Mathematical Quantum Field Theory – Gauge Symmetries

This is one chapter in a series on Mathematical Quantum Field Theory.

The previous chapter is 9. Propagators.

The next chapter is 11. Reduced phase space.

10. Gauge symmetries

In this chapter we discuss these topics:

An infinitesimal gauge symmetry of a Lagrangian field theory (def. 10.5 below) is a infinitesimal symmetry of the Lagrangian which may be freely parameterized, hence “gauged”, by a gauge parameter. A Lagrangian field theory exhibiting these is also called a gauge theory.

By choosing the gauge parameter to have compact support, infinitesimal gauge symmetries in particular yield infinitesimal symmetries of the Lagrangian with compact spacetime support. One finds (prop. 10.1 below) that the existence of on-shell non-trivial symmetries of this form is an obstruction to the existence of the covariant phase space of the theory (prop. 8.6).

gauge symmetries

 name meaning def. infinitesimal symmetry of the Lagrangian evolutionary vector field which leaves invariant the Lagrangian density up to a total spacetime derivative def. 6.6 spacetime-compactly supported infinitesimal symmetry of the Lagrangian obstructs existence of the covariant phase space (if non-trivial on-shell) prop. 10.1 infinitesimal gauge symmetry gauge parameterized collection of infinitesimal symmetries of the Lagrangian; for compactly supported gauge parameter this yields spacetime-compactly supported infinitesimal symmetries def. 10.5 rigid infinitesimal symmetry of the Lagrangian infinitesimal symmetry modulo gauge symmetry def. 10.8 generating set of gauge parameters reflects all the Noether identities remark 10.7 closed gauge parameters Lie bracket of infinitesimal gauge symmetries closes on gauge parameters def. 10.26

But we may hard-wire these gauge equivalences into the very geometry of the types of fields by forming the homotopy quotient of the action of the infinitesimal gauge symmetries on the jet bundle. This homotopy quotient is modeled by the action Lie algebroid (def. 10.21 below). Its algebra of functions is the local BRST complex of the theory (def. 10.28) below.

In this construction the gauge parameters appear as auxiliary fields whose field bundle is a graded version of the gauge parameter-bundle. As such they are called ghost fields. The ghost fields may have infinitesimal gauge symmetries themselves which leads to ghost-of-ghost fields, etc. (example 10.32) below.

It is these auxiliary ghost fields and ghost-of-ghost fields which will serve to remove the obstruction to the existence of the covariant phase space for gauge theories, this we arrive at in Gauge fixing, further below.

gauge parameters and ghost fields

 symbol meaning def. ##\mathcal{G} \overset{gb}{\to} \Sigma## gauge parameter bundle def. 10.5 ##c^\alpha \in C^\infty(\mathcal{G})## coordinate function on gauge parameter bundle ##\epsilon \in \Gamma_\Sigma(\mathcal{G})## gauge parameter ##\mathcal{G}[1]## gauge parameter bundle regarded as graded manifold in degree 1 expl. 10.28 ##C \in \Gamma_\Sigma(\mathcal{G}[1])## gost field history ##\underset{deg = 1}{\underbrace{c^\alpha}} \in C^\infty(\mathcal{G}[1])## ghost field component function ##\underset{deg = 1}{\underbrace{c^\alpha_{,\mu_1 \cdots \mu_k}}} \in C^\infty(J^\infty_\Sigma(\mathcal{G}[1]))## ghost field jet component function ##\phantom{A}## ##\overset{(2)}{\mathcal{G}} \overset{gb}{\to} \Sigma## gauge-of-gauge parameter bundle expl. 10.32 ##\overset{(2)}{c}^\alpha \in C^\infty(\overset{(2)}{\mathcal{G}})## coordinate function on gauge-of-gauge parameter bundle ##\overset{(2)}{\epsilon} \in \Gamma_\Sigma(\mathcal{G})## gauge-of-gauge parameter ##\overset{(2)}{\mathcal{G}}[2]## gauge-of-gauge parameter bundle regarded as graded manifold in degree 1 ##\overset{(2)}{C} \in \Gamma_\Sigma(\mathcal{G}[1])## gost-of-ghost field history ##\underset{deg = 2}{\underbrace{\overset{(2)}{c}{}^\alpha}} \in C^\infty(\overset{(2)}{\mathcal{G}}[2])## ghost-of-ghost field component function ##\underset{deg = 2}{\underbrace{{\overset{(2)}{c}}{}^\alpha_{,\mu_1 \cdots \mu_k}}} \in C^\infty(J^\infty_\Sigma(\overset{(2)}{\mathcal{G}}[2]))## ghost-of-ghost field jet component function

The mathematical theory capturing these phenomena is the higher Lie theory of Lie-∞ algebroids (def. 10.22 below).

compactly supported infinitesimal symmetries obstruc the covariant phase space

As an immediate corollary of prop. 6.17 we have the following important observation:

Proposition 10.1. (spacetime-compactly supported and on-shell non-trivial infinitesimal symmetries of the Lagrangian obstruct the covariant phase space)

Let ##(E,\mathbf{L})## be a Lagrangian field theory over a Lorentzian spacetime.

If there exists a single infinitesimal symmetry of the Lagrangian ##v## (def. 6.6) such that

1. it has compact spacetime support (def. 7.31)
2. it does not vanish on-shell (50) (so not a trivial one, example 10.2)

then there does not exist any Cauchy surface (def. 8.1) for the Euler-Lagrange equations of motion (def. 5.24) outside the spacetime support of ##v##.

Proof. By prop. 6.17
the flow along ##\hat v## preserves the on-shell space of field histories. Now by the assumption that ##\hat v## does not vanish on-shell implies that this flow is non-trivial, hence that it does continuously change the field histories over some points of spacetime, while the assumption that it has compact spacetime support means that these changes are confined to a compact subset of spacetime.

This means that there is a continuum of solutions to the equations of motion whose restriction to the infinitesimal neighbourhood of any codimension-1 suface ##\Sigma_p \hookrightarrow \Sigma## outside of this compact support coincides. Therefore this restriction map is not an isomorphism and ##\Sigma_p## not a Cauchy surface for the equations of motion.

Notice that there always exist spacetime-compactly supported infinitesimal symmetries that however do vanish on-shell:

Example 10.2. (trivial compactly-supported infinitesimal symmetries of the Lagrangian)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) over Minkowski spacetime (def. 2.17), so that the Lagrangian density is canonically of the form

$$\mathbf{L} = L \, dvol_\Sigma$$

with Lagrangian function ##L \in \Omega^{0,0}_\Sigma(E) = C^\infty(J^\infty_\Sigma(E))## a smooth function of the jet bundle (characterized by prop. 4.6).

Then every evolutionary vector field (def. 6.2) whose coefficients which is proportional to the Euler-Lagrange derivative (48) of the Lagrangian function ##L##

$$v \; := \; \frac{\delta_{EL} L }{\delta \phi^a} \kappa^{[a b]} \, \partial_{\phi^a} \;\in\; \Gamma_E^{ev}( T_\Sigma E )$$

by smooth coefficient functions ##\kappa^{a b}##

$$\kappa^{[a b]} \;\in\; \Omega^{0,0}_\Sigma(E)$$

such that

1. each ##\kappa^{a b}## has compact spacetime support (def. 7.31)
2. ##\kappa## is skew-symmetric in its indices: ##\kappa^{[a b]} = – \kappa^{[b a]}##

is an implicit infinitesimal gauge symmetry (def. link ).

This is so for a “trivial reason” namely due to that that skew symmetry:

\begin{aligned} \mathcal{L}_{\hat v} \mathbf{L} & = \iota_{\hat v} \delta \mathbf{L} \\ &= \iota_{\hat v} ( \delta_{EL}\mathbf{L} – d \Theta_{BFV} ) \\ & = \iota_\epsilon \frac{\delta_{EL}L}{\delta \phi^a} \delta \phi^a + d \iota_{\hat v}\Theta_{BFV} \\ & = \underset{= 0}{ \underbrace{ \left( \frac{\delta_{EL} L }{\delta \phi^a} \right) \left( \frac{\delta_{EL} L }{\delta \phi^b} \right) \kappa^{[a b]} } } \, dvol_\Sigma \;+\; d \iota_{\hat v} \Theta_{BFV} \\ & = d \iota_{\hat v} \Theta_{BFV} \end{aligned}

Here the first steps are just recalling those in the proof of Noether’s theorem I (prop. 6.7) while the last step follows with the skew-symmetry of ##\kappa##.

Notice that this means that

1. the Noether current (77) vanishes: ##J_{\hat v} = 0##;
2. the infinitesimal symmetry vanishes on-shell (41): ##\hat v \vert_{\mathcal{E}} = 0##.

Therefore these implicit infinitesimal gauge symmetries are called the trivial infinitesimal gauge transformations.

(e.g. Henneaux 90, section 2.5)

Proposition 10.1 implies that we need a good handle on determining whether the space of non-trivial compactly supported infinitesimal symmetries of the Lagrangian modulo trivial ones is non-zero. This obstruction turns out to be neatly captured by methods of homological algebra applied to the local BV-complex (def. 7.43):

Example 10.3. (cochain cohomology of local BV-complex)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) whose field bundle ##E## is a trivial vector bundle (example 3.4) and whose Lagrangian density ##\mathbf{L}## is spacetime-independent (example 5.14), and let ##\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}## be a constant section of the shell (57).

By inspection we find that the cochain cohomology of the local BV-complex ##\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}## (def. 7.43) has the following interpretation:

In degree 0 the image of the BV-differential coming from degree -1 and modulo ##d##-exact terms

$$im\left( \Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma(E,\varphi)) \overset{s_{BV}}{\to} \Omega^{0,0}_\Sigma(E,\varphi)/im(d) \right)$$

is the ideal of functions modulo ##im(d)## that vanish on-shell. Since the differential going from degree 0 to degree 1 vanishes, the cochain cohomology in this degree is the quotient ring

\begin{aligned} H^0\left(\Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}_{BV}}\vert d\right) & \simeq \Omega^{0,0}_{\Sigma,cp}(E,\varphi)\vert_{\mathcal{E}}/im(d) \end{aligned}

of functions on the shell ##\mathcal{E}## (105).

In degree -1 the kernel of the BV-differential going to degree 0

$$ker\left( \Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma(E,\varphi)) \overset{s_{BV}}{\to} \Omega^{0,0}_\Sigma(E,\varphi)\right)$$

is the space of implicit infinitesimal gauge symmetries (def. link ) and the image of the differential coming from degree -2

$$im\left( \Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi) \wedge_{\Omega^{0,0}_{\Sigma,cp}(E,\varphi)} \Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi) \overset{s_{BV}}{\longrightarrow} \Gamma_{\Sigma,cp}(J^\infty_\Sigma T_\Sigma E,\varphi) \right)$$

is the trivial implicit infinitesimal gauge transformations (example 10.2).

Therefore the cochain cohomology in degree -1 is the quotient space of implicit infinitesimal gauge transformations modulo the trivial ones:

 $$\label{NegativeOneCohomologyBV} H^{-1}\left( \Omega^{0,0}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}} \right) \simeq \frac{ \left\{ \text{implicit infinitesimal gauge transformations} \right\} } { \left\{ \text{ trivial implicit infinitesimal gauge transformations} \right\} }$$ (152)
Proposition 10.4. (local BV-complex is homological resolution of the shell iff there are no non-trivial compactly supported infinitesimal symmetries)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) whose field bundle ##E## is a trivial vector bundle (example 3.4) and whose Lagrangian density ##\mathbf{L}## is spacetime-independent (example 5.14) and let ##\Sigma \times \{\varphi\} \hookrightarrow \mathcal{E}## be a constant section of the shell (57). Furthermore assume that ##\mathbf{L}## is at least quadratic in the vertical coordinates around ##\varphi##.

Then the local BV-complex ##\Omega^{0,0}_\Sigma(E,\varphi)\vert_{\mathcal{E}_{BV}}## of local observables (def. 7.43) is a homological resolution of the algebra of functions on the infinitesimal neighbourhood of ##\varphi## in the shell (example 5.14), hence the canonical comparison morphisms (109) is a quasi-isomorphism precisely if there is no non-trivial (example 10.2) implicit infinitesimal gauge symmetry (def. link ):

$$\left( \Omega^{0,0}_{\Sigma}(E,\varphi)\vert_{\mathcal{E}_{BV}} \overset{\simeq}{\longrightarrow} \Omega^{0,0}_{\Sigma}(E,\varphi)\vert_{\mathcal{E}} \right) \;\Leftrightarrow\; \left( \array{ \text{there are no non-trivial} \\ \text{compactly supported infinitesimal symmetries} } \right) \,.$$

Proof. By example 10.3 the vanishing of compactly supported infinitesimal symmetries is equivalent to the vanishing of the cochain cohomology of the local BV-complex in degree -1 (152).

Therefore the statement to be proven is equivalently that the Koszul complex of the sequence of elements

$$\left( \frac{\delta_{EL} L}{\delta \phi^a} \in \Omega^{0,0}_{\Sigma,\varphi}(E) \right)_{a = 1}^s$$

is a homological resolution of ##\Omega^{0,0}_\Sigma(E,\varphi)\vert_{\mathcal{E}}##, hence has vanishing cohomology in all negative degrees, already if it has vanishing cohomology in degree -1.

By a standard fact about Koszul complexes (this prop.) a sufficient condition for this to be the case is that

1. the ring ##\Omega^{0,0}_{\Sigma}(E,\varphi)## is the tensor product of ##C^\infty(\Sigma)## with a Noetherian ring;
2. the elements ##\frac{\delta_{EL} L }{\delta \phi^a}## are contained in its Jacobson radical.

The first condition is the case since ##\Omega^{0,0}_{\Sigma}(E,\varphi)## is by definition a formal power series ring over a field tensored with ##C^\infty(\Sigma)## (by this example). Since the Jacobson radical of a power series algebra consists of those elements whose constant term vanishes (see this example), the assumption that ##\mathbf{L}## is at least quadratic, hence that ##\delta_{EL}\mathbf{L}## is at least linear in the fields, guarantees that all ##\frac{\delta_{EL}L}{\delta \phi^a}## are contained in the Jacobson radical.

Prop. 10.4
says what gauge fixing has to accomplish: given a local BV-BRST complex we need to find a quasi-isomorphism to another complex which is such that it comes from a graded Lagrangian density whose BV-cohomology vanishes in degree -1 and hence induces a graded covariant phase space, and such that the remaining BRST differential respects the Poisson bracket on this graded covariant phase space.

infinitesimal gauge symmetries

Prop. 10.1 says that the problem is to identify the presence of spacetime-compactly supported infinitesimal symmetries that are on-shell non-trivial. One way they may be identified is if infinitesimal symmetries appear in linearly parameterized collections, where the parameter itself is an arbitrary spacetime-dependent section of some fiber bundle (hence is itself like a field history), because then choosing the parameter to have compact support yields an infinitesimal symmetry of the Lagrangian with compact spacetime support (remark 10.6 below).

In this case we speak of a gauge parameter (def. 10.5 below). It turns out that in most examples of Lagrangian field theories of interest, the compactly supported infinitesimal symmetries all come from gauge parameters this way. Therefore we now consider this case in detail.

Definition 10.5. (infinitesimal gauge symmetries)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1).

Then a collection of infinitesimal gauge symmetries of ##(E,\mathbf{L})## is

1. a vector bundle ##\mathcal{G} \overset{gb}{\longrightarrow} \Sigma## over spacetime ##\Sigma## of positive rank, to be called a gauge parameter bundle ;
2. a bundle morphism (def. 1.6) ##R## from the jet bundle of the fiber product ##\mathcal{G} \times_\Sigma E## with the field bundle (def. 4.1) to the vertical tangent bundle of ##E## (def. 1.13):$$\array{ J^\infty_\Sigma( \mathcal{G} \times_\Sigma E ) && \overset{R}{\longrightarrow} && T_\Sigma E & \overset{i}{\hookrightarrow} & T_\Sigma (\mathcal{G} \times_\Sigma E) \\ & \searrow && \swarrow \\ && E }$$

such that

1. ##R## is linear in the first argument (in the gauge parameter);
2. ##i \circ R## is an evolutionary vector field on ##\mathcal{G} \times_\Sigma E## (def. 6.2);
3. ##R## is an infinitesimal symmetry of the Lagrangian (def. 6.6) in the second argument.

We may express this equivalently in components in the case that the field bundle ##E## is a trivial vector bundle with field fiber coordinates ##(\phi^a)## (example 3.4) and also ##\mathcal{G}## happens to be a trivial vector bundle

$$\mathcal{G} = \Sigma \times \mathfrak{g}$$

where ##\mathfrak{g}## is a vector space with coordinate functions ##\{c^\alpha\}##.

Then ##R## may be expanded in the form

 $$\label{CoordinateExpressionForGaugeParameterized} R \;=\; \left( c^\alpha R^a_\alpha + c^\alpha_{,\mu} R^{a \mu}_\alpha + c^\alpha_{,\mu_1 \mu_2} R^{a \mu_1 \mu_2}_\alpha + \cdots \right) \partial_{\phi^a} \,,$$ (153)

where the components

$$R^{a \mu_1 \cdots \mu_k}_\alpha = R^{a \mu_1 \cdots \mu_k}_\alpha\left( (\phi^b), (\phi^b_{,\mu}), \cdots \right) \;\in\; \Omega^{0,0}_\Sigma(E) = C^\infty(J^\infty_\Sigma(E))$$

are smooth functions on the jet bundle of ##E##, locally of finite order (prop. 4.6), and such that the Lie derivative of the Lagrangian density along ##R(e)## is a total spacetime derivative, which by Noether’s theorem I (prop. 6.7) mean in components that

$$\left( c^\alpha R^a_\alpha + c^\alpha_{,\mu} R^{a \mu}_\alpha + c^\alpha_{,\mu_1 \mu_2} R^{a \mu_1 \mu_2}_\alpha + \cdots \right) \frac{\delta_{EL} \mathbf{L}}{\delta \phi^a} \;=\; \frac{d}{d x^\mu} J^\mu_{R} \,.$$

(e.g. Henneaux 90 (3))

The point is that infinitesimal gauge symmetries in particular yield spacetime-compactly supported infinitesimal gauge symmetries as in prop. 10.1:

Remark 10.6. (infinitesimal gauge symmetries yield spacetime-compactly supported infinitesimal symmetries of the Lagrangian)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) and ##\mathcal{G} \overset{gb}{\to} \Sigma## a bundle of gauge parameters for it (def. 10.5) with gauge parametrization

$$J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E \,.$$

Then for every smooth section ##\epsilon \in \Gamma_\Sigma(\mathcal{G})## of the gauge parameter bundle (def. 1.7) there is an induced infinitesimal symmetry of the Lagrangian (def. 6.6) given by the composition of ##R## with the jet prolongation of ##\epsilon## (def. 4.2)

$$R(\epsilon) \;\colon\; J^\infty_\Sigma(E) = \Sigma \times_\Sigma J^\infty_\Sigma(E) \overset{(j^\infty_\Sigma(\epsilon),id)}{\longrightarrow} J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E \,.$$

In terms of the components (153) this means that

$$R(\epsilon) \;=\; \left( \epsilon^\alpha R^a_\alpha + \frac{\partial^2 \epsilon^\alpha}{\partial x^\mu} R^{a \mu}_\alpha + \frac{\partial \epsilon^\alpha}{\partial x^\mu \partial x^\nu} R^{a \mu_1 \mu_2}_\alpha + \cdots \right) \,,$$

where now

$$\frac{\partial^k \epsilon^\alpha}{\partial x^{\mu_1} \cdots \partial x^{\mu_k}} \;=\; \frac{\partial^k \epsilon^\alpha}{\partial x^{\mu_1} \cdots \partial x^{\mu_k}}((x^\mu))$$

are the actual spacetime partial derivatives of the gauge parameter section (which are functions of spacetime).

In particular, since ##\mathcal{G} \overset{gb}{\to} \Sigma## is assumed to be a vector bundle, there always exists gauge parameter sections ##\epsilon## that have compact support (bump functions). For such compactly supported ##\epsilon## the infinitesimal symmetry ##R(\epsilon)## is spacetime-compactly supported as in prop. 10.1.

The following remark 10.7 and def. 10.8 introduce some useful terminology:

Remark 10.7. (generating set of gauge transformations)

Given a Lagrangian field theory, then a choice of gauge parameter bundle ##\mathcal{G} \overset{gb}{\to} \Sigma## with gauge parameterized infinitesimal gauge symmetries ##J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E## (def. 10.5) is indeed a choice and not uniquely fixed.

For example given any such bundle one may form the direct sum of vector bundles ##\mathcal{G} \oplus_\Sigma \mathcal{G}’## with any other smooth vector bundle ##\mathcal{G}’## over ##\Sigma##, extend ##R## by zero to ##\mathbb{G}’##, and thereby obtain another gauge parameterized of infinitesimal gauge symmetries

$$J^\infty_\Sigma((\mathcal{G}’ \oplus_\Sigma \mathcal{G}) \times_\Sigma E) \overset{(0,R)}{\longrightarrow} T_\Sigma E \,.$$

Conversely, given any subbundle ##\mathcal{G}’ \hookrightarrow \mathcal{G}##, then the restriction of ##R## to ##\mathcal{G}’## is still a gauge parameterized collection of infinitesimal gauge symmetries.

We will see that for the purpose of removing the obstruction to the existence of the covariant phase space, the gauge parameters have to capture all Noether identities (prop. 10.9). In this case one says that the gauge parameter bundle ##\mathcal{G} \overset{gb}{\to} \Sigma## is a generating set.

Definition 10.8. (rigid infinitesimal symmetries of the Lagrangian)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) and let ##J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E## be infinitesimal gauge symmetries (def. 10.5) whose gauge parameters form a generating set (remark 10.7).

Then the vector space of rigid infinitesimal symmetries of the Lagrangian is the quotient space of the infinitesimal symmetries of the Lagrangian by the image of the infinitesimal gauge symmetries:

$$\left\{ \text{rigid infinitesimal symmetries} \right\} \;=\; \left\{ \text{infinitesimal symmetries} \right\} \,/\, \left\{ \text{infinitesimal gauge symmetries} \right\} \,.$$

The following is a way to identify infinitesimal gauge symmetries:

Proposition 10.9. (Noether’s theorem II — Noether identities)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) and let ##\mathcal{G} \overset{gb}{\to} \Sigma## be a vector bundle.

Then a bundle morphism of the form

$$J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E$$

is a collection of infinitesimal gauge symmetries (def. 10.5) with local components (153)

$$R \;=\; \left( c^\alpha R^a_\alpha + c^\alpha_{,\mu} R^{a \mu}_\alpha + c^\alpha_{,\mu_1 \mu_2} R^{a \mu_1 \mu_2}_\alpha + \cdots \right) \partial_{\phi^a}$$

precisely if the Euler-Lagrange form ##\delta_{EL}\mathbf{L}## (prop. 5.12) satisfies the following conditions:

$$\left( R^{a}_\alpha \frac{\delta_{EL}\mathbf{L}}{\delta \phi^a} \frac{d}{d x^\mu} \left( R^{a \mu}_\alpha \frac{\delta_{EL}\mathbf{L}}{\delta \phi^a} \right) + \frac{d^2}{d x^{\mu_1} d x^{\mu_2}} \left( R^{a \mu_1 \mu_2}_\alpha \frac{\delta_{EL}\mathbf{L}}{\delta \phi^a} \right) \cdots \right) \;=\; 0 \,.$$

These relations are called the Noether identities of the Euler-Lagrange equations of motion (def 5.24).

Proof. By Noether’s theorem I, ##R## is an infinitesimal symmetry of the Lagrangian precisely if the contraction (def. 1.20) of ##R## with the Euler-Lagrange form (prop. 5.12) is horizontally exact:

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

From (153) this means that

 \label{NoetherIExpressionForInfinitesimalGaugeSymmetry} \begin{aligned} d J_{\hat R} & = \iota_{R} \delta_{EL} \mathbf{L} \\ & = \underset{k \in \mathbb{N}}{\sum} c^\alpha_{,\mu_1 \cdots \mu_k} R^{a \mu_1 \cdots \mu_k}_\alpha \frac{\delta_{EL} \mathbf{L}}{\delta \phi^a} \\ & = \underset{A}{ \underbrace{ c^\alpha \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R^{a \mu_1 \cdots \mu_k}_\alpha \frac{\delta_{EL} \mathbf{L}}{\delta \phi^a} \right) } } + d K \,, \end{aligned} (154)

where in the last step we used jet-level integration by parts to move the total spacetime derivatives off of ##c^\alpha##, thereby picking up some horizontally exact correction term, as shown.

This means that the term ##A## over the brace is horizontally exact:

 $$\label{NoetherIdentityTermIsHorizontallyExact} c^\alpha \underset{k \in \mathbb{N}}{\sum} (-1)^k \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \left( R^{a \mu_1 \cdots \mu_k}_\alpha \frac{\delta_{EL} \mathbf{L}}{\delta \phi^a} \right) \;=\; d(…)$$ (155)

But now the term on the left is independent of the jet coordinates ##\epsilon^\alpha_{,\mu_1 \cdots \mu_k}## of positive order ##k \geq 1##, while the horizontal derivative increases the dependency on the jet order by one. Therefore the term on the left is horizontally exact precisely if it vanishes, which is the case precisely if the coefficients of ##c^\alpha## vanish, which is the statement of the Noether identities.

Alternatively we may reach this conclusion from (155) by applying to both sides of (155) the Euler-Lagrange derivative (48) with respect to ##c^\alpha##. On the left this yields again the coefficients of ##c^\alpha##, while by the argument from example 5.22 it makes the right hand side vanish.

As a corollary we obtain:

Proposition 10.10. (conserved charge of infinitesimal gauge symmetry vanishes)

The conserved current (def. 6.6)

$$J_R \;\in\; \Omega^{p,0}_\Sigma(E \times_\Sigma \mathcal{G})$$

which corresponds to an infinitesimal gauge symmetry ##R## (def. 10.5) by Noether’s first theorem (prop. 6.7), is up to a term which vanishes on-shell (50)

$$K \;\in\; \Omega^p_\Sigma(E \times_\Sigma \mathcal{G}) \phantom{AA}\,, \phantom{AA} K\vert_{\mathcal{E}^\infty} = 0 \,,$$

not just on-shell-conserved, but off-shell-conserved, in that its total spacetime derivative vanishes identically:

$$d( J_R – K ) \;=\; 0 \,.$$

Moreover, if the field bundle as well as the gauge parameter-bundles are trivial vector bundles over Minkowski spacetime (example 3.4) then ##J_R## is horizontally exact on-shell (50)

$$J_R \vert_{\mathcal{E}^\infty} = d(…) \,.$$

In particular the conserved charge (prop. 8.13)

$$Q_R \;:=\; \tau_{\Sigma_p}(J_R) \;\in\; C^\infty\left( \Gamma_{\Sigma_p}(E)_{\delta_{EL}\mathbf{L} = 0} \right)$$

corresponding to an infinitesimal gauge symmetry vanishes on every codimension one submanifold ##\Sigma_p \hookrightarrow \Sigma## of spacetime (without boundary, ##\partial \Sigma_p = \emptyset##):

$$Q_R = 0 \,.$$

Proof. Take ##K## to be as in equation (154):

$$d J_R = A + d K \,.$$

By the construction there, ##K## manifestly vanishes on the prolonged shell ##\mathcal{E}^\infty## (50), being a sum of total spacetime derivatives of terms proportional to the components of the Euler-Lagrange form.

By Noether’s second theorem (prop. 10.9) we have ##A = 0## and hence

$$d(J_R – K) = 0 \,.$$

Now if the field bundle and gauge parameter bundle are trivial, then prop. 4.14 implies that

 $$\label{DecompositionOfGaugSymmetryConservedCurrent} J_R – K = d(…) \,.$$ (156)

By restricting this equation to the prolonged shell and using that ##K\vert_{\mathcal{E}^\infty} = 0##, it follows that ## J_R \vert_{\mathcal{E}^\infty} = d(…)##.

This implies ##Q_R = 0## by prop. 4.13 and Stokes’ theorem (prop. 1.25).

This situation has a concise cohomological incarnation:

Example 10.11. (Noether’s theorems I and II in terms of local BV-cohomology)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) over Minkowski spacetime ##\Sigma## of dimension ## p + 1##, and let ##\mathcal{G} \overset{gb}{\to} \Sigma## be a gauge parameter bundle (def. 10.6) which is closed (def. 10.26). Assume that both are trivial vector bundles (example 3.4) with field coordinates as in prop. 11.19.

Then in the local BV-complex (def. 7.43) we have:

The ##(s_{BV} + d)##-closure of an element in total degree ##p## is characterizes as the direct sum of an evolutionary vector field which is an infinitesimal symmetry of the Lagrangian and theconserved current that corresponds to it under Noether’s first theorem (prop. 6.7).

Moreover, such a pair is ##(s_{BV} + d)##-exact precisely if the infinitesimal symmetry of the Lagrangian is in fact an infinitesimal gauge symmetry as witnessed by Noether’s second theorem (prop. 10.9).

Proof. An element of the local BV-complex in degee ##p## is the direct sum of a horizontal differential form of degree ##p## with the product of a horizontal form of degree ##(p+1)## times a function proportional to the antifields:

$$\array{ \{J_v\} && \\ && \\ && \{ v^a \phi^\ddagger_a dvol_\Sigma\} }$$

Its closure means that

$$\array{ \{J_v\} &\overset{d}{\longrightarrow}& \{ \overset{= 0}{\overbrace{ d J_v – \iota_v \delta_{EL}\mathbf{L} }} \} \\ && \uparrow\rlap{s_{BV}} \\ && \{ v^a \phi^\ddagger_a dvol_\Sigma\} }$$

where the equality in the top right corner is euqation

It being exact means that

$$\array{ \left\{ … \right\} & \overset{d}{\longrightarrow} & \left\{ J_R = K + d(…) \right\} &\overset{d}{\longrightarrow}& \left\{ d J_R \right\} \\ && \uparrow \\ && \left\{ K^{a \mu} \phi^\ddagger_a \iota_{\partial_\mu} dvol_\Sigma \right\} }$$

where now the equality in the second term from the left is equation (156) for conserved currents corresponding to infinitesimal gauge symmetries (prop. 10.10).

We will need some further technical results on Noether identities:

Definition 10.12. (Noether operator)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1) over Minkowski spacetime ##\Sigma## of dimension ## p + 1##, and let ##\mathcal{G} \overset{gb}{\to} \Sigma## be a gauge parameter bundle (def. 10.6) which is closed (def. 10.26). Assume that both are trivial vector bundles (example 3.4) with field coordinates as in prop. 11.19.

A Noether operator ##N## is a differential operator (def. 4.7) from the vertical cotangent bundle of ##E## (example 1.13) to the trivial real line bundle

$$N(\omega) \;=\; \underset{k \in \mathbb{N}}{\sum} N^{a \mu_1 \cdots \mu_k} \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \omega_a$$

such that it annihilates the Euler-Lagrange form (prop. 5.12):

$$\underset{k \in \mathbb{N}}{\sum} N^{a \mu_1 \cdots \mu_k} \frac{d^k}{d x^{\mu_1} \cdots d x^{\mu_k}} \frac{\delta_{EL} L}{\delta \phi^a} \;=\; 0 \,.$$

Given For ##v## an evolutionary vector field which is an infinitesimal symmetry of the Lagrangian (def. 6.6), we define a new differentia opeator ##v \cdot N## by

 $$\label{MultiplyingANoetherOperatorWithAnInfinitesimalSymmetry} (v \cdot N)^{a \mu_1 \cdots \mu_k} \;:=\; \widehat{v}\left( N^{a \mu_1 \cdots \mu_k} \right) \;-\; N^a \circ (\mathrm{D}_v)^\ast_a \,,$$ (157)

where ##\widehat{v}## denotes the prolongation of the evolutionary vector field ##v## (prop. 6.3) and where ##(\mathrm{D}_v)^\ast## denotes the formally adjoint differential operator (def. 4.9) of the evolutionary derivative of ##v## (def. 6.12).

Proposition 10.13. (Lie algebra action of infinitesimal symmetries of the Lagrangian on Noether operators)

The operation (157) exhibits a Lie algebra action of the Lie algebra of infinitesimal symmetries of the Lagrangian (prop. 6.4) on Noether operators (def. 10.12), in that

1. ##v \cdot N## is again a Noether operator;
2. ##v_1 \cdot (v_2 \cdot N) – v_2 \cdot (v_1 \cdot N) = [v_1, v_2] \cdot N##.

Moreover, if ##\rho## denotes the map which identifies a Noether identity with an infinitesimal gauge symmetry by Noether’s second theorem (def. 10.9) then

 $$\label{LieActionOnNoetherOperatorGivesLieBracketUnderNoetherTheorem} \rho \left( v \cdot N \right) \;=\; \left[ v, \rho(N)\right] \,,$$ (158)

where on the right we have again the Lie bracket of evolutionary vector fields from (prop. 6.4).

Proof. For the first statement observe that by the product law for differentiation> we have

\begin{aligned} 0 & = \widehat{v}\left( N(\delta_{EL} L) \right) \\ & = \widehat{v} \left( \underset{k \in \mathbb{N}}{\sum} N^{a \mu_1 \cdots \mu_k} \right) \left( N^a \circ (\mathrm{D}_v)_a^b\left( \frac{\delta_{EL} L}{\delta \phi^a} \right) \right) \,, \end{aligned}

where on the right we used (80).

Here are examples of infinitesimal gauge symmetries in Lagrangian field theory:

Example 10.14. (infinitesimal gauge symmetry of electromagnetic field)

Consider the Lagrangian field theory ##(E,\mathbf{L})## of free electromagnetism on Minkowski spacetime ##\Sigma## from example 5.6. With field coordinates denoted ##((x^\mu), (a_\mu))## the Lagrangian density is

$$\mathbf{L} \;=\; \tfrac{1}{2} f_{\mu \nu} f^{\mu \nu} \, dvol_\Sigma \,,$$

where ##f_{\mu \nu} := a_{\nu,\mu}## is the universal Faraday tensor from example 4.4.

Let ##\mathcal{G} := \Sigma \times \mathbb{R}## be the trivial line bundle, regarded as a gauge parameter bundle (def. 10.6) with coordinate functions ##((x^\mu), c)##.

Then a gauge parametrized evolutionary vector field (153) is given by

$$R \;=\; c_{,\mu} \partial_{a_\mu}$$

with prolongation (prop. 6.3)

 $$\label{EMProlongedSymmetryVectorField} \widehat R \;=\; c_{,\mu} \partial_{a_\mu} + c_{,\mu \nu} \partial_{a_{\mu,\nu}} + \cdots \,.$$ (159)

This is because already the universal Faraday tensor is invariant under this flow:

\begin{aligned} \widehat {R} f_{\mu \nu} &= \tfrac{1}{2} c_{,\mu’ \nu’} \partial_{a_{\mu’,\nu’}} \left( a_{\nu, \mu} – a_{\mu,\nu} \right) \\ & = \tfrac{1}{2} \left( c_{,\nu\mu} – c_{,\mu \nu} \right) \\ & = 0 \,, \end{aligned}

because partial derivatives commute with each other: ##c_{,\mu \nu} = c_{,\nu \mu}## (29).

Equivalently, the Euler-Lagrange form

$$\delta_{EL}\mathbf{L} \;=\; \frac{d}{d x^\mu }f^{\mu \nu} \delta a_\nu \, dvol_\Sigma$$

of the theory (example 5.18), corresponding to the vacuum Maxwell equations (example 5.29), satisfies the following Noether identity (prop. 10.9):

$$\frac{d}{d x^\mu} \frac{d}{d x^\nu} f^{\mu \nu} = 0 \,,$$

again due to the fact that partial derivatives commute with each other.

This is the archetypical infinitesimal gauge symmetry that gives gauge theory its name.

More generally:

Example 10.15. (infinitesimal gauge symmetry of Yang-Mills theory)

For ##\mathfrak{g}## a semisimple Lie algebra, consider the Lagrangian field theory of Yang-Mills theory on Minkowski spacetime from example 5.7, with Lagrangian density

$$\mathbf{L} \;=\; \tfrac{1}{2} f^\alpha_{\mu \nu} f_\alpha^{\mu \nu}$$

given by the universal field strength (31)

$$f^\alpha_{\mu \nu} \;:=\; \tfrac{1}{2} \left( a^\alpha_{[\nu,\mu]} + \tfrac{1}{2} \gamma^\alpha_{\beta \gamma} a^\beta_{[\mu} a^\gamma_{\nu]} \right) \,.$$

Let ##\mathcal{G} := \Sigma \times \mathfrak{g}## be the trivial vector bundle with fiber ##\mathfrak{g}##, regarded as a gauge parameter bundle (def. 10.6) with coordinate functions ##((x^\mu), c^\alpha)##.

Then a gauge parametrized evolutionary vector field (153) is given by

$$R \;=\; \left( c^\alpha_{,\mu} \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu}$$

with prolongation (prop. 6.3)

 $$\label{OnMinkowskiInfinitesimalGaugeSymmetryForYangMills} \widehat{R} \;=\; \left( c^\alpha_{,\mu} – \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu} \;+\; \left( c^\alpha_{,\mu \nu} – \gamma^\alpha_{\beta \gamma} \left( c^\beta_{,\nu} a^\gamma_\mu + c^\beta a^\gamma_{\mu,\nu} \right) \right) \partial_{a^\alpha_{\mu,\nu}} \;+\; \cdots \,.$$ (160)

We compute the derivative of the Lagrangian function along this vector field:

\begin{aligned} \widehat{R} \left( \tfrac{1}{2} f^\alpha_{\mu \nu} f_\alpha^{\mu \nu} \right) & = \left( R f^\alpha_{\mu \nu} \right) f_\alpha^{\mu \nu} \\ & = \left( R \left( a_{\nu,\mu} + \tfrac{1}{2}\gamma^\alpha_{\beta \gamma} a^\beta_{\mu} a^\gamma_{\nu} \right) \right) f_\alpha^{\mu \nu} \\ & = \left( c^\alpha_{,\nu \mu} \gamma^\alpha_{\beta \gamma} \left( c^\beta_{,\mu} a^\gamma_\nu + c^\beta a^\gamma_{\nu,\mu} \right) + \gamma^\alpha_{\beta \gamma} \left( c^\beta_{,\mu} \gamma^\beta_{\beta’ \gamma’} c^{\beta’} a^{\gamma’}_\mu \right) a^\gamma_{\nu} \right) f_\alpha^{\mu \nu} \\ & = – \gamma^{\alpha}_{\beta \gamma} c^\beta \underset{ = 2 f^\gamma_{\mu \nu} }{ \underbrace{ \left( a^\gamma_{\nu,\mu} + \gamma^\gamma_{\beta’ \gamma’} a^{\beta’}_\mu a^{\gamma’}_\nu \right) } } f_\alpha{}^{\mu \nu} \\ &= 2 \gamma_{\alpha \beta \gamma} c^\alpha f^\beta_{\mu \nu} f^{\gamma \mu \nu} \\ & = 0 \,. \end{aligned}

Here in the third step we used that ##c^\alpha_{,\nu \mu} = + c^\alpha_{,\mu \nu}## (29), so that its contraction with the skew-symmetric ##f_\alpha^{\mu\nu}## vanishes, and in the last step we used that for a semisimple Lie algebra ##\gamma_{\alpha \beta \gamma} := k_{\alpha \alpha’} \gamma^{\alpha’}{}_{\beta \gamma}## is totally skew symmetric.

So the Lagrangian density of Yang-Mills theory is strictly invariant under these infinitesimal gauge symmetries.

Example 10.16. (infinitesimal gauge symmetry of the B-field)

Consider the Lagrangian field theory of the B-field on Minkowski spacetime from example 5.8, with field bundle the differential 2-form-bundle ##E = \wedge^2_\Sigma T^\ast \Sigma## with coordinates ##((x^\mu), (b_{\mu \nu}))## subject to ##b_{\mu \nu} = – b_{\nu \mu}##; and with Lagrangian density

$$\mathbf{L} \;=\; \tfrac{1}{2} h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} \, dvol_\Sigma$$

for

$$h_{\mu_1 \mu_2 \mu_3} = b_{[\mu_1 \mu_2, \mu_3]}$$

the universal B-field strength (example 4.5).

Let ##\mathcal{G} := T^\ast \Sigma## be the cotangent bundle (def. 1.16), regarded as a gauge parameter bundle (def. 10.6) with coordinate functions ##((x^\mu), (c_\mu))## as in example 3.6.

Then a gauge parametrized evolutionary vector field (153) is given by

$$R \;=\; c_{\mu,\nu} \partial_{b_{\mu \nu}}$$

with prolongation (prop. 6.3)

 $$\label{InfinitesimalGaugeSymmetryForBFieldOnMinkowskiSpacetime} \widehat R \;=\; c_{\mu,\nu} \partial_{b_{\mu \nu}} + c_{\mu,\nu \rho} \partial_{b_{\mu \nu, \rho}} + \cdots$$ (161)

In fact this leaves the Lagrangian function invariant, in direct higher analogy to example 10.14:

\begin{aligned} \widehat{R} \tfrac{1}{2} h_{\mu_1 \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} & = \left( \widehat{R} b_{\mu_1 \mu_2, \mu_3} \right) h^{\mu_1 \mu_2 \mu_3} \\ & = c_{\mu_1, \mu_2 \mu_3} h^{\mu_1 \mu_2 \mu_3} \\ & = 0 \end{aligned}

due to the symmetry of partial derivatives (29).

$$h_{,\mu}\partial_{c_{\mu}} + h_{,\mu \nu}\partial_{c_{\mu,\nu}}$$

$$R_\alpha^{a, \mu} = c_{\mu,\nu} R^{\mu, \nu}_{\mu’ \nu’} \partial_{b_{\mu’ \nu’}} \,.$$

While so far all this is in direct analogy to the case of the electromagnetic field (example 10.14), just with field histories being differential 1-forms now replaced by differential 2-forms, a key difference is that now the gauge parameterization ##R## itself has infinitesimal gauge symmetries:

Let

 $$\label{SecondOrderGaugeParameterBundleForBFieldOnMinkowskiSpacetime} \array{ \overset{(2)}{\mathcal{G}} &:=& \Sigma \times \mathbb{R} \\ {}^{\overset{(2)}{gb}}\downarrow && \downarrow^{\rlap{pr_1}} \\ \Sigma &=& \Sigma }$$ (162)

be the trivial real line bundle with coordinates ##((x^\mu), \overset{(2)}{c})##, to be regarded as a second order infinitesimal gauge-of-gauge symmetry, then

$$\overset{(2)}{R} \;:=\; \overset{(2)}{c}_{,\mu} \partial_{c_\mu}$$

with prolongation

 $$\label{SecondOderGaugeSymmetryOfBFieldOnMinkowski} \widehat{\overset{(2)}{R}} \;:=\; \overset{(2)}{c}_{,\mu} \partial_{c_\mu} + \overset{(2)}{c}_{,\mu \nu} \partial_{c_{\mu,\nu}} + \cdots$$ (163)

has the property that

 \label{NoetherIdentitySecondOrderForBFieldOnMinkowskiSpacetime} \begin{aligned} \widehat{\overset{(2)}{R}} (R) &= \overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial c_{\mu,\nu}} \left( c_{\mu’,\nu’} \partial_{b_{\mu’ \nu’}} \right) \\ & = \overset{(2)}{c}_{,\mu \nu} \partial_{b_{\mu \nu}} \\ & 0 \,. \end{aligned} (164)

We further discuss these higher gauge transformations below.

Lie algebra actions and Lie algebroids

We have seen above infinitesimal gauge symmetries implied by a Lagrangian field theory, exhibited by infinitesimal symmetries of the Lagrangian. In order to remove the obstructions that these infinitesimal gauge symmetries cause for the existence of the covariant phase space (via prop. 10.1 and remark 10.6) we will need (discussed below in Gauge fixing) to make these symmetries manifest by hard-wiring them into the geometry of the type of fields. Mathematically this means that we need to take the homotopy quotient of the jet bundle of the field bundle by the action of the infinitesimal gauge symmetries, which is modeled by their action Lie algebroid.

Here we introduce the required higher Lie theory of Lie ∞-algebroids (def. 10.22 below). Further below we specify this to actions by infinitesimal gauge symmetries to obtain the local BRST complex of a Lagrangian field theory (def. 10.28) below.

The following discussion introduces and uses the tremendously useful fact that (higher) Lie theory may usefully be dually expressed in terms of differential graded-commutative algebra (def. 10.17 below), namely in terms of “Chevalley-Eilenberg algebras“. In the physics literature, besides the BRST-BV formalism, this fact underlies the D’Auria-Fré formulation of supergravity (“FDAs“, see the convoluted history of the concept). Mathematically the deep underlying phenomenon is called the “Koszul duality between the Lie operad and the commutative algebra operad“, but this need not concern us here. The phenomenon is readily seen in direct application:

Before we proceed, we make explicit a structure wich we already encountered in example 3.39.

1. a cochain complex ##A_\bullet## of super vector spaces, hence for each ##n \in \mathbb{Z}##1 a super vector space ##A_n = (A_n)_{even} \oplus (A_n)_{odd}##;
1. a super-degree preserving linear map$$d \;\colon\; A_{n} \longrightarrow A_{n+1}$$

such that

$$d \circ d = 0$$

1, an associative algebra-structure on ##A := \underset{n \in \mathbb{Z}}{\oplus} A_n##

such that for all ##a_1, a_2 \in A## with homogenous bidegree ##a_i \in (A_{n_a})_{\sigma_a}## we have the super sign rule

1. ##a b = (-1)^{n_a n_b} (-1)^{\sigma_a \sigma_b} \, b a##
2. ##d(a b) = (d a) b + (-1)^{n_1} a (d b)##.

A homomorphism between two differential graded-commutative superalgebras is a linear map between the underlying super vector spaces which preserves both degrees, and respects the product as well as the differential ##d##.

We write ##dgcSAlg## for the category of differential graded-commutative superalgebra.

For the super sigsn rule appearing here see also e.g. Castellani-D’Auria-Fré 91 (II.2.106) and (II.2.109), Deligne-Freed 99, section 6.

Example 10.18. (de Rham algebra of super differential forms is differential graded-commutative superalgebra)

For ##X## a super Cartesian space, def. 3.37 (or more generally a supermanifold, def. 3.43) the de Rham algebra of super differential forms from def. 3.39

$$(\Omega^\bullet(X), d)$$

is a differential graded-commutative superalgebra (def. 10.17) with product the wedge product of differential forms and differential the de Rham differential.

We will recognize the dual incarnation of this in higher Lie theory below in example 10.25.

Proposition 10.19. (Lie algebra in terms of Chevalley-Eilenberg algebra)

Let ##\mathfrak{g}## be a finite dimensional super vector space equipped with a super Lie bracket ##[-,-] \colon \mathfrak{g} \otimes \mathfrak{g} \to \mathfrak{g}##. Write ##\mathfrak{g}^\ast## for the linear dual map of the Lie bracket. Then on the Grassmann algebra ##\wedge^\bullet \mathfrak{g}^\ast## (which is ##\mathbb{Z} \times \mathbb{Z}/2##-bigraded as in def. 3.39) the graded derivation ##d_{CE}## of degree ##(1,even)##, which on ##\mathfrak{g}^\ast## is given by ##[-,-]^\ast## constitutes a differential in that ##(d_{CE})^2 = 0##. The resulting differential graded-commutative algebra is called the Chevalley-Eilenberg algebra

$$CE(\mathfrak{g}) \;:=\; \left( \wedge^\bullet \mathfrak{g}^\ast \,, d_{CE} = [-,-]^\ast \right) \,.$$

In components:

If ##\{c_\alpha\}## is a linear basis of ##\mathfrak{g}##, so that the Lie bracket is given by the structure constants ##(\gamma^\alpha{}_{\beta \gamma})## as

$$[c_\beta, c_\gamma] = \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} c_\gamma$$

and if ##\{c^\alpha\}## denotes the corresponding dual basis, then ##\wedge^\bullet \mathfrak{g}^\ast## is equivalently the differential graded-commutative superalgebra (def. 10.17) generated from the ##c^\alpha## in bi-degree ##(1,\sigma)##, where ##\sigma \in \mathbb{Z}/2## is the super-degree of ##c_\alpha## as in def. 3.39 subject to the relation

$$c^\alpha \wedge c^\beta = (-1) (-1)^{\sigma_\alpha \sigma_\beta} c^\beta \wedge c^\alpha$$

and the differential is given by

$$d_{CE} c^\alpha = \gamma^\alpha{}_{\beta \gamma} c^\beta \wedge c^\gamma \,.$$

Notice that by degree-reasons every degree +1 derivation on ##\wedge^\bullet \mathfrak{g}^\ast## is of this form,

$$\left\{ \array{ \text{derivations} \\ \text{of degree}\, (1,even) \\ \text{on} \, \wedge^\bullet \mathfrak{g}^\ast } \right\} \;\;\simeq\;\, \left\{ \array{ \text{super-skew} \\ \text{bilinear maps} \\ \mathfrak{g} \otimes \mathfrak{g} \overset{[-,-]}{\longrightarrow} \mathfrak{g} } \right\}$$

The condition that ##(d_{CE})^2 = 0## is equivalently the (super-)Jacobi identity on the bracket ##[-,-]##, making it an actual (super-)Lie bracket:

 $$\label{JacobiIdentity} (d_{CE})^2 = 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} \gamma^\alpha{}_{\beta [\gamma} \gamma^{\beta}{}_{\beta’ \gamma’]} = 0$$ (165)

(where the square brackets on the right denote super-skew-symmetrization).

Hence not only is ##CE(\mathfrak{g})## a differential graded-commutative superalgebra (def. 10.17) whenever ##\mathfrak{g}## is a super Lie algebra, but conversely super Lie algebra-structure on a super vector space ##\mathfrak{g}## is the same as a differential of degree ##(1,even)## on the Grassmann algebra ##\wedge^\bullet \mathfrak{g}^\ast##.

We may state this equivalence in a more refined form: A homomorphism ##\phi \;\colon\; \mathfrak{g} \longrightarrow \mathfrak{h}## between super vector space is, by degree-reasons, the same as a graded algebra homomorphism ##\phi^\ast \;\colon\; \wedge^\bullet \mathfrak{h}^\ast \longrightarrow \wedge^\bullet \mathfrak{g}^\ast## and it is immediate to check that ##\phi## is a homomorphism of super Lie algebras precisely if ##\phi^\ast## is a homomorpism of differential algebras:

$$d_{CE(\mathfrak{g})} \circ \phi^\ast = \phi^\ast \circ d_{CE(\mathfrak{h})} \phantom{AAA} \Leftrightarrow \phantom{AAA} \phi^{\alpha_1}{}_{\beta_1} \gamma^{\beta_1}_{\mathfrak{g}}{}_{\beta_2 \beta_3} = \gamma^{\alpha_1}_{\mathfrak{h}}{}_{\alpha_2 \alpha_3} \phi^{\alpha_2}{}_{\beta_2} \phi^{\alpha_3}{}_{\beta_3} \,.$$

This is a natural bijection between homomrophism of super Lie algebras and of differential graded-commutative superalgebras (def. 10.17)

$$Hom_{SuperLieAlg}( \mathfrak{g}, \mathfrak{h} ) \;\simeq\; Hom_{dgcSAlg}\left( CE(\mathfrak{h}), CE(\mathfrak{g}) \right) \,.$$

Stated more abstractly this means that forming Chevalley-Eilenberg algebras is a fully faithful functor

$$CE \;\colon\; SuperLieAlg^{fin} \overset{\phantom{AAA}}{\hookrightarrow} dgcSAlg^{op} \,.$$

Notice that prop. 10.19 establishes a dual algebraic incarnation of (super-)Lie algebras which is of analogous form as the dual algebraic characterization of (super-)Cartesian spaces from prop. 1.15 and def. 3.37. In fact both these concepts unify into the concept of an action Lie algebroid:

Definition 10.20. (action of Lie algebra by infinitesimal diffeomorphism)

Let ##X## be a supermanifold (def. 3.43), for instance a super Cartesian space (def. 3.37), and let ##\mathfrak{g}## be a finite dimensional super Lie algebra as in prop. 10.19.

An action of ##\mathfrak{g}## on ##X## by infinitesimal diffeomorphisms, is a homomorphism of super Lie algebras

$$\rho \;\colon \mathfrak{g} \longrightarrow ( Vect(X), [-,-] )$$

to the tangent vector fields on ##X## (example 1.12)

Equivalently — to bring out the relation to the gauge parameterized infinitesimal gauge transformations in def. 10.6 — this is a ##\mathfrak{g}##-parameterized section

$$\array{ \mathfrak{g} \times X && \overset{R}{\longrightarrow} && T X \\ & {\llap{pr_2}}\searrow && \swarrow_{\rlap{p}} \\ && X }$$

of the tangent bundle, such that for all pairs of points ##e_1, e_2## in ##\mathfrak{g}## we have

$$\left[R(e_1,-), R(e_2,-)\right] = R([e_1,e_2],-)$$

(with the Lie bracket of tangent vector fields on the left).

In components:

If ##\{c^\alpha\}## is a linear basis of ##\mathfrak{g}^\ast## with corresponding structure constants ##(\gamma^\alpha{}_{\beta \gamma})## (as in prop. 10.19) and if ##\{\phi^a\}## is a coordinate chart of ##X##, then ##R## is given by

$$R \;=\; c^\alpha R_\alpha^a \frac{\partial}{\partial \phi^a} \,.$$

Now the construction of the Chevalley-Eilenberg algebra of a super Lie algebra (prop. 10.19) extends to the case where this super Lie algebra acts on a supermanifold (def. 10.20):

Definition 10.21. (action Lie algebroid)

Given a Lie algebra action

$$\mathfrak{g} \times X \overset{R}{\longrightarrow} T X$$

of a finite-dimensional super Lie algebra ##\mathfrak{g}## on a supermanifold ##X## (def. 10.20) we obtain a differential graded-commutative superalgebra to be denoted ##CE(X/\mathfrak{g})##

1. whose underlying graded-commutative superalgebra is the Grassmann algebra of the ##C^\infty(X)##-free module on ##\mathfrak{g}^\ast## over ##C^\infty(X)##$$\wedge^\bullet_{C^\infty(X)} (\mathfrak{g}^\ast \otimes C^\infty(X)) \;=\; \underset{ deg = 0 }{ \underbrace{ C^\infty(X) }} \oplus \underset{ deg = 1 }{ \underbrace{ C^\infty(X) \otimes \mathfrak{g}^\ast }} \oplus \underset{ def = 2 }{ \underbrace{ C^\infty(X) \otimes \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast }} \oplus \cdots$$which means that the graded manifold underlying the action Lie algebroid according to remark 10.23 is

 $$\label{ActionLieAlgebroidGradedManifold} X/\mathfrak{g} \;=_{grmfd}\; \mathfrak{g}[1] \times X \,,$$ (166)
2. whose differential ##d_{CE}## is given
1. on functions ##f \in C^\infty(X)## by the linear dual of the Lie algebra action$$d_{CE} f := \rho(-)(f) \in C^\infty(X) \otimes \mathfrak{g}^\ast$$
2. on dual Lie algebra elements ##\omega \in \mathfrak{g}^\ast## by the linear dual of the Lie bracket$$d_{CE} \omega := \omega([-,-]) \;\in \; \mathfrak{g}^\ast \wedge \mathfrak{g}^\ast \,.$$

In components:

Assume that ##X = \mathbb{R}^n## is a super Cartesian space with coordinate functions ##(\phi^a)## and let ##\{c_\alpha\}## be a linear basis for ##\mathfrak{g}## with dual basis ##(c^\alpha)## for ##\mathfrak{g}^\ast## and structure constants ##(\gamma^\alpha){}_{\beta \gamma}## as in prop. 10.19 and with the Lie action given in components ##(R_\alpha^a)## as in def. 10.20. Then the differential is given by

\begin{aligned} d_{CE(X/\mathfrak{g})} c^\alpha & = \tfrac{1}{2} \gamma^\alpha{}_{\beta \gamma} \, c^\beta \wedge c^\gamma \\ d_{CE(X/\mathfrak{g})} \phi^a & = R^a_\alpha c^\alpha \end{aligned}

We may summarize this by writing the derivation ##d_{CE(X/\mathfrak{g})}## as follows:

 $$\label{DifferentialOnActionLieAlgebroid} d_{CE(X/\mathfrak{g})} \;=\; c^\alpha R_\alpha^a \frac{\partial}{\partial \phi^a} \;+\; \tfrac{1}{2} \gamma^{\alpha}{}_{\beta \gamma} c^\beta c^\gamma \frac{\partial}{\partial c^\alpha} \,.$$ (167)

That this squares to zero is equivalently

$$(d_{CE(X/\mathfrak{g})})^2 = 0 \phantom{AAA} \Leftrightarrow \phantom{AAA} \array{ \phantom{and} \, \text{Jacobi identity} \\ \text{and} \, \text{action property} }$$

Hence as before in prop. 10.19 the Lie theoretic structure is faithfully captured dually by differential graded-commutative superalgebra.

We call the formal dual of this dgc-superalgebra the action Lie algebroid ##X/\mathfrak{g}## of ##\mathfrak{g}## acting on ##X##.

The concept emerging by this example we may consider generally:

Definition 10.22. (super–Lie ∞-algebroid)

Let ##X## be a supermanifold (def. 3.43) (for instance a super Cartesian space, def. 3.37) and write ##C^\infty(X)## for its algebra of functions. Then a connected super Lie ∞-algebroid ##\mathfrak{a}## over ##X## of finite type is a

1. a sequence ##(\mathfrak{a}_k)_{ k = 1 }^\infty## of free modules of finite rank over ##C^\infty(X)##, hence a graded module ##\mathfrak{a}_\bullet## in degrees ##k \in \mathbb{N}##; ##k \geq 1##
2. a differential ##d_{CE}## that makes the graded-commutative algebra ##Sym_{C^\infty(X)}(\mathfrak{a}^\ast_\bullet)## into a cochain differential graded-commutative algebra (hence with ##d_{CE}## of degree +1) over ##\mathbb{R}## (not necessarily over ##C^\infty(X)##), to be called the Chevalley-Eilenberg algebra of ##\mathfrak{a}##:
 $$\label{CEAlgebra} CE(\mathfrak{a}) \;:=\; \left( Sym_{C^\infty(X)}(\mathfrak{a}^\ast_\bullet) \,,\, d_{CE} \right) \,.$$ (168)

If we allow ##\mathfrak{a}_\bullet## to also have terms in non-positive degree, then we speak of a derived Lie algebroid. If ##\mathfrak{a}_\bullet## is only concentrated in negative degrees, we also speak of a derived manifold.

With ##C^\infty(X)## canonically itself regarded as a differential graded-commutative superalgebra, there is a canonical dg-algebra homomorphism

$$CE(\mathfrak{a}) \longrightarrow C^\infty(X)$$

which is the identity on ##C^\infty(X)## and zero on ##\mathfrak{a}^\ast_{\neq 0}##.

(We discuss homomorphism between Lie ∞-algebroid below in def. 11.1.)

Remark 10.23. (Lie algebroids as differential graded manifolds)

Definition 10.22 of derived Lie algebroids is an encoding in higher algebra (homological algebra, in this case) of a situation that is usefully thought of in terms of higher differential geometry.

To see this, recall the magic algebraic properties of ordinary differential geometry (prop. 1.15)

Together these imply that we may think of the graded algebra underlying a Chevalley-Eilenberg algebra as being the algebra of functions on a graded manifold

$$\cdots \times \mathfrak{a}_2 \times \mathfrak{a}_1 \times X \times \mathfrak{a}_{-1} \times \cdots$$

which is infinitesimal in non-vanishing degree.

The “higher” in higher differential geometry refers to the degrees higher than zero. See at Higher Structures for exposition. Specifically if ##\mathfrak{a}_\bullet## has components in negative degrees, these are also called derived manifolds.

Example 10.24. (basic examples of Lie algebroids)

Two basic examples of Lie algebroids are:

1. For ##X## any supermanifold (def. 3.43), for instance a super Cartesian space (def. 3.37) then setting ##\mathfrak{a}_{\neq 0 } := 0## and ##d_{CE} := 0## makes it a Lie algebroid in the sense of def. 10.22.
2. For ##\mathfrak{g}## a finite-dimensional super Lie algebra, its Chevalley-Eilenberg algebra (prop. 10.19) ##CE(\mathfrak{g})## exhibits ##\mathfrak{g}## as a Lie algebroid in the sense of def. 10.22. We write ##B\mathfrak{g}## or ##\ast/\mathfrak{g}## for ##\mathfrak{g}## regarded as a Lie algebroid this way.
3. For ##X## and ##\mathfrak{g}## as in the previous items, and for ##R \colon \mathfrak{g} \times X \to T X## a Lie algebra action (def. 10.20) of ##\mathfrak{g}## on ##X##, then the dgs-superalegbra ##CE(X/\mathfrak{g})## from def. 10.21 defines a Lie algebroid in the sense of def. 10.22, the action Lie algebroid.In the special case that ##\mathfrak{g} = 0## this reduces to the first example, while for ##X = \ast## this reduces to the second example.

Here is another basic examples of Lie algebroids that will plays a role:

Example 10.25. (horizontal tangent Lie algebroid)

Let ##\Sigma## be a smooth manifold or more generally a supermanifold or more generally a locally pro-manifold (prop. 4.6). Then we write ##\Sigma/T\Sigma## for the Lie algebroid over ##X## and whose Chevalley-Eilenberg algebra is generated over ##C^\infty(X)## in degree 1 from the module

$$\mathfrak{a}_1^\ast := (\Gamma(T \Sigma))^\ast \simeq\Gamma(T^\ast \Sigma) = \Omega^1(\Sigma)$$

of differential 1-forms and whose Chevalley-Eilenberg differential is the de Rham differential, so that the Chevalley-Eilenberg algebra is the de Rham dg-algebra of super differential forms (example 10.18)

$$CE( \Sigma/T\Sigma ) := (\Omega^\bullet(\Sigma), d_{dR}) \,.$$

This is called the tangent Lie algebroid of ##\Sigma##. As a graded manifold (via remark 10.23) this is called the “shifted tangent bundle” ##T[1] \Sigma## of ##X##.

More generally, let ##E \overset{fb}{\to} \Sigma## be a fiber bundle over ##\Sigma##. Then there is a Lie algebroid ##J^\infty_\Sigma(E)/T\Sigma## over the jet bundle of ##E## (def. 4.1) defined by its Chevalley-Eilenberg algebra being the horizontal part of the variational bicomplex (def. 4.11):

$$CE\left( J^\infty_\Sigma(E)/T\Sigma \right) \;:=\; \left(\Omega^{\bullet,0}_\Sigma(E), d\right) \,.$$

The underlying graded manifold of ##J^\infty_\Sigma(E)/T\Sigma## is the fiber product ##J^\infty_\Sigma(E)\times_\Sigma T[1]\Sigma## of the jet bundle of ##E## with the shifted tangent bundle of ##\Sigma##.

There is then a canonical homomorphism of Lie algebroids (def. 11.1)

$$\array{ J^\infty_\Sigma(E)/T\Sigma \\ \downarrow \\ \Sigma/T\Sigma }$$

local off-shell BRST complex

With the general concept of Lie algebra action (def. 10.20) and the corresponding action Lie algebroids (def. 10.21) and more general Lie ∞-algebroids in hand (def. 10.22) we now apply this to the action of infinitesimal gauge symmetries (def. 10.5) on field histories of a Lagrangian field theory, but we consider this locally, namely on the jet bundle. The Chevalley-Eilenberg algebra of the resulting action Lie algebroid (def. 10.21) is known as the local BRST complex, example 10.28 below.

The Lie algebroid-perspective on BV-BRST formalism has been made explicit in (Barnich 10).

Definition 10.26. (closed gauge parameters)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1). Then a gauge parameter bundle ##\mathcal{G} \overset{gb}{\to} \Sigma## parameterizing infinitesimal gauge symmetries (def. 10.5)

$$J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E$$

is called closed if it is closed under the Lie bracket of evolutionary vector fields (prop. 6.4) in that there exists a morphism (not necessarily uniquely)

 $$\label{ClosedGaugeParametersBracket} [-,-]_{\mathcal{G}} \;\colon\; J^\infty_\Sigma( \mathcal{G} \times_\Sigma \mathcal{G} \times_\Sigma E ) \longrightarrow J^\infty_\Sigma(\mathcal{G} \times_\Sigma E)$$ (169)

such that

$$\left[ R(-) , R(-)\right] \;=\; R([-,-]_{\mathcal{G}}) \,,$$

where on the left we have the Lie bracket of evolutionary vector fields from prop. 6.4.

Beware that ##[-,-]_{\mathcal{G}}## may be a function of the fields, namely of the jet bundle of the field bundle ##E##. Hence for closed gauge parameters ##[-,-]_{\mathcal{G}}## in general defines a Lie algebroid-structure (def. 10.22).

Notice that the collection of all infinitesimal symmetries of the Lagrangian by necessity always forms a (very large) Lie algebra. The condition of closed gauge parameters is a condition on the choice of parameterization of the infinitesimal gauge symmetries, see remark 10.7.

Recall the general concept of a Lie algebra action from def. 10.20. The following realizes this for the action of closed infinitesimal gauge symmetries on the jet bundle of a Lagrangian field theory.

Example 10.27. (action of closed infinitesimal gauge symmetries on fields)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1), and let ##\mathcal{G} \overset{gb}{\to} \Sigma## be a bundle of gauge parameters (def. 10.5) paramaterizing infinitesimal gauge symmetries

$$J^\infty_\Sigma(\mathcal{G} \times_\Sigma E) \overset{R}{\longrightarrow} T_\Sigma E$$

which are closed (def. 10.26), via a bracket ##[-,-]_{\mathcal{G}}##.

By passing from these evolutionary vector fields ##R## (def. 6.2) to their prolongations ##\widehat{R}##, being actual vector fields on the jet bundle (prop. 6.3), we obtain a bundle morphism of the form

$$\array{ J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma (E) && \overset{\widehat{R(e)}}{\longrightarrow} && T_\Sigma J^\infty_\Sigma(E) \\ & \searrow && \swarrow \\ && J^\infty_\Sigma(E) }$$

and via the assumed bracket ##[-,-]_{\mathcal{G}}## on gauge parameters this exhibits Lie algebroid structure on ##J^\infty_\Sigma(\mathcal{G}) \times_\Sigma J^\infty_\Sigma(E) \overset{pr_2}{\to} J^\infty_\Sigma(E)##.

In the case that ##\mathcal{G} = \mathfrak{g} \times \Sigma## is a trivial vector bundle, with fiber ##\mathfrak{g}##, then so is its jet bundle

$$J^\infty_\Sigma(\mathfrak{g} \times \Sigma) = \mathfrak{g}^\infty \times \Sigma \,.$$

If moreover the bracket (169) on the infinitesimal gauge symmetries is independent of the fields, then this induces a Lie algebra structure on ##\mathfrak{g}^\infty## and exhibits an Lie algebra action

$$\array{ \mathfrak{g}^\infty \times J^\infty_\Sigma E && \overset{\widehat{R(e)}}{\longrightarrow} && T_\Sigma J^\infty_\Sigma(E) \\ & \searrow && \swarrow \\ && J^\infty_\Sigma(E) } \,.$$

of the gauge parameterized infinitesimal gauge symmetries on the jet bundle of the field bundle by infinitesimal diffeomorphisms.

Example 10.28. (local BRST complex and ghost fields for closed infinitesimal gauge symmetries)

Let ##(E,\mathbf{L})## be a Lagrangian field theory (def. 5.1), and let ##\mathcal{G} \overset{gb}{\longrightarrow} \Sigma## be a bundle of irreducible closed gauge parameters for the theory (def. 10.6) with bundle morphism

$$\array{ J^\infty_\Sigma( \mathcal{G} \times_\Sigma E ) && \overset{R}{\longrightarrow} && T_\Sigma E \\ & \searrow && \swarrow \\ && E } \,.$$

Assuming that the gauge parameter bundle is trivial, ##\mathcal{G} = \mathfrak{g} \times \Sigma##, then by example 10.27 this induces an action ##\hat R## of a Lie algebra ##\mathfrak{g}^\infty## on ##J^\infty_\Sigma E## by infinitesimal diffeomorphisms.

The corresponding action Lie algebroid ##J^\infty_\Sigma(E)/\mathfrak{g}^\infty## (def. 10.21) has as underlying graded manifold (remark 10.23)

$$\mathfrak{g}^\infty[1] \times J^\infty_\Sigma(E) \;\simeq\; J^\infty_\Sigma( \mathcal{G}[1] \times_\Sigma E )$$

the jet bundle of the graded field bundle

$$E_{BRST} \;:=\; E \times_\Sigma \mathcal{G}[1]$$

which regards the gauge parameters as fields in degree 1. As such these are called ghost fields:

$$\left\{ \text{ghost field histories} \right\} \;:=\; \Gamma_\Sigma( \mathcal{G}[1] ) \,.$$

Therefore we write suggestively

$$E/\mathcal{G} \;:=\; J^\infty_\Sigma(E)/\mathfrak{g}^\infty$$

for the action Lie algebroid of the gauge parameterized implicit infinitesimal gauge symmetries on the jet bundle of the field bundle.

The Chevalley-Eilenberg differential of the BRST complex is traditionally denoted

$$s_{BRST} := d_{CE} \,.$$

To express this in coordinates, assume that the field bundle ##E## as well as the gauge parameter bundle are trivial vector bundles (example 3.4) with ##(\phi^a)## the field coordinates on the fiber of ##E## with induced jet coordinates ##((x^\mu), (\phi^a), (\phi^a_{\mu}), \cdots)## and ##(c^\alpha)## are ghost field coordinates on the fiber of ##\mathcal{G}[1]## with induced jet coordinates ##((x^\mu), (c^\alpha), (c^\alpha_\mu), \cdots)##.

Then in terms of the corresponding coordinate expression for the gauge symmetries ##R## (153) the BRST differential is given on the fields by

$$s_{BRST} \, \phi^a \;=\; c^\alpha_{,\mu_1 \cdots \mu_k} \underset{k \in \mathbb{N}}{\sum} R^{a \mu_1 \cdots \mu_k}_{\alpha}$$

and on the ghost fields by

$$s_{BRST} \, c^\alpha = \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} c^\beta c^\gamma \,,$$

and it extends from there, via prop. 6.3, to jets of fields and ghost fields by (anti-)commutativity with the total spacetime derivative.

Moreover, since the action of the infinitesimal gauge symmetries is by definition via prolongations (prop. 6.3) of evolutionary vector fields (def. 6.2) and hence compatible with the total spacetime derivative (73) this construction descends to the horizontal tangent Lie algebroid ##J^\infty_\Sigma(E)/T\Sigma## (example 10.25) to yield

$$E/(\mathcal{G}\times_\Sigma T \Sigma) \;:=\; \left(J^\infty_\Sigma(E)/T\Sigma\right)/\mathfrak{g}^\infty$$

The Chevalley-Eilenberg differential on ##E/(\mathcal{G}\times_\Sigma T \Sigma)## is

$$d – s_{BRST}$$

The Chevalley-Eilenberg algebra of functions on this differential graded manifold (168) is called the off-shell local BRST complex.

Definition 10.29. (global BRST complex)

We may pass from the off-shell local BRST complex (def. 10.28) on the jet bundle to the “global” BRST complex by transgression of variational differential forms (def. 7.32):

Write ##Obs(E \times_\Sigma \mathcal{G}[1])## for the induced graded off-shell algebra of observables (def. 7.38). For ##A \in \Omega^{p+1,\bullet}_\Sigma(E \times_\Sigma \mathcal{G}[1])## with corresponding local observable ##\tau_\Sigma(A) \in LocObs_\Sigma(E \times_\Sigma \mathcal{G}[1])## its BRST differential is defined by

$$s_{BRST} \tau_{\Sigma}(A) \;:=\; \tau_{\Sigma}(s_{BRST} A)$$

and extended from there to ##Obs(E \times_\Sigma \mathcal{G}[1])## as a graded derivation.

Examples of local BRST complexes of Lagrangian gauge theories

Example 10.30. (local BRST complex for free electromagnetic field on Minkowski spacetime)

Consider the Lagrangian field theory of free electromagnetism on Minkowski spacetime (example 5.6) with its gauge parameter bundle as in example 10.14.

By (159) the action of the BRST differential is the derivation

$$s_{BRST} \;=\; c_{,\mu} \frac{\partial}{\partial a_\mu} + c_{, \mu \nu} \frac{\partial}{\partial a_{\mu, \nu}} + \cdots \,.$$

In particular the Lagrangian density is BRST-closed

\begin{aligned} s_{BRST} \mathbf{L} & = s_{BRST} f_{\mu \nu} f^{\mu \nu} dvol_\Sigma \\ & = c_{,\mu \nu} f^{\mu \nu} dvol_\Sigma \\ & = 0 \end{aligned}

as is the Euler-Lagrange form (due to the symmetry ##c_{,\mu \nu} = c_{,\nu \mu}## (29) and in contrast to the skew-symmetry ##f_{\mu \nu} = – f_{\nu \mu}##).

Example 10.31. (local BRST complex for the Yang-Mills field on Minkowski spacetime)

For ##\mathfrak{g}## a semisimple Lie algebra, consider the Lagrangian field theory of Yang-Mills theory on Minkowski spacetime from example 5.7, with Lagrangian density

$$\mathbf{L} \;=\; \tfrac{1}{2} f^\alpha_{\mu \nu} f_\alpha^{\mu \nu}$$

given by the universal field strength (31)

$$f^\alpha_{\mu \nu} \;:=\; \tfrac{1}{2} \left( a^\alpha_{[\nu,\mu]} + \tfrac{1}{2} \gamma^\alpha_{\beta \gamma} a^\beta_{[\mu} a^\gamma_{\nu]} \right) \,.$$

Let ##\mathcal{G} := \Sigma \times \mathfrak{g}## be the trivial vector bundle with fiber ##\mathfrak{g}##, regarded as a gauge parameter bundle (def. 10.6) with coordinate functions ##((x^\mu), c^\alpha)## and consider the gauge parametrized evolutionary vector field (153)

$$R \;=\; \left( c^\alpha_{,\mu} \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu}$$

from example 10.15.

We claim that these are closed gauge parameters in the sense of def. 10.26, hence that the local BRST complex in the form of example 10.28 exists.

To see this, observe that, by def. 10.21 the candidate BRST differential needs to be of the form (160) plus the linear dual of the Lie bracket ##[-,-]_{\mathcal{G}}^\ast##

$$s_{BRST} \;=\; \left( \left( c^\alpha_{,\mu} \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu} \;+\; \text{prolongation} \right) + ([-,-]_{\mathcal{G}})^\ast \,.$$

Moreover, by def. 10.21 we may equivalently make an Ansatz for ##([-,-]_{\mathcal{G}})^\ast## and if the resulting differential ##s_{BRST}## squares to zero, as this dually defines the required closure bracket ##[-,-]_\mathcal{G}##.

We claim that

 $$\label{OffShellYangMillsOnMinkowskiBRSTOperator} s_{BRST} \;:=\; \widehat{ \left( c^\alpha_{,\mu} – \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \frac{\partial}{\partial a^\alpha_\mu} } + \widehat{ \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} \, c^\beta c^\gamma \frac{\partial}{\partial c^\alpha} } \,,$$ (170)

where the hat denotes prolongation (prop. 6.3). This is the local (jet bundle) BRST differential for Yang-Mills theory on Minkowski spacetime.

Proof. We need to show that (170) squares to zero. Consider the two terms that appear:

$$(s_{BRST})^2 = \left[ \widehat{ \left( c^\alpha_{,\mu} \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu} } \;,\; \widehat{ \left( c^{\alpha’}_{,\mu} \gamma^{\alpha’}_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^{\alpha’}_\mu} } \right] \;+\; 2 \left[ \widehat{ \left( c^\alpha_{,\mu} \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu} } \;,\; \widehat{ \tfrac{1}{2}\gamma^\alpha{}_{\beta \gamma} \, c^\beta c^\gamma \frac{\partial}{\partial c^\alpha} } \right] \,.$$

The first term is

\begin{aligned} \left[ \widehat{ \left( c^\alpha_{,\mu} \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu} } \;,\; \widehat{ \left( c^{\alpha’}_{,\mu} \gamma^{\alpha’}_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^{\alpha’}_\mu} } \right] & = 2 \gamma^{\alpha’}_{\beta \gamma} \widehat{ c^\beta \left( c^\gamma_{,\mu} \gamma^\gamma_{\beta’ \gamma’} c^{\beta’} a^{\gamma’}_\mu \right) \frac{\partial}{\partial a^{\alpha’}_\mu} } \\ & = 2 \gamma^{\alpha’}_{\beta \gamma} \widehat{ c^\beta c^\gamma_{,\mu} \frac{\partial}{\partial a^{\alpha’}_\mu} } + 2 \gamma^{\alpha’}_{\beta \gamma} \gamma^\gamma_{\beta’ \gamma’} \widehat{ c^\beta c^{\beta’} a^{\gamma’}_\mu \frac{\partial}{\partial a^{\alpha’}_\mu} } \\ & = 2 \gamma^{\alpha’}_{\beta \gamma} \widehat{ c^\beta c^\gamma_{,\mu} \frac{\partial}{\partial a^{\alpha’}_\mu} } + \gamma^{\alpha’}_{\beta \gamma} \gamma^\gamma_{\beta’ \gamma’} \widehat{ \left( c^\beta c^{\beta’} a^{\gamma’}_\mu c^{\beta’} c^{\beta} a^{\gamma’}_\mu \right) \frac{\partial}{\partial a^{\alpha’}_\mu} } \\ & = 2 \gamma^{\alpha’}_{\beta \gamma} \widehat{ c^\beta c^\gamma_{,\mu} \frac{\partial}{\partial a^{\alpha’}_\mu} } + \gamma^{\alpha’}_{\beta \gamma} \gamma^\gamma_{\beta’ \gamma’} \widehat{ \left( c^\beta c^{\gamma’} a^{\beta’}_\mu c^{\beta’} c^{\beta} a^{\gamma’}_\mu \right) \frac{\partial}{\partial a^{\alpha’}_\mu} } \\ & = 2 \gamma^{\alpha’}_{\beta \gamma} \widehat{ c^\beta c^\gamma_{,\mu} \frac{\partial}{\partial a^{\alpha’}_\mu} } + \gamma^{\alpha’}_{\beta \gamma} \gamma^\gamma_{\beta’ \gamma’} \widehat{ c^{\gamma’} c^{\beta’} a^{\beta}_\mu \frac{\partial}{\partial a^{\alpha’}_\mu} } \\ & = 2 \gamma^{\alpha’}_{\beta \gamma} \widehat{ c^\beta c^\gamma_{,\mu} \frac{\partial}{\partial a^{\alpha’}_\mu} } + \gamma^{\alpha’}_{\gamma \beta} \gamma^\beta_{\beta’ \gamma’} \widehat{ c^{\beta’} c^{\gamma’} a^{\gamma}_\mu \frac{\partial}{\partial a^{\alpha’}_\mu} } \end{aligned}

Here first we expanded out, then in the second-but-last line we used the Jacobi identity (165) and in the last line we adjusted indices, just for convenience of comparison with the next term. That next term is

$$\left[ \widehat{ \left( c^\alpha_{,\mu} \gamma^\alpha_{\beta \gamma} c^\beta a^\gamma_\mu \right) \partial_{a^\alpha_\mu} } \;,\; \gamma^\alpha{}_{\beta \gamma} \, \widehat{c^\beta c^\gamma \frac{\partial}{\partial c^\alpha}} \right] = 2 \gamma^\alpha_{\beta \gamma} \widehat{ c^\beta_{,\mu} c^\gamma \frac{\partial}{\partial a^\alpha_\mu} } \gamma^\alpha_{\beta \gamma} \gamma^\beta_{\beta’ \gamma’} \widehat{ c^{\beta’} c^{\gamma’} a^\gamma_\mu \frac{\partial}{\partial a^\alpha_\mu} } \,,$$

where the first summand on the right comes from the prolongation.

This shows that the two terms cancel.

Example 10.32. (local BRST complex for B-field on Minkowski spacetime)

Consider the Lagrangian field theory of the B-field on Minkowski spacetime from example 5.8, with field bundle the differential 2-form-bundle ##E = \wedge^2_\Sigma T^\ast \Sigma## with coordinates ##((x^\mu), (b_{\mu \nu}))## subject to ##b_{\mu \nu} = – b_{\nu \mu}##; and with Lagrangian density.

By example 10.16 the local BRST complex (example link ) has BRST differential of the form

$$c_{\mu, \nu} \frac{\partial}{\partial b_{\mu \nu}} + c_{\mu,\nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}} + \cdots \,.$$

In this case this enhanced to an Lie 2-algebroid by regarding the second-order gauge parameters (162) in degree 2 to form a graded field bundle

$$\underset{ \{\overset{(2)}{c}\} }{ \underbrace{ \overset{(2)}{\mathcal{G}}[2] }} \times_\Sigma \underset{\{c_\mu\}}{ \underbrace{ \mathcal{G}[1] } } \times_\Sigma \underset{ (b_{\mu \nu}) }{ \underbrace{ E }} \;=\; \mathbb{R}[2] \times T^\ast \Sigma [1] \times_\Sigma E$$

by adding the ghost-of-ghost field ##(\overset{(2)}{c})## (163) and taking the local BRST differential to be the sum of the first order infinitesimal gauge symmetries (161) and the second order infinitesimal gauge-of-gauge symmetry (163):

$$s_{BRST} \;=\; \left( c_{\mu, \nu} \frac{\partial}{\partial b_{\mu \nu}} + c_{\mu,\nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}} + \cdots \right) + \left( \overset{(2)}{c}_{,\mu} \frac{\partial}{\partial c_\mu} + \overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial c_{\mu,\nu}} + \cdots \right) \,.$$

Notice that this indeed still squares to zero, due to the second-order Noether identity (164):

\begin{aligned} \left( s_{BSRT} \right)^2 & = \left[ \overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial c_{\mu,\nu}}, c_{\mu, \nu} \frac{\partial}{\partial b_{\mu \nu}} \right] \;+\; \left[ \overset{(2)}{c}_{,\mu \nu_1 \nu_2} \frac{\partial}{\partial c_{\mu,\nu_1 \nu_2}}, c_{\mu, \nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}} \right] \\ & = \underset{ = 0 }{ \underbrace{ \overset{(2)}{c}_{,\mu \nu} \frac{\partial}{\partial b_{\mu \nu}} }} \;+\; \underset{ = 0 }{ \underbrace{ \overset{(2)}{c}_{,\mu \nu_1 \nu_2} \frac{\partial}{\partial b_{\mu \nu_1, \nu_2}} }} \;+\; \cdots \\ & = 0 \,. \end{aligned}

This concludes our discussion of infinitesimal gauge symmetries, their off-shell action on the jet bundle of the field bundle and the corresponding homotopy quotient exhibited by the local BRST complex. In the next chapter we discuss the homotopy intersection of this construction with the shell: the reduced phase space.

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