Equivariant BV formalism
Equivariantly closed cocycle in the Cartan model
For all cocycles coming from half-densities, Eq. (22) is satisfied with:
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As in 〚Special cocycles〛,
suppose that 
is an embedding of 
in 
.
Eq. (23) becomes (cp Eqs. (13) and (14)):
is an embedding of in .
Eq. (23) becomes (cp Eqs. (13) and (14)):
Then, equivariantly closed cocycle in the Cartan model is given by:
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Our notations here differ from our previous papers;
was called 
in Section 4 of Mikhailov:2016myt
and 
in Section 6 of Mikhailov:2016rkp. Here is the summary of notations:
was called in Section 4 of Mikhailov:2016myt
and in Section 6 of Mikhailov:2016rkp. Here is the summary of notations:here |
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Section 4 of Mikhailov:2016myt |
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Section 6 of Mikhailov:2016rkp |
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Deformations
If 
and solve
Eqs. (32) and (33) with some 
, then
and 
solve
with 
. This allows us to assume, without loss of generality, that 
,
i.e. to consider representations of 
rather than .
Or, we can fix 
for some fixed 
. Let us fix the half-density,
delegating the deformations into 
.
Consider the deformations of the embedding 
keeping
fixed.
keeping
fixed.
Notice that
describes the action of symmetries on the BV phase space. We do not want to deform the action of symmetries.
Eq. (32) implies that a small variation 
satisfies:
satisfies:
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Those 
which are in the image of 
correspond to trivial deformations. This means that
the cohomologies of the operator 
, considered in Alekseev:2010gr,
in our context compute infinitesimal deformations of the equivariant half-density.
which are in the image of
correspond to trivial deformations. This means that
the cohomologies of the operator , considered in Alekseev:2010gr,
in our context compute infinitesimal deformations of the equivariant half-density.