### On Noether’s Second (BV, Part VI)

#### Posted by Urs Schreiber

One aim of

R. Fulp, T. Lada, J. Stasheff
*Noether’s Variational Theorem II and the BV formalism*

math/0204079

was to

[…] restore […] an emphasis [on] the relevance of Noether’s theorem in […] the BV approach

Namely it is Noether’s second theorem (see page 6 of the above article) for Lagrangian theories which is reincarnated equivalently in the BV statement that

the space of ghosts is canonically isomorphic to that of anti-ghosts.

Meaning that

For every Noether identity there is a symmetry. And vice versa.

In terms of the little toy example (which is not that toy-ish, actually, rather skeletalized, I think), which I talked about last time (see also parts I, II, III, IV), this means that in our little complex

$\array{ 0 &\to& \mathrm{ker}(d S(\cdot)) &\hookrightarrow& \Gamma(T X) &\stackrel{d S(\cdot)}{\to}& C^\infty(X) &\stackrel{\rho}{\to}& C^{\infty}(X) \otimes g &\to& 0 \\ \\ && anti-ghosts && anti-fields && fields && ghosts \\ \\ && Tate && Koszul && && Chevalley-Eilenberg \\ \\ && Noether identities && && && symmetries \\ \\ && deg -2 && deg -1 && deg 0 && deg 1 }$ which is induced entirely from a smooth function $S : X \to \mathbb{R}$ on a manifold $X$, we have a canonical isomorphism between the first and the last term $\mathrm{ker}(d S (\cdot)) \simeq C^\infty \otimes g \,.$ And this canonical isomorphism is, I think, Noether’s second theorem in this context.

And I’ll claim: this is here nothing but a special case of *Cartan’s magic formula* (or whatever you call that).

For that to make sense, I’ll first need to say mor precisely how the $g$ in $C^\infty(X) \otimes g$ is defined in the first place.

Instead of plunging into the jet space gymnastics performed by Fulp, Lada and Stasheff, I shall here follow the discussion of

P.O. Kazinski, S.L. Lyakhovich, A.A. Sharapov
*Lagrange structure and quantization*

hep-th/0506093

which pretends throughout that the space of fields is a smooth, finite dimensional manifold. The relation we are after is their equation (10) on p. 7 (take notice of the paragraph right beneath that!).

All I want to do here is actually point out the proof of that simple statement, using slightly more invariant language than used there.

**Definition (Equations of motion, Symmetries, Noether identities)** The *equation of motion* induced by $S$ is
$d S = 0
\,.$
A local *symmetry* of $S$ is a vector field $v \in \Gamma(T X)$ preserving the equations of motion
$L_{\epsilon v} (d S) \; = 0$
($L_v$ denotes the Lie derivative) for all $\epsilon \in C^\infty(X)$. A *Noether identity* is a vector field $v$ preserving the function $S$ itself
$L_v S = 0
\,.$

It seems too trivial to point it out here, but I shall do anyway: people usually think of a Noether identity as a vector field $v$ such that $d S (v) = 0 \,.$

Then if you write the components of $d S$ as $E_a$ in some basis, a Noether identity becomes $v^a E_a = 0 \,.$ In that form you may more easily recognize it in the existing component-ridden literature.

Now

**Simple proposition.** The space of symmetries is canonically isomorphic to that of Noether identities.

If $L_v d S = 0$ then $L_{\epsilon v} d S = \epsilon L_v d S + d \epsilon \wedge v(S)$. For this to vanish for all $\epsilon$ we need $v(S)$ to vanish.

**Definition (ghosts and anti-ghosts)** Suppose that the space of symmetries is generated, over $C^\infty(X)$, from a subspace
$g \subset \Gamma(T X)$
on which the bracket of vector fields closes. Then $g$ is the *Lie algebra of symmetries*, and we identify the space of symmetries with
$C^\infty(X) \otimes g
\,.$
BV practitioners will call the elements of $g$ the *ghosts*.

By the above theorem this induces a similar decomposition of the space of Noether identites. The Noether identites corresponding to the elements in $g$ BV practicioners will call the *anti-ghosts*.

(Again, keep in mind that the full BV formalism is designed to handle much more general cases than that. In particular, its whole *raison d’être* is the case where the symmetries do *not* form a Lie algebra on the nose. But that shall not concern us right now.)

## Re: On Noether’s Second (BV, Part VI)

Sorry to correct you but from v being a symmetry of dS does not follow that fv is also a symmetry of dS for any smooth function f. Or maybe i understood the equation L_v dS=0 wrong.