1 Ghosts 2 Integration measure 3 Lift of symmetries to BRST configuration space

Construction

#### 1Ghosts

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 :
 (1)

Functions on satisfy:

#### 2Integration measure

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 :
 (2)
Notice the following properties of left shifts:
1. They commute with

2. 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.

3. The measure on is invariant.