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for ease of linking I have given antibracket its own little entry (it used to just redirect to BV-BRST complex).
I had also given local antibracket an little entry of its own. Possibly these two should be merged…
I have a couple of questions regarding the local antibracket as defined here: https://ncatlab.org/nlab/show/geometry+of+physics+–+A+first+idea+of+quantum+field+theory#LocalJetBundleAntibracket
Is the local antibracket supposed to be defined on all forms in the variational bicomplex? If it wasn’t, it does not seem to be possible to define the BV-BRST variational bicomplex (or Shaparov’s variational tricomplex, https://arxiv.org/abs/1506.04652) by using that bracket. However, the definition given on the nLab does only speak about the local antibracket of volume forms. Also the (in-)dependence on the volume form is not completely clear. To get rid of any dependence: Are the antifields supposed to be twisted with the dual of the canonical bundle?
In what kind of structure does the antibracket fit? Is it part of a kind of Poisson-infinity-algebra? It would be nice if it was possible to give the local antibracket just on generators and where every other application of the local antibracket can be reduced to the generator case by algebraic rules.
Thanks!
Marc
That’s right, the local antibracket is defined on the top-degree horizontal forms. That’s as it should for the local antibracket to yield the usual global antibracket under transgression.
That this is well defined is manifest in the version of the local antibracket that involves no partial derivatives, just the Euler-Lagrange derivatives (and this version is equal to the other version which actually gives the nilpotent local BV-differential by a horizontally exact bit.)
And yes, in a discussion not tied to the assumption of a spacetime with a given volume form (as the notes are for efficiency of exposition) all dual bundles that appear throughout need to be replaced by densitized dual bundles.
Thanks for the quick reply.
In formula (188) of your notes, the differential $s$ is defined using the local antibracket. If I understand the discussion that follows correctly, $s$ is applied not only to forms of top degree but to forms of any horizontal degree. Thus, at least the second entry of the (unprimed) antibracket should be able to take forms of any horizontal degree, shouldn’t it? This leads naturally to the question whether the first entry could also be a form of any degree (so that graded symmetry in the case of the primed bracket can again be discussed).
Yes, but that then does depend on the choice of trivialization of the density bundle, I suppose.
I have some hints for the variational tri-complex in the notes (i guess I call it the BV-variational bicomplex or something), but I got distracted by other bits of the notes before I could develop that more fully. I’d need to look into the article by Shaparov that you point to.
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