BV from BRST

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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

with

notations:

does not include , while includes both and

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 :

(11)

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''.

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