Taking apart the AdS sigma model
The standard action given by Eq. (25) depends on the worldsheet complex structure and is
polynomial in the pure spinor variables. In the BV formalism, it corresponds to a specific
choice of the Lagrangian submanifold. We can change the action to a physically equivalent one,
by adding BRST quartets and/or deforming the Lagrangian submanifold. We can ask ourselves,
what is the simplest formulation of the theory, in the BV language, preserving the
symmetries of 
? (Of course, the notion of “being the simplest” is somewhat
subjective.) In this paper we gave an example of such a “minimalistic” formulation:
? (Of course, the notion of “being the simplest” is somewhat
subjective.) In this paper we gave an example of such a “minimalistic” formulation:
Here 
are the BV Hamiltonians of the left shift, Eq. (42).
The relation of Eq. (67) to the original BV action (40) is through adding BRST quartet
(Section Regularization) and canonical transformations (Eqs. (33), (49), (52)). Subjectively, Eq. (67)
is the simplest way of presenting the worldsheet Master Action for .
are the BV Hamiltonians of the left shift, Eq. (42).
The relation of Eq. (67) to the original BV action (40) is through adding BRST quartet
(Section Regularization) and canonical transformations (Eqs. (33), (49), (52)). Subjectively, Eq. (67)
is the simplest way of presenting the worldsheet Master Action for .The Master Action (67) does not depend on the worldsheet metric. The dependence on the worldsheet metric (through the complex structure) comes later when we choose the Lagrangian submanifold.
The way Eq. (67) is written, it seems that 
is completely
decoupled from 
and 
. But the transition functions on overlapping charts, described
in Section Gluing charts, do mix the two sets of fields.
The Master Action (67) is non-polynomial in , because of 
.