    \\
    & \phantom{=}
    +
    \frac{\partial \beta^\ast }{\partial \phi^a}
    \cdot \alpha
    -
    \frac{d}{d x^\mu}
    \left(
       \frac{\partial \beta^\ast }{\partial \phi^a_{,\mu}}
       \cdot
       \alpha
    \right)
    +
    \frac{d}{d x^\mu d x^\nu}
    \left(
       \frac{\partial \beta^\ast  }{\partial \phi^a_{,\mu \nu}}
       \cdot
       \alpha
    \right)
    -
    \cdots
    \\
    & =
    (\mathrm{D}_\alpha)_a^\ast(\beta^\ast)
    +
    (\mathrm{D}_{\beta^\ast})_a^\ast(\alpha)
  \end{aligned}
$$

=--

+-- {: .num_prop #EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint}
###### Proposition
**([[evolutionary derivative]] of [[Euler-Lagrange forms]] is [[formally self-adjoint differential operator|formally self-adjoint]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) over [[Minkowski spacetime]] (def. \ref{MinkowskiSpacetime}) and regard the [[Euler-Lagrange derivative]]

$$
  \delta_{EL}\mathbf{L}
  \;=\;
  \delta_{EL}L \wedge dvol_\Sigma
$$

(from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) as a field-dependent section of the [[vertical cotangent bundle]]

$$
  \delta_{EL}L
  \;\in\;
  \Gamma_{J^\infty_\Sigma(E)}(T^\ast_\Sigma E)
$$

as in example \ref{EvolutionaryVectorFieldsAsFieldDependentSections}. Then the corresponding [[evolutionary derivative]] field-dependent [[differential operator]] $D_{\delta_{EL}L}$ (def. \ref{FieldDependentDifferentialOperatorDerivative})  is [[formally self-adjoint differential operator|formally self-adjoint]] (def. \ref{FormallyAdjointDifferentialOperators}):

$$
  (D_{\delta_{EL}L})^\ast
  \;=\;
  D_{\delta_{EL}L}
  \,.
$$

(In terms of the [[Euler-Lagrange complex]], remark \ref{EulerLagrangeComplex}, this says that the [[Helmholtz operator]] vanishes on the image of the [[Euler-Lagrange operator]].)

=--

([Olver 93, theorem 5.92](evolutionary+derivative#Olver93)) The following proof is due to [[Igor Khavkine]].

+-- {: .proof}
###### Proof

By definition of the [[Euler-Lagrange form]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}) we have

$$
  \frac{\delta_{EL} L }{\delta \phi^a}
  \delta \phi^a
  \,
  \wedge
  dvol_\Sigma
  \;=\;
  \delta L \,\wedge dvol_\Sigma \;+\; d(...)
  \,.
$$

Applying the [[variational derivative]] $\delta$ (def. \ref{VariationalBicomplexOnSecondOrderJetBundleOverTrivialVectorBundleOverMinkowskiSpacetime}) to both sides of this equation yields

$$
  \left(\delta \frac{\delta_{EL} L }{\delta \phi^a}\right)
  \wedge
  \delta \phi^a
  \,
  \wedge
  dvol_\Sigma
  \;=\;
  \underset{= 0}{\underbrace{\delta \delta L}} \wedge dvol_\Sigma
  \;+\;
  d(...)
  \,.
$$

It follows that for $v,w$ any two [[evolutionary vector fields]] the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of their prolongations $\hat v$ and $\hat w$ (def. \ref{EvolutionaryVectorFieldProlongation}) into the [[variational differential form|differential 2-form]] on the left is

$$
  \left(
    \delta \frac{\delta_{EL} L }{\delta \phi^a}
    \wedge
    \delta \phi^a
  \right)(v,w)
  =
    w^a
    (\mathrm{D}_{\delta_{EL}})_a(v)
    -
    v^b(\mathrm{D}_{\delta_{EL}})_b(w)
  \,,
$$

by inspection of the definition of the [[evolutionary derivative]] (def. \ref{FieldDependentDifferentialOperatorDerivative}).
Moreover, their contraction into the differential form on the right is

$$
  \iota_{\hat v}
  \iota_{\hat w}
  d(...)
  \;=\;
  d(...)
$$

by the fact (prop. \ref{EvolutionaryVectorFieldProlongation}) that contraction with prolongations of evolutionary vector fields anti-commutes with the [[total spacetime derivative]] (eq:ProlongedEvolutionaryVectorFieldContractionAnticommutedWithHorizontalDerivative).

Hence the last two equations combined give

$$
  w^a
  (\mathrm{D}_{\delta_{EL}})_a(v)
  -
  v^b(\mathrm{D}_{\delta_{EL}})_b(w)
  \;=\;
  d(...)
  \,.
$$

This is the defining condition for $\mathrm{D}_{\delta_{EL}}$ to be [[formally self-adjoint differential operator]] (def. \ref{FormallyAdjointDifferentialOperators}).

=--

$\,$

Now we may finally prove that an [[infinitesimal symmetry of the Lagrangian]] is also an infinitesimal symmetry of the [[Euler-Lagrange equations|Euler-Lagrange]] [[equations of motion]]:

+-- {: .num_prop #InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}
###### Proposition
**([[infinitesimal symmetries of the Lagrangian]] are also [[infinitesimal symmetries]] of the [[equations of motion]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]]. If an [[evolutionary vector field]] $v$ is an [[infinitesimal symmetry of the Lagrangian]] then the [[flow]] along its prolongation $\hat v$ preserves the [[prolonged shell]] $\mathcal{E}^\infty \hookrightarrow J^\infty_\Sigma(E)$ (eq:ProlongedShellInJetBundle) in that the [[Lie derivative]] of the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L}$ along $\hat v$ vanishes on $\mathcal{E}^\infty$:

$$
  \mathcal{L}_{\hat v}\mathbf{L} = d(...)
  \phantom{AAA}
  \Rightarrow
  \phantom{AAA}
  \mathcal{L}_{\hat v} \, \delta_{EL}\mathbf{L}\vert_{\mathcal{E}^\infty} = 0
  \,.
$$

=--

+-- {: .proof}
###### Proof

Notice that for any vector field $\hat v$ the [[Lie derivative]] (prop. \ref{CartanHomotopyFormula})$\mathcal{L}_{\hat v}$ of the [[Euler-Lagrange form]] $\delta_{EL}\mathbf{L} = \frac{\delta_{EL}L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma$ differs from that of its component functions $\frac{\delta_{EL}L}{\delta \phi^a} dvol_\Sigma$ by a term proportional to these component functions, which by definition vanishes on-shell:

$$
  \mathcal{L}_{\hat v} \left( \frac{\delta_{EL} L}{\delta \phi^a} \delta \phi^a \wedge dvol_\Sigma \right)
  \;=\;
  \underset{
    = \hat v\left( \frac{\delta_{EL}L}{\delta \phi^a} \right)
  }{
  \underbrace{
  \left(
    \mathcal{L}_{\hat v}
     \frac{\delta_{EL}L}{\delta \phi^a}
  \right)
  }
  }
  \delta \phi^a
  \wedge
  dvol_\Sigma
  +
  \underset{
    = 0 \, \text{on} \, \mathcal{E}^\infty
  }{
  \underbrace{
    \frac{\delta_{EL}L}{\delta \phi^a}
  }
  }
    \left(
    \mathcal{L}_{\hat v} \delta \phi^a
    \right)
    \wedge dvol_\Sigma
$$

 But the Lie derivative of the component functions is just their plain derivative. Therefore it is sufficient to show that

$$
  \hat v
  \left(
    \frac{\delta_{EL} L}{\delta \phi^a}
  \right)
  \vert_{\mathcal{E}^\infty}
  \;=\;
  0
  \,.
$$

Now by [[Noether's theorem|Noether's theorem I]] (prop. \ref{NoethersFirstTheorem}) the condition $\mathcal{L}_{\hat v} = d \tilde J_{\hat v}$ for an [[infinitesimal symmetry of the Lagrangian]] implies that  the contraction (def. \ref{ContractionOfFormsWithVectorFields}) of the [[Euler-Lagrange form]] with the corresponding [[evolutionary vector field]] is a [[total spacetime derivative]]:

$$
  \iota_{\hat v} \, \delta_{EL}\mathbf{L}
  \;=\;
  d J_{\hat v}
  \,.
$$

Since the [[Euler-Lagrange derivative]] vanishes on [[total spacetime derivative]] (example \ref{TrivialLagrangianDensities}) also its application on the contraction on the left vanishes. But via example \ref{EvolutionaryVectorFieldsAsFieldDependentSections} that contraction is a pairing of field-dependent sections as in prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}. Hence we use this proposition to compute:

$$
  \label{TowardsProofThatSymmetriesPreserveTheShell}
  \begin{aligned}
    0
    & =
    \frac{\delta_{EL} \left( v \cdot \delta_{EL} L\right) }{ \delta \phi^a }
    \\
    & =
    (\mathrm{D}_{v})^\ast_a( \delta_{EL}L )
    +
    (\mathrm{D}_{\delta_{EL}L})^\ast_a(v)
    \\
    & =
    (\mathrm{D}_{v})^\ast_a( \delta_{EL}L )
    +
    (\mathrm{D}_{\delta_{EL}L})_a(v)
    \\
    & =
    (\mathrm{D}_{v})^\ast_a( \delta_{EL}L )
    +
    \hat v\left( \frac{\delta_{EL}L}{\delta \phi^a} \right)
    \,.
  \end{aligned}
$$

Here the first step is by prop. \ref{EulerLagrangeDerivativeIsDerivationViaAdjointFrechetDerivatives}, the second step is by prop. \ref{EvolutionaryDerivativeOfEulerLagrangeFormIsFormallySelfAdjoint} and the third step is
(eq:FrechetDerivativeAsDifferentialOperatorEquality).

Hence

$$
  \begin{aligned}
    \hat v(\delta_{EL}L) \vert_{\mathcal{E}^\infty}
    & =
    -
    (\mathrm{D}_{v})^\ast( \delta_{EL}L ) \vert_{\mathcal{E}^\infty}
    \\
    & =
    0
  \end{aligned}
  \,,
$$

where in the last line we used that on the [[prolonged shell]] $\delta_{EL}L$ and all its horizontal derivatives vanish, by definition.

=--

As a corollary we obtain:

+-- {: .num_prop #FlowAlongInfinitesimalSymmetryOfLagrangianPreservesOnShellSpaceOfFieldHistories}
###### Proposition
**([[flow]] along [[infinitesimal symmetry of the Lagrangian]] preserves [[on-shell]] [[space of field histories]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

For $v$ an [[infinitesimal symmetry of the Lagrangian]] (def. \ref{SymmetriesAndConservedCurrents})
the [[flow]] on the [[space of field histories]] (example \ref{DiffeologicalSpaceOfFieldHistories}) that it induces by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField}
preserves the space of [[on-shell]] field histories (from prop. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime}):

$$
  \array{
    \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
      &\hookrightarrow&
    \Gamma_\Sigma(E)
    \\
    {\mathllap{\exp(\hat v)\vert_{\delta_{EL}\mathbf{L} = 0}  }}
    \uparrow
      &&
    \uparrow {\mathrlap{\exp(\hat v)}}
    \\
    \Gamma_\Sigma(E)_{\delta_{EL}\mathbf{L} = 0}
      &\hookrightarrow&
    \Gamma_\Sigma(E)
  }
$$

=--

+-- {: .proof}
###### Proof

By def. \ref{EulerLagrangeEquationsOnTrivialVectorFieldBundleOverMinkowskiSpacetime} a field history
$\Phi \in \Gamma_\Sigma(E)$ is [[on-shell]] precisely if its [[jet prolongation]] $j^\infty_\Sigma(E)$ (def. \ref{JetProlongation})
factors through the [[shell]] $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ (eq:ShellInJetBundle).
Hence by def. \ref{FlowOfFieldHistoriesAlongEvolutionaryVectorField} the statement is equivalently that
the ordinary flow (prop. \ref{CartanHomotopyFormula}) of $\hat v$ (def. \ref{EvolutionaryVectorFieldProlongation})
on the [[jet bundle]] $J^\infty_\Sigma(E)$ preserves the [[shell]]. This in turn means that it preserves the
vanishing locus of the [[Euler-Lagrange form]] $\delta_{EL} \mathbf{L}$, which is the case by prop. \ref{InfinitesimalSymmetriesOfLagrangianAreAlsoSymmetriesOfTheEquationsOfMotion}.

=--

$\,$

**[[infinitesimal symmetries]] of the [[presymplectic potential current]]**
 {#InfinitesimalSymmetriesOfThePresymplecticPotentialCurrent}

Evidently [[Noether's theorem|Noether's theorem I]] in [[variational calculus]] (prop. \ref{NoethersFirstTheorem})
is the special case for horizontal $p+1$-forms of a more general phenomenon relating
symmetries of variational forms to forms that are closed up to a contraction. The same phenomenon applied instead
to the [[presymplectic current]] yields the following:

+-- {: .num_defn #LieDerivativeVariational}
###### Definition
**(variational Lie derivative)**

Let $E \overset{fb}{\to} \Sigma$ be a [[field bundle]] (def. \ref{FieldsAndFieldBundles})
with [[jet bundle]] $J^\infty_\Sigma(E)$ (def. \ref{JetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

For $v$ a vertical [[tangent vector field]] on the [[jet bundle]] (a variation def. \ref{Variation})
write

$$
  \label{LieDerivativeVariational}
  \mathcal{L}^{var}_{v}
    \;\coloneqq\;
  \delta \circ \iota_v + \iota_v \circ \delta
$$

for the _variational Lie derivative_ along $v$, analogous to [[Cartan's homotopy formula]] (prop. \ref{CartanHomotopyFormula})
but defined in terms of the variational derivative $\delta$ (eq:VariationalDerivative) as opposed to the full [[de Rham differential]].

Then for $v_1$ and $v_2$ two vertical vector fields, write

$$
  [v_1, v_2]^{var} \;\in \; \Gamma( T_{vert} J^\infty_\Sigma(E) )
$$

for the vector field whose contraction operator (def. \ref{ContractionOfFormsWithVectorFields}) is given by

$$
  \begin{aligned}
    \iota_{[v_1,v_2]^{var}}
     & =
    \left[
      \mathcal{L}^{var}_{v_1}, \iota_{v_2}
    \right]
    \\
    & \coloneqq
    \mathcal{L}^{var}_{v_1} \circ \iota_{v_2}
    -
    \iota_{v_2} \circ \mathcal{L}^{var}_{v_1}
  \end{aligned}
  \,,
$$

=--

+-- {: .num_defn #HamiltonianForms}
###### Definition
**([[Hamiltonian vector fields|infinitesimal symmetry of the presymplectic potential]] and [[Hamiltonian differential forms]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}) with [[presymplectic potential current]] $\Theta_{BFV}$ (eq:PresymplecticPotential). Write $\mathcal{E} \hookrightarrow J^\infty_\Sigma(E)$ for the [[shell]] (eq:ShellInJetBundle).

Then:

1. An [[on-shell]] variation $v$ (def. \ref{Variation}) is an _[[infinitesimal symmetry]] of the [[presymplectic current]]_
   or _[[Hamiltonian vector field]]_ if [[on-shell]] (def. \ref{EulerLagrangeOperatorForTivialVectorBundleOverMinkowskiSpacetime})
   its variational Lie derivative along $v$ (def. \ref{LieDerivativeVariational}) is a [[variational derivative]]:

   $$
     (\delta \circ \iota_v + \iota_v \circ \delta)  \Theta_{BFV}  = \delta \tilde H_v
     \phantom{AAA}
     \text{on}\, \mathcal{E}
   $$

   for some variational form $\tilde H_v$.

1. A _[[Hamiltonian differential form]]_ $H$ (or _local Hamiltonian current_) is a variational form on the shell such that there
   exists a variation $v$ with

   $$
     \delta H = \iota_v \Omega_{BFV}
     \phantom{AA}
     \,
     \text{on}\, \mathcal{E}
     \,.
   $$

We write

$$
  \Omega^{p,0}_{\Sigma, Ham}(E)
  \;\coloneqq\;
  \left\{
    (H,v)
    \;\vert\;
    v \, \text{is a variation and}\,
    \iota_v \Omega_{BFV} = \delta H
  \right\}
$$

for the space of pairs consisting of a Hamiltonian differential forms [[on-shell]] and a corresponding variation.

=--

+-- {: .num_prop #HamiltonianDifferentialForms}
###### Proposition
**([[Hamiltonian Noether's theorem]])**

A variation $v$ is an infinitesimal symmetry of the presymplectic potential (def. \ref{HamiltonianForms})
with $\mathcal{L}^{var}_v ( \Theta_{BFV} ) = \delta \tilde H_v$ precisely if

$$
  H_v \coloneqq \tilde H_v - \iota_v \Theta_{BFV}
$$

is a [[Hamiltonian differential form]] for $v$.

=--

+-- {: .proof}
###### Proof

From the definition (eq:LieDerivativeVariational) of $\mathcal{L}^{var}_v$ we have

$$
  \begin{aligned}
    & \mathcal{L}^{var}_v \Theta_{BFV}
    =
    \delta \tilde H_v
    \\
    \Leftrightarrow\;\;
    &
    \delta \iota_v \Theta_{BFV} + \iota_v \underset{= \Omega_{BFV}}{\underbrace{\delta \Theta_{BFV}}} = \delta \tilde H_v
    \\
    \Leftrightarrow\;\;
    &
    \delta \left(
      \tilde H_v - \iota_v \Theta_{BFV}
    \right)
    =
    \iota_v \Omega_{BFV}
    \,,
  \end{aligned}
$$

where we used the definition (eq:PresymplecticCurrent) of $\Omega_{BFV}$ .

=--

$\,$

Since therefore both the [[conserved currents]] from [[Noether's theorem]] as well as
the [[Hamiltonian differential forms]] are generators of infinitesimal [[symmetries]] of certain variational forms
(namely of the [[Lagrangian density]] and of the [[presymplectic current]], respectively) they
form a [[Lie algebra]]. For the conserved currents this is sometimes known as the _[[Dickey bracket]] Lie algebra_.
For the Hamiltonian forms it is the _[[Poisson bracket Lie n-algebra|Poisson bracket Lie p+1-algebra]]_. Since here for simplicity
we are considering just vertical variations, we have just a plain [[Lie algebra]].
The [[transgression of variational differential forms|transgression]] of this Lie algebra of Hamiltonian forms
on the jet bundle to [[Cauchy surfaces]] yields a [[presymplectic structure]] on [[phase space]], this we discuss
[below](#PhaseSpace).

+-- {: .num_prop #LocalPoissonBracket}
###### Proposition
**([[Poisson bracket Lie n-algebra|local Poisson bracket]])**

Let $(E,\mathbf{L})$ be a [[Lagrangian field theory]] (def. \ref{LocalLagrangianDensityOnSecondOrderJetBundleOfTrivialVectorBundleOverMinkowskiSpacetime}).

On the space $\Omega^{p,0}_{\Sigma,Ham}(E)$ pairs $(H,v)$ of [[Hamiltonian differential forms]] $H$ with compatible variation $v$ (def. \ref{HamiltonianForms}) the following operation constitutes a [[Lie bracket]]:

$$
  \label{LocalPoissonLieBracket}
  \left\{(H_1, v_1),\, (H_2, v_2)\right\}
  \;\coloneqq\;
  (\iota_{v_1} \iota_{v_2} \Omega_{BFV},\, [v_1,v_2]^{var})
  \,,
$$

where $[v_1, v_2]^{var}$ is the variational Lie bracket from def. \ref{LieDerivativeVariational}.

We call this the _local Poisson Lie bracket_.

=--

+-- {: .proof}
###### Proof

First we need to check that the bracket is well defined in itself. It
is clear that it is linear and skew-symmetric, but what needs proof is that
it does indeed land in $\Omega^{p,0}_{\Sigma,Ham}(E)$, hence that the following
equation holds:

$$
  \delta \iota_{v_2} \iota_{v_1} \Omega_{BFV}
  \;=\;
  \iota_{[v_1, v_2]^{var}} \Omega_{BFV}
  \,.
$$

With def. \ref{LieDerivativeVariational} for $\mathcal{L}^{var}$ and $[-,-]^{var}$ we compute this as follows:

$$
  \begin{aligned}
    \delta \iota_{v_1} \iota_{v_2} \Omega_{BFV}
    & =
    \tfrac{1}{2} \delta \iota_{v_1} \iota_{v_2} \Omega_{BFV}
    -
    \tfrac{1}{2} (v_1 \leftrightarrow v_2)
    \\
    & =
    \tfrac{1}{2}
    \left(
      \mathcal{L}^{var}_{v_1} \iota_{v_2} \Omega_{BFV}
      -
      \iota_{v_1} \delta \iota_{v_2} \Omega_{BFV}
    \right)
    -
    \tfrac{1}{2} (v_1 \leftrightarrow v_2)
    \\
    & =
    \tfrac{1}{2}
    \left(
      \mathcal{L}^{var}_{v_1} \iota_{v_2} \Omega_{BFV}
      -
      \iota_{v_1} \mathcal{L}^{var}_{v_2} \Omega_{BFV}
      +
      \iota_{v_1} \iota_{v_2} \underset{= 0}{\underbrace{\delta \Omega_{BFV}}}
    \right)
    -
    \tfrac{1}{2} (v_1 \leftrightarrow v_2)
    \\
    & =
    [\mathcal{L}^{var}_{v_2}, \iota_{v_1}] \Omega_{BFV}
    \\
    & =
    \iota_{[v_1, v_2]^{var}} \Omega_{BFV}
    \,.
  \end{aligned}
$$

This shows that the bracket is well defined.

It remains to see that the bracket satifies the [[Jacobi identity]]:

$$
\left\{ (H_1, v_1), \left\{ (H_2, v_2), (H_3,v_3) \right\} \right\}
  \;+\; (cyclic)
   \;=\;
  0
$$

hence that

$$
  \left(  \iota_{v_1} \iota_{[v_2,v_3]^{var}}  \Omega_{BFV} ,\,   [v_1, [v_2, v_2]^{var}]^{var} \right)
  \;+\; (cyclic)
  \;=\; 0
  \,.
$$

Here $ [v_1, [v_2, v_3]^{var}]^{var} + (cyclic) = 0 $ holds because by def. \ref{LieDerivativeVariational} $[v_1,-]^{var}$
acts as a derivation,
and hence what remains to be shown is that

$$
  \iota_{v_1} \iota_{\left([v_2, v_3]^{var}\right)}   \Omega_{BFV} + (cyclic) = 0
$$

We check this by repeated uses of def. \ref{LieDerivativeVariational}, using in addition
that

1. $\delta \iota_{v_i} \Omega_{BFV} = 0$

   (since $\iota_{v_i} \Omega_{BFV} = \delta H_i$ by $v_i$ being Hamiltonian)

1. $\mathcal{L}^{var}_{v_i} \Omega_{BFV} = 0$

   (since in addition $\delta \Omega_{BFV} = 0$)

1. $\iota_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV} = 0$

   (since $\Omega_{BFV} \in \Omega^{p,2}_\Sigma(E)$ is of vertical degree 2, and since all variations $v_i$ are vertical by assumption).

So we compute as follows (a special case of [FRS 13b, lemma 3.1.1](Poisson+bracket+Lie+n-algebra#FRS13b)):

$$
  \begin{aligned}
    0
    & =
    \delta \iota_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV}
    \\
    & =
    \mathcal{L}^{var}_{v_1} \iota_{v_2} \iota_{v_3} \Omega_{BFV}
    -
    \iota_{v_1} \delta \iota_{v_2} \iota_{v_3} \Omega_{BFV}
    \\
    & =
    \iota_{[v_1, v_2]^{var}} \iota_{v_3} \Omega_{BFV}
    +
    \iota_{v_2} \mathcal{L}^{var}_{v_1} \iota_{v_3} \Omega_{BFV}
    -
    \iota_{v_1} \mathcal{L}^{var}_{v_2} \iota_{v_3} \Omega_{BFV}
    +
    \iota_{v_1} \iota_{v_2} \delta \iota_{v_3} \Omega_{BFV}
    \\
    & =
    \iota_{[v_1, v_2]^{var}} \iota_{v_3} \Omega_{BFV}
    +
    \iota_{v_2} \iota_{[v_1,v_3]^{var}} \Omega_{BFV}
    -
    \iota_{v_1} \iota_{[v_2, v_3]^{var}} \Omega_{BFV}
    \\
    & =
    -
    \iota_{v_1} \iota_{[v_2, v_3]^{var}} \Omega_{BFV}
    -
    \iota_{v_2} \iota_{[v_3, v_1]^{var}} \Omega_{BFV}
    -
    \iota_{v_3} \iota_{[v_1, v_2]^{var}} \Omega_{BFV}
    \,.
  \end{aligned}
$$

=--

$\,$

The [[Poisson bracket Lie n-algebra|local Poisson bracket]] [[Lie algebra]] $(\Omega^{p,0}_{\Sigma,Ham}(E), [-,-]^{var})$ from prop.
\ref{LocalPoissonBracket} is but the lowest stage of
a [[higher Lie theory|higher Lie theoretic]] structure called the _[[Poisson bracket Lie n-algebra|Poisson bracket Lie p-algebra]]_.
Here we will not go deeper into this [[schreiber:Higher Structures|higher structure]] (see at _[[schreiber:Higher Prequantum Geometry]]_
for more), but below we will need the following simple shadow of it:

+-- {: .num_lemma #HorizontallyExactFormsDropOutOfLocalLieBracket}
###### Lemma

The horizontally exact Hamiltonian forms constitute a [[Lie ideal]] for the
local Poisson Lie bracket (eq:LocalPoissonLieBracket).

=--

+-- {: .proof}
###### Proof

Let $E$ be a horizontally exact Hamiltonian form, hence

$$
  E = d K
$$

for some $K$. Write $e$ for a [[Hamiltonian vector field]] for $E$.

Then for $(H,v)$ any other pair consisting of a Hamiltonian form and a corresponding
Hamiltonian vector field, we have

$$
  \begin{aligned}
    \iota_v \, \iota_e \, \Omega_{BFV}
    & =
    \phantom{-}\iota_v \, \delta E
    \\
    & =
    \phantom{-}\iota_v \, \delta \, d \, K
    \\
    & =
    - \iota_v \, d \, \delta K
    \\
    & =
    \phantom{-}d \, \iota_v \, \delta \, K
    \,.
  \end{aligned}
$$

Here we used that the horizontal derivative anti-commutes with the
vertical one by construction of the [[variational bicomplex]], and
 that $\iota_e$ anti-commutes with the horizontal derivative $d$ since the variation $e$ (def. \ref{Variation}) is
by definition vertical.

=--

+-- {: .num_example #LocalPoissonBracketForRealScalarField}
###### Example
**([[Poisson bracket Lie n-algebra|local Poisson bracket]] for [[real scalar field]])**

Consider the [[Lagrangian field theory]] for the [[free field|free]] [[real scalar field]] from example \ref{LagrangianForFreeScalarFieldOnMinkowskiSpacetime}.

By example \ref{FreeScalarFieldEOM} its [[presymplectic current]] is

$$
  \Omega_{BFV}
   =
  \eta^{\mu \nu} \delta \phi_{,\mu} \wedge \delta \phi \wedge \iota_{\partial_\mu} dvol_\Sigma
  \,
$$

The corresponding [[Poisson bracket Lie n-algebra|local Poisson bracket algebra]] (prop. \ref{LocalPoissonBracket}) has in degree 0
[[Hamiltonian forms]] (def. \ref{HamiltonianDifferentialForms}) such as

$$
  Q \;\coloneqq\; \phi \,\iota_{\partial_0} dvol_\Sigma \in \Omega^{p,0}(E)
$$

and

$$
  P \;\coloneqq\; \eta^{\mu \nu} \phi_{,\mu} \, \iota_{\partial_\nu} dvol_{\Sigma} \in \Omega^{p,0}(E)
  \,.
$$

The corresponding [[Hamiltonian vector fields]] are

$$
 v_Q = -\partial_{\phi_{,0}}
$$

and

$$
  v_P = - \partial_{\phi}
  \,.
$$

Hence the corresponding [[Poisson bracket Lie n-algebra|local Poisson bracket]] is

$$
  \{P,Q\} = \iota_{v_P} \iota_{v_Q} \omega = \iota_{\partial_0} dvol_\Sigma
  \,.
$$

More generally for $b_1, b_2 \in C^\infty_{cp}(\Sigma)$ two [[bump functions]] then

$$
  \{ b_1 P, b_2 Q \} = b_1 b_2 \iota_{\partial_0} dvol_\Sigma
  \,.
$$

=--

+-- {: .num_example #LocalPoissonBracketForDiracField}
###### Example
**([[Poisson bracket Lie n-algebra|local Poisson bracket]] for [[free field theory|free]] [[Dirac field]])**

Consider the [[Lagrangian field theory]] of the [[free field theory|free]] [[Dirac field]] on [[Minkowski spacetime]] (example \ref{LagrangianDensityForDiracField}), whose [[presymplectic current]] is, according to example \ref{PresymplecticCurrentDiracField}, given by

$$
  \label{RecallPresymplecticCurrentOfDiracField}
  \Omega_{BFV}
  \;=\;
  (\overline{\delta \psi}) \, \gamma^\mu \, (\delta \psi) \, \iota_{\partial_\mu} dvol_\Sigma
  \,.
$$

Consider this specifically in [[spacetime]] [[dimension]] $p + 1 = 4$ in which case the components $\psi_\alpha$ are [[complex number]]-valued (by prop./def. \ref{SpacetimeAsMatrices}), so that the [[tuple]] $(\psi_\alpha)$ amounts to 8 real-valued coordinate functions. By changing complex coordinates, we may equivalently consider $(\psi_\alpha)$ as four coordinate functions, and $(\overline{\psi}^\alpha)$ as another four independent coordinate functions.

Using this coordinate transformation, it is immediate to find the following [[pairs]] of [[Hamiltonian vector fields]] and their [[Hamiltonian differential forms]] from def. \ref{HamiltonianForms} applied to (eq:RecallPresymplecticCurrentOfDiracField)

| [[Hamiltonian vector field]] | [[Hamiltonian differential form]] |
|------------------------------|-----------------------------------|
|  $\phantom{AA} \partial_{\psi_\alpha}$      | $\phantom{AA}\left(\overline{\delta \psi}\gamma^\mu\right)^\alpha\, \iota_{\partial_\mu} dvol_\Sigma$ |
| $\phantom{AA} \partial_{\overline{\psi}_\alpha}$ | $\phantom{AA}\left( \gamma^\mu \psi  \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma$ |

and to obtain the following non-trivial [[Poisson bracket Lie n-algebra|local Poisson brackets]] (prop. \ref{LocalPoissonBracket}) (the other possible brackets vanish):

$$
  \left\{
     \left( \gamma^\mu \psi  \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma
     \,,\,
     \left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma
  \right\}
  \;=\;
  \left(\gamma^\mu\right)_\alpha{}^{\beta} \, \iota_{\partial_\mu} dvol_\Sigma
  \,.
$$

Notice the signs: Due to the odd-grading of the field coordinate function $\psi$,
its variational derivative $\delta \psi$ has bi-degree $(1,odd)$ and the
contraction operation $\iota_{\psi}$ has bi-degree $(-1,odd)$, so that commuting it
past $\overline{\psi}$ picks up _two_ minus signs, a "cohomological" sign due to the differential form
degrees, and a "supergeometric" one (def. \ref{DifferentialFormOnSuperCartesianSpaces}):

$$
  \iota_{\partial_\psi} \overline{\delta \psi} \cdots
  =
  (-1) (-1) \overline{\delta \psi} \,\iota_{\partial_\psi} \cdots
  \,.
$$

For the same reason, the [[Poisson bracket Lie n-algebra|local Poisson bracket]]
is a _[[super Lie algebra]]_ with _symmetric_ super Lie bracket:

$$
  \left\{
     \left( \gamma^\mu \psi  \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma
     \,,\,
     \left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma
  \right\}
  \;=\;
  +
  \left\{
     \left(\overline{\psi}\gamma^\mu\right)^\beta\, \iota_{\partial_\mu} dvol_\Sigma
     \,,\,
     \left( \gamma^\mu \psi  \right)_\alpha \, \iota_{\partial_\mu} dvol_\Sigma
  \right\}
  \,.
$$

=--

$\,$

This concludes our discussion of general [[infinitesimal symmetries of a Lagrangian]]. We pick this up again in the discussion of
_[Gauge symmetries](#GaugeSymmetries)_ below. First, in the [next chapter](#Observables) we discuss
the concept of [[observables]] in [[field theory]].


A first idea of quantum field theory > history

see _[[geometry of physics -- A first idea of quantum field theory]]_


A History of Western Philosophy

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

This page collects material related to

* [[Bertrand Russell]],

  _A History of Western Philosophy_

  1945

on the history of [[philosophy]].

## Chapter 22 -- Hegel

Chapter 22 is on [[Georg Hegel]] and his philosophy as expressed in the _[[Science of Logic]]_ etc.

> {#MysticYouth} In youth he was much attracted to [[mysticism]], and his later views may be regarded, to some extent, as an intellctualizing of what had first appeared to him as a mystic insight.

## Chapter 31 -- The Philosophy of Logical Analysis

([html text](http://www.personal.kent.edu/~rmuhamma/Philosophy/RBwritings/philoLogicalAnaly.htm))
> In philosophy ever since the time of Pythagoras there has been an opposition between the men whose thought was mainly inspired by mathematics and those who were more influenced by the empirical sciences. Plato, Thomas Aquinas, Spinoza, and Kant belong to what may be called the mathematical party; Democritus, Aristotle, and the modern empiricist from Locke onwards, belong to the opposite party. In our day a school of philosophy has arisen which sets to work to eliminate Pythagoreanism from the principles of mathematics, and to combine empiricism with an interest in the deductive parts of human knowledge. The aims of this school are less spectacular than those of most philosophers in the past, but some of its achievements are as solid as those of the men of science.

> The origin of this philosophy is in the achievements of mathematicians who set to work to purge their subject of fallacies and slipshod reasoning. The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibnitz believed in actual infinitesimals, but although this belief suited his metaphysics it has no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came [[Georg Cantor]], who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like [[Hegel]], who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicist. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated.

(Here it is maybe worth remarking that not much later [[infinitesimal objects]] as anticipated by [[Gottfried Leibniz]] were given a neat formal basis in [[mathematics]], by formalisms such as [[synthetic differential geometry]] and [[nonstandard analysis]].)

## References

* Wikipedia, _[A History of Western Philosophy](http://en.wikipedia.org/wiki/A_History_of_Western_Philosophy)_

category: reference


* [[Kuen-Bang Hou (Favonia)]], [[Eric Finster]], [[Dan Licata]], [[Peter LeFanu Lumsdaine]]:

  **A mechanization of the Blakers-Massey connectivity theorem in Homotopy Type Theory**

  LICS'16 (2016) 565–574

  [arXiv:1605.03227](https://arxiv.org/abs/1605.03227)

  [doi:10.1145/2933575.2934545](https://doi.org/10.1145/2933575.2934545)

on a [[formal proof]] of the [[Blakers-Massey theorem]] in [[homotopy type theory]].

> **Abstract.** This paper continues investigations in "[[synthetic homotopy theory]]": the use of [[homotopy type theory]] to give [[proof assistant|machine-checked proofs]] of constructions from [[homotopy theory]]. We present a mechanized proof of the [[Blakers-Massey theorem|Blakers-Massey connectivity theorem]], a result relating the higher-dimensional [[homotopy groups]] of a [[pushout type]] (roughly, a space constructed by gluing two spaces along a shared subspace) to those of the components of the pushout. This theorem gives important information about the pushout type, and has a number of useful corollaries, including the [[Freudenthal suspension theorem]], which has been studied in previous formalizations.
The new proof is more elementary than existing ones in abstract homotopy-theoretic settings, and the mechanization is concise and high-level, thanks to novel combinations of ideas from [[homotopy theory]] and [[type theory]].

## See also

* [[Blakers-Massey theorem]]

* [[connectivity]]

category: reference


A Not-So-Nice Submanifold

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

## Idea ##

As shown in [[evaluation fibration of mapping spaces]] and [[tubular neighbourhoods of mapping spaces]], if we carve out a submanifold of a mapping space by specifying "coincidences", we often get a tubular neighbourhood.  On this page, we shall give an example of a submanifold with no tubular neighbourhood.  The example is simple to describe.  To make it concrete, we shall fix as our source space the circle, $S^1$.  For our target space, we shall take a finite dimensional smooth manifold, $M$.  The full smooth mapping space, $C^\infty(S^1, M)$ is known as the [[smooth loop space]].  For simplicity, let us take _based_ loops within this, which we write as $\Omega M$.  Within that, we consider the space of based smooth maps $S^1 \to M$ which are infinitely flat at the point $1 \in S^1$.  Let us write this as $\Omega_♭ M$.  As we are using based loops, we can identify the tangent space of $M$ at the basepoint with $\mathbb{R}^n$ and so we have a sequence, which is exact by [[Borel's theorem]]:
\[
\Omega_♭ M \to \Omega M \to \prod_{j = 1}^\infty \mathbb{R}^n
\]

It is easy to show that this does not admit a tubular neighbourhood.  If it did, there would be a splitting of the induced map on tangent spaces:
\[
T_\alpha \Omega_♭ M \to \Omega_\alpha M \to \prod_{i = 1}^\infty \mathbb{R}^n
\]
but as the second map is surjective, this cannot split as a splitting map would induce a continuous injection from $\prod_{i=1}^\infty \mathbb{R}^n$ to a normed vector space and that is impossible.


A Polarized View of String Topology

This entry is about the article

* [[Ralph Cohen]], [[Veronique Godin]],

  _A polarized view of string topology_,

  In:

  [[Graeme Segal]], [[Ulrike Tillmann]] (eds.)

  _Topology, Geometry and Quantum Field Theory_,

  LMS, Lecture Notes Series 308, 2004

  [arXiv:math/0303003](http://arxiv.org/abs/math/0303003)

  [doi:10.1017/CBO9780511526398.008](https://doi.org/10.1017/CBO9780511526398.008)

on realization of closed-[[string topology]] operations as an [[HQFT]].

See also the followup

* [[Veronique Godin]], _Higher string topology operations_ ([arXiv:0711.4859](http://arxiv.org/abs/0711.4859))

for more details and the extension of open-closed [[HQFT]].

And see example 4.2.16 and remark 4.2.17, in

* [[Jacob Lurie]], _[[On the Classification of Topological Field Theories]]_

## Related entries

* [[Sullivan chord diagram]]

category: reference


A pseudo metric space, i.e. a set with a metric where the distance between two distinct points can also be 0, need not be Haussdorf

>This is not really the place to ask what seems to be a question. You could ask us on the n-forum or better on [math.stackexchange](http://math.stackexchange.com/) for questions that are high level but not research questions.


a Serre fibration between CW-complexes is a Hurewicz fibration

## Statement

\begin{theorem}
Let $X$ and $Y$ be [[CW-complexes]] and $f:X\to Y$ a [[Serre fibration]].  Then $f$ is in fact a [[Hurewicz fibration]] (at least in a [[convenient category of spaces]]).
\end{theorem}

## References

This was originally proven in

* Mark Steinberger and James West, *Covering homotopy properties of maps between C.W. complexes or ANRs*, Proc. Amer. Math. Soc. 92 (1984), 573-577, [web page with fulltext PDF](https://www.ams.org/journals/proc/1984-092-04/S0002-9939-1984-0760948-6/)

An error was corrected in

* Robert Cauty, *Sur les ouverts des CW-complexes et les fibrés de Serre*, Colloquium Mathematicae (1992) Volume: 63, Issue: 1, page 1-7, [web page with fulltext PDF](https://eudml.org/doc/210130)


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###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
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This page is to provide links related to the text

* [[Geraldine Brady]], [[Todd Trimble]],

  _A string diagram calculus for predicate logic and C. S. Peirce's System Beta_

  ([[BradyTrimbleString.pdf:file]])

on [[string diagram]] calculus for [[indexed monoidal categories]] with [[poset]] fibers (monoidal [[hyperdoctrines]]). This is related to [[Charles Peirce]]'s "System beta". Based on this work, a [[string diagram]] calculus for general [[indexed monoidal categories]] is given in

* {#PontoShulman12} [[Kate Ponto]], [[Michael Shulman]], _Duality and traces for indexed monoidal categories_, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 ([arXiv:1211.1555](http://arxiv.org/abs/1211.1555))

category: reference


A study in derived algebraic geometry

This entry is about the book (in progress)

* [[D. Gaitsgory]], [[N. Rozenblyum]], _A study in derived algebraic geometry_, [web](http://www.math.harvard.edu/~gaitsgde/GL/).

It develops some aspects of the theory of [[derived algebraic geometry]] with the [[categorical geometric Langlands conjecture]] in mind.

## See also

* [[Dennis Gaitsgory]], _[[Notes on geometric Langlands]]_, [web](http://www.math.harvard.edu/~gaitsgde/GL/).

* [[D. Gaitsgory]] et. al., _Geometric representation theory_, graduate seminar, Fall 2009--Spring 2010, [web](http://www.math.harvard.edu/~gaitsgde/grad_2009/).

## Contents

* Preface

* Introduction to Part I: Preliminaries

* Chapter I.1: Some higher algebra

An introduction to $\infty$-categories and review of Lurie's books

* Chapter I.2: Basics of derived algebraic geometry

Introduces (derived) [[prestacks]], [[stacks]], [[schemes]] and [[Artin stacks]].

* Chapter I.3: Quasi-coherent sheaves on prestacks

Introduces and studies the basic properties of the category of quasi-coherent sheaves on a prestack.

* Introduction to Part II: Ind-coherent sheaves

* Chapter II.1: _Ind-coherent sheaves on schemes_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/IndCohSch.pdf)

Introduces and studies elementary properties of the [[stable (infinity,1)-category]] of [[ind-coherent sheaves]] on [[derived schemes]].

* Chapter II.2: _The !-pullback and base change_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/corrtext.pdf).

Discusses how the system of [[stable (infinity,1)-categories]] of [[ind-coherent sheaves]] satisfies the formalism of [[six operations]].

* Chapter II.3: _Relation between QCoh and IndCoh_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/QCohandIndCoh.pdf).

Discusses the relationship between the [[stable (infinity,1)-categories]] of [[quasi-coherent sheaves]] and [[ind-coherent sheaves]].

* Introduction to Part III: Inf-schemes

* Chapter III.1: _Deformation theory_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/deftext.pdf).

Sets up [[derived deformation theory]].

* Chapter III.2: _(Ind)-inf-schemes_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/InfSchtext.pdf).

Introduces [[inf-schemes]], which are algebro-geometric objects that include [[derived schemes]] and de Rham [[prestacks]] of schemes

* Chapter III.3: _Ind-coherent sheaves on (ind)-inf-schemes_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/InfSchtext.pdf).

Extends the formalism of [[ind-coherent sheaves]] to [[inf-schemes]] and discusses functoriality.

* Chapter III.4: _An application: crystals_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/crystext.pdf).

Studies the [[six operations]] for [[D-modules]].

* Introduction to Part IV: Formal geometry

* Chapter IV.1: _Lie algebras and co-commutative co-algebras_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/Lie.pdf).

Sets up the theory of [[Lie algebras]] from the point of view of [[Koszul-Quillen duality]].

* Chapter IV.2: _Formal moduli_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/formalmoduli.pdf).

Reinterprets [[Jacob Lurie]]'s theory of [[formal moduli problems]] using the language of [[inf-schemes]].

* Chapter IV.3: _Formal groups and Lie algebras_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/exp.pdf).

Explains how to pass between [[Lie algebras]] and [[formal groups]] within the framework of [[derived algebraic geometry]].

* Chapter IV.4: _Lie algebroids_, [pdf](http://www.math.harvard.edu/~gaitsgde/GL/algebroids.pdf).

Introduces [[Lie algebroids]] and studies various aspects of [[infinitesimal geometry]].

* Chapter IV.5: Infinitesimal differential geometry

Studies various aspects of infinitesimal geometry, such as the n-th infinitesimal neighborhood.

* Introduction to Part V: Categories of correspondences

* Chapter V.1: The $(\infty,2)$-category of correspondences

Introduces the formalism of correspondences.

* Chapter V.2: Extension theorems for category of correspondences

Extends the theory of IndCoh from schemes to inf-schemes.

* Chapter V.3: The (symmetric) monoidal structure on the category of correspondences

Shows how the formalism of correspondences encodes Serre duality

* Introduction to Part A (appendix on $(\infty,2)$-categories)

* Chapter A.1: Basics of $(\infty,2)$-categories

Defines $(\infty,2)$-categories and introduces some basic constructions

* Chapter A.2: Straightening and Yoneda $(\infty,2)$-categories

Constructs the straightening/unstraightening procedures and the Yoneda embedding

* Chapter A.3: Adjunctions in $(\infty,2)$-categories

Studies the procedure of passing to the adjoint 1-morphism

* References


A Survey of Cohomological Physics

* [[Jim Stasheff]], _[[StasheffCohomPhysicsSurvey.pdf:file]]_, 2009

See also

* {#Stasheff16} [[Jim Stasheff]], _Higher homotopy structures, then and now_, talk at _[Opening workshop](https://www.mpim-bonn.mpg.de/node/6356)_ of _[Higher Structures in Geometry and Physics](https://www.mpim-bonn.mpg.de/node/5883)_, MPI Bonn 2016 ([[StasheffHomotopyStructuresReview.pdf:file]])

#Content#

[[Jim Stasheff]] has spent much of his work on identifying and studying _cohomological_ (as in: [[homological algebra]]) and _homotopical_ (as in: [[homotopy theory]]) structures in [[physics]].

This includes notably the invention and study of homotopy-coherent structures such as [[A-infinity operad]]s, [[A-infinity algebra]]s and [[L-infinity-algebra]]s and their application to [[BV theory]], [[string field theory]]  (longer list of links should eventually go here).

The above survey lists key concepts and collects references to further literature.

#Higher categorical background#

Essentially using variants of the [[Dold-Kan correspondence]] one may regard many of these [[differential graded algebra]] structures as [[higher category theory|higher categorical]] structures. For instance the [[pretriangulated dg-category|pretriangulated dg-categories]] formed by homotopy coherent structures are [[presentable (infinity,1)-category|presentations]] for [[stable (infinity,1)-category|stable (infinity,1)-categories]].

In this sense "cohomological physics" is understood as part of [[n-categorical physics]].

category: reference


A Survey of Elliptic Cohomology

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#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

+-- {: .standout}

This entry is about the text

* [[Jacob Lurie]], _A Survey of Elliptic Cohomology_, in:  _Algebraic Topology_, Abel Symposia Volume 4, 2009, pp 219-277 ([pdf](http://www.math.harvard.edu/~lurie/papers/survey.pdf), [doi:10.1007/978-3-642-01200-6_9](https://doi.org/10.1007/978-3-642-01200-6_9))

It

* reviews basics of [[elliptic cohomology]]

* indicates the construction of the [[tmf]]-[[spectrum]]

* and discusses this in the context of [[equivariant cohomology]].

The central theorem is

* the realization of the moduli space of [[elliptic curve]]s as a [[structured (∞,1)-topos]]

* with a structure [[(∞,1)-sheaf]] of [[E-∞ ring]]-valued functions

* such that the [[tmf]]-[[spectrum]] is the [[E-∞ ring]] of global sections of this structure sheaf.

=--

See also _[[Elliptic Cohomology I]]_ and _[[Chromatic Homotopy Theory]]_.

#Table of Contents#

1. [Summary](#Summary)

   The following entry has some paragraphs that summarize central ideas.

   1. [gluing all elliptic cohomology theories to the tmf spectrum](#gluingallellipticcohomologytheoriestothetmfspectrum)

   1. [interpretation in terms of higher topos theory](#interpretationintermsofhighertopostheory)

1. Partial surveys

   These links point to pages that contain notes on aspects of the theory that are in the style of and originate from [a seminar on A Survey of Elliptic Cohomology](http://nd.edu/~rgrady/elliptic_seminar.html):

   1. [[A Survey of Elliptic Cohomology - cohomology theories]]

   1. [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

   1. [[A Survey of Elliptic Cohomology - E-infinity rings and derived schemes]]

   1. [[A Survey of Elliptic Cohomology - elliptic curves]]

   1. [[A Survey of Elliptic Cohomology - equivariant cohomology]]

   1. [[A Survey of Elliptic Cohomology - derived group schemes and (pre-)orientations]]

   1. [[A Survey of Elliptic Cohomology - A-equivariant cohomology]]

   1. [[A Survey of Elliptic Cohomology - the derived moduli stack of derived elliptic curves]]

   1. [[A Survey of Elliptic Cohomology - towards a proof]]

   1. [[A Survey of Elliptic Cohomology - compactifying the derived moduli stack]]

   1. [[A Survey of Elliptic Cohomology - descent ss and coefficients]]

1. towards geometric models

    These links point to pages that have an exposition of the Stolz-Teichner program for constructing [[geometric models for elliptic cohomology]].

* Outline of the constructions and statements

  * [[Axiomatic field theories and their motivation from topology]]

  * [[(1,1)-dimensional Euclidean field theories and K-theory]]

  * [[(2,1)-dimensional Euclidean field theories and tmf]]

* Definitions

  * [[bordism categories following Stolz-Teichner]]

  * [[(2,1)-dimensional Euclidean field theories]]

* [[Geometric Models for Elliptic Cohomology|Geometric models for elliptic cohomology -- Axiomatic field theories and their motivation from topology]]

#Contents#

Here is the table of contents of the Survey reproduced. Behind the links are linked keyword lists for relevant terms.

* table of content
{:toc}

## Summary
 {#Summary}

The text starts with showing or recalling that

* the collection of all [[elliptic cohomology]] theories
and

* the gluing of their representing [[spectrum|spectra]] into the single [[tmf]] spectrum

is best understood in terms of global sections of the structure sheaf of functions on the refinement of the moduli [[space]] of all elliptic curves to a [[structured (∞,1)-topos]].

Then it uses this [[higher topos theory|higher topos theoretic]] [[derived algebraic geometry]] perspective to analyze further properties of [[elliptic cohomology]] theories, in particular their refinements to [[equivariant cohomology]].

### Gluing all elliptic cohomology theories to the tmf spectrum {#gluingallellipticcohomologytheoriestothetmfspectrum}

The triple of [[generalized (Eilenberg-Steenrod) cohomology]] theories

1. periodic ordinary [[integral cohomology]]

1. complex [[K-theory]]

1. [[elliptic cohomology]]

constitutes the collection of all possible [[generalized (Eilenberg-Steenrod) cohomology]] theories with the extra [[stuff, structure, property|property]] that they are

* [[multiplicative cohomology theory|multiplicative]]

and

* [[periodic cohomology theory|periodic]].

It so happens that all multiplicative periodic generalized Eilenberg-Steenrod cohomology theories $A$ are characterized by the [[formal group]] (an [[infinitesimal space|infinitesimal]] [[group]]) whose [[ring]] of functions is the [[cohomology ring]] $A(\mathbb{C}P^\infty)$ obtained by evaluating $A$ on the complex projective space $\mathbb{C}P^\infty \simeq \mathcal{B} U(1)$ -- the [[classifying space]] for complex [[line bundle]]s -- and whose group product is induced from the morphism $\mathbb{C}P^\infty \times \mathbb{C}P^\infty \to \mathbb{C}P^\infty$ that representes the [[tensor product]] of complex [[line bundle]]s.

There are precisely three different types of such formal groups:

* the additive formal group (a single one)

* the multiplicative formal group (a single one)

* a formal group defined by an [[elliptic curve]] (many).

The first case corresponds to periodic [[integral cohomology]]. The second corresponds to complex [[K-theory]]. Each element in the third family corresponds to one flavor of [[elliptic cohomology]].

It is therefore natural to subsume all elliptic cohomology theories into one single cohomology theory. This is the theory called [[tmf]].

It turns out that the right way to formalize what "subsume" means in the above sentence involves formulating the way in which an [[elliptic cohomology]] theory is associated to a given [[elliptic curve]] in the correct [[higher category theory|higher categorical]] language:

The collection of all 1-dimensional [[elliptic curve]]s forms a generalized [[space]] $M_{1,1}$ -- a [[stack]] -- defined by the property that it is the [[classifying space]] for elliptic curves in that elliptic curves over a ring $R$ correspond to classifying maps $\phi : Spec R \to M_{1,1}$.

Then the classical assignment of an [[elliptic cohomology]]  theory to an [[elliptic curve]] is an assignment

$$
  \{\phi : Spec R \to M_{1,1}\}
  \to
  CohomologyTheories
  \,.
$$

We may think of maps $Spec R \to M_{1,1}$ as picking certain subsets of the generalized [[space]] $M_{1,1}$ and of morphisms

$$
  \array{
     Spec(R) &&\to&& Spec(R')
     \\
     & \searrow && \swarrow
     \\
     && M_{1,1}
  }
$$

as maps between such subsets. Hence the assignment of cohomology theories to elliptic curves is much like a [[sheaf]] of cohomology theories on the moduli [[space]] ([[stack]]) of [[elliptic curve]]s.

In order to _glue_ all elliptic cohomology theories in some way one would like to take something like the [[category of elements]] of this [[sheaf]], i.e. its [[homotopy limit]]. In order to say what that should mean, one has to specify the suitable nature of the codomain, the collection of "all cohomology theories".

As emphasized at [[generalized (Eilenberg-Steenrod) cohomology]], the best way to do this is to identify a generalized (Eilenberg-Steenrod) cohomology theory with the [[spectrum]] that represents it. It is and was well known how to do this for each elliptic curve separately. What is not so clear is how this can be done coherently for all elliptic curves at once: we need a lift of the above cohomology-theory-valued sheaf to a sheaf of representing [[spectrum|spectra]]

$$
  \array{
    && Spectra
    \\
    & {}^{?}\nearrow & \;\;\;\downarrow^{represent}
    \\
    \{\phi : Spec R \to M_{1,1}\}
    &\to&
    CohomologyTheories
  }
  \,.
$$

In this generality this turns out to be a hard problem. But by definition here we are really interested just in the special case where all cohomology theories in question are [[multiplicative cohomology theory|multiplicative cohomology theories]] and where hence all spectra in question are [[commutative ring spectrum|commutative ring spectra]]

$$
  \array{
    && CommRingSpectra
    \\
    & {}^{O_{M^{der}}}\nearrow & \downarrow
    \\
    \{\phi : Spec R \to M_{1,1}\}
    &\to&
    MultiplicativeCohomologyTheories
  }
  \,.
$$

As indicated, this problem does turn out to have a solution: [[Paul Goerss|Goerss]], [[Mike Hopkins|Hopkins]] and [[Haynes Miller|Miller]] showed that the desired lift denoted $O_{M^{der}}$ above exists -- the _[[Goerss-Hopkins-Miller theorem]]_

Accordingly, one can then obtain the [[tmf]] spectrum as the [[homotopy limit]] of this sheaf of [[E-∞ ring]]s $O_{M^{der}}$. Recall from the discussion at [[limit in a quasi-category]] that such a homotopy limit computes global sections. It is an $\infty$-version of computing sections in a [[Grothendieck construction]], really, as described there.

### Interpretation in terms of higher topos theory {#interpretationintermsofhighertopostheory}

What is noteworthy about the above construction is that, as the notation above suggests, sheaves of [[E-infinity ring]]s generalize sheaves of rings as thery are familiar from the theory of [[ringed space]]s, where they are called **structure sheaves**.

Accordingly, the morphism $O_{M^{der}}$ makes the moduli [[space]] of elliptic curves into a [[structured (∞,1)-topos]].

This perspective embeds the theory of [[elliptic cohomology]] and of the [[tmf]] spectrum as an application into the general context of [[higher topos theory]] and [[derived algebraic geometry]].

## equivariant elliptic cohomology {#equivariantellipticcohomology}

* [[equivariant elliptic cohomology]]

## Contents
 {#Contents}

### 1. Elliptic Cohomology {#EllipticCohomology}

#### 1.1 Cohomology Theories {#CohomologyTheories}

* [[cohomology]]

  * [[generalized (Eilenberg-Steenrod) cohomology]]

  * [[integral cohomology]]

  * [[K-theory]]

  * [[elliptic cohomology]]

* [[multiplicative cohomology theory]]

* [[periodic cohomology theory]]

* [[topological modular form]]

#### 1.2 Formal Groups from Cohomology Theories {#FormalgroupsFromCohomologyTheories}

* [[formal group]]

#### 1.3 Elliptic Cohomology {#EllipticCohomology}

* [[elliptic curve]]

* [[elliptic cohomology]]

* [[E-∞ ring]]

### 2 Derived Algebraic geometry {#DerivedAlgebraicCohomology}

* [[derived algebraic geometry]]

  * [[higher topos theory]]

  * [[higher algebra]]

#### 2.1 $E_\infty$ rings

* [[commutative algebra in an (∞,1)-category]]

* [[E-∞ ring]]

* [[commutative ring spectrum]]

#### 2.2 Derived Schemes

* [[structured (∞,1)-topos]]

* [[derived scheme]]

### 3 Derived Group Schemes and Orientations

* [[group scheme]]

#### 3.1 Orientations of the Multiplicative Group

#### 3.2 Orientations of the Additive Group

#### 3.3 The Geometry of Preorientations

#### 3.4 Equivariant $A$-Cohomology for Abelian Groups

#### 3.5 The Nonabelian Case

### 4 Oriented Elliptic Curves

#### 4.1 Construction of the Moduli Stack

#### 4.2 The Proof of Theorem 4.4: The Local Case

#### 4.3 Elliptic Cohomology near $\infty$

### 5 Applications

#### 5.1 2-Equivariant Elliptic Cohomology

* [[equivariant elliptic cohomology]]

#### 5.2 Loop Group Representations

* [[loop group]]

#### 5.3 String Orientation

* [[string structure]]

* [[sigma-orientation]]

#### 5.4 Higher Equivariance

#### 5.5 Elliptic Cohomology and Geometry

## Further references
 {#furtherreferences}

Lots of literature on [[modular form]]s is collected at

* [[Nora Ganter]], [Topological modular forms literature list](http://www.math.uiuc.edu/~ganter/talbot/index.html)

An introduction to and survey of the [[Goerss-Hopkins-Miller-Lurie theorem]] is in

* [[Paul Goerss]], _Topological modular forms (after Hopkins, Miller, and Lurie)_ Séminaire BOURBAKI Mars 2009 61ème année, 2008-2009, no 1005(2009)([arXiv](http://arxiv.org/abs/0910.5130))

which has grown out of

* [[Topological Algebraic Geometry - A Workshop]].

A good bit of details is in

* [[David Gepner]], _[[Homotopy topoi and equivariant elliptic cohomology]]_, 1999 ([pdf](http://www.ms.unimelb.edu.au/~nganter/talbot/gepner.thesisformat.pdf))