Tangent space to the double coset
1 Identification of the tangent space
The tangent space to 
at the point 
can be
identified as the space of all functions 
on 
modulo those functions
which can be extended to the first infinitesimal neighborhood of
as a function 
satisfying:
at the point can be
identified as the space of all functions on modulo those functions
which can be extended to the first infinitesimal neighborhood of
as a function satisfying:
It turns out that this space very much depends on 
and on 
. Let us
even just restrict ourselves to those which can be obtained from the
“standard” Lagrangian submanifold 
by canonical transformations.
Then, without loss of generality, we can say that the is and
the problem is parametrized just by .
and on . Let us
even just restrict ourselves to those which can be obtained from the
“standard” Lagrangian submanifold by canonical transformations.
Then, without loss of generality, we can say that the is and
the problem is parametrized just by .2 Quantum limit
Let us choose the half-density:
Let us choose to be . In this case the problem (2) has solution
for any 
. Indeed, we can simply choose 
to be independent of 
:
to be . In this case the problem (2) has solution
for any . Indeed, we can simply choose to be independent of :
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3 Classical limit
In the classical limit we have:
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(notice that the of Eq. (3) can be considered the opposite limit).
In this case, the equation (2) reads:
of Eq. (3) can be considered the opposite limit).
In this case, the equation (2) reads:
At the zeroth order in the -expansion, Eq. (4) implies:
-expansion, Eq. (4) implies:
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Therefore, the necessary condition for the continuation to exist is that
is BRST-closed on-shell.
is BRST-closed on-shell.
Notice that
can not be interpreted as a deformation of the BV action, because it has ghost number
; the action should have ghost number zero.
4 Conclusion
In the quantum theory the BRST structure is locally rigid. However, as the quantum BV theory is poorly defined anyway, we probably need the expansion around the classical limit.
In the classical limit the tangent space to
is parametrized by
BRST-exact expressions modulo those BRST-exact expressions which are zero on-shell.