One starts with the “classical action” which is invariant under some gauge symmetry.
Let be the “classical” space of fields, e.g. for Yang-Mills theory the fields are .
Suppose the gauge symmetry is , with the Lie algebra .
Then we should introduce ghost fields, geometrically:
fields and ghosts
where the action of is via right shift on :
We use the coordinates on (denoting “” the coordinate on the fiber ) and on .
Notice that this commutes with .
We can always find a representative with (i.e. choose ).
Then, with the standard notation :
Functions on satisfy:
We assume that comes with some integration measure:
This should be understood as an integration measure, i.e. a density of weight (rather than a function of ).
The product of this measure with the canonical measure on gives us a measure on
which we will call . Notice that preserves this measure.
This can be proven as follows. For any function :
because comes from the canonical odd vector field on .
3Lift of symmetries to BRST configuration space
Original gauge symmetries can be lifted to the BRST field space as left shifts on :
Notice the following properties of left shifts:
They commute with
Infinitesimal symmetries of the form Eq. (2) are actually -exact.
Indeed, they come from infinitesimal left shifts on , and comes from the de Rham differential.