BV
We will now apply the technique developed in the previous sections to the BV formalism.
Let 
be the Lie algebra of functions on the BV phase space with flipped statistics.
Its elements are 
where 
is a function on the BV phase space and 
the suspension:
be the Lie algebra of functions on the BV phase space with flipped statistics.
Its elements are where is a function on the BV phase space and the suspension:
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The Lie bracket is given by the odd Poisson bracket.
Half-densities as a representation of 
The space of half-densities on the BV phase space is a representation of :
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We are now in the context of 〚PDFs from representations of 
〛.
Now 
is , 
is the space of half-densities,
is 
and 
is 
—
Correlation functions as a Lie superalgebra cocycle
Correlation function defines a linear map:
where 
means symmetrized tensor product
(examples of sign rules are in 〚Supercommutative algebra and its dual coalgebra〛).
means symmetrized tensor product
(examples of sign rules are in 〚Supercommutative algebra and its dual coalgebra〛).Proposition 3
Eq. (29) defines an injective map from the space of half-densities to the space of
cochains of with values in functionals on Lagrangian submanifolds;
to every half-density 
corresponds a cochain given by Eq. (29).
This map is an intertwiner of the actions of the differential Lie superalgebra .
In particular, if satisfies the Quantum Master Equation:
with values in functionals on Lagrangian submanifolds;
to every half-density corresponds a cochain given by Eq. (29).
This map is an intertwiner of the actions of the differential Lie superalgebra .
In particular, if satisfies the Quantum Master Equation:
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then Eq. (29) defines a cocycle of with coefficients
in the space of functionals on Lagrangian submanifolds.
with coefficients
in the space of functionals on Lagrangian submanifolds.Proof follows from Proposition 2.
The image of this map consists of the cochains satisfying the following locality property.
Given 
, if for some 
and 
then 
.
It is important for us, that this subset is preserved by the canonical transformations,
i.e. by the action of on its cocycles.
Cocycles with
coefficients in , defined by Eq. (29),
can be interpreted as closed differential forms on 
, by the construction
of 〚Mapping to PDFs〛. We take:
, defined by Eq. (29),
can be interpreted as closed differential forms on , by the construction
of 〚Mapping to PDFs〛. We take:
