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:
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 :
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〛).
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:
then Eq. (29) defines a cocycle of
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: