Special case: BV structures coming from BRST
1 BRST formalism
This is a special case. We start from the which is invariant under some
symmetry which we refer to as gauge symmetry.
Then one introduces the additional “ghost” field .
The nilpotent symmetry is defined in the following way:
2 Master Action
Given this BRST data, we can construct :
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This solves the Master Equation .
3 Base form
To construct the base form, we have to first identify . It turns out that we can choose to coincide, as Lie algebra, with the underlying gauge symmetry of the BRST data. This corresponds to the following :
where are generators of the underlying gauge symmetry of the BRST formalism.
In physics we usually consider the “gauge fixed” BRST action. This corresponds in BV formalism to choosing a Lagrangian submanifold so that the restricted is nondegenerate. Our does not stabilize this Lagrangian submanifold, therefore they are not symmetries of the action in the usual sense. Instead, they change the action by adding to it BRST-exact terms.
The base form corresponding to this choice of is:
where is the horizonthal projection of , using some connection,
and is the curvature of this connection.
This can be applied to both bosonic string and NSR superstring. Indeed, as we now explain,
they fall into this class of ``theories obtained from BRST''.